Transcript
University of Pennsylvania University of Pennsylvania
ScholarlyCommons ScholarlyCommons
Publicly Accessible Penn Dissertations
Spring 2010
Dark Matter Halo Mergers and Quasars Dark Matter Halo Mergers and Quasars
Jorge Moreno University of Pennsylvania, jmoreno@haverford.edu
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Part of the Cosmology, Relativity, and Gravity Commons, and the External Galaxies Commons
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Dark Matter Halo Mergers and Quasars Dark Matter Halo Mergers and Quasars
Abstract Abstract The formation and evolution of galaxies and the supermassive black holes they harbor at their nuclei depends strongly on the merger history of their surrounding dark matter haloes. First we developed a semi-analytic algorithm that describes the merger history tree of a halo. The following tests were performed: the conditional mass function, the time and mass distributions at formation, and the time distribution of the last major merger. We provide a model for the creation rate of dark matter haloes, informed by both coagulation theory and the modified excursion set approach with moving barriers. A comparison with N-body simulations shows that our square-root barrier merger rate is significantly better than the standard extended Press-Schechter rate used in the literature. The last chapter is dedicated to a simple model of quasar activation by major mergers of dark matter haloes. The model consists of two main ingredients: the halo merger rate describing triggering, and a quasar light curve, which tracks the evolution of individual quasars. In this model, the mass of the seed black hole at triggering is assumed to be a fixed fraction of its mass at the peak luminosity. The light curve has two components: an exponential ascending phase and a power-law descending phase that depends on mass of the host. We postulate a self-regulation condition between the peak luminosity of the active galactic nucleus (AGN) and the mass of the host halo at triggering. This type of model for quasar evolution is at the heart of the latest semianalytic models (SAMs) of galaxy formation and it is therefore definitely worth studying in some detail. By carefully revisiting some of the main issues linked to this approach, we were able to derive several interesting and physically meaningful constraints regarding black hole evolution. We expect simple, yet observationally-constrained models like ours to play a central role in future models of galaxy formation.
Degree Type Degree Type Dissertation
Degree Name Degree Name Doctor of Philosophy (PhD)
Graduate Group Graduate Group Physics & Astronomy
First Advisor First Advisor Ravi K. Sheth
Keywords Keywords galaxies, black holes, quasars, haloes
Subject Categories Subject Categories Cosmology, Relativity, and Gravity | External Galaxies
This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/177
DARK MATTER HALO MERGERS AND QUASARS
Jorge Moreno
A DISSERTATION
in
Physics and Astronomy
Presented to the Faculties of the University of Pennsylvania inPartial Fulfillment of the Requirements for the Degree of Doctor of
Philosophy
2010
Supervisor of Dissertation
Ravi K. Sheth, Professor of Physics and Astronomy
Graduate Group Chairman
Ravi K. Sheth, Professor of Physics and Astronomy
Dissertation Committee
Gary Bernstein, Professor of Physics and Astronomy
Adam Lidz, Assistant Professor of Physics and Astronomy
Andrea J. Liu, Professor of Physics and Astronomy
Gordon T. Richards, Assistant Professor of Physics
Dedication
This thesis is dedicated to my wife Viridiana, and our children Camila and Mateo.
You are the best thing a person can have. Thank you for your love, patience and
encouragement during these difficult years and for giving meaning to my life.
ii
Acknowledgments
To break the tradition, I wish to thank the most important people in my life first:
my wonderful wife Viridiana Moreno and our children Camila and Mateo, to whom
this thesis is dedicated. I would also like to thank my parents Martha Soto and
Juan Moreno for doing everything in their power so I had the opportunities they
could not have. Also to my brothers Hector and Juan Ernesto, for always having
my back. Likewise, I would like to thank Rafael Moreno, Irene Gomez, Vianney
Moreno, Demian Cortes, Jacqueline Ibarrola, Sandra Arvezu, little Paulo – and of
course, the rest of the family (too many to mention individually).
It is my pleasure to thank my thesis advisor, Ravi Sheth. My father always says
that the two most important decisions in a person’s life is choosing who they are
going to marry, and choosing their career. I would replace the latter with “choosing
the right Ph.D advisor”. I personally could not have chosen a better mentor. There
are not enough pages, or years, to show my gratitude to Ravi. I also want to
thank Francesco Shankar and Carlo Giocoli for all their support, encouragement,
and the wonderful opportunity of writing papers together. Also I wish to thank
iii
my other collaborators: Bepi Tormen, David Weinberg, Raul Angulo, Cheng Li,
Martin Crocce, and Federico Marulli. I thank Beth Willman for her encouragement,
guidance, and friendship during these last few difficult months.
I wish to thank Gary Bernstein, Adam Lidz, Andrea Liu, Gordon Richards, and
of course, Ravi Sheth, for agreeing to serve as committee members, and for tak-
ing the time and dedication to read this rather long thesis. Also to my professors
for imparting their knowledge and advice, including Gary Bernstein, Mariangela
Bernardi, Bhuvnesh Jain, Mark Devlin, Masao Sato, Vijay Balasubramanian, Burt
Ovrut, Mirjam Cvetic, Mark Gulian, Randy Kamien, Joe Kroll, Phil Nelson, Paul
Langacker, and Fay Ajzenberg-Selove. To CONACyT-Mexico, and Grant 2002352
from the US-Israel BSF for funding. Also to colleagues at Haverford College for their
constant support, especially Walter Smith, Steve Boughn, and Bruce Partridge for
their mentoring; and Beth Willman, Stephon Alexander, Peter Love, Anna Sajina,
Chema Diego, Adi Zolotov, and Aurelia Gomez Unamuno for their friendship. To
folks at Cinvestav for making my move to Penn possible, including my former advi-
sor Hugo Compean, and my professors Arnulfo Zepeda, Juan Jose Godina, Denjoe
O’Connor, Tonatiuh Matos, Eloy Ayon, Mauricio Carvajal, Gerardo Herrera, Piotr
Kielanowski, Masiej Pzranowski, Gabriel Lopez, Jorge Hirsch, Jose Mendez, and
Miguel Angel Perez. But mostly to Guillermo Moreno, Jerzy Plebanski and Au-
gusto Garcia, may they rest in peace. Also, my friends Ramon Castaneda, Armando
Perez, Jose Manuel Lara Bauche, Argelia Bernal, Carlos Chavez, Jorge Mercado,
iv
Juan Barranco, Idrish Huet, Eduard de la Cruz Burelo, Ildefonso Leon, Americo
Rodriguez and Pedro Podesta. To my friends at Penn, including Joey Hyde, Mike
Fischbein, Andres Plazas, Zhaoming Ma, Jacek Guzik, Matt Martino, Graziano
Rossi, Dan Swetz, Ed Chapin, Amitai Bin-Nun, Tsz Yan Lam, Laura Marian,
Reiko Nakajima, Federica Bianco, Greg Dobler, Josko Kirigin, Alex Borisov, Hite
and Willis Geffert, Dutch Ratliff, and many others for the great years together and
their friendship. I wish to thank all my teachers for encouraging me, especially Mrs.
and Mr. Blackwood, Mr. Erdman, Ms. Faridian, Mr. Guember, Ms. Kahn, Mr.
Lange, Ms. Hooper, Ms. Komatsu, Ms. Martin, Ms. O’Connell, Ms. Scharf, Mr.
Silverman, Mr. Taylor, Ms. Votto, and maestras Eva, Edith and Maria Luisa. And
a few good friends that have supported me during the years: Noah Bray-Ali, Adam
Ferrell, Alex Prieto and Rosa Elia Victorio.
Parts of these work were done with the direct help, suggestions, and inputs of
Silvia Bonoli, Marcella Brusa, Anca Constantin, Onsi Fakhouri, Joey Hyde, Avi
Loeb, Eyal Neistein, Robert Smith and Yue Shen. Many others, listed below,
contributed indirectly in many ways.
I wish to thank the following people for the insightful conversations, comments,
questions, encouragement and hospitality during my ‘talk tour’ where presenta-
tions of this work were delivered. Although it is a rather long list, the input from
every single one of you has contributed to make me a better scientist. At Cin-
vestav: Eduard de la Cruz Burelo, Miguel Garcia, Isaac Hernandez, Miguel Angel
v
Perez, Tonatiuh Matos, Arnulfo Zepeda, At UNAM: Vladimir Avila-Reese, Anto-
nio Peimbert, Octavio Valenzuela. At INAOE: Vahram Chavushyan, David Hughes,
Hector Ibarra, Omar Lopez Cruz, Divakara Mayya, Alfredo Montana. At Colgate:
Thomas Balonek, Jeff Bary. At Williams: Marek Demianski, Karen Kwitter, Steve
Souza. At Texas: Guillermo Blanc, Amy Forestell, Martin Gaskell, Karl Gebhardt,
Donghui Jeoung, Jarrett Johnson, Jun Koda, Eiichiro Komatsu, Randi Ludwig, Yi
Mao, Irina Marinova, Milos Milosavljevic, Paul Shapiro, Matashoshi Shoji, Chalence
Timer, Berverly Wills. At Harvard: Federica Bianco, Laura Blecha, Anca Con-
stantin, Claude-Andre Faucher-Guiguere, Ryan Hickox, Avi Loeb, Ryan O’Leary,
Adam Lidz, Matt Mcquinn, Ramesh Narajan, Jonathan Pritchard, Jaiyul Yoo,
Matthias Zaldarriaga. At UC Santa Barbara: Nicola Bennert, Peng Oh, Tommaso
Treu. At Stanford: Marcelo Alvarez, Michael Busha, Brian Gerke, Nelima Sehgal,
Louie Strigari, Risa Wechsler, Chen Zheng. At UC Berkeley: Kevin Bundy, Jor-
dan Carlson, Joanne Cohn, Marina Cortes, Roland De Putter, Marc Davis, Shirley
Ho, Phil Hopkins, Alexie Leauthaud, Eric Linder, Chung-Pei Ma, Reiko Nakajima,
Jonathan Pober, Linda Strubbe, Tristan Smith, George Smoot, Andrew Wetzel,
Martin White, Jun Zhang. At Caltech: Lotty Ackerman, Shin’ichiro Ando, Esfan-
diar Alizadeh, Andrew Benson, Sean Carroll, Adrienne Erickcek, Steve Furlanetto,
Chris Hirata, Elisabeth Krause, Benjamin Wandelt. At UC Irvine: Aaron Barth,
Betsy Barton, Misty Bentz, James Bullock, David Buote, Helene Flohic, Chris Pur-
cell, Devdeep Sarkar, Kyle Stewart, Erik Tollerud, Carol Thornton, Chris Trinh,
vi
Gaurang Yodh, Joanelle Walsh. At Cal Poly Pomona: Nina Abramzon, Harvey Leff,
Roger Morehouse, Alex Rudolph, Alex Small. At UC San Diego: Alison Coil, David
Kirkman, Thomas Murphy, Art Wolfe. At the Carnegie Institution of Washington:
Julio Chaname, Nick Moskowitz. At Wesleyan: Erin Arai, Tyler Desjardins, Bill
Herbs, Roy Kilgard, Amy Langford, Edward Moran, Seth Redfield, Katy Wyman.
At Heidelberg: Matthias Bartelmann, Carlo Giocoli, Matteo Maturi, Faviola Molina.
At SISSA: Michael Cook, Luigi Danese, Paolo Salucci. At MPA: Raul Angulo, Silvia
Bonoli, Michael Boylan-Kolchin, Qi Guo, Carlos Hernandez Montegudo, Guinevere
Kauffmann, Cheng Li, Andrea Merloni, Eyal Neistein, Francesco Shankar, Simon
White, Jesus Zavala. At Groningen: Muhammad Abdul Latif, Amina Helmi, J. P.
Perez Beaupuits, Laura Sales, Marco Spaans, Esra Tigrak, Rien van de Weyn-
gaert, Saleem Zaroubi. At Los Alamos National Laboratories: Suman Bhattacharya,
Stirling Colgate, Daniel Holz, Daniel Whalen. At Columbia: Greg Bryan, Zoltan
Haiman, Kristen Menou, Takamistu Tanaka. At Michigan: Eric Bell, Joel Breg-
man, Carlos Cuhna, Gus Evrard, Oleg Gnedin, Kayhan Gultekin, Elena Rasia,
Mateusz Ruszkowski, Min-Su Shin, Monica Varulli, Marta Volonteri, Marcel Zemp.
At Rutgers: Chuck Keeton, Viviana Acquaviva. At Vanderbilt: Andreas Berlind,
Kelly Holley-Bockelmann, Cameron McBride. Also to folks I have met at meetings,
including Paula Aguirre, Karla Alamo, Kaushi Bandara, Maria Beltran, Ami Choi,
David Cohen, Sanghamitra deb, Roland Dunner, Debra Elmegreen, Adiv Gonzalez,
Mandeep Gill, Lucia Guaita, Atakan Gurkan, Sarah Hansen, Luis Ho, Eric Jensen,
vii
Claudia Lagos, Felipe Marin, Kim McLeod, Joel Mendoza, Irving Morales, Fran-
cisco Mueller, Federico Pelupessy, Enrico Ramirez Ruiz, Mario Riquelme, Meghan
Roscioli, Douglas Rudd, Anil Seth, Allyson Sheffield, Greg Sivakoff, Roberto Soria
and Kendrik Smith. For answering my questions via e-mail, I wish to acknowledge
Esfandiar Alizadeh, Andrew Benson, Julie Comerford, T. J. Cox, Onsi Fakhouri,
Chris Hayward, William Keel, Eyal Neistein, Ryan O’Leary, Antonio Riotto, Marco
Spaans and Benjamin Wandelt. Lastly, to David Charbonneau, Guy Consolmagno,
Rocky Kolb and Alex Rudolph for paying me a visit in my office, despite your busy
schedule. I have been very fortunate to present my results at so many places, and
meet so many interesting people in my field, which has really shown me the human
side of this wonderful enterprise. I hope I did not leave anybody out, and I truly
look forward to bumping into you again in the future.
Doing physics and astronomy has been one of the most exciting, fascinating,
and challenging endevors in my life. I am indebted to Viridiana Moreno, the love of
my life, for her constant support. This thesis is decated to you, and to our children
Camila and Mateo for all these years together and for the years to come.
viii
ABSTRACT
DARK MATTER HALO MERGERS AND QUASARS
Jorge Moreno
Ravi Sheth, Advisor
The formation and evolution of galaxies and the supermassive black holes they
harbor at their nuclei depends strongly on the merger history of their surrounding
dark matter haloes. First we developed a semi-analytic algorithm that describes the
merger history tree of a halo. The following tests were performed: the conditional
mass function, the time and mass distributions at formation, and the time distri-
bution of the last major merger. We provide a model for the creation rate of dark
matter haloes, informed by both coagulation theory and the modified excursion
set approach with moving barriers. A comparison with N-body simulations shows
that our square-root barrier merger rate is significantly better than the standard
extended Press-Schechter rate used in the literature. The last chapter is dedicated
to a simple model of quasar activation by major mergers of dark matter haloes. The
model consists of two main ingredients: the halo merger rate describing triggering,
and a quasar light curve, which tracks the evolution of individual quasars. In this
model, the mass of the seed black hole at triggering is assumed to be a fixed fraction
of its mass at the peak luminosity. The light curve has two components: an expo-
nential ascending phase and a power-law descending phase that depends on mass of
ix
the host. We postulate a self-regulation condition between the peak luminosity of
the active galactic nucleus (AGN) and the mass of the host halo at triggering. This
type of model for quasar evolution is at the heart of the latest semianalytic models
(SAMs) of galaxy formation and it is therefore definitely worth studying in some
detail. By carefully revisiting some of the main issues linked to this approach, we
were able to derive several interesting and physically meaningful constraints regard-
ing black hole evolution. We expect simple, yet observationally-constrained models
like ours to play a central role in future models of galaxy formation.
x
Contents
1 Prologue 1
2 Introduction 7
2.1 In the beginning ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The smooth expanding universe. . . . . . . . . . . . . . . . . . . . . 132.3 Perturbing the universe. . . . . . . . . . . . . . . . . . . . . . . . . 172.4 The spherical collapse model . . . . . . . . . . . . . . . . . . . . . . 212.5 The power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 The growth of structure . . . . . . . . . . . . . . . . . . . . . . . . 282.7 The excursion set theory . . . . . . . . . . . . . . . . . . . . . . . . 372.8 The unconditional mass function. . . . . . . . . . . . . . . . . . . . 442.9 The conditional mass function . . . . . . . . . . . . . . . . . . . . . 482.10 Ellipsoidal collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.11 Moving barrier models . . . . . . . . . . . . . . . . . . . . . . . . . 542.12 Comparison with N-body simulations . . . . . . . . . . . . . . . . . 60
3 Merger history trees 64
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2 Mass histories and merger trees . . . . . . . . . . . . . . . . . . . . 683.3 The Monte Carlo merger tree . . . . . . . . . . . . . . . . . . . . . 743.4 A conditional scaling symmetry . . . . . . . . . . . . . . . . . . . . 763.5 The progenitor mass function . . . . . . . . . . . . . . . . . . . . . 783.6 The distribution of formation redshifts . . . . . . . . . . . . . . . . 843.7 The mass distribution at formation . . . . . . . . . . . . . . . . . . 893.8 The last major merger . . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Halo creation 96
4.1 Creation of dark matter haloes . . . . . . . . . . . . . . . . . . . . . 964.2 Mass history in the excursion set theory . . . . . . . . . . . . . . . 101
4.2.1 Mass history for different barriers . . . . . . . . . . . . . . . 1024.2.2 Creation times from Bayes’ rule . . . . . . . . . . . . . . . . 1064.2.3 Monte Carlo test and self-similarity . . . . . . . . . . . . . . 107
4.3 Creation time distribution . . . . . . . . . . . . . . . . . . . . . . . 110
xi
4.3.1 Distribution of creation redshifts . . . . . . . . . . . . . . . 1114.3.2 Self-similarity in halo creation . . . . . . . . . . . . . . . . . 1134.3.3 Why it doesn’t work in general . . . . . . . . . . . . . . . . 1154.3.4 Why it is a useful approximation in practice . . . . . . . . . 1194.3.5 Conditional distribution of creation redshifts . . . . . . . . . 120
4.4 Halo creation rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4.1 The unconditional rate . . . . . . . . . . . . . . . . . . . . . 1264.4.2 The time-normalized creation rate . . . . . . . . . . . . . . . 1294.4.3 The conditional rate . . . . . . . . . . . . . . . . . . . . . . 130
4.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Merger-induced Quasars 134
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.1 The halo merger rate . . . . . . . . . . . . . . . . . . . . . . 1475.2.2 The light curve . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.3 Theoretical halo merger rates . . . . . . . . . . . . . . . . . . . . . 1635.3.1 Extended Press-Schechter . . . . . . . . . . . . . . . . . . . 1635.3.2 Ellipsoidal Collapse . . . . . . . . . . . . . . . . . . . . . . . 1655.3.3 Comparison with N-body simulations . . . . . . . . . . . . . 167
5.4 Implemention of the model . . . . . . . . . . . . . . . . . . . . . . . 1725.4.1 The luminosity function . . . . . . . . . . . . . . . . . . . . 1725.4.2 The large scale bias . . . . . . . . . . . . . . . . . . . . . . . 1735.4.3 Testing the Fry formula . . . . . . . . . . . . . . . . . . . . 175
5.5 Measuring the clustering of faint AGNs and low-redshift quasars . . 1795.5.1 The clustering of faint AGNs . . . . . . . . . . . . . . . . . 1795.5.2 The clustering of low-redshift quasars . . . . . . . . . . . . . 182
5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.6.1 The luminosity function . . . . . . . . . . . . . . . . . . . . 1865.6.2 Quasar clustering . . . . . . . . . . . . . . . . . . . . . . . . 1915.6.3 The predicted black hole mass function . . . . . . . . . . . . 199
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2035.7.1 Further constraint from high-z X-ray counts. . . . . . . . . . 2035.7.2 Scaling relations . . . . . . . . . . . . . . . . . . . . . . . . . 2085.7.3 Comparison with previous models . . . . . . . . . . . . . . . 2155.7.4 Assumptions, input parameters, and degeneracies . . . . . . 2215.7.5 Black holes masses at triggering . . . . . . . . . . . . . . . . 225
5.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A Diffusion theory 237
A.1 Diffusion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237A.1.1 Constant barriers . . . . . . . . . . . . . . . . . . . . . . . . 237
xii
A.1.2 Square root barriers . . . . . . . . . . . . . . . . . . . . . . 241
B Merger tree algorithm 248
B.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248B.1.1 The branch: from random walks to mass histories . . . . . . 248B.1.2 The tree: connecting the branches . . . . . . . . . . . . . . . 251
B.2 The main branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C Coagulation and fragmentation 255
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256C.2 The binary merger-fragmentation model . . . . . . . . . . . . . . . 256
C.2.1 Linear polymers . . . . . . . . . . . . . . . . . . . . . . . . . 257C.2.2 Branched polymers . . . . . . . . . . . . . . . . . . . . . . . 260
C.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265C.3.1 General linear kernel . . . . . . . . . . . . . . . . . . . . . . 269C.3.2 Multiplicative kernel . . . . . . . . . . . . . . . . . . . . . . 271
C.4 r−mer initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 273C.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Bibliography 354
xiii
List of Tables
3.1 The conditional scaling symmetry. . . . . . . . . . . . . . . . . . . . 82
5.1 The clustering of low luminosity AGNs. . . . . . . . . . . . . . . . . 180
xiv
List of Figures
2.1 The linear power spectrum. . . . . . . . . . . . . . . . . . . . . . . 262.2 The variance in smoothed overdensities. . . . . . . . . . . . . . . . 272.3 Linearly enhanced overdensities. . . . . . . . . . . . . . . . . . . . . 292.4 The overdensity field smoothed at two scales. . . . . . . . . . . . . 312.5 Two pictures of gravitational growth. . . . . . . . . . . . . . . . . . 332.6 The spherical collapse threshold. . . . . . . . . . . . . . . . . . . . . 352.7 The smoothed overdensity field at various scales. . . . . . . . . . . . 392.8 The smoothed overdensity field with increasing variance. . . . . . . 412.9 The first crossing of a barrier by a random walk. . . . . . . . . . . . 432.10 The extended Press-Schechter argument. . . . . . . . . . . . . . . . 462.11 The two barrier problem. . . . . . . . . . . . . . . . . . . . . . . . . 492.12 Different collapse barriers. . . . . . . . . . . . . . . . . . . . . . . . 572.13 The unconditional mass function. . . . . . . . . . . . . . . . . . . . 62
3.1 The merger history tree of a halo with mass M at redshift Z. . . . . 653.2 A random walk and its associated mass history. . . . . . . . . . . . 693.3 The same as Figure 3.2, but with square-root barriers. . . . . . . . 713.4 Intersecting barriers with GIF2 time snapshots. . . . . . . . . . . . 733.5 A branch at ‘continuous’ and ‘snapshot’ redshifts. . . . . . . . . . 753.6 The progenitor mass fraction and mass function, with M/m⋆ = 6. . 793.7 Same as Figure 3.6, with M/m⋆ = 0.6. . . . . . . . . . . . . . . . . 803.8 Same as Figure 3.6, with M/m⋆ = 0.06. . . . . . . . . . . . . . . . . 813.9 The conditional scaling symmetry. . . . . . . . . . . . . . . . . . . . 833.10 Scaled distribution of formation redshifts. . . . . . . . . . . . . . . . 863.11 Distribution of formation redshifts. . . . . . . . . . . . . . . . . . . 883.12 Distribution of mass at formation. . . . . . . . . . . . . . . . . . . . 903.13 Close-up of Figure 3.12 near the peak. . . . . . . . . . . . . . . . . 913.14 The redshift distribution of the most recent major merger. . . . . . 94
4.1 The mass history associated with a random walk using a constantbarrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Same as Figure 4.1, but with a square-root barrier. . . . . . . . . . 1054.3 The creation time distribution in self-similar form. . . . . . . . . . . 109
xv
4.4 Distribution of creation redshifts for a number of bins in halo mass. 1124.5 Universality of the distribution of halo creation times. . . . . . . . . 1144.6 The mass history associated with a random walk and linear barriers. 1164.7 The creation time distribution associated with linear barriers in self-
similar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.8 Conditional distribution of creation redshifts. Symbols and style as
in Figure 4.4. We plot m = m⋆/10 (left panels) and m = m⋆/100(right panels) conditioned to end up in haloes of mass M = M⋆ (toppanels) and M = 10m⋆ (bottom panels). In all cases, T denotes thepresent time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.9 Halo creation rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.10 Conditional creation rates. . . . . . . . . . . . . . . . . . . . . . . . 132
5.1 The quasar light curve. . . . . . . . . . . . . . . . . . . . . . . . . . 1615.2 Halo merger rates divided by mass function vs. mass ratio. . . . . . 1685.3 Ellipsoidal collapse merger rates. . . . . . . . . . . . . . . . . . . . 1705.4 Ellipsoidal collapse merger rates with linear ξ. . . . . . . . . . . . . 1715.5 Testing the Fry formula. . . . . . . . . . . . . . . . . . . . . . . . . 1785.6 The bias of low-luminosity type II AGNs. . . . . . . . . . . . . . . . 1815.7 The bias of low-redshift quasars. . . . . . . . . . . . . . . . . . . . . 1845.8 Predicted bolometric luminosity function at different redshifs. . . . 1875.9 Predicted luminosity function from a model with with and without
mass dependence in the light curve. . . . . . . . . . . . . . . . . . . 1895.10 Predicted bias as a function of luminosity at different redshifts. . . . 1925.11 Comparison among models characterized by different delay times. . 1955.12 Comparison between reference and alternative models (with less mas-
sive hosts and minor mergers. . . . . . . . . . . . . . . . . . . . . . 1985.13 Predicted black hole mass function at different redshifts. . . . . . . 2005.14 Predicted black hole mass functions for the reference and alternative
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.15 Predicted X-ray number counts and bolometric luminosity functions. 2045.16 Predicted MBH-σstar relation at different redshift and different values
of the ratio between circular velocity Vc and virial velocity Vvir. . . . 2105.17 Predicted MBH-Mstar relation at different redshift, as labeled, ob-
tained by matching our reference model MBH-M relation with theempirical Mstar-M relation obtained from cumulative number match-ing techniques.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.18 Relevant black hole masses involved in a halo merger of haloes. . . . 227
A.1 Diffusion with a constant absorbing barrier. . . . . . . . . . . . . . 239A.2 Exact square-root barrier solution. . . . . . . . . . . . . . . . . . . 242
B.1 A sample merger history tree of a halo. . . . . . . . . . . . . . . . . 249
xvi
B.2 Main branch algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 253
C.1 Haloes as branched polymers. . . . . . . . . . . . . . . . . . . . . . 262
xvii
Chapter 1
Prologue
Understanding how the universe and its constituents evolve has been one of the
longest quests in human history. The last century saw how this quest not only
enticed astronomers, but physicists as well. Our current understanding states that
slightly overdense regions in the early universe were enhanced by gravity, and
became virialized. These gravitationally-bound dark matter haloes merged with
one another, building more massive haloes in a hierarchical fashion. This process
strongly affects the assembly and nature of galaxies. Understanding galaxy forma-
tion theoretically is one of the most exciting challenges in physics today – and I
believe that it should be regarded as important as understanding critical phenom-
ena in condensed matter physics, or the unification of fundamental interactions in
high-energy physics.
Aside from observations, a great deal of knowledge can be inferred with the
1
aid of N-body simulations. From the early numerical work of Toomre & Toomre
(1972) to the Millennium Simulation of Springel & Hernquist (2005), N-body mod-
eling of gravitational growth has seen unprecedented developments. Unfortunately,
our understanding from an analytic point of view has not increased at the same
pace. One of the aims of the present thesis is to gain insight on the assembly of
gravitationally-bound dark matter haloes using analytic and semi-analytic meth-
ods. Such methods not only complement current N-body studies, but allow us to
formulate the language necessary to interpret such simulations.
The excursion set formalism, which is based on the properties of gravitational
instability, successfully describes many properties related to the abundance of dark
matter haloes. This formulation takes advantage of what is known about Brown-
ian motion random walks to gain insight. The original version describes collapse
spherically. A revised version, which incorporates ellipsoidal collapse, provides a
major improvement. In this work we advocate a model of collapse expressed in
terms of a ‘square-root’ moving barrier. This simple model correctly describes the
abundance of haloes (the mass function) and it predicts a scaling relation for those
haloes conditioned to belong to more massive haloes at a later time.
Background material, including the excursion set formalism, is introduced in
Chapter 2. Chapter 3 is dedicated to the formulation of a merger tree algorithm
based on this square-root barrier model. The assembly of haloes through mergers
is equivalent to a branching process if time is reversed. One particular feature
2
that distinguishes this algorithm from others in literature is that mass (not time) is
discretized. This property facilitates the study of many problems. Predictions of the
conditional mass function, and the distribution of redshifts and masses at formation
are tested against analytic formulas and measurements in N-body simulations. We
also test the redshift distribution of the last major merger (one where the ratio of
the masses involved is at least one third).
A description of the times at which these mergers happen – the so-called creation
time distribution – is formulated in the context of ellipsoidal collapse with moving
barriers. An interesting ‘Bayesian’ link between this quantity and the mass function
is discussed in Chapter 4. We are the first to extend this link to the conditional case.
The creation of haloes by mergers of smaller objects can be described in terms of
the Smoluchowski coagulation equation. This formalism has been applied in many
areas of physics and astronomy (see e.g., Benson, 2008, and references therein).
A formulation for halo mergers in a very special case – with white-noise initial
conditions and spherical collapse – has been available for a number of years. We
use this result as a guide to extract the creation term from the time-derivative of the
mass function with general initial conditions and ellipsoidal collapse. More details
on this formalism, its relation to polymer growth, and a discussion of fragmentation
are found in Appendix C. This is a clear example of how knowledge from other
fields can yield fruitful results. We conclude this chapter with a discussion of how
halo creation is related to the concept of halo formation described in Chapter 3.
3
Mergers of haloes (and their associated galaxies) have a huge impact on cer-
tain phenomena, including sudden bursts of star formation, and quasars. After
a merger takes place, gas is funneled towards the supermassive black hole at the
center. This phase of fast accretion is eventually quenched when the associated
luminosity is powerful enough to unbind the reservoir of gas feeding the black hole.
In Chapter 5, we develop a model that convolves the halo merger rate with a uni-
versal quasar light curve consisting of three phases: an ascending phase describing
efficient accretion, a peak luminosity controlled by a postulated self-regulation con-
dition, and a mass-dependent descending phase describing the shut down of the
quasar. The reference model, and possible extensions are discussed and constrained
by observations including the AGN luminosity function, the clustering of quasars,
the local black hole mass function, and X-ray number counts. Implications of the
model on scaling relations between the nuclear black hole and properties of the
hosts are addressed.
This thesis is the result of investigations carried during the last few years. The
presentation here does not match the chronological order in which results were ob-
tained, nor it presents all the attempts considered. This thesis is very different
from many others in astronomy. It picks and borrows ideas from random Brownian
processes and diffusion theory; polymer coagulation; and it makes use of analytic,
Monte Carlo, and N-body techniques. Its final chapter on quasars is the first step
towards many possible astrophysical applications of the ideas presented here. Our
4
quasar model is very powerful, despite its simplicity, and matches (or has the poten-
tial of matching with some modifications) several independent sets of observations.
The results presented in this thesis can be found in a number of research papers:
- An improved model for the formation times of dark matter haloes.
C. Giocoli, J. Moreno, R. K. Sheth & G. Tormen (2006).
MNRAS, 376, 977-983.
- Merger history trees of dark matter haloes in moving barrier models.
J. Moreno, C. Giocoli & R. K. Sheth (2008).
MNRAS, 391, 1729-1740
- Dark matter halo creation in moving barrier models.
J. Moreno, C. Giocoli & R. K. Sheth (2009).
MNRAS, 397, 299-310
- Models of reversible coagulation and fragmentation.
R. K. Sheth & J. Moreno (2010).
J. of Phys. A, submitted.
- Merger-induced quasars, light curves and their host haloes
F. Shankar, J. Moreno, D. H. Weinberg, R. K. Sheth, Martin Crocce,
Cheng Li, R. E. Angulo, and Federico Marulli (2010).
MNRAS, in preparation.
5
This work would not have been possible without the guidance, patience, and
support of my thesis advisor, Ravi Sheth, who is certainly the most influential
person in my career. I am indebted to Carlo Giocoli for providing the N-body
simulation data used in Chapters 3 and 4, for all his constant help, encouragement,
and for clarifying all my questions. The mentoring of Francesco Shankar on the
subject of quasars and his input in Chapter 5 was invaluable. This thesis is a
result of a collaboration with these three individuals, and it required the help of
many others, including Bepi Tormen, David Weinberg, Martin Crocce, Cheng Li,
Raul Angulo and Federico Marulli. In particular, the input and expertise of the
first two individuals in this list improved the quality of our work greatly. On the
other hand, the last four provided invaluable analyses of the passive evolution of
clustering in simulations (e.g., the ‘Fry formula’) and the clustering of faint type 2
AGNs and quasars at low redshift. Nevertheless, I take full responsibility on the
results presented here. It is my hope that you find this thesis and the problems
addressed here enjoyable and interesting.
6
Chapter 2
Introduction
The aim of this chapter is to introduce the necessary background ideas in cosmology
for this thesis. It is not our intention to present a comprehensive treatise, which
can be found in several excellent books that treat these topics in more detail. Some
of them, which were consulted at some point or another, include Peebles (1980),
Peacock (1999), Dodelson (2003) and Carroll (2004). A brief introduction to the
excursion set formalism with spherical and ellipsoidal collapse is included. An
introduction to quasars and supermassive black holes is deferred to Chapter 5.
2.1 In the beginning ...
We live in a special time in history because, for the first time, mankind has a
quantitative understanding of the universe and its evolution. The basic picture is
that the universe is homogeneous and isotropic. Observations reveal that, at large
7
enough scales, it looks pretty much the same everywhere, with no special directions.
Moreover, it is reasonable to assume that observers at distant locations would reach
similar conclusions. This view overthrows the ancient Aristotelian-Ptolemic belief
that we are at the center of the universe, with the heavens rotating around us.
Galileo’s discovery of Jupiter’s moons and the phases of Venus were among the
first pieces of evidence against that belief. The Copernican principle, which states
that the universe does not revolve around us, was put on firm ground. Perhaps
the most triumphant result in this direction is Newton’s unification of the motions
of terrestrial and heavenly bodies. Planets and apples are described by the same
natural laws.
Since that time, theories and observations have improved dramatically. In the
1700’s, Kant and Herschel realized that our sun is not the center of the universe
either, but merely a member of a group of stars called the Milky Way. It is tempting
to believe that this galaxy comprises the whole universe. But not only stars and
planets fill the sky; the presence of smudges – which Messier dubbed ‘nebulae’ –
caused great concern. Were these part of the galaxy, or faraway objects? Eventually,
Leavitt’s observations of cepheid variable stars helped measure the distance to these
objects once and for all. With this, it was established that these nebulae are in fact
their own separate galaxies. This was followed by the advent of spectroscopy, which
showed that distant stars and galaxies are made of the same elements found here on
Earth and in our Sun. The universe now seems immensely bigger than previously
8
thought, but not vastly different from our local neighborhood.
The Copernican principle came into question once more, when Hubble observed
that the spectra of distant galaxies are highly redshifted, meaning that these galaxies
are racing away from us. Did this mean that our galaxy is at the center? He also
noticed that the radial velocities of these galaxies were proportional to their distance
from us. Such behavior is allowed in a universe that is expanding everywhere – one
that in principle could reconcile Hubble’s law with the Copernican idea. If so,
neither our solar system nor our galaxy are at the center of the universe – the
universe is expanding everywhere and it has no center!
Observational discovery is often accompanied by theoretical breakthroughs. Ein-
stein replaced the Newtonian notion of absolute space and stated that gravity was
a manifestation of a dynamical space-time. Friedman and Lemaitre championed
the idea of an expanding universe as a solution to Einstein’s equations with ho-
mogeneity and isotropy. If the universe is really expanding, one could conclude
that in a distant past everything was closer together. Furthermore, that there must
have been an initial moment in which the universe came into being. Evidently,
this pleased the Church, but not those who believed in an eternally old universe.
Einstein advocated a static universe, one where space and time were homogeneous;
one that obeyed the ‘perfect’ Copernican principle. To this end, he introduced an
‘ad-hoc’ cosmological constant to his field equations, something he would later refer
to as his life’s ‘biggest blunder’. Even after Hubble reported that galaxies were
9
receding from us, Hoyle and other skeptics proposed that the universe could be
both eternal and expanding if matter was spontaneously created as space stretched.
To some extent, this contrived steady-state solution seemed reasonable, in view of
the fact that no direct evidence of a primordial ‘big bang’ was available. Big bang
theory predicts that the early universe was in a very hot and dense state, and that
a background cosmic glow must remain. As this controversy continued, Peebles
and Dicke of Princeton went forth to detect this radiation. Everything changed
when Penzias and Wilson, a few miles away, announced the discovery of a cosmic
microwave background (CMB) with primeval origins.
The universe we live in is very homogeneous and isotropic. The distribution
of galaxies is roughly the same everywhere. Even the temperature of the CMB is
nearly the same in every direction. The same cannot be said about its evolution. As
the universe expands, it cools and becomes more diffuse, experiencing several phase
transitions as a result. Our understanding of the first 10−43 seconds is somewhat
fuzzy, although a theory that describes fundamental particles as modes of extended
stringy objects seems promising. By 10−34 seconds, inflationary expansion followed,
exponentionally growing the primordial universe to great proportions. At this point,
it is believed that the strong interaction decoupled from grand unification, making
the first quarks and leptons. At 10−13 seconds, quarks bound up into protons
and neutrons. These and other particles interacted in this primordial soup, and
the lightest elements were synthesized. This plasma was so hot and dense that
10
photons scattered off electrons, preventing the formation of atoms. As the universe
expanded, this plasma cooled. The first atoms formed (recombined), light streamed
freely, and the universe became transparent. The wavelength of such light has been
stretched by the cosmic expansion, to the order of microwaves. This is the cosmic
microwave background (CMB) predicted by the Big Bang theory, and observed by
Penzias and Wilson.
Science took centuries to get to this stage, with an expanding universe that
is homogeneous and isotropic; but nature had more surprises in store. The first
is that the particles we are made of account for only about 4 percent of the con-
tents of the universe. Most of the matter consists of dark matter, which reveals
itself gravitationally in the motions of the stars and globular clusters around the
galaxy, galaxies in clusters, and gravitational lenses. The current version of the
Standard Model of particle physics does not include dark matter. Supersymme-
try could provide a viable candidate, and there is hope that the Large Hadronic
Collider at CERN or the CDMSII experiment in Minnesota could shed light on
its properties. Embarrasingly enough, matter itself only contributes to about 30
percent of the universe. Our knowledge of the other 70 percent, which causes the
universe expansion to accelerate, is even more limited. Some call it ‘dark energy’,
while others associate it with the energy of the vacuum, a cosmological constant.
Another problem is that, although the universe looks rather smooth at large scales,
it is certainly not so at smaller scales. Galaxies aggregrate in clusters, walls, and
11
filaments in the so-called cosmic web. A possible explanation is found in the theory
of inflation, which predicts that quantum fluctuations were exponentially enlarged
into density perturbations of the smooth universe – the very seeds of the large scale
structure we see today. It is reassuring to know that recent observations of the
CMB by space missions, such as the Cosmic Background Explorer (COBE) and
the Wilkinson Microwave Anisotropy Probe (WMAP), indicate that this smooth
background has tiny fluctuations of one part in 105.
Our story begins here. As the universe became matter-dominated and transpar-
ent, another transition began. The tiny perturbations were enhanced by gravity,
and became virialized objects. These objects, known as haloes, merged with other
haloes, making larger and larger objects. Their associated potential wells became
deeper and the gas within cooled and turned into stars. Aside from the primordial
elements, heavier atoms are cooked in stars. Some stars exploded into supernovae,
dumping their heavy metals into the interstellar medium. New stars formed, recy-
cled this gas into more elements, making the formation of planets and life possible.
Galaxies collided or simply became bound with others into galaxy clusters. Such
collisions funneled cool gas to the centers, triggering sudden bursts of star for-
mation. These mergers also helped supermassive black holes at the centers grow,
allowing their surrounding accretion disk to shine at immense luminosities that
can be observed at cosmological distances. Quasars, as these events are known,
are strong evidence that the universe is evolving. Understanding the interactions,
12
cannibalism, gastronomy and morphology of galaxies is a quite complex challenge.
Nevertheless, many properties can be explained by following the hierarchical assem-
bly of the dark matter haloes that contain them. This process, in turn, is strongly
determined by the properties of the initial fluctuations and the evolution of the
background universe.
But the story does not end there. If our theories are correct, our universe is about
to undergo another such phase transition, into an accelerated state dominated by a
cosmological constant (or dark energy). Cosmic acceleration will prevent the largest
regions from becoming bound, and the existing structures will become isolated in
this forever expanding universe. The structure, evolution and fate of our universe
is indeed a fascinating subject. We now introduce the basic tools necessary for
this description. For the sake of clarity, we postpone our introduction to the broad
subject of quasars and supermassive black holes for Chapter 5.
2.2 The smooth expanding universe.
In the theory of General Relativity, the metric gµν describes the geometry of space-
time. In particular, a line element is given by
ds2 = gµνdxµdxν , (2.1)
where µ, ν = 0, 1, 2, 3 and repeated indices get summed (Einstein’s convention). For
instance, for a homogeneous-isotropic universe, this is given by
13
ds2 = c2dt2 − a2(t)[ dr2
1− κr2+ r2(dθ2 + sin2 θdφ2)
]
, (2.2)
where c is the speed of light, (r, θ, φ) are the spherical spatial coordinates and a(t)
will be referred to as the scale factor. This is known as the Friedman-Robertson-
Walker (FRW) metric. Here the global geometry of the universe (and its fate) is
encapsulated in the curvature constant: κ > 0 (closed), κ = 0 (flat) or κ < 0 (open).
In this theory, the metric is a dynamical quantity, which obeys the Einstein field
equations:
Rµν −1
2Rgµν =
8πG
c3Tµν + Λgµν , (2.3)
where Rµν is the Riemann tensor, R is the Ricci scalar, G is Newton’s gravitational
constant. Spacetime is determined by the contents of the universe, encoded in Tµν ,
the stress-energy tensor. Notice that we have explicitly included the cosmological
constant Λ. For an ideal isotropic-homogeneous fluid with density ρ and pressure
p, from the Einstein equations one can obtain
( a
a
)2
=8πG
3ρ(a)− κc2
a2+
Λ
3. (2.4)
This is known as the Friedman equation, and it describes the evolution of the scale
factor a in time. Λ is often associated with the vacuum energy. Defining
ρΛ =Λ
8πG(2.5)
allows us to write the Friedman equation as
H2(a) =8πG
3
∑
j
ρj(a)− κc2
a2. (2.6)
14
where the Hubble parameter is defined by H(a) = a/a, which is determined by κ
and the densities of the different components. Each of these constituents evolves
differently with a. Explicitly, density is composed of relativistic matter (radiation),
ordinary matter, and vacuum energy, and it evolves as
ρ(a) =∑
i
ρi(a) = ρr(a) + ρm(a) + ρΛ(a) =ρr,0
a4+
ρm,0
a3+ ρΛ,0, (2.7)
where the quantities with the ‘0’ subscript are evaluated at the present time. It is
conventional to set the present time scale factor to unity (a0 = 1) and to express
the Hubble constant today as H0 = 100h km s−1 Mpc, with h = 0.7 as the accepted
value. The general solution is not simple. For a flat universe with matter only,
known as an Einstein de Sitter (EdS) cosmology, solving the Friedman equation is
fairly straightforward. Here, ρ(a) = ρm(a) = ρm,o/a3, a(t) = (t/t0)
2/3 and H(t) =
2/(3t). We will often make reference to this cosmology.
A flat κ = 0 is also known as a critical universe. In such a case, one can define
the critical density as
ρcrit(a) =3H2
8πG. (2.8)
Dividing equation (2.6) by ρcrit(a) and setting
Ωj(a) =ρj(a)
ρcrit(a), Ωκ(a) = − κc2
a2H2, (2.9)
the Friedman equation reads
1 =∑
j
Ωj(a) + Ωκ(a). (2.10)
15
The current values of the cosmological parameters today are (Ωm,0, ΩΛ,0, Ωκ,0, Ωr,0) ≃
(0.3, 0.7, 0, 10−4). In a flat cosmology with neglegible Ωr, the quantities Ωm(a) and
ΩΛ(a) are easily obtained. First one needs to rewrite the critical density as
ρcrit(a) = ρcrit,0
( H
H0
)2
= ρcrit,0
(Ωm,0
a3+ ΩΛ,0
)
, (2.11)
from which
Ωm(a) =Ωm,0/a
3
ΩΛ,0 + Ωm,0/a3, ΩΛ(a) =
ΩΛ,0
ΩΛ,0 + Ωm,0/a3(2.12)
immediately follow.
The Friedman equation describes the evolution of a(t). Now suppose that light
is emitted from a distant galaxy with wavelength λem. As the universe expands,
this wavelength gets stretched. If peculiar (local) motions are neglegible relative to
the bulk expansion, and we measure a wavelenght λobs on earth, then the difference
between these two is given by the redshift, defined as
z =λobs − λem
λem
=λobs
λem
− 1 =aobs
aem
− 1. (2.13)
Here, aobs and aem are the values of the scale factor at the observing and emitting
times, respectively. If one observes today light emitted at a time t < t0, then
a(t) =1
1 + z. (2.14)
Redshift, like the expansion factor, is a common proxy for cosmic time.
16
2.3 Perturbing the universe.
The homogeneous-isotropic FRW metric describes the universe well at large scales.
At smaller scales, we see that tiny inhomogeneities in the initial density field have
evolved gravitationally into the highly non-linear structures we see today: the cos-
mic web. Therefore the mean matter density in the Friedman equation, now denoted
by ρ0(t), needs to be replaced by a density field ρ(~x, t).1 If matter is treated as an
ideal fluid, then ρ obeys the following dynamical equations:
• Euler’s equation:
D~v
dt= −∇~x φ− ∇~x p
ρ. (2.15)
• The continuity equation:
Dρ
dt= −ρ∇~x · ~v. (2.16)
• Poisson’s equation:
∇2~x φ = 4πGρ. (2.17)
Here ~v(~x, t) is the velocity field, φ(~x, t) is the gravitational potential, p(~x, t) is the
pressure field,
D
dt=
∂
∂t+ ~v · ∇~x (2.18)
denotes the convective derivative and the ~x-index in ∇~x is set to emphasize that we
are working on physical (rather than comoving) coordinates.
1In this section, the ‘0’-subscript denotes ‘unperturbed’ quantities.
17
Suppose we write
ρ(~x, t) = ρ0(t) + δρ(~x, t), (2.19)
p(~x, t) = p0 + δp(~x, t), (2.20)
~v(~x, t) = ~v0 + δ~v(~x, t), (2.21)
φ(~x, t) = φ0 + δφ(~x, t). (2.22)
In the absence of δρ, δp, δ~v and δφ, the unperturbed quantities also obey the
above three fluid equations. Assuming the perturbations are small, one can linearize
the fluid equations, obtaining:
d(δ~v)
dt= −[(δ~v) · ∇~x]~v0 −∇~x (δφ)− ∇~x(δp)
ρ0
, (2.23)
d(δρ)
dt= −ρ0∇~x · ~v, (2.24)
∇2~x(δφ) = 4πG (δρ), (2.25)
where
d
dt=
∂
∂t+ ~v0 · ∇~x. (2.26)
In comoving coordinates, ∇~x = (1/a)∇, δ~v = a~u, and the fluid equations become:
d~u
dt+ 2H~u = −∇(δφ)
a2− ∇(δp)
a2ρ0
, (2.27)
dδ
dt= −∇ · ~u, (2.28)
∇2(δφ) = 4πGa2ρ0 δ, (2.29)
18
where
δ(~x, t) ≡ ρ(~x, t)− ρ0
ρ0
=(δρ)
ρ0
(2.30)
is the overdensity contrast.
Combining the divergence of the Euler equation with the time derivative of the
continuity equation, it can be shown that δ obeys
δ + 2Hδ = (4πGρ0δ +c2s∇2
a2)δ, (2.31)
where c2s = δp/δρ is the equation of state of the fluid. The Fourier transform of this
differential equation is
¨δ + 2H ˙δ = (4πGρ0 −c2sk
2
a2)δ, (2.32)
where
δ(~k, t) ≡∫
d3x δ(~x, t) ei~k·~x. (2.33)
The coefficient of ˙δ in equation (2.32) may be thought of as a friction factor, or Hub-
ble drag. The coefficient of δ can be interpreted as the competition of two effects:
gravity and gas pressure. These two effects balance out at a physical wavelength
λJ =2πa
k= cs
√
π
Gρ0
, (2.34)
called the Jeans wavelength.
Suppose the solution is separable: δ(~k, t) = D(t)δ(~k, ti), where ti is some initial
time. Inserting this into equation (2.32), one gets
D + 2HD− 4πGρ0D = −c2sk
2
a2. (2.35)
19
Notice that the left-hand side of the equation is k-independent. For this ansatz to
work, the right-hand side must also be independent of k. This is only true for scales
with λ = 2π/k ≫ λJ. Theories of structure formation favor the assumption that
dark matter is cold (pressureless), making cs neglegible. In other words, gravity
dominates for all the scales of interest. In this case, one is left with
D + 2HD− 4πGρ0D = 0. (2.36)
The solution to this second-order linear differential equation is a linear combination
of two factors: D+(t) (the growth factor) and D−(t) (the decay factor). Except for
the very earliest moments, the decay factor is neglegible and we will ignore it. From
this point on, D+ will be denoted as D.
For an EdS cosmology, recall that a ∝ t2/3. Therefore, H = 2/(3t) and 4πGρ0 =
(3/2)H2 = 2/(3t2). Equation (2.36) then becomes
D +4D
3t− 2D
3t2= 0. (2.37)
Assuming D(t) ∝ tα, then α obeys a quadratic equation, given by
3α2 + α− 2 = 0, (2.38)
The positive root, associated with the growing mode, is α = 2/3. That is, D(t) ∝
t2/3 ∝ a(t) in an Einstein de Sitter universe.
Equation (2.36) has no analytic solution for every cosmology, but Carroll et al.
(1992) provide an accurate approximation for flat cosmologies, given by
D(a) ≃ 5
2aΩm(a)
[
Ωm(a)4/7 − ΩΛ(a) +(
1 +Ωm(a)
2
)(
1 +ΩΛ(a)
70
)]−1
. (2.39)
20
For alternative expressions, we refer the reader to Heath (1977); Lightman &
Schechter (1990); Lahav et al. (1991); Navarro et al. (1997) and Percival et al.
(2000). See also Silveira & Waga (1994); Wang & Steinhardt (1998); Basilakos
(2003) and Percival (2005) for generalizations.
2.4 The spherical collapse model
Gravity tends to enhance slightly overdense regions. For sufficiently early times,
density fluctuations evolve as δ(~x, t) = δ(~x, ti)D(t)/D(ti). As soon as δ(~x, t) → 1,
linear theory breaks down, and one usually has to rely on N-body simulations to de-
scribe the evolution thereafter. Nevertheless, under certain assumptions, simplified
models that capture the essential features of nonlinear gravitational growth have
been formulated. The spherical collapse model is one of them.
Imagine two spherical regions, one unperturbed and the other perturbed, con-
taining the same mass m. The radius of the first one, denoted by Rb(t), will follow
the background expansion and evolve according to the Friedman equation. The
perturbed region (with overdensity δ) will follow the background universe early on.
Eventually, self-gravity will cause it to decouple from the overall expansion. Its
radius, denoted by Rp(t), will reach a maximum value, turn around, and collapse.
This region can be thought of as a sub-universe with a density larger than crit-
ical (i.e., a closed sub-universe). Since both regions contain the same mass, the
21
overdensity of the perturbed region can be written as
1 + δ(t) =R3
b(t)
R3p(t)
. (2.40)
Finding solutions for any cosmology is non-trivial. Gunn & Gott (1972) worked out
the case for an Einstein de Sitter (EdS) cosmology, and Lacey & Cole (1993) solved
it for an open universe (also see Navarro et al., 1997). Solutions that incorporate Λ
are found in Barrow & Stein-Schabes (1984); Lahav et al. (1991); Barrow & Saich
(1993); Eke et al. (1996); Kitayama & Suto (1996); Navarro et al. (1997); Percival
et al. (2000) and Percival (2005). For a similar treatment with underdense regions
(associated with voids), see Sheth & van de Weygaert (2004).
For the sake of illustration, we present the case with an EdS background cos-
mology. The solution for the perturbed region can be written parametrically as
Rp(θ) = A(1− cos θ), t(θ) = B(θ − sin θ), (2.41)
where 0 < θ ≤ 2π and the constants obey A3/B2 = Gm. The maximum radius
is reached when θ = π and collapse happens at θ = 2π. At turn around (θ = π),
t = πB. In principle, when θ = 2π, Rp = 0 and the region collapses to an infinitely
dense point. We will denote this time as tcoll = 2πB. In reality, dissipative physics
intervenes and random motions turn the region into a virialized object.
We will be particularly interested in the evolution at early times (the linear
regime). At early times (small θ),
Rp(θ) = A(θ2
2!− θ4
4!+ . . .
)
, t(θ) = B(θ3
3!− θ5
5!+ . . .
)
. (2.42)
22
Eliminating θ, we obtain
Rp(t) = (Gm)1/3t2/3 62/3
20
[
1− 62/3
20
(2πt
tcoll
)2/3
+ . . .]
= Rb(t)[
1− 1
20
(12πt
tcoll
)2/3
+ . . .]
.
(2.43)
The term in brackets can be interpreted as a first order correction to the background
evolution. Taking the cube of Rp/Rb and substituting into equation (2.41) we obtain
that, to leading order,
δ(t) =3(12π)2/3
20
( t
tcoll
)2/3
=3(12π)2/3
20
D(t)
D(tcoll). (2.44)
At collapse, such sphere will have an overdensity
δc ≡ δ(tcoll) =3(12π)2/3
20≃ 1.686. (2.45)
This critical threshold will be key to select virialized objects from the (linear-
extrapolated) initial overdensity field.
2.5 The power spectrum
In the cold dark matter (CDM) scenario, different Fourier modes of the perturbation
field evolve independently. In order to describe the evolution of these perturbations,
all we need is the growth factor and the initial perturbation field δ(~x) at some initial
time ti. In practice, it is impossible to know this field exactly at all points. Hope is
not lost if one acknowledges that theories of gravitational clustering are stochastic
in nature, not deterministic. That is, one does not need to follow every particle to
23
learn about the large scale structure of the universe. Instead, one may study this
process statistically, by treating δ(~x) as a Gaussian random variable. In this case,
the δ(~x)-distribution is completely specified by its two-point (correlation) function:
δ(2)(~x, ~x′) ≡< δ(~x)δ(~x′) >= δ(2)(|~x− ~x′|), (2.46)
which depends only on |~x− ~x′| if the universe is isotropic. It is common to work in
terms of the Fourier transform of this quantity:
< δ(~k)∗δ(~k′) >= (2π)3δD(|~k − ~k′|)P (k), (2.47)
where δD is the Dirac delta and P (k) = P (|~k|) is known as the power spectrum.
We are interested in the power spectrum early on in the matter-dominated universe
(which we call the initial power spectrum). This is related to the primordial power
spectrum p(k) as follows:
P (k) = T 2(k)p(k) (2.48)
where T (k) is known as the transfer function. Here we will assume a Harrison-
Zeldovich primordial power spectrum: p(k) ∝ kn (Harrison, 1970; Zeldovich, 1972),
with n = 1 being the preferred value in inflationary theories (Guth, 1981; Guth
& Pi, 1982). All the physical processes that turn the primordial into the initial
power spectrum are encapsulated in the transfer function. Several accurate fits
for this quantity exist in the literature (Bardeen et al., 1986; Efstathiou et al.,
1992; Sugiyama, 1995; Eisenstein & Hu, 1999). In this work we use the outputs
of cmbfast (Seljak & Zaldarriaga, 1996). Figure 2.1 shows the resulting power
24
spectrum (solid blue curve), along with a featureless scale-free power spectrum
P (k) ∝ k−2 (dashed red line).
So far we have only given the k-dependence of the power spectrum. We now
explain how the normalization is obtained. Consider a spherical region of radius R
at time ti. The mean overdensity in this region is
δR(~x) =
∫
d3y WR(|~x− ~y|)δ(~y), (2.49)
where WR is some spherically symmetric window (filter) function. Given this, the
variance in overdensity within spherical regions of radius R is
S(R) = σ2(R) ≡< |δR(~x)|2 > =
∫
dk3
(2π)3W 2(kR)P (k), (2.50)
where we have used equation (2.47). Here, W (kR) is the Fourier transform of WR.
The intuitive choice for the window function is
W(TH)R (r) =
(43πR3)−1 if r ≤ R;
0 if r > R,
a top-hat in real space. We will see that it is more convenient to use a so-called
sharp k-space window function, given by
WR(r) = (3/r3)(sin r − r cos r), (2.51)
whose Fourier transform is a top-hat.
To normalize the power spectrum it is common practice to use σ8, the rms of
the linearly-extrapolated density fluctuations (equation 2.50) in spherical regions
25
Figure 2.1: The linear power spectrum. The solid blue curve is the cmbfast output,
and the dashed red line is an approximation that scales like k−2.
26
Figure 2.2: The variance in smoothed overdensities. The solid blue curve is derived
from the cmbfast power spectrum. A power spectrum that scales as k−2 predicts
that S(m) scales as m−1/3 (dashed red line). Both use a sharp-k space filter function
(equations 2.51) with σ8 = 0.9 normalization.
27
of radius 8 Mpc h−1 (equation 2.50). The accepted value we use is σ8 = 0.9. The
power spectra in Figure 2.1 are normalized to this value.
For sufficiently small initial perturbations, it is safe to assign a mass m =
(4π/3)R3ρ0(1 + δR) ≃ (4π/3)R3ρ0 to regions of radius R, allowing us to write
S(m) = S(m(R)). We plot this quantity in Figure 2.2 (solid blue curve). For a
scale-free power spectrum ∝ k−2, it can be shown that S(m) ∝ m−1/3 (dashed red
line). S(m) with a sharp-k space filter will play a key role later in this chapter. For
fits to this quantity, see Kitayama & Suto (1996); Navarro et al. (1997) and Taruya
& Suto (2000). In this work we compute S(m) numerically.
2.6 The growth of structure
The pioneering work of Press & Schechter (1974), followed by the excursion set
formalism of Bond et al. (1991), provides a simple prescription for estimating the
abundance of virialized objects, the so-called dark matter haloes. The key is to ac-
knowledge that the growth of structure via gravitational instability can be described
by studying the statistics of the initial field of density perturbations.
Consider the initial density fluctuation field, filtered with a sharp-k space window
function of size R containing a mass m. This is illustrated in Figure 2.3; see the
field δm(~x, ti) in light red near the bottom. Here ~x denotes the position of the filter’s
center. As we vary this center across the universe, we get very different values of δm.
On average, δm is zero and the various fluctuations are distributed in a Gaussian
28
Figure 2.3: Linearly enhanced overdensities. The smoothed ovedensity field at an
initial time ti (light red); linearly extrapolated to t (dark red) and t0 (black). Here ~x
denotes the centers of the smoothing filters and the dotted horizontal line represents
the spherical collapse threshold δc (equation 2.45).
29
fashion about this value. Thus, the probability of drawing a particular value δm
between δ and δ + dδ is given by
Q(δ|Sm)d δ =1√
2πSm
exp(
− δ2
2Sm
)
d δ, (2.52)
where Sm = Sm(ti) is the variance of this smoothed field at t = ti.
The next step is to linearly extrapolate the initial (smoothed) overdensity field:
δm(~x, ti) → δm(~x, t) = δm(~x, ti)D(t)/D(ti). The field near the middle of Figure 2.3
(in dark red) is extrapolated to some t > ti, while the one at the top (in black) is
extrapolated to the present. These are identical to the initial field, up to a mul-
tiplicative factor. Furthermore, they are also drawn from a Gaussian distribution,
but now the variance is Sm(t) = Sm(ti)D2(t)/D2(ti). Some very strong assump-
tions are being made here. First of all, the non-linear evolution of regions beyond
δ ∼ 1 is treated linearly. Secondly, extrapolated fluctuations can be very large, and
underdensities will go well beneath −1. This is physically incorrect, strictly speak-
ing. In reality, the distribution of non-linear evolved overdensities will be highly
non-Gaussian. Nevertheless, we will see that such an oversimplified model does an
exceptional job at predicting the abundances of virialized objects.
Overdensity fluctuations will be smaller on average, if a bigger filter is used
(Figure 2.2 and equation 2.52). This is illustrated in Figure 2.4, where the thin light
green curve represents the perturbation field smoothed with a scale m, whereas that
smoothed with a larger scale M > m is shown as the thick dark green curve. In
this figure, both fields are linearly evolved to the present. The horizontal dotted
30
Figure 2.4: The overdensity field smoothed at two scales. Two smoothing scales
containing mass m (thin light green) and M > m (thick dark green) are used on
the same initial overdensity field, which is then extrapolated to the present. The
dotted horizonal line is the spherical collapse threshold, which is pierced more often
when the smaller filter is used.
31
line denotes the critical collapse threshold, δc ≃ 1.686, predicted by the spherical
collapse model (Section 2.4). Notice that the smoothed field δm(~x, t0) is more likely
to pierce this threshold at more points than δM(~x, t0), the field with the larger
filter. As we will see in the next chapter, this property is essential in determining
the abundance of haloes of a given mass.
At early times, fluctuations tend to be much more smaller than the spherical
collapse threshold. As the field (smoothed at any given scale) is linearly extrapo-
lated to later times, it will be above this threshold at more places. An equivalent
picture is to simply evolve the initial overdensity field (smoothed at different scales)
to the present time and raise the collapse barrier as
δc(t) = δc0D(t0)
D(t), (2.53)
where δc0 ≡ δc(t = t0) ≃ 1.686. This picture is illustrated in Figure 2.5 (compare
with Figure 2.3).
In the first picture, δc is fixed and the variance in the smoothed initial field is
linearly amplified in time as Sm(t) = Sm(ti)D2(t)/D2(ti). In the second picture, the
variance is fixed at Sm = Sm(ti)D2(t0)/D
2(ti) and the spherical collapse threshold
depends on time as δc(t) = δc0D(t0)/D(t). We will see that the abundance of haloes
can be conveniently described in terms of a ‘self-similar’ variable defined as
ν(m, t) ≡ δ2c
Sm
(2.54)
where the time dependence is determined by the picture chosen (in Sm in the first
picture or in δc in the second picture). Notice that in both pictures, the self-similar
32
Figure 2.5: Two pictures of gravitational growth. In Figure 2.3 we fix δc and raise
the fluctuations linearly. Here we fix the fluctuations at their linear value today
and raise the threshold back in time. The two pictures are equivalent.
33
variable evolves as
ν(m, t) =D2(ti)
D2(t)ν(m, ti). (2.55)
Hereafter, we will only use the second picture. Ultimately we will be interested
in the abundance of haloes of mass m at some redshift z. The key ingredients
are S(m) and δc(z). Figure 2.2 shows the variance of initial density fluctuations,
linearly extrapolated to the present time, as a function of m. Figure 2.6 shows the
predictions of the spherical collapse overdensity in EdS and ΛCDM cosmologies as
a function of z. For an Einstein de Sitter cosmology, the critical collapse threshold
depends on redshift as
δc(z) = δc0D(0)
D(z)=
δc0
a(z)= δc0(1 + z). (2.56)
For ΛCDM cosmology, Navarro et al. (1997) found that a good numerical approxi-
mation to this quantity is given by
δc(z) =3(12π)2/3
20Ωm(z)0.0055 D(0)
D(z). (2.57)
This is shown in Figure 2.6 for ΛCDM (solid blue curve) and EdS cosmology (dashed
red line). For more rigorous treatement of spherical collapse in the presence of Λ,
we refer the reader to the papers mentioned in Section 2.5.
We have introduced all the background results necessary for the excursion set
theory of gravitational clustering. Before we continue, a few comments are in order.
A central assumption in this work is that the initial fluctuations are Gaussian
(equation 2.52). In general, the statistics of a random field are encoded in its point
34
Figure 2.6: The spherical collapse threshold. The red dotted line corresponds to
EdS cosmology, whereas the blue solid curve represents ΛCDM.
35
functions (e.g., < δ~k >,< δ~k δ~k ′ >,< δ~k δ~k ′ δ~k ′′ >, . . . ). For a Gaussian random
field, the odd n-point functions are null, while the even n-point functions are simple
products of the 2-point function (i.e., < δ~k δ~k ′ >∝ P (k)). In other words, all the
details are contained in the power spectrum. Standard inflationary cosmologies
predict that the initial fluctuations are indeed nearly Gaussian (Maldacena, 2003;
Acquaviva et al., 2003; Creminelli, 2003; Lyth & Rodrıguez, 2005; Seery & Lidsey,
2005) – see Lo Verde et al. (2008) for a review. Recently, a variety of inflationary
models that predict non-Gaussianity have been proposed (see Bartolo et al., 2004,
and references therein). One consequence of non-Gaussianity is that the higher
n-point functions are non-zero. In other words, the different k-modes (the δ~k’s) are
correlated, and the simple description given in equation (2.50) becomes non-trivial.
Nevertheless, recent authors have pointed out that observable non-Gaussianity only
affects the largest, rarest objects (Sadeh et al., 2007; Grossi et al., 2007; Dalal et
al., 2008; Afshordi & Tolley, 2008; Pillepich et al., 2008). For this reason, in this
work we will assume that the initial field of density fluctuations are well-described
by a Gaussian random field.
Another assumption that may be brought into question is the current gravity
model: General Relativity. Several authors have attempted to modify standard
gravity, often with different motivations in mind. For instance, some seek to re-
move the need of dark matter (Milgrom, 1983; Bekenstein, 2004), the cosmological
constant (Dvali et al., 2000; Lue et al., 2004), dark energy (Mannheim, 1990; Di-
36
aferio & Ostorero, 2008) – while others are interested in unifying dark matter and
dark energy (Kamenshchik et al., 2001; Makler et al., 2003; Sen & Scherrer, 2005;
Scherrer & Sen, 2008; Bertacca et al., 2008). In modified gravity, one may no longer
assume the validity of Birkoff’s theorem. Namely, the evolution of a spherical patch
is governed by both internal and external gravitational contributions. The net ef-
fect of this is that the growth of fluctuation depends on scale – that is, the growth
factor in equation (2.36) becomes k-dependent (Shirata et al., 2005). Moreover,
while a region with zero overdensity remains that way relative to the background in
regular gravity, this is no longer true in modified gravity. In other words, whether
or not a patch in the initial field collapses at a later time does not only depend
on its density, but on its mass itself. Recently, Martino et al. (2008) proposed a
method for estimating how halo abundances are altered in modified gravity. This
formulation of spherical collapse requires one to work with the initial density field
directly, given by ‘picture one’ in Figure 2.5 (they show that using ‘picture two’
gives the wrong result). In their construction, nevertheless, only the most massive
and rarest haloes are affected. In this work, only regular gravity will be considered.
2.7 The excursion set theory
The mass function of dark matter haloes can be estimated by using the spherical
collapse model and the statistics of the (linearly-extrapolated) initial overdensity
field, smoothed at different scales with a sharp-k space window function. The larger
37
the scale, the smaller the variance of fluctuations (Figure 2.2). Intuitively one can
think of this as follows. For the very largest scales, smoothed fluctuations are very
small, and the densities of such large regions approach the background density of the
universe. For these smoothing scales, the distribution of overdensities approaches
a Dirac delta. If the same field is smoothed at smaller scales, perturbations from
the mean become more evident. With this higher resolution, a greater diversity in
overdensities results, making the width of their distribution (the variance) broader.
Figure 2.4 illustrates this, showing the same initial field as a function of ~x, times
D(t0)/D(ti), smoothed at two different scales. On average, the peaks and troughs
of the smoothed field with the smaller filter tend to be more pronounced than those
with the larger filter.
Now pick a point ~x at random early in the universe (at time t = ti). Pick a very
large sphere and smooth this field with a sharp-k space filter. Gradually shrink
the radius of this filter, and smooth the field to increasingly smaller regions. Then
linearly extrapolate the result to the present. One such process is shown in Fig-
ure 2.7. For this particular point, the smoothed overdensity enclosed in the filter’s
radius tends to increase as the region gets smaller. However, for some portions the
opposite happens, indicating that a dense region at one scale can itself be embedded
in a rather underdense region, a void.
For comparison, Figure 2.3 shows the smoothed field at only two scales and
several ~x. Figure 2.7, on the other hand, shows to only one point ~x, and it shows
38
Figure 2.7: The smoothed overdensity field at various scales. The jagged line re-
pressents the run of smoothing the initial overdensity field with a sharp-k space
filter in spheres of increasing radius R around a randomly chosen point ~x. The
dotted horizontal lines are the spherical collapse thresholds at two redshifts.
39
the value of the smoothed field at all scales. Choosing a different point ~x in the
universe would yield a different jagged trajectory. For this particular location, we
ended up with an overdensity as R → 0. In other words, this point is found in an
overdense patch of the initial density field. Had we picked a point in an underdense
patch, the jagged line would have ended at a negative height at small R. In both
situations, the jagged lines deviate more from zero (in either direction), on average,
as the radius of the smoothing filter decreases.
In Figure 2.7, we have included the present-time critical collapse threshold, and
that associated with a higher redshift (dotted horizontal lines). Take, for instance,
the lower of the two. Notice that the jagged line crosses this barrier at several
points. Identifying all the scales as collapsed objects would miscount regions that are
embedded in larger collapsed regions. This is the so-called ‘cloud-in cloud’ problem
(Epstein, 1983; Bardeen et al., 1986; Peacock & Heavens, 1990; Jedamzik, 1995;
Sheth, 1995; Avelino & Viana, 2000). The excursion set formalism of Bond et al.
(1991) solves this by selecting only the largest of such regions as the collapsed object.
In other words, the largest scale that crosses the barrier carries with it information
about the mass of the virialized halo surrounding the point ~x. Notice that the
higher the barrier (z > 0), the smaller the value of R is at which this happens. In
other words, haloes tend to be less massive at earlier times, in agreement with the
hierarchical picture of gravitational clustering.
So far we have discussed smoothed overdensities in terms of the radius of the
40
Figure 2.8: The smoothed overdensity field with increasing variance. This is the
same as Figure 2.7, but in terms of the variance S, which increases with decreasing
R. The jagged line is a Brownian motion random walk.
41
filter. A more natural choice is to use the variance S = σ2(R) itself, which truly
describes how fluctuations deviate from zero at each scale. This process is described
in Figure 2.8. Since large scales correspond to small values of S, virialized regions are
identified with the first place where the jagged line upcrosses the collapse barrier.
The excursion set approach maps the fraction of random trajectories that first
upcross a barrier of height δc(z) between S and S+dS with the fraction of mass at
redshift z in haloes with mass between m and m+dm:
f(Sm|δc(z))d Sm = f(m|z)d m. (2.58)
This situation is illustrated in Figure 2.9, where the red solid circle indicates where
the random trajectory first crosses the barrier of height δc(z). Very few random
walks will upcross this barrier immediately – most of them will wander around
before crossing. Since S is a monotonically decreasing function of mass, most of
the mass will be in low-mass haloes. At higher redshift, the height of the barrier
increases, making it more difficult for walks to cross it. This means that most haloes
will, on average, have smaller masses. This is consistent with the idea that smaller
haloes formed before massive ones.
Finding this crossing distribution is nontrivial. Nevertheless, having a sharp-k
space filter makes matters much easier. With such a filter, the action of replacing
the smoothing scale as R → R+dR amounts to changing the size of the top-hat
filter in Fourier space from some kTH to kTH+dk. The Fourier modes that contribute
to this new scale come only from a thin spherical shell of width dk. This means
42
Figure 2.9: The first crossing of a barrier by a random walk. The jagged trajectory
denotes a Brownian random walk and the horizontal dotted line is a constant barrier.
The solid red circle denotes the location where the walk first upcrosses the barrier.
43
that the new contribution carries no information on the smaller scales, making
consecutive scales uncorrelated. In other words, the steps in these trajectories
are uncorrelated, and such trajectories are true Brownian random walks. Random
walks – also known as excursion sets – have been studied in great detail for many
years in pure mathematics (where they are known as Wiener processes, Grimmett &
Stirzaker, 2001), and have been applied to subjects as diverse as genetics, gambling,
colloids, neurology, stellar dynamics, and finance. Bachelier (1900) was the first to
compute the first crossing distribution of a constant barrier. The next section is
devoted to the derivation of this result.
2.8 The unconditional mass function.
In this section we follow Chandrasekhar (1943) and Bond et al. (1991), who ex-
ploit the symmetries of the constant barrier to find the solution. A more elegant
treatment, in terms of diffusion theory, is offered in Appendix A. The formalism
presented here shows the correct way to compute the halo mass function found
originally by Press & Schechter (1974) with heuristic arguments. In this work, we
will refer to results presented in this section and the next as the ’extended Press-
Schechter’ (ePS) theory.
Let F (> m′|z) denote the cumulative fraction of mass in collapsed objects of
mass greater than m′ at redshift z. The differential fraction of m-haloes at z is
f(m|z)dm =∂
∂m′F (> m′|z)
∣
∣
∣
m′=m. (2.59)
44
In terms of random walk variables, this is given by
f(Sm|δc(z))dSm =∂
∂S ′F (< S ′|δc(z))
∣
∣
∣
S′=Sm
. (2.60)
Now we show how F (< S ′|δc(z)) is obtained in the extended Press-Schechter theory.
First consider the probability that a trajectory is located between δ and δ+dδ after
S steps (one such trajectory is shown in Figure 2.8). This is a standard stochastic
problem, whose solution has been known for a number of years (e.g., Chandrasekhar,
1943; Grimmett & Stirzaker, 2001), and it is given by
Q(δ|S)dδ =1√2πS
exp(
− δ2
2S
)
dδ, (2.61)
(compare with equation 2.52). Next, we need to compute the fraction of these
walks that cross a barrier of constant height before having performed S ′ steps. For
instance, the black and the black-gray walks in Figure 2.10 satisfy this criterion
(they cross the barrier at S < S ′), while the red one does not. All the walks that
contribute to F (< S ′|δc(z)) cross the δc(z) threshold at some S < S ′. These walks
can be classified into two groups, those with δ ≥ δc at S ′ (e.g., the black walk) and
those with δ < δc (e.g., the black-gray walk).
The cumulative fraction of walks in the first category is given by
Fup(< S ′|δc(z)) =
∫ ∞
δc(z)
dδ Q(S ′, δ). (2.62)
This is because all the walks that have δ ≥ δc at S ′, have certainly crossed the
threshold (with height δc) at some smaller S < S ′.
45
Figure 2.10: The extended Press-Schechter argument. Three random walks with
0 < S ≤ S ′ and a constant barrier of height δc(z) (horizontal dotted line) are
shown. One never crosses the barrier (in red), one crosses it and continues above it
(in black), and one that pierces it and continues beneath it (in black-gray).
46
Obtaining the contribution to F (< S ′|δc(z)) from walks belonging to the second
category is a bit tricky. Recall that, by virtue of the sharp-k space filter, consecutive
steps in a walk are independent. This means that from any point onward, a walk
has no memory of the previous steps. At the origin, a random walk is equally likely
to wander in the positive or in the negative direction. The same is true at any
other point. In particular, consider the point where the black-gray walk crosses
the barrier (the point where it changes color). The fraction of walks going up (in
black) or down (in gray) beyond S is the same: Fdown = Fup. Since the cumulative
fraction of walks in the first (second) category is Fup (Fdown), the cumulative fraction
of walks that have crossed the threshold at some S < S ′ is simply
F (< S ′|δc(z)) = Fup + Fdown = 2×∫ ∞
δc(z)
dδ Q(S ′, δ). (2.63)
This quantity includes all the walks that have crossed the barrier at some S < S ′,
without regard of whether they are above or beneath δc(z) at S ′.
Substituting Q in equation (2.61), one obtains that
F (< S ′|δc(z)) = erfc( δc(z)√
2πS ′
)
. (2.64)
Finally, substituting this expression into equation (2.60), we obtain that the fraction
of mass in m-haloes at redshift z is
f(Sm|δc(z))dSm =
√
δ2(z)
2πSm
exp(
− δ2(z)
2Sm
)dSm
Sm
. (2.65)
This is the famous Press-Schechter result, and it gives an estimate of the fraction
of mass in haloes with mass between m and m+dm at redshift z.
47
2.9 The conditional mass function
The Press-Schechter mass function, which gives the comoving number density of
haloes with mass in the (m,m+dm) at redshift z, is given by
n(m|t)d m =ρ
mf(m|z)d m =
ρ
mf(Sm|δc(z))
∣
∣
∣
dSm
dm
∣
∣
∣, (2.66)
where f(Sm|δc(z))dSm is the fraction of walks that first upcross a barrier of height
δc(z) between Sm and Sm+dSm (equation A.6 and Figure 2.9).
Suppose that instead we were only interested in the mass function of those haloes
of mass m at redshift z conditioned to end up in bound objects of mass M > m at
a more recent redshift Z < z. The conditional mass function N(m|z,M,Z)dm de-
scribes this. Some authors refer to N as the progenitor mass function, where m are
the progenitors that form a final object of mass M by merging. Just like the uncon-
ditional n(m|z)dm was obtained from f(m|z)dm; the conditional N(m|z,M,Z)dm
can be obtained from f(m|z,M,Z)dm, the fraction of mass in m-haloes at z con-
ditioned to be in M -haloes at Z.
Bower (1991) was the first to extend the Press-Schechter heuristic argument to
the conditional case (hence the word ‘extended’ in ePS). In the excursion set theory,
this can be computed by setting
f(m|z,M,Z) dm = f(Sm|δc(z), SM , δc(Z)) dSm, (2.67)
where the quantity on the right-hand side is the first crossing distribution of a
barrier of height δc(z) by random walks with origin at (SM , δc(Z)). Figure 2.11
48
Figure 2.11: The two barrier problem. The crossing distribution of a barrier at
height δc(z) by random walks originating at (SM , δc(Z)) is associated to the condi-
tional mass function of m-haloes at Z found in haloes of mass M > m at Z < z.
This becomes a one barrier problem by shifting coordinates: δ → a = δ−δc(Z) and
S → s = S − SM .
49
illustrates this. The configuration is identical to Figure 2.9, except for a simple
shift of coordinates:
S → s ≡ S − SM ,
δ → a ≡ δ − δc(Z). (2.68)
This single barrier has height ac = δc(z)− δc(Z), and its first crossing distribution
is:
f(s|ac)ds =
√
a2c
2πsexp
(
− a2c
2s
)ds
s. (2.69)
Therefore, the conditional crossing distribution is
f(Sm|δc(z), SM , δc(Z))dSm =
√
(δc(z)− δc(Z))2
2π(Sm − SM)exp
(
− (δc(z)− δc(Z))2
2(Sm − SM)
) dSm
Sm − SM
.
(2.70)
One may also write
f(m|z,M,Z) dm = f(ν) dν =
√
ν
2πe−
ν2dν
ν. (2.71)
where we have replaced
ν =δ2c (z)
Sm
→ (δc(z)− δc(Z))2
Sm − SM
=a2
c
s. (2.72)
In other words, the ePS conditional mass function is self-similar.
Lastly, N represents the mean number density of m-haloes at redshift z con-
ditioned to belong to M -haloes at a later Z. This means that we can write the
conditional f in terms of N as
f(m|z,M,Z)d m =m
MN(m|z,M,Z)d m. (2.73)
50
In other words, the conditional mass function is
N(m|z,M,Z)d m =M
mf(m|z,M,Z)d m. (2.74)
In summary, the excursion set approach extracts information from the initial
overdensity field and predicts statistical properties of collapsed haloes at later times.
One key ingredient is the spherical collapse threshold, which allows us to map halo
abundances to first crossing distributions of a constant barrier (Figure 2.9 and 2.11).
This set of properties is commonly known as the extended Press-Schechter theory
(ePS). We will see that the assumption that the spherical collapse prescription can
be generalized, thereby improving the excursion set predictions of halo abundance.
2.10 Ellipsoidal collapse
For many years, the extended Press-Schechter formalism appeared to agree well
with N-body simulations (Efstathiou & Rees, 1988; Efstathiou et al., 1988; White
& Frenk, 1991; White et al., 1993). However, as simulations improved in size and
resolutions, the limitations of this theory became evident (Lacey & Cole, 1994; Gelb
& Bertschinger, 1994). For instance, it was noted that the Press-Schechter mass
function underpredicts the abundance of haloes of high mass. The opposite trend
is true in the low-mass end. In particular, Sheth & Tormen (1999) found that
f(ν)dν = A
√
qν
2πe−
qν2 [1 + (qν)−p]
dν
ν, (2.75)
51
with (A, p, q) = (0.332, 0.3, 0.707), provides a better description of the mass func-
tion2 (in this context, see also Jenkins et al., 2001). Similar criticisms were made by
Tormen (1998) for the conditional mass function. These findings sparked a thorough
revision of the ePS theory, and its underlying assumptions.
The excursion set approach assumes that all the information regarding the abun-
dance of collapse objects is encoded in the initial overdensity field. Such objects
are selected with a spherical collapse threshold, found by following the gravitational
evolution of an isolated region. In reality, regions are affected by their surroundings,
and collapse is triaxial (Doroshkevich, 1970; Bardeen et al., 1986). It is therefore
more appropriate to assume that the growth of structures is determined by the
initial shear field (Hoffman, 1986, 1988; Dubinski, 1992; Bertschinger & Jain, 1994;
Audit & Alimi, 1996; Audit et al., 1997). The gravitational evolution of ellipsoids
has been studied by several authors (Lynden-Bell, 1964; Lin et al., 1965; Fujimoto,
1968; Zeldovich, 1970; Icke, 1973; White & Silk, 1979; Barrow & Silk, 1981; Lem-
son, 1993; Hui & Bertschinger, 1996; Bond & Myers, 1996; Jing & Suto, 2002) –
see Desjacques (2008) for a review. In such models, the deformation (shear) ten-
sor is defined as Dij ≡ ∂i∂jφ = Dji, where φ is the gravitational potential and D
has six independent components. The natural generalization of the excursion set
formalism presented earlier would be to replace the random walk the overdensity
performs with shrinking smoothing scale with six such random walks, one for each
2The constant A here is a normalization factor, not a free parameter.
52
component of D (Chiueh & Lee, 2001; Sheth & Tormen, 2002; Sandvik et al., 2007).
A much simpler prescription is proposed by Sheth, Mo & Tormen (2001), who find
that triaxial dynamics can be effectively described by a mass-dependent ellipsoidal
collapse barrier. Incorporating ellipsoidal collapse into the excursion set theory
merely requires replacing the spherical collapse barrier δc(z) for δec(m, z). We now
discuss this prescription in more detail.
For a Gaussian random field smoothed on a scale S, Doroshkevich (1970) pro-
vides a formula for the probability of having a given set of (ordered) eigenvalues
of the deformation tensor: p(λ1, λ2, λ3)dλ1dλ2dλ3 (see also Monaco, 1995, 1997a,b;
Audit et al., 1997; Lee & Shandarin, 1998). However, it is more practical to describe
an ellipsoid in terms of (e, p, δ), the initial ellipticity, prolateness and overdensity –
which are combinations of (λ1, λ2, λ3) (Bardeen et al., 1986). Particular cases are
an ellipsoid with one axis of symmetry (p = 0) and a sphere (e, p = 0). For given
e and p, the collapse threshold δec(e, p) can be found. Sheth, Mo & Tormen (2001)
propose that, to a good approximation,
δec(e, p)
δc
= 1 + β[
5(e2 ± p2)δ2ec(e, p)
δ2c
]γ
, (2.76)
where (β, γ) = (0.47, 0.615), and the plus (minus) is used for negative (positive) p.
In general, one may only solve equation (2.76) for δec numerically. To obtain the
desired δec(m, z), Sheth, Mo & Tormen (2001) set p to its mean value and e to its
most probable value. From this, one must compute p(e, p|δ)de dp, the distribution
of e and p for a given δ (Bardeen et al., 1986). The average prolateness is p = 0.
53
The ellipticity distribution peaks at e = (√
S/δ2)/√
5 when p = 0. Thus, the
ellipsoidal collapse threshold at redshift z is δec(e, p, z). Setting p = p = 0 and
e = e = (1/√
S/δ2ec)/√
5 in equation (2.76), they obtain
δec(m, z) = δc(z)
1 + β[ S
δ2c (z)
]γ
. (2.77)
A few comments are in order. First note that all the redshift dependence is
contained in δc(z), the spherical collapse threshold. For large masses, S → 0 and
δec → δc; massive regions are barely affected by their surroundings and their collapse
is spherical. For small masses, S is large and δec(m, z) increases. Physically, this
mean that small regions are more affected by their surroundings and therefore must
be much denser to keep themselves together and be able to collapse. The following
section will show how this simplified version of ellipsoidal collapse can be readily
incorporated into to the excursion set approach (Sheth, Mo & Tormen, 2001).
2.11 Moving barrier models
Recall that in the excursion set approach, the problem of estimating halo abun-
dances is mapped to one of estimating the distribution of the number of steps a
Brownian-motion random walk must take before it first crosses a barrier of specified
height. For instance, the Press-Schechter mass function is associated with barriers of
constant height δc – such model assumes that haloes form from a spherical collapse.
In constrast, the mass function associated with the simplified version of ellipsoidal
54
collapse (Sheth, Mo & Tormen, 2001) is actually related to barriers whose height
increase monotonically with the number of steps:
B(S, δc) =√
qδc
1 + β
[
S
qδ2c
]γ
, (2.78)
This barrier is identical to δec(S, δc) in equation (2.77), except for the additional
constant q (Sheth & Tormen, 2002) (see below). In the context of diffusion theory,
this is called a ‘moving’ barrier (see Section A.1.2).
Having a barrier that increases with S makes it more difficult for random walks
to cross it. The net effect of this is to decrease the abundance of low-mass objects.
Replacing δc → √qδc with q < 1 decreases the height of the barrier for all values
of S. The effect of this is that it is easier for random walks to cross the barriers at
low S, associated with high masses. By choosing this ‘ad-hoc’ constant correctly,
the abundance of massive objects can be such that the number of small ones is
not increased significantly. This solves the discrepancy between ePS and N-body
simulations. We will see that the value of q in the Sheth & Tormen (1999) formula
(equation 2.75) works well.
This free parameter can be considered a weakness of the theory. There are
two possible explanations. First, Sheth & Tormen (2002) point out that N-body
simulations need to rely on one free parameter to identify dark matter haloes. For
instance, the ‘spherical overdensity’ method groups particles in spherical regions
into collapsed objects if such regions are denser than some critical density (typically
55
ρSO ≃ 200ρ). There is some freedom in how ρSO is chosen.3 Others use a ‘friends-of-
friends’ algorithm to connect particles so that each is at least at some fixed distance
from another in the group (typically this distance bFOF ≃ 0.2× the mean inter-
particle distance). In this method too, the choice of ‘FOF’ distance is somewhat
arbitrary. The second explanation, also discussed by Sheth, Mo & Tormen (2001),
is somewhat more technical. The excursion set theory predicts the abundance
of collapse objects from the statistics of random walks associated with random
positions in the initial density field. However, it is more realistic to assume that
haloes collapse around the peaks in the initial density field, and not just around
any random position (see also Bond & Myers, 1996). Calculating quantities like the
mass function from the more realistic approach is a rather difficult task. Sheth, Mo
& Tormen (2001) argue that the net effect of using the first approach is that the
predicted halo masses are underestimated. This effect is, in principle, encoded in
the q parameter. For our present purposes, q will be treated as a free parameter.
Figure 2.12 shows the different barriers for different collapse models (see below)
at two redshifts. The solid dots are included for easy comparison with Figure 2.11.
The red (dotted) horizonal lines represent the constant barriers associated with ePS
at the two redshifts. Here, (q, β, γ) = (1, 0, 0). The green (long-dashed) curves show
the ellipsoidal collapse barrier in equation (2.78), which has (0.707, 0.47, 0.615). A
few comments regarding the nature of this barrier are in order. Because γ > 1/2,
3We must emphasize that this is not the same as the ePS approach, where linear-extrapolations
of the initial overdensity field are used.
56
Figure 2.12: Different collapse barriers. Barriers associated with spherical (constant
– red dotted lines), and ellipsoidal (γ > 1/2 – green long-dashed curves) at different
redshifts are shown along with the square-root barriers (γ = 1/2 – blue dashed
curves). The filled circles are included for easy comparison with Figure 2.11; these
show that the mass functions (unconditional and conditional) will change with
different choice of barriers for a given random walk.
57
not all walks are guaranteed to cross it, because the rms of the walks scales as
√S (Sheth & Tormen, 2002), which increases slower than S0.615. Another problem
is that the ellipsoidal collapse barriers associated with two different times may
intersect; of course, this never happens for the spherical collapse barriers. Sheth &
Tormen (2002) suggest that this intersection of barriers may represent the possibility
that haloes can fragment. We will be interested in models which assume that
fragmentation never occurs. This is one of the reasons why we study the limiting
case of ‘square-root’ barriers for which γ = 1/2:
B(S, δc) =√
qδc + β√
S. (2.79)
Such barriers are shown in blue (short-dashed), and their predicted halo abundances
are very similar to those in N-body simulations if one sets (q, β, γ) = (0.55, 0.5, 0.5)
(see Figure 2.13 in Section 2.12). Aside for the value of q, this barrier is very similar
to the γ = 0.615 barrier. Notice that this barrier is the sum of two factors, a redshift-
dependent one and a mass-dependent one. All the redshift information is encoded
in the barrier’s ‘δ-intercept’; and the barriers at different redshifts are identical,
up to a vertical shift. These properties are also true for the constant barrier, but
not for the Sheth & Tormen (2002) barrier with γ > 1/2. Also, this ‘square-root’
model is particularly interesting, in view of the fact that an analytic solution to
the first crossing distribution of square-root barriers is available (Breiman, 1967).
(See Section A.1.2 for expressions and a derivation of the exact solution to the
diffusion equation with square-root barriers.) Analytic expressions for the first
58
crossing distribution of barriers with γ = 1 and γ = 2 are also known (Schroedinger,
1915; Groeneboem, 1989). Unfortunately, an exact solution for general γ does not
exist – but see Zhang & Hui (2006) for an integral-equation solution.
For general barriers, Sheth & Tormen (2002) provide a series approximation
that works reasonably well for a large range of regimes, and it is given by
f(S|δc)dS =|T (S)|√
2πSexp
[
− B(S, δc)2
2S
]
dS
S, (2.80)
where T (S) includes the first few terms of the Taylor expansion of B(S, δc):
T (S) =5∑
n=0
(−S)n
n!
∂nB(S, δc)
∂Sn.
They point out that, when γ = 0.615, setting n = 5 is sufficient to reach reasonable
accuracy. We will see that this is true when γ = 0.5 as well. Notice that this
approximation is exact for the constant and linear barrier (Sheth, 1998).
The barrier in equation (2.78) can be written as
B(S, δc)√qδc
= 1 + β(qν)−γ. (2.81)
where ν = δ2c/S is the self-similar variable. The mass functions arising from such
barriers are also self-similar. The series approximation (equation 2.80) becomes
f(ν)dν =
√
qν
2πe−
qν2
[1+ β(qν)γ
]2 [1 +βα
(qν)γ]dν
ν, (2.82)
where α is the result of the Taylor sum; and it is given by α ≃ (0.094, 0.2461) for
the models with γ = (0.615, 0.5), respectively.
59
Similarly, Sheth & Tormen (2002) provide an approximate formula for the con-
ditional crossing distribution. This is given by
f(Sm|δc(z), SM , δc(Z))dSm =|T (Sm|SM)|
√
2π(Sm − SM)exp
[
− (Bm(z)−BM(Z))2
2(Sm − SM)
]
dSm
Sm − SM
,
(2.83)
where the Taylor-series factor is
T (Sm|SM) =5∑
n=0
(SM − Sm)n
n!
∂n
∂Smn
[
Bm(z)−BM(Z)]
. (2.84)
and we have denoted Bm(z) = B(Sm, δc(z)), etc., (compare to equation 2.70 with
constant barriers). We will now compare these findings with measurements in N-
body simulations.
2.12 Comparison with N-body simulations
In this section, we compare the unconditional mass function for the various mod-
els presented above, against N-body simulations. A similar comparison with the
conditional mass function will be postponed until Chapter 3.
In this thesis we use the GIF2 N-body simulation, which follows the evolution
of 4003 particles in a periodic cubic box 110h−1Mpc on a side in a flat ΛCDM back-
ground cosmology with parameters (Ωm,0, σ8, h, Ωb,0h2, n) = (0.3, 0.9, 0.7, 0.0196, 1)
(Gao et al., 2004b). We use these cosmological parameters everywhere, unless stated
otherwise. Fifty simulation snapshots were output, equally spaced in log(1+ z). At
each snapshot, haloes were identified using the spherical overdensity (SO) criterion,
60
adopting for virial mass the definition of Eke et al. (1996) (i.e., with virial density at
∼ 324ρ at redshift zero). The particle mass is mp = 1.73× 109h−1M⊙ and only ob-
jects with at least ten particles are considered. m⋆(z), defined by δ2c (z) = S(m⋆(z)),
is the typical mass scale at redshift z (that for which ν = 1). It is common prac-
tice to express halo masses in terms of m⋆ = m⋆(z = 0) (the z-dependence is
suppressed for the present time). For this cosmology and initial power spectrum,
m⋆ = 8.7×1012h−1M⊙ ≃ 5030mp. The simulation data and halo catalogs are avail-
able at http://www.mpa-garching.mpg.de/Virgo. The N-body data used in this
work was acquired by the Giocoli-Tormen pipeline in Padova, Italy. See Giocoli et
al. (2008a) for more details regarding the post-processing of the simulation.
In Figure 2.13 we plot the Sheth & Tormen (1999) mass function (long-dashed
black curve) and the different barrier predictions against the N-body simulation
data (filled black circles). Notice the improvement over ePS (red dotted curve) when
γ = 0.615 is used (dashed dark-green curve). The square-root barrier also gives an
excellent description in this range of masses. Moreover, the Sheth & Tormen (2002)
series (equation 2.80 – short-dashed blue curves) agrees well with the exact square-
root solution (solid magenta curve), given in Section A.1.2 below. The bottom panel
shows the ratio with respect to the Sheth & Tormen (1999) formula.
For the sake of simplicity, we have only presented the mass function at the
present time. This is valid because, as shown by Sheth & Tormen (1999), the mass
function is self-similar and it can be rescaled for other redshifts. These authors
61
Figure 2.13: The unconditional mass function. Top panel: N-body data are the
filled black circles and the long-dashed black curve is the Sheth & Tormen (1999)
prediction. The green (dashed) and blue (short-dashed) are the series approxi-
mations to the γ = (0.615, 0.5) barriers, respectively. The ePS mass function is
presented in red (dotted) and the exact square-root barrier solution in magenta
(solid). Bottom panel: All data and curves divided by the Sheth & Tormen (1999)
formula.
62
acknowledge that the halo mass function is not expected to be self-similar exactly.
This is confirmed by recent precision investigations (e.g., Warren et al., 2006; Reed
et al., 2007; Lukic et al., 2007; Tinker et al., 2008), but the deviations are small
enough for our purposes to be ignored.
63
Chapter 3
Merger history trees
In the early matter-dominated universe, overdensities get enhanced by gravity,
which later become virialized dark matter haloes. Small haloes form first and merge
with others, creating more massive haloes as a result. Looking backwards in time,
this merger picture is equivalent to a branching process, and the history of a final
halo is contained in a merger tree (Figure 3.1). Our main objective in this chapter
is to show that the excursion-set formalism can be used to reconstruct the merger
history tree of a halo. We provide an algorithm based on the ellipsoidal collapse
model with square-root barriers, and several tests are performed. Some technical
details on the implementation of this algorithm can found in Appendix B.
64
Figure 3.1: The merger history tree of a halo with mass M at a final redshift Z.
Redshift z increases in the vertical direction, and the masses of the merging pieces
are proportional to the width of the branches. The horizontal lines represent sample
redshift snapshots.
65
3.1 Motivation
Most models of galaxy formation in a hierarchical universe assume that the merger
history of the surrounding dark matter halo plays an important role in determining
the properties of a galaxy (White & Rees, 1978; Blumenthal et al., 1984; White
& Frenk, 1991) – (see Baugh, 2006; Avila-Reese, 2006, for reviews). Although
halo merger histories can be measured using N-body simulations, these can be
time consuming and computationally intensive (Springel & Hernquist, 2005). This
has fueled considerable study of the formation and merger histories of dark matter
haloes from a excursion-set Monte Carlo perspective. Monte Carlo merger trees have
the advantage of being fast and one may easily probe mass regimes inaccessible to
current N-body simulations. Moreover, unlike N-body experiments, the cosmology
and initial conditions may be easily modified.
Fast algorithms for generating halo merger trees, in which haloes were assumed
to form from a spherical collapse, were developed in the 1990s (Kauffmann & White,
1993; Somerville & Kolatt, 1999; Sheth & Lemson, 1999; Cole et al., 2000) – see
Zhang et al. (2008b) for a comparison. However, recall that spherical collapse over-
predicts (underpredicts) the abundance of haloes in the low (high) mass regime.
To address these issues, Sheth & Tormen (1999) extended the excursion set frame-
work to include ellipsoidal collapse (Sheth, Mo & Tormen, 2001; Sheth & Tormen,
2002). This clearly showed that merger-trees which assume spherical collapse are
inadequate.
66
Hiotelis & Del Popolo (2006) and Zhang et al. (2008b) describe merger tree
algorithms which extend some of the older algorithms to incorporate aspects of
the ellipsoidal collapse results. In addition, a number of new algorithms have re-
cently been published (Parkinson et al., 2008; Neistein & Dekel, 2008a); although
efficient and accurate, such methods side-step the idea of ellipsoidal collapse alto-
gether. Moreover, these methods are calibrated to match N-body simulations, and
are therefore limited by the accuracy and scope of these simulations.
The most significant difference between the algorithm we derive here and all the
others described above (full N-body simulations included) is that it takes discrete
steps in mass rather than time. The horizontal lines in Figure 3.1 show how tradi-
tional merger trees are built. Each of these discrete ‘snapshots’ contains the pro-
genitors of the final halo at the corresponding redshifts. If one takes the limit where
the separation between these snapshots is small, the full merger history tree can
be recovered. Then one typically uses the progenitor mass function to break haloes
into their progenitors at each step. In the present approach, on the other hand,
redshift snapshots do not enter the picture. Instead, we generate random walks
directly, and extract ‘mass history’ information from them. This way a branch is
built and subsequent branches can be connected. In this approach, the number of
steps S of the random walk – or equivalently the mass – is discretized. This feature
allows us to study a number of problems which are more difficult to address with
the other methods. We now proceed to introduce the notion of ‘mass history’ in
67
the spherical and ellipsoidal collapse contexts.
3.2 Mass histories and merger trees
Figure 3.2 illustrates how the mass growth history of an object is encoded in the
excursion set approach if objects form from a spherical collapse (also see Figure 1 of
Lacey & Cole, 1993). The jagged line shows a random walk which starts from the
origin: (S, δ) = (0, 0). Imagine drawing a horizontal line with height δc0 = 1.686 and
marking the smallest value of S at which the walk intersects this barrier of constant
height (recall that δc0 corresponds to the present time and δc > δc0 corresponds to
higher redshifts). The dotted horizontal line denotes such a barrier. Then increase
the height of this barrier, and record how this value of S changes as δ increases. Such
mass history points are depicted as dark red filled circles in Figure 3.2. The dashed
lines show that S will occasionally jump from a small value to a larger one. Since
S is a proxy for mass, and δc for time, such a jump is a proxy for an instantaneous
change in mass: a merger. Note that the random walk steps under such jumps are
not part of the mass history (e.g., the gray portion with Sm < S < Sm′).
The key to our merger tree algorithm is to recognize that these jumps mean
that there are a set of other walks which one might associate with this one – one
for each jump. One such walk is illustrated by the second jagged curve, which
starts at about the middle of the panel. If the jump from Sm to Sm′ occurred when
the barrier height was δc(z), then this other walk starts from (Sm−m′ , δc(z)). The
68
Figure 3.2: A random walk and its associated mass history. The dark-red filled
circles represent the history of a halo of mass M at redshift z = 0. A merger
(m′,m − m′) → m at redshift z is depicted by the Sm → Sm′ jump (in gray) at
height δc(z). A new branch associated with (m−m′) is connected at (Sm−m′ , δc(z)).
The light-red filled circles denote the mass history of this object.
69
‘merger history’ associated with this new branch is represented by the light-shade
red filled circles in Figure 3.2. For every such jump, a new random walk must be
drawn. For each jump within each of those new walks, the same process applies
– more walks must be drawn. In summary, the bundle of such walks encodes the
entire merger history of a present-day object. Notice that jumps can occur at any
z – there is no constraint that they happen at discrete times. However, if one is
interested in the mass function of progenitors at some fixed z, one simply reads-off
the list of values of S at which this bundle of walks first cross δc(z).
So far, we have discussed how to generate trees in the spherical collapse model.
Figure 3.3 shows the same walk as before, but now the mass growth history associ-
ated with the walk is given by its intersection with square-root barriers of gradually
increasing height. This shows clearly that the jumps in mass, and the times at
which they occur, are modified. But the overall logic remains the same. Each
jump gives rise to a new walk from (Sm−m′ , B(Sm−m′ , δc(z))), where B is given by
equation (2.79).
The natural generalization would be to incorporate the original γ > 1/2 barrier
of Sheth, Mo & Tormen (2001). One feature of this model is that barriers with
different values of δc intersect (Sheth & Tormen, 2002). This is because the barrier
in equation (2.78) increases faster with S when δc is small. Consider two barriers
with δ-intercepts given by δc1 and δc2 (with δc2 > δc1). These two barriers cross at
70
Figure 3.3: The same random walk as in Figure 3.2, but now with square-root
rather than constant barriers, illustrating that the mass accretion history depends
on the barrier shape. In our algorithm, the new object with mass (m−m′) is now
connected at (Sm−m′ ,√
qδc(z) + β√
Sm−m′).
71
some S = S× that satisfies
β(S×/q)γ =
√q(δc2 − δc1)
1/δ2γ−1c1 − 1/δ2γ−1
c2
, where 2γ − 1 > 0. (3.1)
The value S× depends on δc1 and δc2. For instance, if δc2 & δc1, the denominator in
the above expression will be much smaller than the numerator, and S× will be large.
Likewise, if δc1 >> δc1, the denominator will be much larger than the numerator,
and S× will be small.
For merger tree algorithms that rely on time snapshots, having barriers that
intersect can be problematic (e.g., Hiotelis & Del Popolo, 2006; Zhang et al., 2008b).
For instance, consider two barriers B1 and B2 with δc1 and δc2 respectively (with
δc2 > δc1 and z2 > z1). Also, consider a point (SM , B2), with SM > S×, and
the first crossing distribution of the B1 barrier by random walks originating from
that point. This distribution is associated to the conditional mass function of the
progenitors of the M -halo. Notice that the redshift of this progenitors is lower than
that of the parent halo. This can be interpreted as a halo of mass M at redshift z2,
which fragments into its progenitors at z1. There is nothing particularly problematic
about fragmenting haloes – and this phenomenon is certainly seen in simulations
(Tormen, 1998; Fakhouri & Ma, 2008a). However, it is not clear that the intersection
of moving barriers correctly describes fragmentation in N-body simulations. This
problem can be avoided altogether by choosing the time snapshots close enough so
that S× > Sdust. For example, consider the time snapshots in the GIF2 simulation.
For every two consecutive snapshots, there is a value of S× associate to each pair.
72
Figure 3.4: Intersecting barriers with GIF2 time snapshots. The black dots are
located at each (S×, δc(z)), where δc(z) refers to the redshift snapshots in the GIF2
simulation and S× are the points where the barriers (with γ > 1/2) associated with
a pair of consecutive snapshots intersect (see equation 3.1). The vertical dotted line
is located at S = Sdust and the horizontal dotted line is located at δc0.
73
Figure 3.4 illustrates this. Notice that all the values of S× > Sdust (where the
vertical dotted line is located at Sdust). In fact, Figure 2.2, shows that the masses
associated with the crossing points in Figure 3.4 are all ≪ 107M⊙h−1. Thus, for
the problems studied in this thesis, the mass regime where intersection of barriers
occurs is never probed.1
Nevertheless, avoiding the issue of intersecting barriers simplifies matters sig-
nificantly. For instance, recall that the algorithm considered here does not rely on
time snapshots. Instead, for each point along each random walk generated, one
must be able to associate a value of m and z. If barriers with γ > 1/2 are used,
this mapping is not unique. This is one of the main reasons we choose the limiting
case with γ = 1/2. Moreover, we will see that non-intersecting moving barriers will
play a key role in our formulation of halo creation (Chapter 4).
3.3 The Monte Carlo merger tree
To compare our merger histories with those in the GIF2 simulation, we generated
2000 realizations of our tree for each final halo mass bin M of interest. In all cases,
the minimum mass considered was mdust = M/1000, and the merger histories of
haloes with mass below this were not followed (we call this minimum mass the
‘branching-mass resolution’). We used random walks with 105 steps in between SM
1See, however, Angulo & White (2009). These authors probe the S ≃ 700 regime, corresponding
to M ≃ 10−3M⊙, where the physics of free-streaming of neutralinos matters.
74
Figure 3.5: A branch at ‘continuous’ and ‘snapshot’ redshifts. The blue filled cirles
show the mass history retrieved from a random walk (‘Cont’). The black circles
show the result of constraining this history to the redshift snapshots of the GIF2
simulation (‘Snap’).
75
and Sdust to ensure that the mass change between the steps was less that mdust.
Having a small step size is essential to faithfully reproduce random walks in a
computer. Moreover, if the step size is too large, we run the risk of missing branches.
See Appendix B for more details on the implementation of our Monte Carlo tree.
Recall that our tree does not take discrete steps in time. Nevertheless, for fair
comparison with the measurements from the GIF2 simulation, the tree data were
stored in the same discrete redshift bins as were output from the simulation. We use
the word ‘Cont’ to denote the original tree data and the word ‘Snap’ for the data
stored in redshift snapshots. Figure 3.5 shows the result of storing a given branch
using these two prescriptions. For instance, near z = 7, the ‘snapshot’ version shows
a jump in mass, whereas the ‘continuous’ one sees two. Sections 3.7 and 3.8 study
some merger-related quantities which are sensitive to the differences between these
two ways of storing trees (Figures 3.12, 3.13 and 3.14). Before testing our tree’s
prediction of the conditional mass function, we will discuss a scaling symmetry
inherent to the square-root barrier model.
3.4 A conditional scaling symmetry
Recall that the unconditional mass function is associated to the first crossing dis-
tribution associated with walks which start from the origin: (S, δ) = (0, 0). When
constant barriers are used, this can be expressed self-similarly as
f(m|z)dm = f(S|δc)dS = f(ν)dν, (3.2)
76
where ν ≡ δ2c/S. The conditional mass function of m-haloes at z that end up
in bound objects of mass M > m at Z < z is given by f(m|z,M,Z)dm =
f(Sm|δ1, SM , δ0)dSm, where where δ1 = δc(z), δ0 = δc(Z). In other words, the
conditional mass function is associated with the first crossing distribution of a bar-
rier of height δ1 by random walks with origin at (SM , δ0). Because a straight-line
is straight whatever the origin of the coordinate system, the conditional mass func-
tion, in the spherical collapse model has the same functional form as that of the
unconditional mass function, provided one sets ν = (δ1 − δ0)2/(Sm − SM).
However, for the square-root barrier, a walk which starts from (√
q δ0+β√
SM , SM)
must cross a barrier of shape B =√
q(δ1 − δ0) + β√
Sm − SM + SM (see Sec-
tion A.1.2). This is not quite of the same form as equation (2.79). As a result,
the conditional mass function is not simply a rescaled version of the unconditional
one. Rather, in this model,
f(m|z,M, z0) dm = f(Sm/SM |ηc) d(Sm/SM), (3.3)
where
ηc ≡δ1 − δ0√
SM
= ηβ − β. (3.4)
Thus, final haloes of different masses will have similar progenitor mass functions
when expressed in terms of Sm/SM , provided they have similar values of ηc. While
this scaling is like that for the constant barrier model, in the square-root barrier
model, the progenitor mass function is not a function of the combination ν2 =
η2c/(Sm/SM − 1). This is interesting, because Sheth & Tormen (2002) have shown
77
that the conditional mass function in simulations is not well-fit by a function of
ν. In what follows, we will present evidence that it is, however, a function of ηc
and Sm/SM separately, so the qualitatively different scaling associated with the
square-root barrier is indeed seen in simulations.
3.5 The progenitor mass function
Figures 3.6-3.8 show the progenitor mass fractions and mass functions at five dif-
ferent redshifts (z = 0.5, 1, 2, 3, 5), for haloes identified at Z = 0 with final masses
given by M/m⋆ = 0.06, 0.6 and 6 (recall that m⋆ ≃ 8.7 × 1012M⊙h−1 in this cos-
mology). The corresponding values of ηc (equation 3.4) are shown in each panel. In
all three figures, filled circles show measurements in the GIF2 simulation, and open
circles show results from our square-root trees. We probe the m < mp regime with
our trees to verify consistency with analytic excursion-set predictions (the smooth
curves in all the panels). The short-dashed red curve shows the constant barrier
(β, q) = (0, 1) prediction associated with spherical collapse. The solid magenta
curves show the exact square-root barrier solution with (β, q, γ) = (0.5, 0.55, 0.5).
Dashed blue curves show the considerably simpler approximation to the solution
which is due to Sheth & Tormen (2002); this approximation is excellent over the
entire range of interest. The long-dashed dark-green curves show this same ap-
proximation for the ellipsoidal collapse barrier: (β, q, γ) = (0.707, 0.47, 0.615). The
square-root barrier prediction agrees well with the γ = 0.615 curve, except in the
78
Figure 3.6: The progenitor mass fraction (left) and mass function (right) at redshifts
z = (0.5, 1, 2, 3, 5), for haloes of mass M/m⋆ = 0.06 at z = 0. Filled circles show
measurements in the GIF2 simulation, and open circles are from our square-root
trees. The smooth solid (magenta) and dashed curves (blue) show the exact square-
root barrier solution, and the series approximation, respectively. The long-dashed
curve (dark green) shows the ellipsoidal collapse model with γ > 1/2, and the short-
dashed curve is the constant barrier prediction. Values of the scaling parameter ηc
(equation 3.4) are also shown (see Figure 3.9).
79
Figure 3.7: Same as Figure 3.6, but with M/m⋆ = 0.6.
80
Figure 3.8: Same as Figures 3.6, but with M/m⋆ = 6.
81
M/m⋆ z ηc — M/m⋆ z ηc
0.06 1 0.3 — 6 0.5 0.31
0.06 2 0.65 — 6 1 0.66
0.6 3 1.45 — 6 2 1.44
Table 3.1: The conditional scaling symmetry. Pairs in with similar ηc (equation 3.4)
in Figures 3.6-3.8 are listed here and plotted in Figure 3.9.
high-mass regime. This discrepancy becomes evident when ηc > 1 and it is amplified
with increasing ηc.
Before we ask how our merger tree algorithm compares with simulations, we note
that it produces progenitor mass functions that are well-described by the theory
curves over a wide range of masses and redshifts. At high redshifts, our tree data
lie slightly below the theory curves at both high and low m/M , and slightly above
in between, although where the cross-over points occur depends on z and M . In
all other regimes, our Monte Carlo trees match the square-root barrier predictions.
Any additional disagreement with the GIF 2 simulation measurements (compare
open and filled circles) is due to limitations of the γ = 1/2 model.
Finally, recall that the square-root and constant barrier models make specific
predictions for how the conditional mass functions should scale with final halo mass
and time. Table 3.5 lists pairs with similar ηc, yet quite distinct values of M
and z. Figure 3.9 compares the associated conditional mass functions. The black
82
Figure 3.9: The conditional scaling symmetry. Different combinations of M and z
with similar ηc (equation 3.4 and Table 3.5). N-body simulation measurements and
the Sheth & Tormen (2002) result with γ > 1/2 are shown.
83
squares and long-dashed black lines refer to the left-hand side of Table 3.5 (low
final masses), whereas the gray triangles and short-dashed gray lines refer to the
right-hand side (high final masses). Notice that the curves are remarkably similar
to one another, as are the symbols. This is true despite the fact that the values of ηc
are not perfectly identical, and that f(m, z|M,Z)dm = f(Sm/SM |ηc)d(Sm/SM) ≃
f(m/M |ηc)d(m/M). The results for low-mass haloes (black squares) are truncated
at higher m/M than they are for larger M (gray triangles), simply because only
haloes with at least ten particles are considered. Evidently, the conditional mass
functions are indeed functions of ηc and Sm/SM separately, rather than of the
combination ν.
3.6 The distribution of formation redshifts
Following Lacey & Cole (1993), a halo is said to have ‘formed’ when it first acquires
half of its final mass. For a given halo mass M (at a final redshift Z), there is a dis-
tribution of formation redshifts, denoted by p(zF |M,Z)dzF . We now show that this
distribution is intimately related to the progenitor mass function N(m, z|M,Z)dm.
Consider the merger history tree of an M -halo (e.g., Figure 3.1). At different red-
shifts, this halo has a set of progenitors. However, by conservation of mass, if one of
the progenitors has mass m > M/2, there may only be one such halo. Computing
the cumulative formation-redshift distribution up to zF is equivalent to counting
the number progenitors of that have been formed by that epoch (with m > M/2).
84
Namely,
P (> zF ) ≡∫ ∞
zF
dz p(z|M,Z) =
∫ M
M/2
dm N(m, zF |M,Z). (3.5)
Lacey & Cole (1993) realized that for spherical collapse, it is more convenient to
cast this equation in terms of
ω ≡ δc(z)− δc(Z)√
S(M/2)− S(M), and S =
S(m)− S(M)
S(M/2)− S(M). (3.6)
In these variables, they find that for white-noise initial conditions,
P (> ω) =
√
2
πωe−ω2/2 + (1− ω2)erfc
( ω√2
)
. (3.7)
Therefore, the distribution of formation redshifts is
p(zF |Z,M)dzF = p(ωF )dωF = −dωFd
dωP (> ω)
∣
∣
∣
ω=ωF
= 2ωF erfc(ωF√
2
)
dωF . (3.8)
For general Gaussian initial conditions, this can only be computed numerically.
Nevertheless, as pointed out by Lacey & Cole (1993), the dependence on the initial
power spectrum is very weak. We now compare these predictions with our merger
trees and with N-body simulations.
The filled circles in Figure 3.10 show the scaled formation redshift distributions
for haloes with masses in the range 0.9M ≤ M ≤ 1.1M with log10(M/m∗) =
1, . . . ,−1.5 in steps of −0.5 in the GIF2 simulation. We use the first snapshot
when at least half the mass is in a single progenitor at the formation time, and
make no attempt to interpolate our simulation formation redshifts between these
discretely spaced output times (Harker et al., 2006; Giocoli et al., 2007). Recall that
85
Figure 3.10: Scaled distribution of formation redshifts. Filled black circles show
simulation data, open blue circles and red triangles show results from the square-
root and constant barrier trees. Smooth curves show equation (3.8) with q = 1,
0.707 and 0.55 (short-dashed red, long-dashed green, and dashed blue), correspond-
ing to the predicted distribution for constant barriers (spherical collapse), moving
(ellipsoidal collapse) and square-root barriers.
86
we do not discretize redshift in our tree, so the question of interpolation does not
arise. The open circles and triangles show the corresponding formation time distri-
butions from our square-root and constant barrier trees. The same distribution, but
now in terms redshift is presented in Figure 3.11. Notice that for all models, Monte
Carlos and the GIF2 data, the formation time distribution peaks at higher red-
shifts (and are broader) when the final masses are smaller. In other words, massive
objects formed recently, whereas less massive ones are typically older.
For the ellipsoidal collapse model, Giocoli et al. (2007) showed that the formation
redshift is well-described by equation (3.8) if one replaces ωF → √qωF . The smooth
curves show this with q = 1, 0.707 and 0.55 (short-dashed red, long-dashed green,
and dashed blue), which represent the (constant, γ = 0.615, and square-root) barrier
predictions. For higher values of q, the peaks are located at lower redshifts and the
widths of the curves decrease. Recall that, strictly speaking, equation (3.8) only
holds for white-noise initial conditions. Nevertheless, as pointed out Lacey & Cole
(1993), it remains a reasonable approximation to the CDM case. Furthermore, note
that it provides an excellent description of the formation times generated by our
trees. However, no choice of q provides particularly good agreement with the GIF2
simulation, a discrepancy noted by previous authors (Lin et al., 2003; Hiotelis &
Del Popolo, 2006; Giocoli et al., 2007). This is likely a consequence of the excursion
set assumption that different steps in the walk are uncorrelated (Sheth & Tormen,
2002). Brownian motion random walks with this property are said to be Markovian.
87
Figure 3.11: Distribution of formation redshifts. Same as Figure 3.10, but in terms
of redshift.
88
To induce non-Markovianity, it is necessary to replace the sharp-k space filter, which
is a rather non-trivial matter. See Pan et al. (2008) and references therein for how
one might improve on this.
3.7 The mass distribution at formation
The previous section studied halo formation, where formation was defined as the
first time that the mass of one of the progenitors exceeds half the total. Therefore,
this mass can have any value between 1/2 and 1 times the final mass, and one can
study the distribution of masses at, and just prior to, formation. The excursion set
constant barrier model makes a prediction for this distribution (Nusser & Sheth,
1999). The mass distribution at formation is expected to be
p(µ) dµ =2
π
√
1− µ
2µ− 1
dµ
µ2, where 1/2 ≤ µ ≤ 1, (3.9)
and µ ≡ m/M , and the distribution just before formation is
q(µ) dµ =1
π(1− µ)
(
√
µ
1− 2µ−√
1− 2µ) dµ
µ2, (3.10)
where 1/4 ≤ µ ≤ 1/2. We have found that, to a very good approximation,
µ p(µ) → µ q(µ) if one replaces µ → 1/4µ (solid and dashed curves in Figure 3.12,
respectively).
Although these expressions were derived assuming a white-noise power spec-
trum, they are expected to be relatively independent of P (k). Sheth & Tormen
(2004) showed that they did indeed match numerical simulations well for different
89
Figure 3.12: Distribution of the mass at formation for several final masses. The left
half of each panel shows the mass just prior to formation, whereas the right half
shows the mass just after formation. Filled circles show simulation data, open circles
and triangles are from the square-root and constant barrier trees. The solid curve
shows µq(µ) (right half) and µp(µ) (left half) (equations 3.10 and 3.9 respectively).
The dashed curve shows these same expressions with µ→ 1/4µ.
90
Figure 3.13: Same as Figure 3.12, but showing only the region around m/M = 1/2.
The peak in the simulations (filled symbols) is less pronounced than in the merger
trees (jagged lines). Open circles show the result of sampling the merger trees at
the same redshifts as the simulation snapshots: this makes a dramatic difference
around 0.49 ≤ µ ≤ 0.51, suggesting that the sharp cusp predicted by the theory will
also be present in simulations with sufficiently closely spaced outputs. The smaller
discrepancies further from the peak remain.
91
cosmologies and initial power spectra. Figure 3.12 shows that they also work well
for square-root barriers.
The agreement between the theory curves (smooth curves) and our Monte Carlo
trees (jagged lines, labeled ‘Cont’) is excellent. All panels show that agreement with
simulation data is also quite good. However, there is a systematic discrepancy: the
cusp at µ = 1/2 appears to be less pronounced in the simulation, with correspond-
ingly lower tails. A similar discrepancy was seen by Sheth & Tormen (2004), who
suggested that the fact that the simulations only provide discrete snapshots in time
may be smoothing out the peak. By sampling our trees at the simulation snapshots
(open symbols, labeled ‘Snap’), we have attempted to model the magnitude of this
effect. Figure 3.13 illustrates that the cusp has indeed been smoothed, but this is
a dramatic effect only around 0.49 ≤ µ ≤ 0.51. The discrepancies further from the
peak remain (Figure 3.12).
3.8 The last major merger
Mergers of galaxies with similar masses are expected to produce strong short-lived
periods of star formation (i.e., starbursts) and quasars (see Chapter 5). Moreover,
recent numerical studies suggest that the mass ratio of the galaxies involved plays
an important role: merger-induced bursts occur when the galaxies have similar
masses (Gao et al., 2004a; Springel & Hernquist, 2005; Cox et al., 2008). Moreover,
it has been suggested that a galaxy’s Hubble type is strongly correlated with its last
92
major merger (e.g., Toomre & Toomre, 1972; Barnes, 1988; Hernquist, 1992, 1993).
(See Maller et al., 2006, for a discussion of the redshifts associated with the last
major mergers of galaxies). Understanding such mergers requires understanding
the mergers their host haloes undergo. Consider a merger (m′,m′ −m)→ m, with
m′ < m−m′. For ease of comparison with Parkinson et al. (2008), we will define a
‘major’ merger as one with mass ratio µ ≡ m′/(m−m′) ≥ 1/3. The filled circles in
Figure 3.14 show the redshift distribution of the last major merger onto the main
branch. (The last major merger does not necessarily happen on the main branch.
However, Figure 3 of Parkinson et al. (2008) suggests that, in most cases, it does.
Presumably, this is because the assembly of haloes in recent times is dominated by
mergers.) Curves show measurements in our full trees (‘Cont’), and open symbols
show the result of only sampling the trees at the GIF2 simulation outputs (‘Snap’).
For the discretely-sampled data, only mergers involving haloes with at least ten
particles are considered. Note that the anomalously low data point that is second
from the left in all panels appears to be an effect of seeing the tree at discrete
snapshots – the smooth curves show no such dip. This feature is also present in the
simulation analysed by Parkinson et al. (2008); we expect it to disappear if more
finely spaced snapshots are analysed.
For high masses, the data from the square-root trees peak at about the same
redshifts as the simulations; the constant barrier, spherical collapse trees peak at
lower redshifts. This improvement relative to the spherical collapse case is similar
93
Figure 3.14: The redshift distribution of the most recent major merger, for several
final masses. A major merger is defined as one in which the minor component
has at least 1/3 of the mass of the major component. Filled circles show the
GIF2 simulation data, open symbols show the corresponding measurements in the
same ‘snapshot’ versions of our trees, and smooth curves show the ‘continuous’
distributions which would be seen with arbitrarily closely spaced output times.
94
to that in the modified galform trees of Parkinson et al. (2008). However, our
square-root trees tend to lie above the simulation at low redshifts, and below at
higher redshifts. The discrepancy with simulation becomes increasingly worse at
small masses, although it is possible that the GIF2 results for our two smallest mass
bins are not reliable – the high redshift mergers involve haloes with few particles.
Because we require haloes to have more than 10 particles, we are likely to miss
major mergers once the typical mass becomes of this order.
95
Chapter 4
Halo creation
The previous chapter concentrated on describing the assembly of a dark matter halo
as a product of mergers. A numerical algorithm was introduced and its predictions
were evaluated. The aim of this chapter is to estimate analytically the rate at which
haloes are ‘created’ by mergers of smaller haloes.
4.1 Creation of dark matter haloes
The aim of this section we introduce the notion of halo ‘creation’. Let n(m|t) dm
denote the number density of haloes with mass in the range dm about m at time
t. As a result of mergers, dn/dt is the sum of two competing effects - the number
of objects in a given mass bin increases if smaller mass objects merge to form an
object of precisely this mass - an event we call creation - or the number decreases
as objects of this mass merge with others, thus depleting the number in the bin -
96
an event we call destruction. Thus, the time derivative of the halo mass function is
the difference of the creation and destruction rates:
dn/dt = C(m, t)−D(m, t). (4.1)
Given n(m|t), it is easy enough to take the time derivative; the problem is to
separate dn/dt into its two contributions. Roughly speaking, low mass objects
may have undergone significant mergers in the past but they are not being created
in merging events any more - their evolution is expected to be dominated by the
destruction term. In contrast, extremely massive objects are undergoing substantial
merging activity at the current time, and the time derivative of the halo mass
function should be a good estimator of the creation rate of these objects. But
quantifying the general case requires a richer model.
Early work used the Press & Schechter (1974) form for n(m|t), and advocated
equating the ‘positive’ term in dn/dt with the creation rate, and the ‘negative’
term with the destruction rate (e.g., Haehnelt et al., 1998). But this is clearly
not a solution at all, since it provides no rule for how to determine what one
should correctly equate with ‘positive’. For example, if dn/dt = P − N , there
is no particular reason why one could not have written the right hand side as
(P − ǫ) − (N − ǫ). The second part of this chapter is devoted to extracting the
creation rate C from dn/dt. See Blain & Longair (1993a,b), Sasaki (1994) and
Kitayama & Suto (1996) for other attempts to solve this problem.
Before addressing the creation term C (the number density of mergers per Gyr),
97
in the first part of this chapter we discuss the distribution of times c(t|m) when
these creation events take place (i.e., c ≡ C/∫
C dt is the rate normalized by the
total number of creation events that will ever occur). Although c and C differ
only by normalization constant, it turns out that c is somewhat easier to model.
This is because the excursion set formalism from which the Press-Schechter mass
function can be derived (Bond et al., 1991; Lacey & Cole, 1993), carries with it a
prescription for computing c(t |m), the distribution of creation times (Percival &
Miller, 1999). In this case, the functional form of c(t |m) is very similar to that of
f(m|t) ≡ m n(m|t)/ρ, where ρ is the mean comoving background density.
Since that time, interest has shifted to functional forms for n(m|t) which more
closely approximate the abundances measured in numerical simulations (e.g., Sheth
& Tormen, 1999; Warren et al., 2006; Reed et al., 2007; Lukic et al., 2007; Tinker
et al., 2008) – see Chapter 2 for a discussion. So it is interesting to ask how
the creation time distributions are modified. Percival et al. (2000) argue that the
relation between the functional forms of c(t|m) and f(m|t) should survive these
modifications, and show that this does indeed provide a good description of halo
creation in simulations. However, although they use intuition from the excursion set
approach to motivate their arguments, their method side-steps the generalization of
the excursion set approach from which the modified mass functions may be derived
– this is the ellipsoidal collapse ‘moving’ barrier approach (Sheth, Mo & Tormen,
2001; Sheth & Tormen, 2002). The first goal of this chapter is to calculate the
98
creation time distribution self-consistently within the moving barrier excursion set
approach. We do find that c and f are simply related, but that this is actually
extremely fortuitous – Section 4.3.3 demonstrates that this scaling does not hold
generally.
Getting the normalization constant which relates c to the creation rate C(m, t) is
a more challenging problem. Percival et al. (2000) obtained this quantity by match-
ing the creation time distribution to the rate measured in N-body simulations in
the low redshift regime (see also Percival & Miller, 1999), but they acknowledge
that they have no theory for the normalization factor. One possible solution to this
problem is to explore the evolution of the halo population in terms of coagulation
theory, where the creation and destruction terms are estimated separately (Smolu-
chowski, 1916, 1917). Early applications to galaxy formation and dark-matter halo
interactions include Silk & White (1978), Cavaliere et al. (1991a), Cavaliere et al.
(1991b), Cavaliere et al. (1992), Cavaliere & Menci (1993), Sheth & Pitman (1997)
and Menci et al. (2002). For a more recent treatment, see Benson et al. (2005) and
Benson (2008). For white-noise initial conditions, both the Smoluchowski and the
Press-Schechter excursion-set expressions for n(m|t) agree, so, for this case, the cre-
ation and destruction rates are known (Sheth & Pitman, 1997). However, obtaining
the rates for more general initial conditions, or for the modified mass functions that
are of more current interest, remains unsolved. The second goal of the present
chapter is to provide a model for the creation rate of dark matter haloes that is
99
informed by both coagulation theory and the modified excursion set approach with
moving barriers.
We also study the problem of how halo creation is modified if it is known that
the merging haloes are bound up in objects of mass M at some later time T .
The excursion set theory provides a way to compute the conditional mass func-
tion N(m|t,M, T ). We show how the conditional distribution of creation times
c(t|m,M, T ) is related to f(m|t,M, T ) = (m/M)N(m|t,M, T ). For the conditional
rate, the problem is to separate dN/dt into creation and destruction components.
Sheth (2003) argues that this conditional distribution may be the basis for under-
standing the phenomenon known as down-sizing (also see Neistein et al., 2006).
A few final remarks regarding the different uses of the term ‘halo creation’ are
in order. The first part of the chapter focuses on the creation time distribution,
c(t|m), which can be derived within the excursion-set formalism. The second part
focuses on the creation rate, C(m, t), the first term in the coagulation equation
(equation 4.1). The former is a normalized time distribution, while the latter is not.1
Another source of confusion is that halo ‘creation’ is distinct from halo ‘formation’;
following Lacey & Cole (1993) (and Chapter 3), the latter is typically defined as
the time that an object first reaches half its current mass. See Giocoli et al. (2007)
for an explicit calculation showing how creation and formation are related.
1The normalized distribution is denoted with lower case c, while the un-normalized rate is
denoted with capital C.
100
4.2 Mass history in the excursion set theory
Recall that in the excursion set approach, the problem of estimating the halo abun-
dances is mapped to one of estimating the distribution of the number of steps a
Brownian-motion random walk must take before it first crosses a barrier of speci-
fied height (Bond et al., 1991). In this approach, the height of the barrier plays a
crucial role. The Press-Schechter mass function is associated with barriers of con-
stant height - such barriers arise naturally in models in which haloes form from a
spherical collapse model. In constrast, the more accurate mass functions may be
related to ellipsoidal-collapse moving-barriers (equation 2.78).
Recall too that for the γ > 1/2 model of Sheth & Tormen (2002), not all
walks are guaranteed to cross the barrier, and two distinct barriers intersect. To
avoid these two problems, we introduced the ‘square-root’ barriers with γ = 1/2
(equation 2.79).
The dependence on S of the square root barrier means that it is more like the
ellipsoidal than spherical collapse barrier (which has constant height, independent
of S). However, there is one important respect in which the square root model
is very like the constant one. Consider the barriers associated with two different
times. For square-root barriers, the difference between the barrier heights is
B(S, δc2)−B(S, δc1) = (δc2 + β√
S)− (δc1 + β√
S)
= δc2 − δc1. (4.2)
101
Notice that this difference is independent of S. This is also (trivially) true for
constant barriers, but it is not true for any other values of γ. In this respect, the
excursion set model based on square-root barriers is extremely special. This will be
important later.
4.2.1 Mass history for different barriers
Figure 4.1 illustrates the relation between Brownian motion random walks and
the mass growth history of an object. The jagged line shows an example of a
random walk — this walk represents the run of smoothed overdensity around a
randomly chosen position in the initial fluctuation field, as the region over which
the overdensity is smoothed changes from large (left) to small (right). (The plot
actually shows the initial overdensity evolved to the present time using linear theory
— it differs from the initial overdensity by a multiplicative constant.) The initial
overdensities are all small compared to unity, so one may associate a mass with
each smoothing scale: this mass is larger for the larger smoothing scales.
Consider a horizontal line, and consider the places where it first intersects the
random walk, as the height of this line, this barrier, is raised. Clearly, this position
shifts to the right as the barrier is raised (red filled circles) — mass decreases as
redshift increases. Whereas the halo abundance problem corresponds to fixing the
barrier height δc (to illustrate, the height of dotted line corresponds to δc0 = 1.686)
and asking for the distribution of S values at which the barrier is first crossed,
102
Figure 4.1: The mass history associated with a random walk (jagged line) using a
constant barrier. Time increases as δ decreases and mass decreases as S increases.
The filled circles on the random walk denote the history, and the horizonal jumps
denote mergers. For reference, the horizontal dotted line denotes the barrier asso-
ciated with the present.
103
the halo creation problem corresponds to asking for the distribution of δc values
for a fixed S. We will use f(S|δc) dS to denote the first crossing distribution, and
c(δc|S) dδc to denote the distribution of creation times, where, for haloes of a given
mass m, c(δc|S) dδc = c(t |m) dt.
Notice that the mass increases relatively smoothly sometimes, and rather abruptly
at others. For instance, in the interval 4.2 . S . 4.8 in Figure 4.1, mass decreases
smoothly. Compare this situation to the sudden jump from S ≃ 4.8 → S ≃ 8.7
(red long-dashed line). Thus, this walk does not contribute to the creation time
distribution for any values of S between 4.8 and 8.3. But it does contribute in the
calculation of halo abundances for every δc.
It is interesting to compare this mass accretion history with that shown in Fig-
ure 4.2. The same walk is shown in both figures. However, now the horizontal dotted
line at δc0 has been replaced by a curve that has δ-intercept at√
qδc0, and increases
with increasing S — this is the square-root barrier (equation 2.79) associated with
the same epoch as the constant one. The comparison clearly shows that the mass
accretion history (blue solid circles) depends on the barrier shape. For instance, the
values with S . 4.3 and S ≃ 8.8, 9.7 and S . 10.4 are no longer included in the
mass accretion history (red solid circles). Moreover, even if a point on the random
walk happens to be part of the mass history in both cases, its associated time is
different under the two barrier prescriptions. To illustrate, the point at S ≃ 4.4 is at
the present for the square-root barrier case and in the past when constant barriers
104
Figure 4.2: Same as Figure 4.1, but with a square-root barrier. The blue soid circles
depict the history associated with the square-root barrier, while the red circles (for
spherical collapse) were kept to emphasize that different barriers predict different
mass histories for a given random walk. For reference, the horizontal dotted curve
denotes the barrier associated with the present.
105
are used. As a result, the halo mass function and the creation-times distribution
are modified. One of the goals in this thesis is to quantify these changes.
4.2.2 Creation times from Bayes’ rule
Consider the joint probability that a random walk first upcrosses a constant barrier
(with δ-intercept between δc and δc + dδc) between S and S + dS. Using Bayes’
Theorem, this can be written as
P (S, δc)dSdδc = f(S|δc)f(δc)dSdδc = c(δc|S)c(S)dδcdS,
For constant barriers, Percival & Miller (1999) argue that the δc-prior must be
uniform. First they note that all walks must have a creation event for any barrier,
regardless of its height (see Figure 4.1, left panel). Moreover, for any two equal-
sized intervals dδc1 and dδc2, the probability that such a creation event exists must
be equal. This is because the steps in the walk are uncorrelated, implying that any
point along the walk can be regarded as the starting point of a new walk. Therefore,
the walk is not altered at different values of δc and the probability of crossing two
different barriers at some point is the same. In other words, the δc-prior is given by
f(δc)dδc =dδc
∆δc
, (4.3)
where the constant ∆δc is infinite, since δc ∈ [0,∞).
Given f(δc)dδc and f(S|δc)dS, one can marginalize the joint distribution in δc.
106
This yields
c(S)dS = dS
∫ ∞
0
dδc[P (S, δc)] =dS
2A√
S∆δc
, (4.4)
where A = (√
π/2, 2, 2.08, 1.893) respectively for each of the constant, exact square-
root, square-root series approximation and the Sheth & Tormen (1999) result (see
Chapter 2 for a review). Inserting (4.3) and (4.4) in Bayes’ formula gives
c(δc|S)dδc = 2A√
Sf(S|δc)dδc. (4.5)
Let us write this expressions in terms of δ2c/S. In this work we will denote this
variable as ν if δc is fixed and as νc if S is fixed. Thus, f(S|δc) = (νc/S)f(νc) (S is
fixed) and c(δc|S) = (2νc/δc)c(νc), from which
c(νc)dνc = A√νcf(νc)dνc. (4.6)
Thus, for constant barriers, there is a remarkably simple relation between the cre-
ation distribution c and the first crossing distribution f .
4.2.3 Monte Carlo test and self-similarity
Barriers of the form given in equation (2.78) are self-similar, in the sense that
if δc is increased by a factor κ, then so is√
S. As a result, the first crossing
distribution f(S|δc) and the distribution of halo creation times c(δc|S) are both
simply functions of δ2c/S. In other words, if equation (2.78) describes ellipsoidal
collapse, then f(S|δc)dS = f(ν)dν and c(δc|S)dδc = c(νc)dνc. In the previous
107
section we showed that
c(νc) dνc = A√νc f(νc) dνc (4.7)
for the constant (A =√
π/2) and square root (A ≃ 2) barriers. This simple
relationship is one of the central results of this chapter, as is the warning that it
does not hold in general. To illustrate, Section 4.3.3 uses a system of linear barriers
in where this simple result does not apply.
To test equation (4.7) for constant and square-root barriers, we generated 105
random walks with 104 steps between S = 0 and S = S(mp) ≃ 28. Then we
stored the corresponding mass histories for the constant and square-root barriers
(e.g., solid circles in Figures 4.1-4.2). Every (S, δc) point along the history has a
corresponding νc. To study how the creation time distribution depends on S, we
could have chosen the subset of walks which have the correct value of S, and then
found the distribution of νc = δ2c/S values for those walks. However, the self-similar
scaling above means that c(νc) should be the same for all S. As a result, there is
no need to select a subset in S before binning in νc. If we simply bin up all the νc
values, whatever the associated values of S, then we can compare the result with
the predicted c(νc).
The symbols in Figure 4.3 show the creation time distributions for the constant
(triangles) and square-root (circles) barriers: the distribution associated with the
square-root barrier is broader and peaks at slightly higher νc. The curves show the
predicted creation time distributions (equation 4.7); they are in excellent agreement
108
Figure 4.3: The creation time distribution in self-similar form. The variable νc de-
notes δ2c/S at fixed mass. Triangles and circles show the distribution measured from
an ensemble of random walks with constant and square-root barriers respectively.
Dotted and solid (dashed) lines show the associated predictions. The long dashed
curve shows the result of inserting the Sheth & Tormen (1999) form for f(ν)dν into
equation (4.7).
109
with the measurements. Equation (4.7) shows that these creation time distributions
depend on the functional form of the first crossing distribution f(ν). For the case of
square-root barriers, we show the Breiman (1967) exact but complicated expression
for f(ν), and a much simpler approximation for it from Sheth & Tormen (2002).
The two curves are almost indistinguishable.
The use of barriers which scale self-similarly (equation 2.78) was motivated
by the observation that, when expressed as a function of ν, halo abundances in
simulations could be scaled to a universal form (Sheth & Tormen, 1999). We have
added the long-dashed curve in the Figure, which shows the result of using the
Sheth & Tormen (1999) functional form for f(νc) in equation (4.7); it is almost
indistinguishable from the curves associated with the square-root barrier.
4.3 Creation time distribution
In Section 4.2 we discussed the creation time distribution from the excursion set
point of view. Having shown that the analytic expressions accurately reproduce
our Monte Carlo measurements, we now study if they provide a good description of
halo creation in N-body simulations. Simulated haloes are labeled as having been
‘created’ if at least half of their particles were not observed in a more massive halo
at an earlier time.
110
4.3.1 Distribution of creation redshifts
The creation time distributions we measure in simulations and shown as filled circles
in Figure 4.4 are normalized to unity. However, the simulations only sample δc at
epochs before the present time, whereas the theory curves assume that 0 ≤ δc ≤ ∞.
Therefore, for haloes of mass m, we set
c(z|m) dz =c(δc|Sm)
∫∞
δc0dδ′c c(δ′c|Sm)
∣
∣
∣
∣
dδc
dz
∣
∣
∣
∣
dz, (δc ≥ δc0). (4.8)
These are the curves in Figure 4.4. The open triangles and circles show the creation
times measured in constant and square-root random walk ensembles sampled at the
same redshifts as the simulations. We studied bins of size dlog10 m = 0.2 in mass
centered at log10(m/m⋆) = 0.5 to −3 in steps of −0.5.
A couple of remarks are in order. Only the haloes with the highest redshift
in each mass bin were treated as newly-created. These measurement were tested
with different bin sizes (not shown), yielding similar results. One limitation is that if
dlog10 m is too small, most bins are empty, and the data does not follow a continuous
curve. One should not take dlog10 m to be too large – in particular, the mass bins
for the different values of m should be disjoint. Our choice of dlog10 m satisfied
both criteria. Lastly, we found that our choice of redshift bin dlog10(1 + z) = 0.05
was sufficiently large to capture enough creation events and sufficiently small for
comparison with the different theory curves.
The solid circles in the Figure denote the N-body measurement, where only
haloes with m > 10mp are considered (notice that the lowest-mass panel has no
111
Constant Exact Sqrt Approx Sqrt
ST-99
MC (Const) MC (Sqrt) N-Body
Figure 4.4: Distribution of creation redshifts for a number of bins in halo mass.
Filled circles show measurements in the simulations, and open triangles and circles
show analogous measurements made from sampling our constant and square-root
barrier random walk ensembles similarly to the simulations. Dotted curve shows the
prediction associated with a constant barrier; the exact square-root solution and its
series approximation are the solid and dashed curves; long-dashed curve shows the
result of inserting the Sheth & Tormen (1999) form into equation (4.7).
112
black filled circles). In all cases, improvement over the location of the peaks is seen
when the square-root barrier is used. However, these curves are slightly broader
than those traced by the simulation data, making the height of these normalized
distributions lower.
4.3.2 Self-similarity in halo creation
The excursion set model suggests that if the halo mass function f(ν)dν can be scaled
to a self-similar form, then the creation time distribution c(νc)dνc is also self-similar.
To test this we have scaled the values of δc(z) associated with each mass bin in the
simulations to νc = δc(z)2/S(m), and measured the resulting distribution of νc.
However, because all mass bins sample the same range in δc, they sample different
ranges in νc. We account for this by dividing the measured distribution of νc by
a normalization factor given by∫∞
νc0dν ′
c c(ν ′c), where νc0 = δ2
c0/S(m) and c(νc) is
associated with the Sheth & Tormen (1999) formula.
Figure 4.5 shows the result: different symbols show the rescaled distributions
associated with the various masses. Note that they do indeed appear to trace out a
universal curve. The various smooth curves show the constant barrier, square-root
barrier, and Sheth & Tormen (1999) based predictions. The symbols approximately
split the difference between the constant and square-root barrier models.
113
Figure 4.5: Universality of the distribution of halo creation times. The same theory
curves as in Figure 4.3 are shown. Different symbols show results for different halo
masses in the GIF2 simulation data, as indicated.
114
4.3.3 Why it doesn’t work in general
One of the key results of this work is equation (4.7), which relates the creation times
distribution to the mass function in a simple way. It is tempting to assume that
this holds for every mass function (e.g., Percival et al., 2000). The purpose of this
section and the next is to offer a counter-example, and to show why the general
case is more complicated. The next section discusses why this result works well for
constant and square-root barriers.
Consider the linear barrier
B(S, δc) = δc
[
1− β( S
δ2c
)]
. (4.9)
This is special case of the barrier in equation (2.78) with γ = 1 (we have suppressed
the parameter q). Notice that we have deliberately replaced β → −β in the above
expression. In this discussion we will only consider the β > 0 case to avoid barriers
that intersect. Moreover, if β > 0, all walks are guaranteed to cross. The exact
solution is known, and it is given by
f(ν)dν =
√
ν
2πe−
ν2(1−β
ν)2 dν
ν, (4.10)
where ν = δ2c/S (fixed S) (Schroedinger, 1915; Sheth, 1998). If equation (4.3) holds,
then the creation time distribution should be given by
c(νc)dνc = A√νcf(νc)dνc, where A =
√
π
2e−
β2 , (4.11)
and νc = δ2c/S (fixed δc).
115
Figure 4.6: The mass history associated with a random walk (jagged line) and linear
barriers. The brown filled circles on the random walk denote the creation events.
Notice that the barriers become steeper as δc increases, and the separation between
any two barriers increases with increasing S. For reference, the dotted line in the
lower left represents the linear barrier associated with δc = δc0.
116
To test this, we performed a Monte Carlo simulation of the mass histories as-
sociated with random walks where linear barriers (with β = 1) are used to select
creation events. Figure 4.6 shows one sample mass history of this ensemble. Com-
pare this with Figure 4.1, which shows the same process, but with constant and
square-root barriers. The jagged line is a random walk, and the brown solid circles
are the associated history. The long dashed lines denote jumps in the history. For
completeness we have included the linear barrier with δc = δc0, depicted as a dotted
line in the lower left. Figure 4.7 shows our Monte Carlo data (brown open squares).
The solid black curve is the prediction in equation (4.11). This disagreement inval-
idates the claim that c(νc)dνc is always proportional to√
νcf(νc)dνc.
Before moving on, note that the Monte Carlo data in Figure 4.7 follow a smooth
curve that is quite different from that associated with constant or square-root bar-
riers (Figure 4.3). The main difference is that the latter peak at some intermediate
value of νc, whereas the former decreases monotonically with νc. This is because it
is unlikely for a random walk to upcross a constant barrier (or a square-root barrier)
after a few steps. In other words, creation events with S ≪ δ2c (i.e., νc ≪ 1) are very
unlikely. Similarly, it is unlikely that a walk survives for many steps without being
absorbed by a constant barrier. That is, creation events with S ≫ δ2c (i.e., νc ≫ 1)
are unlikely. For the linear barrier, the ensemble of creation events is dominated
by points with small δc (i.e., small νc). This is because the linear barrier becomes
steeper as δc decreases. Since the height of a linear barrier decreases with S, it
117
Figure 4.7: The creation time distribution associated with linear barriers in self-
similar form. The variable νc denotes δ2c/S at fixed mass. The squares show the
distribution measured from an ensemble of random walks with linear barriers. The
solid curves show the associated predictions – assuming that equation (4.3) applies
(the ‘Bayesian’ result). The discrepancy between the theory prediction and the
data indicates that care must be taken when using the result in equation (4.7).
118
becomes easier for walks to upcross a barrier as δc decreases. This even allows for
cases where creation events are selected from points along a walk with δ < 0. Such
cases would be impossible for the constant and the square-root barrier models.
In principle, the creation time distribution with linear barriers can be computed
analytically, without using the Bayesian approach presented here (e.g., Karlin &
Taylor, 1975). Since the mass function associated with linear barriers does not
resemble halo abundances in N-body simulations, we do not pursue this any further
(but see Sheth, 1998, for interesting applications of this barrier).
4.3.4 Why it is a useful approximation in practice
If equation (4.6) is incorrect in general, then why then did it work so well in for con-
stant and square-root barriers (see e.g., Figure 4.3)? The first step is to recognize
that, because of the property highlighted by equation (4.2), a uniform distribution
in δc is appropriate for the family of square-root barriers of interest to us (equa-
tion 2.79), and so, for square-root barriers, equation (4.6) is exact. Our Monte
Carlo simulations shown in the main text confirm that this is indeed the case.
The second step is to note that, for barriers with general γ, the difference be-
tween two barriers carries additional factors of δc (e.g., equation 2.78). As a result,
assumption (4.3) is no longer valid. E.g., for linear barriers,
B(S, δc2)−B(S, δc1) = δc2 − δc1 − β( S
δc2
− S
δc1
)
6= δc2 − δc1. (4.12)
This property makes the linear barrier considerably different from the constant and
119
square-root barriers. Figure 4.7 shows that the distance between any two linear
barriers increases with increasing S (compare with Figure 4.1), so the δc-prior is
not uniform.
It is important to emphasize that the central conclusion of this and the previous
subsection – that the assumptions behind equation (4.7) do not hold in general –
do not depend on the fact that the linear barrier decreases in height with S. E.g.,
barriers of the form given by equation (2.78) with β > 0 and 0 < γ < 1/2 increase
with S. However,
B(S, δc2)−B(S, δc1) 6= δc2 − δc1; (4.13)
the separation between any two barriers increases with S, so the δc-prior is not
uniform in this case either. For these barriers too, equation (4.6) is incorrect.
Nevertheless, for γ close to 0 or 1/2, equation (4.6) should provide a reasonable
approximation. This is the fundamental reason why barriers with γ = 0.6, or of the
form required to give the Sheth & Tormen (1999) formula as the first crossing dis-
tribution, are likely to have creation time distributions which are well approximated
by equation (4.6).
4.3.5 Conditional distribution of creation redshifts
So far we have discussed the unconditional creation-time distribution c(t |m) and
its relation to f(m|t). In this section we study the creation time distribution,
c(t |m,T,M), of m-haloes at time t conditioned to be bound up in M -haloes at
120
a later time T . We also discuss how this quantity is related to f(m|t,M, T ), the
fraction of mass in m-progenitors at time t of a final halo of mass M at time T .
The latter is derived from the excursion-set theory by setting
f(m|t,M, T )dm = f(S|δ1, S0, δ0)dS, (4.14)
where S = S(m), S0 = S(M), δ1 = δc(t) and δ0 = δc(T ). The right-hand term
is the crossing distribution of a barrier B(S, δ1) by random walks with origin at
(S0, B(S0, δ0)). In Figure 4.1, this amounts to shifting the origin from (0, 0) to
(S0, δ0) (left panel) or to (S0,√
qδ0 + β√
S0) (right panel). In this work, the condi-
tional mass function is denoted by N(m|t,M, T )dm = (M/m)f(m|t,M, T )dm.
In essence, conditioning is equivalent to finding the (unconditional) crossing
distribution of a barrier
B(s, δ1, δ0) = B(s + S0, δ1)−B(S0, δ0), s = S − S0. (4.15)
In the constant barrier problem, B = δ1 − δ0. This means one simply replaces all
the unconditional expressions given previously with δc(z) → δ1 − δ0 and S → s =
S − S0. As a result, the only change occurs in the self-similar variable ν (and νc):
δ2c/S → (δ1 − δ0)
2/(S − S0) (Lacey & Cole, 1993; Percival & Miller, 1999; Sheth,
2003).
The square-root barrier is slightly more complicated:
B(s, ac) = ac + β√
s + S0, ac = δ1 − δ0 − β√
S0. (4.16)
Because equation (4.16) is not quite the same functional form as equation (2.79), the
121
first crossing distribution is not simply a suitably rescaled version of the uncondi-
tional distribution. Rather, it is a function of ηβ ≡ ac/√
S0 and s/S0. Nevertheless,
the logic one follows to arrive at the creation time distribution is the same. In
particular, the conditional versions of c and f are simply related:
c(ηβ|S/S0)dηβ = A(S/S0)f(S/S0|ηβ)d(S/S0), (4.17)
where now the A factor is a function of S/S0. This is another central result of this
chapter (compare to equation 4.7).
Let us derive this result for constant and square-root barriers. First notice that
the conditional crossing distribution can be written as
f(s/S0|ηβ)d(s/S0) = g(s/S0, ηβ)d(s/S0)
s/S0 + 1. (4.18)
where two forms of g are given next (Breiman, 1967; Sheth & Tormen, 2002). First,
for the exact square-root barrier solution:
g(s/S0, ηβ) =∑
λ
eηβ2/4Dλ(ηβ)lλ(−β)
(s/S0 + 1)λ/2. (4.19)
Likewise, for the Sheth & Tormen (2002) series approximation:
g(s/S0, ηβ) =
∣
∣
∣ηβ + β
√
s/S0 + 1[
1 + α(s/S0)
]
∣
∣
∣
√
2πs/S0
× exp
− (ηβ + β√
s/S0 + 1)2
2s/S0
s/S0 + 1
s/S0
, (4.20)
where
α(s/S0) =5∑
n=1
αn
s/S0 + 1, α 1 = −1
2, and αn = (1− 3
2n)αn−1.
122
The joint distribution of s/S0 and ηβ is given by
P (s/S0, ηβ)d(s/S0)dηβ = f(s/S0|ηβ)f(ηβ)d(s/S0)dηβ
= c(ηβ|s/S0)c(s/S0)d(s/S0)dηβ. (4.21)
Following equation (4.3) for δ1 and δ0 and using the fact that S0 is fixed, it can be
shown that the ηβ-prior is uniform:
f(ηβ)dηβ =dηβ
∆η(4.22)
where ∆η is an infinite constant. Marginalizing over the joint distribution in ηβ we
obtain
c(s/S0)d(s/S0)G
s/S0 + 1
d(s/S0)
∆η, (4.23)
where
G(s/S0) =
∫ ∞
0
dηβ g(s/S0, ηβ). (4.24)
Inserting equations (4.18), (4.22) and (4.23) in Bayes’ rule, we obtain
c(ηβ|s/S0)dηβ =g
Gdηβ. (4.25)
Comparing (4.18) to (4.25), we find that
c(ηβ|s/S0)dηβ = A(s/S0)f(s/S0|ηβ)dηβ, (4.26)
where
A(s/S0) = (s/S0 + 1)/G. (4.27)
123
Constant Exact Sqrt Approx Sqrt
Constant Exact Sqrt Approx Sqrt
MC (const)MC (Sqrt)N-Body
MC (const)MC (Sqrt)N-Body
Figure 4.8: Conditional distribution of creation redshifts. Symbols and style as
in Figure 4.4. We plot m = m⋆/10 (left panels) and m = m⋆/100 (right panels)
conditioned to end up in haloes of mass M = M⋆ (top panels) and M = 10m⋆
(bottom panels). In all cases, T denotes the present time.
124
Figure 4.8 shows the conditional distribution of creation redshifts for m/m⋆ =
(0.1, 0.01) that end up in haloes of mass M/m⋆ = (1, 10) today. Filled symbols show
the GIF2 measurements, open circles show our Monte Carlos with square root bar-
rier, and open triangles show Monte Carlos with a constant barrier. Smooth curves
show the corresponding predictions – notice that they are in excellent agreement
with the Monte Carlos.
The simulation bin sizes dlog10 m and dlog10(1+z) were used as in the uncondi-
tional case. Selecting haloes to be conditioned to belong to a final M -halo reduces
the number of creation events significantly. We selected haloes bound to end-up
in clumps with mass in a bin of size dlog10 M = 0.5. As in the unconditional case
(Figure 4.4), the distributions peak at higher redshifts and are slightly broader for
lower m (compare left and right panels). The same trends are seen as one increases
the final M (compare top and bottom panels).
In general, the moving barrier based curves provide a much better description
of the simulations, although the agreement is by no means perfect. For example,
the square root barrier tends to produce distributions which are slightly broader
than those in the N-body simulation. This effect is more pronounced in the right-
hand side panels. A similar effect was found in Chapter 3 for the formation time
distribution.We speculate that the discrepancy there was due to non-Markovian
effects. We refer the interested reader to Pan et al. (2008) for a discussion of this
topic.
125
4.4 Halo creation rates
Extracting the creation rate from dn/dt (or dN/dt) is a non-trivial problem. In
this section we make use of halo coagulation theory to estimate this quantity.
4.4.1 The unconditional rate
In the coagulation formalism (Smoluchowski, 1917), the halo mass function n(m|t)
obeys
dn(m|t)dt
= C(m, t)−D(m, t), (4.28)
where
C(m, t) =
∫ m
0
K(m′,m−m′; t)
2n(m′|t)n(m−m′|t)dm′, (4.29)
is our creation term, and the destruction term is
D(m, t) =
∫ ∞
0
K(m,m′; t)n(m|t)n(m′|t)dm′. (4.30)
In these expressions, the coagulation kernel K(m,m′; t) is symmetric in m and
m′.
Few analytic solutions to Smoluchowski’s equation exist. However, when the
kernel is additive in mass, then the associated mass function is given by the Press-
Schechter formula for white-noise initial conditions (Silk & White, 1978; Sheth
& Pitman, 1997). Of course, white-noise is a bad approximation to the initial
conditions in the CDM models of current interest. Moreover, we have shown that
ellipsoidal collapse gives a better description of halo abundances and creation times
126
than Press-Schechter (spherical collapse). Nevertheless, let us discuss the expression
obtained for the creation term in that special case.
White-noise initial conditions can be regarded as the continuum limit (large m
and small δc) of a discrete system initially with Poisson initial conditions. Therefore,
the creation rate in the discrete Smoluchowski equation is
C(m, t) =m−1∑
m′=1
K(m′,m−m′; t)
2n(m′|t)n(m−m′|t). (4.31)
Sheth & Pitman (1997) showed that the above equation can be written as
C(m, t) = n n(m|t)m− 1
1 + δc
∣
∣
∣
dδc
dt
∣
∣
∣
m−1∑
m′=1
pm′|m−m′ , (4.32)
where n is the mean number-density of particles. If we picture haloes as a collection
of particles held together by (m − 1) non-intersecting bonds (branched polymers),
pm′|m−m′ gives the probability of obtaining an m′-halo and and (m − m′)-halo by
deleting one random bond in the m-halo. The sum over a normalized probability is
trivial, so the creation term is simply
C(m, t) = n n(m|t)m− 1
1 + δc
∣
∣
∣
dδc
dt
∣
∣
∣. (4.33)
In the continuum limit, this becomes
C(m, t) = ρ m n(m|t)∣
∣
∣
dδc
dt
∣
∣
∣(4.34)
Figure 4.9 compares this assumption with the measured creation rates in the
simulations. The simulation measurements in Figures 4.9 and 4.4 are the same,
127
Constant Exact Sqrt Approx Sqrt
ST-99 N-Body
Figure 4.9: Halo creation rates. Symbols, line styles and choices of mass as in
Figure 4.4. As in Figure 4.4, the lowest panel on the right shows not data because
this case is beneath the resolution of the N-body simulation.
128
except that in the latter, data is normalized in time. Notice that the heights of the
curves increase with decreasing mass. This reflects that fact that, in hierarchical
models, more small haloes are created during the history of the Universe than are
massive halos (which are only created at later times). The constant barrier model
works well for massive haloes, but it overpredicts the creation rate of less massive
haloes – showing a similar discrepancy as in the Press-Schechter mass function in
that mass regime. In all cases, the square-root barrier and the Sheth & Tormen
(1999) creation rates match N-body results reasonably well.
4.4.2 The time-normalized creation rate
In this section, we show that creation rate C(m, t) (equation 4.34, Section 4) is
related to the creation time distribution c(t|m) (equation 4.7, Section 3) in a simple
way. First, notice that c(t|m) can be written in terms of νc = δ2c/S. That is,
c(t|m) = c(δc|S)∣
∣
∣
dδc
dt
∣
∣
∣= c(νc)
∣
∣
∣
dνc
dδc
∣
∣
∣
∣
∣
∣
dδc
dt
∣
∣
∣=
2νc
δc
c(νc)∣
∣
∣
dδc
dt
∣
∣
∣, (4.35)
where we have used the fact that |d ln νc/d ln δc| = 2.
Now, following equation (4.7), the above expression can be written as
c(t|m) =2νc
δc
A√νcf(νc)∣
∣
∣
dδc
dt
∣
∣
∣. (4.36)
On the other hand,
mn(m|t)ρ
= f(m|t) = f(S|δc)∣
∣
∣
dS
dm
∣
∣
∣=
ν
Sf(ν)
∣
∣
∣
dS
dm
∣
∣
∣, (4.37)
where we have used the fact that |d ln ν/d ln S| = 1.
129
Our prescription for the creation rate (equation 4.34) is
C(m, t) = ρ2mn(m|t)ρ
∣
∣
∣
dδc
dt
∣
∣
∣= ρ2 ν
Sf(ν)
∣
∣
∣
dS
dm
∣
∣
∣
∣
∣
∣
dδc
dt
∣
∣
∣. (4.38)
Taking the ratio of (4.38) and (4.36), we obtain
C(m, t)
c(t|m)=
ρ2 νSf(ν)
∣
∣
∣
dSdm
∣
∣
∣
∣
∣
∣
dδcdt
∣
∣
∣
2νc
δcA√νcf(νc)
∣
∣
∣
dδcdt
∣
∣
∣
=ρ2
2A√S
∣
∣
∣
dS
dm
∣
∣
∣≡ g(m), (4.39)
where we have dropped the distinction in notation between ν and νc. Simply put,
C(m, t) = g(m)c(t|m). (4.40)
Integrating both sides with respect to time,
∫
C(m, t)dt =
∫
g(m)c(t|m)dt = g(m)
∫
c(t|m)dt = g(m), (4.41)
since c(t|m) is, by definition, a time-normalized distribution. Thus,
c(t|m) =C(m, t)
g(m)=
C(m, t)∫
C(m, t)dt. (4.42)
In words, c(t|m) can be obtained from C(m, t) by normalizing the latter in time.
4.4.3 The conditional rate
We assume that the same prescription can be applied for the conditional case – i.e.,
the rate of creation of haloes of mass m at time t conditioned to belong to M haloes
at a later time T > t. In the discrete Poissonian case, the conditional rate is given
by
C(m, t|M,T ) = n N(m|t,M, T )m− 1
1 + δc
∣
∣
∣
dδc
dt
∣
∣
∣, (4.43)
130
(Sheth, 2003). In the continuum limit, this becomes
C(m, t |M,T ) = ρ m N(m, t |M,T )∣
∣
∣
dδc
dt
∣
∣
∣. (4.44)
Figure 4.10 shows our results for the same set of m and M as in Figure 4.8. As
in the unconditional case, the simulation measurements in Figures 4.10 and 4.8 are
the same, except that in the latter, data is normalized in time.
For both choices of m, the height of the curves decreases with increasing final
mass M (compare top and bottom panels). This effect is less pronounced for the
smaller m (compare left and right panels). In all panels, curves based on the square-
root barrier provide a more accurate description of the measurements.
Lastly, one interesting fact is that the two rates (equations 4.34 and 4.44) are
related by the following consistency condition:
∫ ∞
m
dM C(m, t |M,T ) n(M |T ) = C(m, t). (4.45)
In other words, the unconditional rate is recovered from the conditional rate by
multiplying by the number density of M -haloes at T and integrating over all possible
M > m. This consistency relation is true in general, not just when the initial
conditions are white-noise.
4.5 Final remarks
An alternative derivation for the creation rate can be obtained by using R(m,m′|t),
the rate of mergers of m-haloes with m′-haloes, creating (m+m′)-haloes as a result.
131
Constant Exact Sqrt
Approx SqrtN-Body
Figure 4.10: Conditional creation rates. Symbols, line styles and choice of masses
as in Figure 4.8.
132
This merger rate has been measured recently in cosmological simulations (Fakhouri
& Ma, 2008a,b) and studied in the spherical (Lacey & Cole, 1993) and ellipsoidal
(Zhang et al., 2008a) collapse versions of the excursion set approach. The problem
in this case is that, with the exception of white-noise initial conditions, the excursion
set theory predicts that R(m,m′|t) 6= R(m′,m|t) (Sheth & Pitman, 1997) (but see
Neistein & Dekel, 2008b, for a possible solution). Even if the merger rate were
defined unambiguously within the excursion set approach with moving barriers,
care must be taken when the integrals in equations (4.29) and (4.30) are computed.
Our prescription of the creation rate circumvents these problems altogether. A third
possible method is to use the merger rate developed by Benson et al. (2005) (see also
Benson, 2008). This method avoids the complications in the excursion set theory by
estimating the coagulation kernel K numerically. Unfortuntely, the answer in this
case depends on the choice of regularization technique. The formulation a complete
model for halo mergers with general initial conditions and with the advantages of
ellipsoidal collapse remains an open, and quite interesting, problem.
133
Chapter 5
Merger-induced Quasars
So far, the focus of this work has been on dark matter haloes and their assembly.
In this chapter, we present a model where quasars are triggered by major mergers
of dark matter haloes. In particular, a detailed comparison among theoretical halo
merger rates is provided. This project is the culmination of this thesis, and it
marks the beginning of the many astrophysics projects that can be studied with
our formalism.
5.1 Background
Our understanding of quasars has increased dramatically since their discovery over
four decades ago (Schmidt, 1963; Hazard et al., 1963). Evidence has shown that
these objects are associated with intense activity at the nuclei of galaxies at cosmo-
logical distances, produced by the intense accretion of gas onto supermassive black
134
holes (SMBHs or BHs) (Salpeter, 1964; Zeldovich & Novikov, 1964; Lynden-Bell,
1969; Bardeen, 1970). Likewise, our understanding of cosmology and structure for-
mation has increased at a similar pace. Our universe is well-described by a ΛCDM
expanding cosmology, where structure assembles hierarchically via mergers of dark
matter haloes. These haloes provide the gravitational potential wells where gas can
cool and fragment, allowing the formation of stars and galaxies (White & Rees,
1978; Blumenthal et al., 1984; White & Frenk, 1991). It is natural to expect that
violent halo mergers are followed by the mergers of the galaxies they harbor. More-
over, it has been suggested that major mergers of gas-rich galaxies are an efficient
mechanism to feed SMBHs. This suggestion is supported by hydrodynamic simula-
tions, where mergers produce gravitational torques, which induce strong gas inflows
into the center (e.g., Hernquist, 1989; Barnes & Hernquist, 1991, 1996). In this sce-
nario, the hierarchical assembly of haloes serves as the stage, or back bone of the
coevolution of galaxies and their nuclear black holes. The problem with this picture
is that while the halo hierarchy continues today, observations show that the peak
of quasar activity (and galaxy formation) has already passed (Osmer, 1982; Warren
et al., 1994; Schmidt et al., 1995; Shaver et al., 1996; Steffen et al., 2003; Ueda et
al., 2003; Barger et al., 2005; Brusa et al., 2009). Reconciling these two seemingly
contradictory pictures requires a deeper understanding of the evolution of BHs and
their host galaxies.
In recent years, dynamical observation have revealed that SMBHs are ubiquitous
135
at the centers of most, if not all local massive galaxies (Richstone et al., 1998).
Moreover, these observations show that tight correlations exist between the mass
of the nuclear black hole and properties of the spheroidal component of the host
galaxy. These include scalings between the mass of the black hole and the bulge’s
stellar mass (Kormendy & Richstone, 1995; Magorrian et al., 1998; Marconi &
Hunt, 2003; Haring & Rix, 2004), surface brightness (early-type) (van der Marel,
1999), light concentration (Graham et al., 2001; Graham & Driver, 2007), luminosity
(Bettoni et al., 2003; McLure & Dunlop, 2002, 2004; Graham, 2007; Bernardi et
al., 2007; Gultekin et al., 2009; Bentz et al., 2009), kinetic energy (Feoli & Mele,
2005, 2007), and velocity dispersion (Ferrarese & Merritt, 2000; Gebhardt et al.,
2000; Tremaine et al., 2002; Shields et al., 2003; Bettoni et al., 2003; McLure &
Dunlop, 2004; Bernardi et al., 2007; Tundo et al., 2007; Gultekin et al., 2009) –
this last correlation is commonly referred to as the ‘MBH-σstar relation’. Beyond
the spheroidal component, scaling relations between the black hole mass and either
the host’s circular velocities (Baes et al., 2003; Ferrarese, 2002) or the mass of
the surrounding dark matter halo (Ferrarese, 2002; Bandara et al., 2009) have also
been proposed. See Novak et al. (2006) for a comparison amongst some of these
scaling relations. These observations suggest that the formation of galaxies and
their nuclear black holes are intimately linked; where the dormant SMBHs in the
local universe are the present-day remnants of distant luminous quasars.
A simple way to constrain the growth of these SMBHs is to link their local
136
abundances to the luminosity evolution of quasars throughout the history of the
universe. By comparing the local black hole mass function (BHMF) to the integral
of the luminosity function (LF) of active galactic nuclei (AGN) at all epochs, it
has been established that SMBHs acquire most of their mass through fast-accreting
‘quasar’ phases (see e.g., Soltan, 1982; Salucci et al., 1999; Yu & Tremaine, 2002;
Aller & Richstone, 2002; Yu & Lu, 2004a,b; Marconi et al., 2004; Merloni, 2004;
Shankar et al., 2004; Barger et al., 2005; Yu et al., 2005; Hopkins et al., 2006a;
Yu & Lu, 2008; Shankar et al., 2009).1 One way to describe the evolution of black
holes is with semi-empirical models, which numerically follows the ‘flow’ of BH mass
induced by accretion by means of the continuity equation (Cavaliere et al., 1971,
1983; Cavaliere & Szalay, 1986; Cavaliere et al., 1988; Cavaliere & Padovani, 1989;
Caditz & Petrosian, 1990; Caditz et al., 1991; Small & Blandford, 1992; Cavaliere &
Vittorini, 2000; Yu & Tremaine, 2002; Yu & Lu, 2004a; Shankar et al., 2009). While
it is true that these models provide very valuable constrains on BH evolution (e.g.,
on the radiative efficiency and the mean Eddington ratio), they do not address the
coevolution of SMBHs and their hosts directly.
In order to connect the evolution of BHs to their hosts, early works use the
Press & Schechter (1974) model, or the more accurate Sheth & Tormen (1999)
model. This is typically done by using the halo mass function (Monaco et al.,
2000; Haiman & Menou, 2000; Hatziminaoglou et al., 2001), its time derivative
1Throughout this work, the term ‘quasar’ refers to AGN in the brightest, fast-accretion phases.
137
(Efstathiou & Rees, 1988; Haehnelt & Rees, 1993; Haiman & Loeb, 1998; Haiman,
2004), or ad-hoc versions of the halo creation rate (Cavaliere & Vittorini, 1998;
Haehnelt et al., 1998; Percival & Miller, 1999; Cavaliere & Vittorini, 2002; Vittorini
et al., 2005; Lapi et al., 2006; Miller et al., 2006). Aside from inconsistencies in the
formalism of the creation rate (see Chapter 4), it is unclear whether the merger of
two haloes with any mass ratio can trigger nuclear activity. In particular, numerical
simulations show that minor mergers of galaxies only affect the outskirts, and are
therefore rather inefficient at transporting fuel towards the center (Cox, 2004; Cox
et al., 2008). This raises one of the central questions addressed in this thesis: do
mergers trigger quasars?
For several years, it has been suspected that mergers of galaxies trigger quasar
activity (Stockton, 1982; Sanders et al., 1988a). Many quasar hosts undergoing
mergers and interactions have been observed at low redshifts (e.g., Heckman et al.,
1984; Hutchings et al., 1988; Canalizo & Stockton, 2001; Kauffmann et al., 2003;
Hutchings, 2003a; Hutchings et al., 2003, 2006; Zakamska et al., 2006; Letawe et al.,
2007; Urrutia et al., 2008). Also, strong Ca II/Mg II absorption lines in quasars are
associated with a disturbed interstellar medium, possibly due to a recent merger
(Wang et al., 2005). Further evidence comes from ultraluminous infrared galaxies
(ULIRGs) (see e.g., Sanders & Mirabel, 1996; Jogee, 2006). Almost all of these
objects appear to be in the last stages of merging (Allen et al., 1985; Joseph &
Wright, 1985; Armus et al., 1987; Kleinmann et al., 1988; Borne et al., 1999; Veilleux
138
et al., 2002; Dasyra et al., 2006a,b), and CO observations show that many of these
galaxies contain huge amounts of gas in their nuclei (Scoville et al., 1986; Sargent
et al., 1987, 1989). Moreover, the presence of ‘warm’ IR spectra (Sanders et al.,
1988b) suggests that ULIRGs are likely to be a dust-enshrouded ‘buried’ quasars,
triggered by galaxy mergers (Sanders et al., 1988a; Genzel et al., 1998; Tran et
al., 1999; Sajina et al., 2007). One would expect, therefore, that all quasar hosts
should have visible imprints of a recent merger. Recently, high-resolution imaging
studies have revealed the existence of spectacular shells in quasar hosts (Canalizo
et al., 2007; Bennert et al., 2008), many of which had been previously catalogued
as ‘bona-fide undisturbed galaxies’ (Dunlop et al., 2003).
An attractive feature of major mergers is that during these events, stars can be
easily scrambled from circular to random orbits, producing a change in morphology
as a result. Indeed, simulations show that the only way to reproduce the kine-
matic properties of elliptical galaxies and bulges is via mergers (Toomre & Toomre,
1972; Toomre, 1977; Barnes, 1988; Hernquist, 1989; Barnes, 1992; Hernquist, 1992;
Schweizer, 1992; Hernquist, 1993; Naab et al., 1999; Naab & Burkert, 2003; Naab
et al., 2006a; Naab & Trujillo, 2006b; Naab et al., 2006c; Bournaud et al., 2005;
Cox et al., 2006; Jesseit et al., 2007). If the spheroidal component of galaxies and
SMBHs are formed by the same mechanism as quasar activity, then it is plausible
that this ‘merger hypothesis’ helps explain the aforementioned tight correlations
between the mass of the black hole and the properties of the host.
139
Aside from feeding the BH, the new reservoir of available cold gas provided by
the merger is capable of producing sudden bursts of star formation. A connection
between AGNs and starburst in merging galaxies has been suggested by several
authors (Canalizo & Stockton, 1997; Stockton et al., 1998; Brotherton et al., 1999)
– see Ho (2005) for a review. For instance, spectroscopic signatures, which are
indicative of post-starburst stellar populations, are present in the spectral energy
distributions (SEDs) of quasars (Brotherton et al., 1999; Canalizo & Stockton, 2001;
Kauffmann et al., 2003; Yip et al., 2004; Jahnke et al., 2004a; Sanchez et al., 2004;
Vanden Berk et al., 2006; Barthel, 2006; Zakamska et al., 2006).
One interesting observation is that since redshift z ∼ 2, the most massive galax-
ies appear to shut down star formation earlier than their less-massive counterparts
(Juneau et al., 2005; Treu et al., 2005; Bundy et al., 2006; Borch et al., 2006;
Cimatti et al., 2006; Bell et al., 2007; Bundy et al., 2008; Cowie & Barger, 2008).
This phenomenon, known as ‘downsizing’ (Cowie et al., 1996), coincides with the
drop in number of quasars at recent epochs – pointing to the possibility that black
hole accretion and star formation are quenched by the same mechanism (Shaver
et al., 1996; Dunlop, 1997; Boyle & Terlevich, 1998). Several authors advocate
AGN feedback as the primary quenching mechanism (e.g., Ciotti & Ostriker, 1997;
Silk & Rees, 1998; Fabian, 1999; Ciotti & Ostriker, 2001; King, 2003; Begelman,
2004). In this scenario, when the nuclear luminosity becomes comparable to the
binding energy of the host, it prevents gas from cooling and turning into stars,
140
while accretion by the black hole is terminated. Invoking supernova feedback as
the primary quenching mode of star formation is another possibility (Dekel & Silk,
1986; Kauffmann & Charlot, 1998). However, this method predicts that the most
massive galaxies should harbor the youngest stellar populations, contrary to ob-
servations. It seems, therefore, that supernovae feedback is insufficient to inhibit
star formation in massive galaxies (Benson et al., 2003; Schawinski et al., 2007),
further enforcing the idea that AGN feedback regulates the coevolution of SMBHs
and their spheroidal hosts.
In particular, the most massive elliptical galaxies tend to harbor old, metal-rich,
quiescent stellar populations (Sandage & Visvanathan, 1978; Bernardi et al., 1998;
Jørgensen et al., 1999; Trager et al., 2000; Terlevich & Forbes, 2002; Caldwell et
al., 2003; Nelan et al., 2005; Thomas et al., 2005) – and so do their precursors,
the hosts of the most luminous quasars (Jahnke et al., 2004b; Vanden Berk et al.,
2006). This is corroborated by observations of the [Mg/Fe] ratio (Faber et al.,
1992; Carollo et al., 1993), which an indicator that the spheroid forms before Type
Ia supernovae explode (Matteucci, 2001). Interestingly, the [Mg/Fe] ratio increases
with the mass of the host (Worthey et al., 1992; Weiss et al., 1995; Kuntschner,
2000; Kuntschner et al., 2001; Thomas et al., 2005; Nelan et al., 2005). Lastly,
using AEGIS X-ray observations, Bundy et al. (2008) recently found that not only
is star formation quenched first in massive galaxies, but also faster! These last few
points will motivate our modeling of quasars in the last stages of activity.
141
Gunn (1979) summarized the quasar phenomenon in terms of three key ingre-
dients: an engine, a fuel supply, and a transport mechanism connecting the two.
Observational evidence supports a picture where the nuclear SMBH is the engine,
gas-rich galaxies provide the fuel, and major mergers are the mechanism. More-
over, a fourth ingredient may be added to Gunn’s list: AGN feedback quenches both
BH accretion and star formation, regulating the coevolution of SMBHs and their
spheroidal hosts as a result. In this picture, supernova feedback plays a secondary
role (Benson et al., 2003; Schawinski et al., 2007). Putting the pieces together into
a coherent picture has taken many years. In parallel, we have witnessed the emer-
gence of several theoretical scenarios of merger-induced quasars. These methods
come in many flavors, and can be classified as follows:
• Analytic models (Wyithe & Loeb, 2002; Hatziminaoglou et al., 2003; Wyithe
& Loeb, 2003; Scannapieco & Oh, 2004; Mahmood et al., 2005; Rhook &
Haehnelt, 2006; Marulli et al., 2006; Shen, 2009b, and the present work).2
With the exception of Shen (2009b), these models typically involve some ver-
sion of the extended Press-Schechter (ePS) merger rate (Lacey & Cole, 1993).
Also, they often invoke simple prescriptions linking the BH mass and the lu-
minosity of the associated quasar – the latter is modeled by means of light
curves with universal shapes.
2See also Roos (1981), Roos (1985a), Roos (1985b), De Robertis (1985) and Carlberg (1990)
for much earlier theoretical attempts to link quasars to mergers.
142
• Semi-analytic models (SAMs) (Cattaneo et al., 1999; Cattaneo, 2001; Kauff-
mann & Haehnelt, 2000; Volonteri et al., 2003; Enoki et al., 2003; Koushiappas
et al., 2004; Yoo & Miralda-Escude, 2004; Bromley et al., 2004; Volonteri &
Rees, 2005; Volonteri et al., 2006; Marulli et al., 2006, 2007; Malbon et al.,
2007; Yoo et al., 2007; Somerville et al., 2008; Tanaka & Haiman, 2008). These
models rely on Monte Carlo realizations of the merger history of dark matter
haloes, along with galaxy formation recipes. Monte Carlo merger trees repro-
duce the merger history to great accuracy, and are flexible enough to allow a
great number of realizations with greater resolution and scope (both in mass
and time) than N-body simulations (see Chapter 3 for a discussion). Alterna-
tively, the MORGANA method (Monaco et al., 2007; Fontanot et al., 2007)
models the merger history of haloes with the Zel’dovich approximation-based
PINOCCHIO code (Monaco et al., 2002a,b; Taffoni et al., 2002), which has
similar advantages as the Monte Carlo merger tree methods.
• Hybrid models (Cattaneo et al., 2005; Bower et al., 2006; Hopkins et. al.,
2006; Li et al., 2007; Marulli et al., 2008; Lagos et al., 2008; Bonoli et al.,
2008). These are very similar to the SAMs, but the galaxy-formation recipes
are incorporated in the post-processing of cosmological N-body simulations
(Springel et al., 2005). These simulations capture all the nonlinear features of
halo assembly – something often missed in the Monte Carlo approach (Gao
et al., 2005; Wechsler et al., 2006; Gao & White, 2007; Wetzel et al., 2007;
143
Angulo et al., 2008).
• Hydrodynamic models (Springel et al., 2005; Di Matteo et al., 2005; Hopkins
et al., 2006a; Robertson et al., 2006; Hopkins et al., 2008a,b; Younger et al.,
2008; Khalatyan et al., 2008; Johansson et al., 2009). These models follow the
merger of individual pairs of gas-rich galaxies in great detail. One limitation
is that these galaxy mergers are not put in a cosmological context, and thus
the nature of the selected participating galaxies and their type of encounters
are not always necessarily representative.
• Direct models (Sijacki et al., 2007; Di Matteo et al., 2008; Colberg & di Matteo,
2008; Booth & Schaye, 2009). These new methods incorporate hydrodynamic
methods into cosmological simulations. One drawback is that testing the
physical inputs requires performing a new run of the full simulation, severely
limiting a thorough study of the impact of the different sub-grid recipes em-
ployed.
The advent of supercomputers in recent years has been key to many of these
approaches. However, although the size and resolution of simulations has increased
dramatically, a full multi-scale approach is not yet available. Even the so-called ‘self-
consistent’ hydrodynamic and direct methods have to rely on separate analytical
prescriptions (e.g., Hoyle & Lyttleton, 1939; Bondi & Hoyle, 1944; Bondi, 1952) to
describe the unresolved circumnuclear region near the SMBH. And while detailed
modeling of accreting BHs has been performed (Gammie, 2001; Lodato & Rice,
144
2004; Shapiro, 2005; Lodato & Rice, 2005; Rice et al., 2005), it is often disconnected
with the potential impact its feedback may have on the stellar evolution of its host
(but see Escala, 2007; Mayer et al., 2008; Levine et al., 2008; Kawakatu & Wada,
2008, for recent attempts in this direction).
In adding computational complexity to a model, flexibility and scope are often
sacrificed. Moreover, our poor knowledge of BH evolution and accretion physics still
prevents us to widely adopt the above techniques in extreme detail. Contrary to this
trend, the spirit of this work is to follow the footsteps of the early analytic methods,
and revisit some of their underlying assumptions. Another issue is that different
SAMs often reach opposite conclusions on BH evolution, even though they match
similar sets of observables (e.g., Lapi et al., 2006; Malbon et al., 2007; Marulli et
al., 2008, see section § 5.7.3). For this reason it is essential to step back a little and
attempt to probe some of the central mechanisms regulating BH evolution with more
transparent and basic recipes. In this work we propose a simple and concise model,
based on very general and widely adopted (and accepted) physical assumptions. We
will show that by matching the predictions of such a basic model with a variety of
different observations, we can identify quite general and important characteristics
of BH evolution, which must be seriously considered by more advanced SAMs.
We assume that quasars are triggered by major mergers of dark matter haloes.
A universal light curve describes the evolution of individual quasars. This light
curve consists of an exponential ascending phase and a mass-dependent power-law
145
descending phase. Physically, right after the halo merger, a torque is exerted in the
host and gas is quickly funnelled into the center of the newly-formed gravitational
potential. In this phase, the BH accretes the fuel very efficiently. However, accretion
cannot last indefinitely, and we assume that it is quenched when the luminosity is
powerful enough to expel this gas or at least to balance its infall, thus inhibiting
further growth. It is precisely the depth of the potential (Vvir) that controls the
peak luminosity of the quasar. We postulate a self-regulating condition relating the
peak luminosity the AGN can reach, and the mass of the host halo at triggering.
This peak luminosity dictates the transition from the ascending to the descending
phase. This is a very general condition adopted by all SAMs and verified, to some
extent, in most numerical simulations (e.g., Di Matteo et al., 2005; Ciotti et al.,
2009; DeBuhr et al., 2009), although several aspects of the actual underlying physics
still remain obscure. In our model, this simple scaling condition encapsulates the
physics responsible of halting nuclear activity. While it is true that the host halo
may grow further through accretion of satellites and diffuse material, this has little
effect on the core of the potential where the spheroid forms (e.g., Zhao et al.,
2003; Diemand et al., 2007; Vass et al., 2009). One key feature in our modeling is
that we assume that quasars inhabiting more massive haloes shut down faster. As
mentioned before, this last ingredient is in line with recent observations indicating
that star formation in massive galaxies is quenched first and faster. Motivated by
this empirical evidence, we will show that shortening the light curves for quasars in
146
massive haloes will help to reproduce the bright-end of the AGN luminosity function
at low redshifts.
5.2 The model
Our model consists of two ingredients: (1) the halo merger rate, which describes
major mergers as the trigger of AGN activity, and (2) the light curve, which de-
scribes the evolution of individual quasars. These two ingredients are described
below.
5.2.1 The halo merger rate
There are a number of theoretical models for the dark matter halo merger rate in the
literature (Lacey & Cole, 1993; Parkinson et al., 2008; Zhang et al., 2008a; Neistein
& Dekel, 2008b; Benson, 2008). These are typically based on either the excursion
set formalism (Bond et al., 1991; Sheth, Mo & Tormen, 2001) or obtained by tuning
to numerical simulations. Section 5.3 includes a detailed comparison among various
excursion-set predictions.
More recently, direct fits to the merger rate in numerical simulations have been
performed (Fakhouri & Ma, 2008a; Genel et al., 2009). It is common practice to
express the merger rate in terms of the sum M = m + m′ and the ratio3 ξ = m/m′
3Note that in previous chapters we have used the notation µ for this quantity. For clarity, we
opted to stick to the original notation used by Fakhouri & Ma (2008a) in this discussion.
147
(or m′/m), where m < m′ (m′ < m):
R(m,m′, z) = R(ξ,M, z)∣
∣
∣
∂(ξ,M)
∂(m,m′)
∣
∣
∣= R(ξ,M, z)(1 + ξ)2/M. (5.1)
This particular form facilitates the selection of major mergers, which may be re-
sponsible for triggering quasars. Moreover, Fakhouri & Ma (2008a) found that the
evolution of R(ξ,M, z) is strongly determined by the mass function (their Figure
6). For this reason, one often focuses instead in the merger rate per unit halo,
R(ξ,M, z)/n(M, z). These authors provide a fit given by
R(ξ,M, z)
n(M, z)= κ0
(M
M
)κ1(ξ
ξ
)−κ2
exp[(ξ
ξ
)κ3](dδc
dz
)κ4
. (5.2)
where δc is the spherical overdensity required for collapse, M = 1.2 × 1012M⊙,
ξ = 0.098, and (κ0, κ1, κ2, κ3, κ4) = (3.0798, 0.083, 2.01, 0.409, 0.371). This fit is a
simple product of functions of M , ξ and z, and is accurate to the 10-20 percent level,
a negligible uncertainty with respect to the error bars in the observables we will
be comparing with. For our purposes, the Sheth & Tormen (1999) mass function
is accurate enough too. As shown in Chapter 2, this is essentially identical to our
square-root barrier model. (In this context, see also the fits by Warren et al., 2006;
Reed et al., 2007; Lukic et al., 2007; Tinker et al., 2008).
5.2.2 The light curve
The light curve consists of three components: an ancending phase of fast accre-
tion, the peak luminosity, and a descending phase where the AGN is gradually
extinguished by its own feedback.
148
The ascending phase
First, we assume that once the major merger takes place (at some triggering time
ttri) the black hole begins to accrete at some constant multiple of the Eddington
rate. The associated bolometric luminosity is
L = λ0LEdd, (5.3)
where the parameter λ0 is the Eddington ratio. The Eddington luminosity is defined
as
LEdd ≡4πGmp
σecMBHc2 ≡ MBHc2
τS
, (5.4)
where G is Newton’s gravitational constant, mp is the mass of the proton, σe is
the electron-photon Thompson scattering cross section, c is the speed of light, and
τS ≃ 0.43 Gyrs. is known as the Salpeter time.4 It is also common to write to above
expression as
LEdd = 1.26× 1038 erg s−1
(
MBH
M⊙
)
≡ l
(
MBH
M⊙
)
. (5.5)
Notice that in this phase, the luminosity scales linearly with the black hole mass:
L =
(
λ0l
M⊙
)
MBH =
(
λ0c2
τS
)
MBH. (5.6)
4This order-of-magnitude result comes from taking an idealized sphere in hydrostatic equilib-
rium, where the pressure force Frad = LEdd/4πr2c associated with the radiation being emitted
balances the gravitational force Fgrav = −GMBHmp/r2 exerted on the material being accreted –
see e.g., Peterson (1997) and Martini (2004) for a review.
149
Black hole growth results from the mass that falls into the black hole that is not
radiated away. If a fraction ǫ of the gravitational potential energy of the infalling
material is converted into radiation, the associated luminosity is
L = ǫMinfallc2, (5.7)
where Minfall is the rate of accretion of infalling material and the parameter ǫ is
known as the radiative efficiency. While a fraction ǫ of the infalling matter is
radiated away, the rest (i.e., 1 − ǫ) feeds the black hole (see e.g., Yu & Tremaine,
2002). In other words, MBH = (1 − ǫ)Minfall, and the luminosity can be expressed
in terms of the black hole’s mass growth rate:
L =ǫ
1− ǫMBHc2 ≡ fMBHc2. (5.8)
Notice that in this phase, the luminosity scales linearly with the black hole mass
growth rate. Combining equations (5.6) and (5.8), one obtains that MBH/MBH =
λ0/fτS, and the black hole mass increases exponentially:
MBH(t) = MBH(ttri) exp(t− ttri
tef
)
, t ≥ ttri, (5.9)
where
tef ≡f
λ0
τS ≃ 0.43( f
λ0
)
Gyr., (5.10)
is the e-folding time. By virtue of equation (5.6), the ascending phase of the light
curve evolves as
Lasc(t) = L(ttri) exp(t− ttri
tef
)
, (5.11)
150
with L(ttri) = λ0lMBH(ttri)/M⊙ (we will discuss the value of MBH(ttri) shortly).
Physically, this exponential growth cannot be sustained indefinitely. Eventually,
this phase is halted at some time tpeak after triggering, where the maximum lumi-
nosity Lpeak = L(tpeak) is reached. At that point, we denote the black hole mass by
MBH, peak = MBH(tpeak).
The peak luminosity
The ascending phase reaches its peak luminosity when the feedback is powerful
enough to unbind gas from the potential well. Subsequently, accretion by the black
hole becomes substantially inefficient, and is eventually terminated. In our model,
the physics of AGN feedback is encapsulated in a self-regulation condition between
the peak luminosity and the mass of the host halo at triggering. Following Wyithe
& Loeb (2002) and Wyithe & Loeb (2003), we postulate this condition to have the
form
Lpeak
1044ergs−1= 7
(
M
1012M⊙h−1
)5/3
(1 + z)5/2 , (5.12)
where M is the mass of the host dark matter halo at the time of triggering.
The role of this condition is to control the peak luminosity at the end of the
ascending phase. Physically, when the peak luminosity is attained, AGN feedback
kicks in, and the light curve switches from the ascending to the descending phase (see
below). Wyithe & Loeb (2002) provide a simple explanation for this self-regulation
condition (see also Silk & Rees, 1998). First, consider the luminosity capable of
151
unbinding the entire energy of the gravitating system, which is proportional to
the ratio of the binding energy (∝ MV 2vir) and the dynamical time (∝ rvir/Vvir),
where rvir is the virial radius and Vvir is the virial velocity. This ratio is given
by Lpeak ∝ MV 3vir/rvir. From the virial relation, GM/rvir ∝ V 2
vir, we can write
Lpeak ∝ V 5vir/G ∝ (M/rvir)
5/2. But the volume r3vir ∝ M/ρ ∝ M/(1 + z)3, and
M/rvir ∝M2/3(1 + z) results.5 Thus, with this order of magnitude calculation, and
assuming that the halo mass at triggering scales with the gravitating mass that
regulates AGN feedback, one obtains
Lpeak ∝M5/3(1 + z)5/2, (5.13)
which is precisely our postulated self-regulation condition.
Recall that we assume that the BH ends its rapid-accretion phase when the
feedback energy is sufficient to unbind, or just energetically balance the infall of,
the surrounding gas reservoir. The potential well Vvir refers to the core of the
halo’s gravitational potential, and it is dominated by the inner regions of the halo
and the newly-formed spheroid (the gravitating system). Note that our feedback-
constrained condition refers to the virial velocity Vvir at triggering. While it is true
that the halo mass may increase between ttri and tpeak – though a second major
merger is unlikely within that (short) time – it has been numerically proven that the
5Strictly speaking, M/rvir ∝M2/3(1+z)ξ(z)1/3, where ξ(z) = (Ωm,0/0.25)(1/Ωm(z))(∆/18π2),
∆ = 18π2 + 82d = 39d2 is the mean density contrast relative to the critical density, and d ≡
Ωm(z) − 1 (Bryan & Norman, 1998; Barkana & Loeb, 2001). Since ξ(z) varies very weakly with
redshift, it is safe to ignore it.
152
central potential well remains rather constant for a long time since the virialization
epoch (e.g., Zhao et al., 2003; Vass et al., 2009). Moreover, minor mergers and
accretion only affect the outskirts (Cox, 2004; Cox et al., 2008), leaving the core of
the potential intact.
Alternatively, some authors argue that feedback momentum (rather than feed-
back energy) may dominate self-regulation (King, 2003; Di Matteo et al., 2005;
Lidz et al., 2006). In that case, Lpeak ∝ v4vir/G ∝ M4/3(1 + z)2, which depends
more weakly on halo mass and redshift than our energy-driven condition. We do
not discuss this model any further below, although we have checked that the main
conclusions of this work do not significantly change when switching to the latter
feedback condition, provided that the appropriate normalization in the Lpeak-M re-
lation (a factor of∼ 2−3 higher) is considered (and some rather small readjustments
in other parameters).
Lastly, in § 5.7.2 we discuss how this self-regulation condition has an impact on
the tight correlations between the nuclear BH and the host spheroid in the local
universe. These correlations have scatter. In our model, we assume that the our
Lpeak −M relation has a scatter of Σ dex (see equation 5.30 below).
The descending phase
This portion of the light curve describes the gradual shutdown of the quasar after
it reaches its peak luminosity. For instance, in hydrodynamic simulations (e.g.,
153
Hopkins et al., 2006b), accretion is halted as the feedback energy couples to the gas
reservoir, leading to a nearly self-similar power-law decay, L(t) ∝ t−α. Motivated
by this, Shen (2009b) recently adopted decaying power-law light curves, where the
parameter α is mass-independent.
Throughout this work, the evolution of SMBHs and their host spheroids has
been emphasized. In particular, the most luminous massive ellipticals tend to be
redder. The color suggests older stellar population, while higher luminosity suggests
that these populations are more metal rich. Moreover, high metallicity is indicative
of efficient periods of star formation, taking place during short periods of time (see
e.g., Lapi et al., 2006, for a discussion). This is further corroborated by Bundy et
al. (2008), who conclude that star formation in massive systems is quenched faster
than in less massive ones. Lastly, even hydrodynamic simulations model a shutdown
phase that depends on mass (Hopkins et al., 2006b; Hopkins & Hernquist, 2009).
Motivated by this, we model our descending phase in terms of a mass-dependent
power-law:
Ldsc(t) = L(tpeak)
(
t
tpeak
)−α(M)
. (5.14)
In our model, the shutdown parameter α(M) is described by the following em-
pirical law:
log α(M) = log α0 + α1 log
(
M
1012.5M⊙
)
ΘH(M − 1012.8M⊙), (5.15)
154
where α0, α1 are free parameters, and ΘH(x) is the Heavyside step function.6 In
our reference model, we use (α0, α1) = (3.5, 0.9). Other parametrizations of this
effect are allowed, provided that the yield the desired behavior that quasars in more
massive host shut down faster.
Major mergers and host masses
Simulations show that minor mergers (those with ξ < 1/10, where ξ ≡ m′/m, m
and m′ are the masses of the merging haloes, and m′ < m) only affect the outskirts
(Cox, 2004; Cox et al., 2008). Likewise, mild mergers (with 1/4 > ξ > 1/10) may
trigger some AGN activity (Hernquist, 1989; Hernquist & Mihos, 1995; Bournaud et
al., 2005), but these are limited to an specific set of orbital geometries (Younger et
al., 2008; Johansson et al., 2009). In this work, we assume that major mergers (with
ξ > 1/4) are the triggers of AGN activity.7 This assumption is further supported by
observations (Dasyra et al., 2006a; Woods et al., 2006), which show that starbursts
and strong inflows of gas are only seen in major mergers.
In the major-merger regime, it is natural to assume that a merger of the haloes
6Formally, the Heavyside step function is defined as
ΘH(x) =
1, if x ≥ 0;
0, if x < 0.
In practice, we use a smooth version, given by (1− e−2x)−1.7In Chapter 3 we used a more strict version of ‘major mergers’ with ξ > 1/3. The operational
difference between the two is largely irrelevant.
155
is followed by a merger of the galaxies within, after some delay time necessary for
the galaxies to sink to the bottom of the potential. However, as also discussed by,
e.g., Cavaliere & Vittorini (2000) and Shen (2009), the time for quasar triggering is
not necessarily associated to the time of the actual merging of the host galaxies, and
actually observations and numerical simulations do indeed favor scenarios where the
tidal forces start acting on the gas reservoirs already at the very initial stages of the
merger (references above). In the following we will therefore assume that quasar
activity is triggered at the moment of the halo merging8, and we will discuss the
consequences of loosening such an assumption in § 5.7.3. Also note that for mild
mergers, the delay times are much longer, and the satellites also suffer from tidal
stripping, which diminishes their ability to trigger any AGN activity (e.g., Colpi et
al., 1999).
Another issue is that not all mergers are gas-rich; and gas-poor ‘dry’ mergers
are certainly not expected to trigger quasars. Theoretical models predict that dry
mergers between ellipticals are seldom, except perhaps at low redshifts (Khochfar
& Burkert, 2003; Ciotti et al., 2007; Stewart et al., 2009; Hopkins et al., 2009). This
assertion is supported by observations (Lin et al., 2008; Bundy et al., 2009), and
will not be addressed in this work.
Lastly, very massive haloes, common at low redshifts, often harbor more than
one galaxy. Significant research on AGNs in galaxy clusters has been done obser-
8This condition is even more valid at higher redshifts, when hosts are smaller at fixed mass,
rendering tidal torques even more effective.
156
vationally (e.g., Houghton et al., 2006; Gebhardt et al., 2007; Georgakakis et al.,
2008a,b; Martini et al., 2009; Heinz et al., 2009) and theoretically (e.g., Thacker et
al., 2006; Scannapieco & Bruggen, 2008; Brueggen & Scannapieco, 2009). In this
work, we do not explore this regime. One reason is that tight correlations between
the SMBHs in central galaxies and other properties of the clusters have not been
found, and therefore our self-regulation condition (equation 5.12) is unlikely to hold
for very massive haloes. For instance, Gebhardt et al. (2007) found that for NGC
1399 (the central galaxy in the Fornax cluster), the mass of of the SMBHs is well
outside the MBH-σstar relation. Haloes more massive that 1013M⊙h−1, tend to be
groups and clusters of galaxies (Cavaliere & Vittorini, 2000; Menci et al., 2005; Vit-
torini et al., 2005). In such cases, major mergers are not likely to trigger significanly
bright AGN activity (Georgakakis et al., 2008b; Croton et al., 2006; Cattaneo et
al., 2007; Okamoto et al., 2008). For these reasons, in our model we constrain the
masses of haloes to be ‘galactic’ (e.g., 1011.5M⊙h−1 ≤ M ≤ 1013M⊙h−1), and one
galaxy per halo is assumed. For a potential way to extend our model, see Hopkins
et al. (2008b), who use halo occupation models (Yang et al., 2003; Yan et al., 2003;
Zheng et al., 2005; Conroy et al., 2006; Vale & Ostriker, 2006) and subhalo mass
functions (Kravtsov et al., 2004; Zentner et al., 2005; van den Bosch et al., 2005)
to populate haloes with more than one galaxy.
157
Universal light curves with two phases
Our theoretical light curve describes how quasars evolve ‘on average’; reality need
not be that simple. Nevertheless, using a universal prescription like ours is actually
quite powerful. Traditional models often adopt a light curve with only an ascending
phase (Salpeter, 1964; Rees, 1984; Small & Blandford, 1992; Haehnelt & Rees, 1993;
Salucci et al., 1999; Cavaliere & Vittorini, 2000; Marconi et al., 2004; Shankar et
al., 2004), only a descending phase (Haehnelt et al., 1998; Haiman & Loeb, 1998;
Kauffmann & Haehnelt, 2000), or assume that quasars shine at a fixed luminosity
for a fixed period of time – the so-called ‘light bulb’ models (Wyithe & Loeb, 2002,
2003). The problem with these prescriptions is that they are fail to match the quasar
bias (Marulli et al., 2006), the local BHMF (Yu & Lu, 2008) or the Eddington-ratio
distribution (Hopkins & Hernquist, 2009). In light of this, Shen (2009b) proposed a
model similar to ours, but with a mass-independent α. The simulations of Hopkins
et al. (2006b) – and the subsequent model by Marulli et al. (2008) – also invoke
a universal two-phase light curve with a descending phase (albeit with a different
mass dependence than in our model).
Let us now briefly discuss our choice of parameters describing the light curve.
First, notice that tef increases with increasing radiative efficienty (equation 5.10).
Physically, allowing more infalling material to turn into radiation slows down the
growth of the black hole. Soltan-like arguments comparing the amount of light
received from past quasars to the mass in local BHMF favors a radiative efficiency
158
of ǫ & 0.1 (Yu & Tremaine, 2002; Aller & Richstone, 2002; Marconi et al., 2004;
Shankar et al., 2004; Yu & Lu, 2004a, 2008; Shankar et al., 2009b). This value is
compatible with standard models of BH accretion (Shakura & Sunyaev, 1973; Rees,
1984; Narayan & Quataert, 2005), and even with more exotic process involving
electromagnetic extraction of energy from spinning BHs (Blandford & Znajek, 1977;
Phinney, 1983; Gammie, 2001; Shapiro, 2005).
The Eddington ratio, on the other hand, allows us to translate the observed
bolometric luminosity into the actual BH mass for a given quasar (equation 5.6).
The higher λ0, the more luminous the quasar is for a given black hole mass. In
this case, the e-folding time is shorter (equation 5.10), and the black hole accretes
faster. In this work we assume that the accretion in the ascending phase is mildly
super-Eddington, which is allowed by theoretical models (Begelman, 1978, 2002;
King, 2002; Ohsuga et al., 2005; Ohsuga & Mineshige, 2007), and below we will
discuss that this choice is actually favored by high-redshift observations.
The black hole in the earliest stages
As we just discussed, more realistic light curves can be more complicated. For
instance, some hydrodynamic simulations show that before the two galaxies sink to
the center of the remnant halo, a mild preliminary accreting phase can triggered by
the ‘first passage’ of the two interacting galaxies (Hopkins et al., 2005). Moreover,
accretion may occur in the two black holes associated with the merging galaxies
159
before BH coalescence. Indeed, sub-Mpc dual AGN have been observed at various
separations, indicating various stages of merging. For instance, the dual AGN in 3C
75 are 7 kpc apart, those in Mrk 463 are ∼ 3.8 kpc apart, while NGC 640 exhibits
active nuclei with a separation of ∼ 700 pc, and those in Arp 299 are separated
by . 50 pc (see Colpi & Dotti, 2009, and references therein). Also, Comerford et
al. (2009) report a dual AGN with separation of ∼ 1.75 kpc, and Chornock et al.
(2009) found another pair at ∼ 5 kpc. Likewise, Foreman et al. (2009) found 85
quasar pairs whose separation distribution peaks at ∼ 30 kpc. Sub-parsec binaries
are harder to observe. Ongoing SDSS surveys have found an increasing fraction of
observed quasar binaries (Hennawi et al., 2006, 2009; Shen et al., 2009c). Binary
quasars are expected to further constrain triggering mechanisms.
In our model, the initial quasar phase is modeled by using a single seed black
hole undergoing extremely fast accretion, which grows from some triggering mass
MBH, tri to some large mass MBH, peak before switching into the descending phase.
Since the bulk of the black hole mass is acquired during the last e-folding time
prior to the peak, we expect activity during this period to be insensitive to the
detailed physics of binary BH coalescence. Choosing the mass at triggering is not a
trivial matter. For instance, while stellar theory predicts seed BH mass ≃ 102−4M⊙
arising from population-III stars (Omukai & Nishi, 1998; Madau & Rees, 2001;
Heger & Woosley, 2002; Ripamonti et al., 2002; Bromm & Loeb, 2003; Pelupessy
et al., 2007; Begelman et al., 2008), the problem of seeding galaxies with BHs at
160
Figure 5.1: The quasar light curve. After triggering, the quasar grows exponentially
for some time tasc = tef ln(µBH). The peak is controlled by the MBH −M relation
(or Lpeak − M relation), where M is the mass of the host dark matter halo at
virialization. The descending phase decays as a power law, with a decaying exponent
that increases monotonically with halo mass.
161
early epochs remains largely unanswered (see e.g, Haiman, 2004). The truth of the
matter is that, at the present moment, our observational window into the early
stages of BH accretion is still very limited. For this reason, we choose the BH mass
at triggering to be a fixed fraction of the BH mass at the peak luminous stages:
MBH, seed ≡ MBH, peak/µBH. A direct consequence of this is that the time spent in
the ascending phase is fixed,
tasc ≡ tpeak − ttri = tef ln(µBH), (5.16)
and so is the number of e-foldings. Notice that in this scenario, tasc ∝ log(µBH)/λ0.
Namely, choosing large BH masses at triggering and high Eddington ratios has the
same effect on tasc. In § 5.6.2 we explore different combinations of log µBH and λ0
and their impact on the LF and quasar clustering. In this context, see also Shen
(2009b), who simply sets tasc at 6.9τS. In principle, a model with mass-dependent
tasc can be formulated. We made various attempts, which introduce breaks and
noise into our outputs of the luminosity function. Given that our knowledge of the
actual distribution of tpeak values is very limited, we adopted a universal duration of
the ascending phase and introduced mass-dependence only in the descending phase
(longer durations for less massive hosts).
In summary, the light curve is given by
L(t) =
L(ttri) exp(
t−ttritef
)
, ttri ≤ t ≤ tpeak;
L(tpeak)(
ttpeak
)−α(M)
, t > tpeak.
where α(M) increases monotonically with halo mass, and the duration of the ascend-
162
ing phase is set by the ratio of black hole mass at the time of maximum luminosity
and the triggering time (tpeak and ttri). Figure 5.1 illustrates the shape of the light
curve for three given values of MBH, peak. Before implementing the model, we will
discuss the different theoretical halo merger rate scenarios.
5.3 Theoretical halo merger rates
Consider the merger of an m-halo with an m′-halo at redshift z, producing a halo
of mass M = m + m′ as a result. The excursion-set theory with constant barriers
(ePS) provides a formula for this quantity. In this section, we show how this rate
is modified when moving barriers (such as our square-root barrier) are considered.
These are then compared against results from N-body simulations.
5.3.1 Extended Press-Schechter
In the excursion-set framework with constant barriers (spherical collapse), the
merger rate is given by
R(m,m′, z) = n(M, z) lim∆z→0
1
∆zN(m, z + ∆z|M, z), (5.17)
where n(M, z) is the (unconditional) mass function of haloes of mass M at redshift
z. The second factor, N(m, z+∆z|M, z), is the mass function of m-haloes at z+∆z
conditioned to be in bound objects of mass M at z. It is common to work in terms
of the ratio R/n because the evolution of R is strongly determined by n (Fakhouri
163
& Ma, 2008a). This ratio is given by
R(m,m′, z)
n(M, z)=
1√2π
M/m
(Sm − SM)3/2
∣
∣
∣
dSm
dm
∣
∣
∣
∣
∣
∣
dδc
dz
∣
∣
∣, (5.18)
where δc(z) is the spherical collapse threshold and S is the variance of the initial
density-perturbation field (linearly-extrapolated to the present), with a (sharp-k
space) window function containing a given mass (see Chapter 2 for a review).
Unfortunately, this prescription is formally self-inconsistent because, for the ini-
tial conditions considered here (encoded in the initial power spectrum), N(m|M) 6=
N(M −m|M). Sheth & Pitman (1997) show that for white-noise initial conditions,
the equality holds (see Chapter 4). In general, however, equation (5.18) predicts
two distinct merger rates, one for m < m′ and one for m > m′. As pointed out by
Benson (2008), one can still use a symmetrized version, such as
Rsym(m,m′, z) =
R(m,m′, z) if m < M/2;
R(m′,m, z) if m ≥M/2,
where R is given by equation (5.18). Alternatively, one could use
Rsym(m,m′, z) =
R(m′,m, z) if m < M/2;
R(m,m′, z) if m ≥M/2.
Zhang et al. (2008a) refer to these as ‘Option A’ and ‘Option B’, respectively. De-
spite this conceptual problem, we find that for all the models considered, ‘Option
A’ does a more reasonable job at reproducing the merger rates in N-body simula-
tions. We refer the reader to Neistein & Dekel (2008b) for a possible solution to
this inconsistency.
164
5.3.2 Ellipsoidal Collapse
In order to generalize the ePS rate in equation (5.18) for ellipsoidal collapse, the
limit of N(m, z + ∆z|M, z) as ∆z → 0 must be computed in that model. Recently,
Zhang et al. (2008a) showed that, while the Sheth & Tormen (2002) approximation
series to general barriers works well in general, it fails in this regime. Likewise, our
recipes for the exact square-root solution have numerical problems in that limit.
Nevertheless, expressions can be obtained by analyzing the barriers directly. (See
Zhang et al., 2008a, whose method we follow for square-root barriers).
Recall that the conditional mass function is obtained by solving the uncondi-
tional crossing distribution of a barrier
B(ac, s) = ac +√
qδ1
(s− S0
qδ21
)γ
(5.19)
where
s = S − S0, and ac ≡√
q(δ1 − δ0)−√
qδ0
( S0
qδ20
)γ
. (5.20)
Here, S = S(m), S0 = S(M), δ1 = δc(z + ∆z), δ0 = δc(z) and (q, β, γ) =
(0.707, 0.45, 0.615). By expanding equation (5.19) in s/S0 and (δ1 − δ0)/δ0, we
find that the leading order terms are given by
B(ac, s) = b0 + b1s + b2s2, (5.21)
After some algebra, one can show that the coefficients can be written as
b0 = G0(δ1 − δ0), b1 =G1√S0
, b2 = − 4G2√2πS
3/20
, (5.22)
165
where we have conveniently defined:
G0 =√
q[1 + (1/2− γ)(2β)(qν)−γ], (5.23)
G1 = γβ(qν)1/2−γ, (5.24)
G2 =
√2π
8γ(1− γ)β(qν)1/2−γ. (5.25)
The crossing distribution of this quadratic barrier is well-approximated by
f(s)ds ≃ b0√2πs
[
exp(
− (b0 + b1s)2
2s
)
−√
2πs3/2
4S0
b2 exp(
− b21s
2
)
]
ds
s. (5.26)
(For details, see Zhang & Hui, 2006; Zhang et al., 2008a). Notice that when b1 = 0
and b2 = 0, this reduces to the Press-Schechter result (equation 2.65). Similarly,
when b2 = 0, this reduces to the exact linear-barrier solution (equation 4.10).
Taking the limit as (δ1 − δ0) → 0, the halo merger rate (modulo the mass
function) is
R(m,m′, z)
n(M, z)=
R(m,m′, z)
n(M, z)
∣
∣
∣
ePS×G(s/S, ν). (5.27)
where ν = δ2c (z)/S(M), and
G(Sm/SM , ν) = G0
1+G2
(Sm
SM
−1)
32[
1+2G1√
π
(Sm
SM
−1)]
exp
− G21
2
(Sm
SM
− 1)
.
(5.28)
The factor of G can be thought of as a ‘perturbation’ to the ePS prediction (equa-
tion 5.18). Notice that all the dependence on redshift is contained only in |dδc/dz|
and in the (G0, G1, G2) parameters (which are functions of ν).
When (q, β, γ) = (1, 0, 0), B reduces to the ePS constant barrier. In this case,
(G0, G1, G2) = (1, 0, 0) and G = 1, as expected. The square-root barrier has
166
(q, β, γ) = (0.55, 0.5, 0.5), and in this case (G0, G1, G2) = (√
q, β/2,√
2πβ/32). No-
tice that in this case the perturbation G = G(s/S) is independent of redshift. See
Parkinson et al. (2008) and Benson (2008) for other expressions of G, which are not
derived from the excursion set theory. We now compare our expressions against a
fit to measurements in N-body simulations.
5.3.3 Comparison with N-body simulations
Figure 5.2 shows the halo merger rate modulo the mass function, (R/n) vs. the mass
ratio (ξ) for various masses, redshifts and models. The different panels show R/n
at redshifts z = (0, 1.2, 2.4, 3.6, 4.8, 6). Each panel shows three bundles of curves,
each bundle corresponding to each of the following halo masses: M = 1013,12,11M⊙
(top-to-bottom). For clarity, the curves with mass M = 1013M⊙ (M = 1011M⊙)
are shifted up (down) by an order of magnitude. The dotted, dashed and long-
dashed curves (in red, blue and green) denote the ePS, the square-root barrier,
and γ > 1/2 prediction of Zhang et al. (2008a). We will maintain these styles
throughout this chapter. The solid black line represents the Fakhouri & Ma (2008a)
fit to the Millennium Simulation (equation 5.2) – and it is truncated at low ξ to
indicate the scope of the simulation. Notice that the cutoff occurs at a higher
value of ξ for lower values of M . This is because, for each M , the minimum value
of ξ is ξmin ≡ Mmin/M . In the Millennium Simulation, only haloes with mass
≥ Mmin = 20Mp are considered, where the particle mass is Mp = 1.2 × 109M⊙.
167
Figure 5.2: Halo merger rates divided by mass function vs. mass ratio. Different
panels show redshifts (z = 0, 1.2, 2.4, 3.6, 4.8, 6) at three masses (bottom-to-top):
M = 1011,12,13M⊙. At each panel, each bundle of curves corresponds to a given
mass M . For clarity, curves with M = 1013M⊙ (1011M⊙) are shifted up (down) by
an order of magnitude. Please see key for each style corresponding to each model.
168
Thus, for smaller halo masses M , the value of ξmin is larger.
Before comparing the different models (Figures 5.3 and 5.4), some remarks are
in order. First of all, notice that in every case, R/n does not evolve strongly with
mass or redshift. As pointed out by Fakhouri & Ma (2008a), this quantity is indeed
‘nearly universal’. In other words, any evolution in the merger rate itself comes
from the mass function. Also, notice that R/n drops strongly with increasing ξ,
indicating that minor mergers occur more frequently than major ones. Let us now
compare the different theoretical predictions.
The curves in Figure 5.3 show R/n, modulo the fit of Fakhouri & Ma (2008a) to
the Millennium Simulation (equation 5.2). The choices of redshift, mass, and styles
are the same as in Figure 5.2, except that here we don’t shift curves corresponding
to different masses. In this case, the theoretical curves are truncated at low ξ
to indicate the scope of the N-body fit. Notice that for each model, the curve
shifts up with increasing mass and increasing redshift (although this trend is hardly
noticeable at high redshifts). Also, in all cases, curves shift up with increasing ξ. For
the most part, theory curves are above (below) the Fakhouri & Ma (2008a) at high
(low) ξ. Exceptions to this trend are found at redshift z = 0, where some ellipsoidal
collapse prediction are below the fit for all values of the mass ratio. Overall, the ePS
curves are the steepest. The Zhang et al. (2008a) prediction follows in steepness,
and our square-root barrier curves are the flattest. The Zhang et al. (2008a) curves
change (shift up) very strongly with increasing mass. This dependence is weaker in
169
Figure 5.3: Ellipsoidal collapse merger rates. Different panels show redshifts
(z = 0, 1.2, 2.4, 3.6, 4.8, 6) at three masses (bottom-to-top): M = 1011,12,13M⊙. All
models are divided by the Fakhouri & Ma (2008a) fit to the Millennium Simulation.
In this case we don’t shift the curves for different masses. Please see key for each
style corresponding to each model.
170
Figure 5.4: The same as Figure 5.3, but in terms of linear (as opposed to logarith-
mic) ξ. This allows us to see the large-ξ regime (associated with major mergers) in
more detail.
171
the ePS curves, and the weakest is our square-root barrier model. In other words,
the square-root barrier curves are closer to one another, the Zhang et al. (2008a)
curves are farther apart, and the separation of ePS curves is somewhere in between.
Ranking which model best resembles the fit to the Millennium simulation de-
pends on what one is actually interested in. In this work, we are interested in major
mergers quite above ξ > 1/10. To this end, we include Figure 5.4, which contains
the same information as Figure 5.3, except that we have plotted ξ linearly, as op-
posed to logarithmically. It is gratifying to see that in all cases our square-root
barrier model (depicted in blue) provides the best match to N-body simulations.
5.4 Implemention of the model
We now show how our model can be implemented to predict the statistics of AGNs
and black holes.
5.4.1 The luminosity function
The luminosity function is defined as the comoving number density of quasars shin-
ing with bolometric luminosity between L and L+dL at redshift zshi. We model
this as
Φ(L, zshi)dL = dL
∫ zmax
zzhi
dztri
∫ M/2
M/5
dm×R(m,M−m, ztri)δD(L−L(zshi)) WΣ[Lpeak,M, ztri].
(5.29)
172
Let us explain this expression physically – which can be interpreted as two
selection effects. First consider all mergers producing haloes of mass M at some
redshift ztri > zshi. The first selection is imposed by the Dirac delta, which only
chooses those M -haloes whose associated AGN light curve L shines in the (L,L+dL)
bin precisely at our redshift zshi of interest. For some, this luminosity will be in
the ascending phase while for others it will be in the descending phase. The second
selection is on the type of mergers involved. The integral from M/4 to M/2 only
chooses those mergers where the ratio of the constituents is m/(M − m) > 1/3.
(Notice that in this notation, m is the smaller of the two.) Lastly, just like the
Magorrian relations have scatter, it is natural to model our self-regulation condition
with some scatter. For this purpose, we assume that the Lpeak −M relation9 has a
log-normal distribution of width Σ = 0.28 dex, given by
WΣ[Lpeak,M, ztri] =1√
2πΣ2exp
[
−(log Lpeak − log Lpeak(M, ztri))2
2Σ2
]
. (5.30)
In this context, see Lapi et al. (2006) and § 5.7.2.
5.4.2 The large scale bias
To further test our models we compute the large scale clustering as a function of
luminosity and redshift via the equation
b(L, zshi) =
∫∞
0dM
∫ zshi
zmaxdz′WΣ[Lpeak,M, z′]R(M, z′)bh(M, zshi; z
′)∫∞
0dM
∫ zshi
zmaxdz′WΣ[Lpeak,M, z′]R(M, z′)
, (5.31)
9Unless it is discussed in the context of WΣ, we will denote the self-regulation condition as
Lpeak −M , as opposed to Lpeak −M .
173
where R(M, z) denotes the integral of R(m,M−m, z) in m from M/5 to M/2. The
evolution of the bias between z′ and zshi is
bh(M, zshi; z′)− 1 =
(
D(z′)
D(zshi)
)
[bh(M, zshi)− 1] (5.32)
as in Fry (1996). Hereafter we refer to this expression as the ‘Fry formula’ (see
the next section for a more thorough discussion). Following Sheth, Mo & Tormen
(2001), the halo bias is given by
bh(M, z) = 1 +1√qδc
[√q(qν) +
√qβ(qν)1−γ − (qν)γ
(qν)γ + β(1− γ)(1− γ/2)
]
, (5.33)
where (q, β, γ) = (0.75, 0.47, 0.615). Here ν(M, z) = δc(z)2/σ2(M) is the square of
the ratio of the critical overdensity required for collapse to the rms density fluctua-
tion on scales containing a mass M . Alternatively, many authors (e.g, Shen, 2009b)
use the Jing (1999) formula. However, the simulations of Shankar et al. (2009c)
show that the Sheth & Tormen (1999) result is more accurate.
We will discuss in § 5.6.2 that although the large systematic and statistical un-
certainties still prevent very accurate quasar clustering measurements, the present
available data are already good enough to set interesting constraints on BH evo-
lution, especially at low and high redshifts. More specifically, we will show that if
we allow the bias to evolve passively (e.g., Fry, 1996) since the triggering epoch ztri
until the time of shining zsh, the high quasar clustering signal measured at z > 3
constraints the delay time tdelay = t(zshi)− t(ztri) to be rather short, thus implying
an initial fast, super-Eddington BH growth. On the other hand, at low redshifts
174
z < 1 and low luminosities L . 1045erg s−1, the flat bias as a function of luminosity,
measured from SDSS, suggests that low-luminosity AGNs are hosted by rather mas-
sive haloes. This constraints models of AGN evolution and implies a lower number
of BHs with MBH< 3× 107M⊙ in the local universe (see § 5.6.3).
5.4.3 Testing the Fry formula
We discuss here the model for the evolution of bias that we used in our model in
more detail, and its validation against N-body simulations. The model, introduced
initially by Fry (1996), results from assuming that objects form by some arbitrary
local process at redshift zf and then move with the streaming velocity caused by
the gravitational potential. If one allows for a biased relation between fluctuations
in the number density of objects and that in the matter density at the time of
formation zf , then the subsequent evolution of this relation to linear order is given
by
b(z)− 1 = D(zf)/D(z)[b(zf)− 1] (5.34)
where D(z) is the linear growth factor and z ≤ zf .
The same result has also been derived in the context of the Press-Schechter for-
malism Mo & White (1996) (provided that the impact of halo mergers is negligible)
or using the Peak-Background split argument of Sheth & Tormen (1999). Moreover
it has been already used in this context in the analysis of QSO clustering (e.g., see
the early work of Croom & Shanks, 1996).
175
To test this model against N-body measurements, we used the Millennium sim-
ulation Springel et al. (2005), that tracked the joint gravitational evolution of a
billion particles within a cubic box-size of Lbox = 500 h−1 Mpc, from the initial con-
ditions at z = 127 until today. The cosmological parameters employed were slightly
different from the ones assumed in this thesis, namely Ωm = 0.26, ΩΛ = 0.74,
Ωb = 0.044, σ8 = 0.78 and h = 0.73 .
As discussed by Tegmark & Peebles (1998), the basic assumption underlying
equation (5.34) is that, at the time of formation, objects are assigned a near perma-
nent ‘observational tag’ and then move satisfying the matter continuity equation.
We thus selected haloes formed at some initial time and tracked their evolution
through the merger history tree of the Millennium run until today, and computed
their clustering amplitude along the way. We chose all FoF groups at z = 3.87
and z = 4.89 in two mass-bins, M ≥ 1012 h−1 M⊙ and M ≥ 3.16× 1012 h−1 M⊙ (at
z = 3.87 we found 40790 and 3477 haloes in each mass bin, while at z = 4.89 there
were 10152 and 471 haloes respectively). We then found their descendants at several
lower redshifts (a “descendant” is defined as the halo that contains most of the 10%
most bound particles of the ‘parent’ halo), and measured their bias by computing
their power spectrum in ratio with the corresponding linear power spectrum, i.e.
b =√
Phh/Plin, at large scales k ≤ 0.2 h Mpc−1.
The mass distribution of the descendants turned to be roughly Gaussian in
log(Mfinal/Minitial), with mean ∼ 1.35 and variance ∼ 0.52 at z = 0 for formation
176
time zf = 4. This implies that these haloes grew by a factor of ∼ 20 in mass
since they were formed until today (for zf = 5 we found a factor of ∼ 40 in mass
increment).
The result for the bias evolution is shown in Fig. 5.5, with different symbols
(circles and squares) corresponding to the measured bias in each mass-bin, as la-
beled. Different line-types (solid and dashed) correspond to the prediction in equa-
tion (5.34) for each formation time. The bottom panel displays the relative differ-
ence with respect to the prediction, showing a remarkably good agreement for both
mass-bins at all z ≤ zf .
Nonetheless, due to the low number density of haloes our measurements are
slightly sensitive to the shot-noise subtraction, and its impact is redshift dependent.
Also these results are a bit sensitive to the range of scales used to compute the bias
(±5% at z = 0 using k ≤ 0.1 h Mpc−1 or k ≤ 0.3 h Mpc−1).
Finally, notice that the comparison performed in this section differs from measur-
ing the bias of all haloes more massive than a given mass threshold, as a function
of redshift. In that case, the evolution of bias evolves much more strongly (e.g,
Matarrese et al., 1997).
177
Figure 5.5: The evolution of bias for haloes formed at redshifts z = 3.87 and
z = 4.89 and their descendants as a function time, for two different halo mass-cuts
(as labeled). The ‘parent’ haloes increase their mass by a factor of ∼ 20 (∼ 40)
from zf ∼ 4 (zf ∼ 5) until today. The bias, defined as b =√
Phh/Plin at large scales,
decreases with evolving time as the inverse of the growth factor, as predicted by
equation (5.34).178
5.5 Measuring the clustering of faint AGNs and
low-redshift quasars
Before presenting how various version of our model are constrained by observed
data, we take a brief detour into some technical aspects related to data we analyzed
and used. In particular, we focus on the clustering of faint AGNs and quasars at
low redshift.
5.5.1 The clustering of faint AGNs
In this section we describe in more detail how we compute the bias associated with
the measurements of low-luminosity AGN by Constantin & Vogeley (2006). The
redshift-space correlation function is commonly quantified in terms of a best-fitting
power law
ξ(s) =
(
s
s0
)−κ
, (5.35)
where s is the comoving separation between the pairs. The parameter s0 repre-
sents the amplitude of clustering, while κ quantifies the ratio of small-to-large-scale
clustering. Table 5.5.1 and Figure 5.6 report the best-fitting values for different
measured samples.
Another common estimator of clustering is the integrated correlation function
(Croom et al., 2005), defined as
ξ(s20) ≡3
s320
∫ s20
s5
ξ(s)s2d s =3
3− κ
[
ξ(s20)−ξ(s5)
64
]
, (5.36)
179
AGN type s0 κ ξ(s20) b
[OI], [OI]/[H α] criteria included:
Emission line 6.33 ± 0.94 1.30 ± 0.16 0.40 ± 0.13 1.20 ± 0.23
Seyferts 5.67 ± 0.62 1.56 ± 0.17 0.29 ± 0.13 1.00 ± 0.27
LINERs 7.82 ± 0.64 1.39 ± 0.09 0.51 ± 0.08 1.38 ± 0.13
Pure LINERs 7.39 ± 0.67 1.33 ± 0.11 0.48 ± 0.10 1.33 ± 0.15
Transition 5.38 ± 0.71 1.35 ± 0.23 0.31 ± 0.17 1.03 ± 0.34
H II galaxies 5.81 ± 0.53 1.28 ± 0.11 0.36 ± 0.09 1.13 ± 0.16
[OI], [OI]/[H α] criteria excluded:
Emission line 6.45 ± 0.24 1.24 ± 0.04 0.42 ± 0.03 1.23 ± 0.06
Seyferts 6.00 ± 0.64 1.41 ± 0.15 0.35 ± 0.12 1.10 ± 0.22
LINERs 7.26 ± 0.61 1.29 ± 0.08 0.47 ± 0.07 1.33 ± 0.11
H II galaxies 5.73 ± 0.24 1.22 ± 0.06 0.37 ± 0.05 1.14 ± 0.11
Table 5.1: The clustering of low luminosity AGNs.
where the right-hand-side follows from equation (5.35). Following Shen et al. (2007),
we integrate from s5 = 5 Mpc h−1 (to minimize non-linear effects) up to s20 = 20
Mpc h−1 (which is large enough for linear theory to apply)10. Table 5.5.1 reports
ξ(s20) for various types of low-luminosity AGN.
10Unlike us, Croom et al. (2005) integrate from 0 Mpc h−1.
180
Figure 5.6: The bias of low-luminosity type II AGNs.
181
The bias is computed as in da Angela et al. (2008):
b(z) =
√
ξ(s20)
ξρ(r20)− 4Ω1.2
m (z)
45− Ω0.6
m (z)
3. (5.37)
Here, ξρ is the correlation function in real space, and
ξρ(r20) ≡3
r320
∫ r20
r5
ξρ(r)r2d r = 0.206, (5.38)
where r5 = 5 Mpc h−1 and r20 = 20 Mpc h−1. Table 5.5.1 and Figure 5.6 show the
bias for these samples, with median redshift z = 0.1.
The corresponding error bars are obtained by using standard propagation of
uncertainty. One can show that
∆ξ(s)
ξ(s)=
√
(
κ
s0
)2
(∆s0)2 +
(
ln
(
s
s0
))2
(∆κ)2,
∆ξ(s20) =3
3− κ
√
(∆ξ(s20))2 +
(
∆ξ(s5)
64
)2
,
and
∆b =1
2
∆ξ(s20)
ξρ(r20)
[
b +Ω0.6
m (z)
3
]−1
. (5.39)
5.5.2 The clustering of low-redshift quasars
We use a sample of 88178 type 2 AGNs, which is publicly available from the
MPA/JHU SDSS DR4 database11, constructed by Kauffmann et al. (2003) using
the SDSS DR4 spectroscopic data. We divide all the AGNs into 10 subsamples
11http://www.mpa-garching.mpg.de/SDSS/DR4/
182
with equal sample size according to their extinction-corrected [O III] luminosity.
We then compute the projected two-point cross-correlation function wp(rp) of each
subsample with respect to a reference sample of galaxies. The reference sample
is composed of ∼ 0.5 million galaxies selected from the Sample dr72 of the New
York University Value Added Galaxy Catalogue (NYU-VAGC)12 (Blanton et al.,
2005), which is based on the SDSS Data Release 7 (DR7; Abazajian et al., 2009).
A random sample is constructed so as to have the same selection effects as the
reference sample. The reference and random samples are cross-correlated with each
of the AGN subsamples, and wp(rp) as a function of the projected separation rp is
defined by the ratio of the two pair counts minus one. Errors of the wp(rp) mea-
surements are estimated using the Bootstrap resampling technique. Details about
our methodology for computing the correlation functions and for constructing the
reference and random samples can be found in Li et al. (2006a) and in Li et al.
(2006b). Finally we convert our wp(rp) measurements to the real-space correlation
function ξ(r), using the Abel transform (see for example Hawkins et al., 2003).
The quasar-galaxy cross-correlation function is
ξQg(r, z) = bQ(r, z)bg(r, z)D2(z)ξρ(r). (5.40)
Similary, the galaxy-galaxy auto-correlation function is
ξgg(r, z) = b2g(r, z)D2(z)ξρ(r). (5.41)
12http://wassup.physics.nyu.edu/vagc/
183
Figure 5.7: The bias of low-redshift quasars. The solid line represents our average
sample bias and the dotted lines represent the estimated error (equation 5.50)
184
One can the solve for the galaxy bias:
bg(r, z) =
√
ξgg(r, z)
D2(z)ξρ(r), (5.42)
and the quasar bias
bQ(r, z) =
(
ξQg(r, z)
ξgg
)
bg(r, z). (5.43)
The associated uncertainties are
∆bg
bg
=∆ξgg
2ξgg
, (5.44)
and
∆bQ
bQ
=
√
(
∆ξQg
ξQg
)2
+
(
∆ξgg
2ξgg
)2
, (5.45)
Our SDSS sample has mean redshift z = 0.1, with growth factor D(z) = 0.819.
Defining
ξgg(r20) ≡3
r320
∫ r20
r5
ξgg(r)r2d r, (5.46)
the mean galaxy bias (Peebles, 1980) can be estimated as
bg =ξgg(r20)
D2(z)ξρ(r20), (5.47)
From our sample, we report
bg(z = 0.1) = 1.207± 1.716. (5.48)
Estimating the mean quasar bias as
bQ =
(
ξQg(r20)
D2(z)ξgg(r20)
)
bg. (5.49)
From our sample, we report
bQ(z = 0.1) = 1.282± 0.192. (5.50)
185
5.6 Results
5.6.1 The luminosity function
Figure 5.8 shows our reference model bolometric luminosity function at different
redshifts, as labeled, compared with the compilation of data from (Shankar et al.,
2009, and references therein). Our model reproduces the bright-end of the AGN
luminosity function at all redshifts. In the alternative framework in which all BHs
of any mass are characterized by the same (mass-independent) light curve, the low-
z bright-end AGN luminosity function is inevitably overproduced (e.g, Wyithe &
Loeb, 2003). As an example, Figure 5.9 compares our reference model (solid lines)
to a model in which no mass dependence in the light curve is allowed (long-dashed
lines). The latter overproduces the bright-end by a factor of & 2 − 3. Also note
that fine-tuning the redshift dependence of the Lpeak-M relation to find a better fit
to the z < 1 bright end of the AGN luminosity function does note overcome the
latter problem.
The success of the model discussed here is actually quite noticeable. As dis-
cussed by several authors (e.g., Cavaliere & Vittorini, 2002; Wyithe & Loeb, 2003;
Scannapieco & Oh, 2004; Vittorini et al., 2005; Marulli et al., 2008; Bonoli et al.,
2008; Shen, 2009b), simultaneously matching the low and high-redshift AGN num-
ber counts through AGN feedback-type Lpeak-M relations, often requires additional
input physics (and parameters) for the models to be successful (such as progressive
186
Figure 5.8: Predicted bolometric luminosity function at different redshifts, as la-
beled. At low redshifts the major-merger model fails to reproduce the faint-end of
the luminosity function, as expected. All the data are from Shankar et al. (2009)
and references therein.
187
gas exhaustion, radio feedback, etc.). Improving on Wyithe & Loeb (2003), Scanna-
pieco & Oh (2004) invoked, for the first time, an additional ‘radio mode’ feedback to
limit the growth of very massive black holes at low redshifts, thus lowering the num-
ber of luminous quasars, and improving the match to the luminosity function. The
more detailed works by Croton et al. (2006) and Marulli et al. (2008) have essen-
tially adopted similar recipes to cope with the low-redshift, high-luminosity portion
of the AGN luminosity function. Similarly, other groups such as Mahmood et al.
(2005) and Shen (2009b) assumed that the maximum halo mass hosting quasars
decreases by a factor of a few at lower redshifts, mimicking a progressive starvation
of massive black holes in massive haloes.
Another interesting feature of our model is that the faint end of the predicted
luminosity function flattens out with increasing redshift (and even turns over for
extremely dim luminosities). If confirmed by future surveys, this behavior has
significant implications for the quasar population. The break claimed by some
groups in the AGN luminosity function at high redshifts (e.g., Small & Blandford,
1992; Willott et al., 2004; Fontanot et al., 2007; Shankar & Mathur, 2007) would be
real, and present at all wavelengths. Also, the obscured fraction of sources should
have a different behavior with respect to luminosity, being constant, or even maybe
decreasing with increasing luminosity (see discussion in Shankar & Mathur, 2007).
If planned deep future surveys, such as PanSTARRS, instead unveil a non-negligible
population of faint AGNs at high redshifts; this will be a direct proof that major
188
Figure 5.9: Predicted luminosity function from a model with no mass dependence
in the light curve (long-dashed lines), and the reference model (solid lines), at two
redshifts, as labeled. Ignoring the mass-dependence in the light curve (see text)
implies growing more numerous massive black holes and, therefore, more luminous
quasars at late epochs.
189
mergers alone cannot be the only trigger of quasar activity, not even at very high
redshifts. Other in-situ and still not well-understood, AGN triggering mechanisms
must be already efficient in fueling BHs at high redshifts.
At z > 2, the model tends to somewhat overpredict the number of faint sources
below the knee of the AGN luminosity function. Allowing for the minimum halo
mass that can host a quasar Mmin to slightly increase with redshift will considerably
decrease the number of faint quasars and also increasing their bias (we will further
discuss this point in § 5.7.1).
We also find that a model like this one characterized by a more short-lived post-
peak phase, yields Eddington ratio distributions at fixed bolometric luminosity well
described by a Gaussian with dispersion around 0.3-0.4 dex, much lower than what
expected when assuming a long descending phase (see discussion in Yu & Lu 2008
and Shen 2009). Taken at face value, the outputs of our model would match well the
observations by AGES (Kollmeier et al. 2006), SDSS (e.g., Netzer & Trakhtenbrot
2007), and 2dF (e.g., Fine et al. 2008), that actually show comparable Eddington
ratio dispersions, although flux limited biases in the observations (e.g., Shen et al.
2008, Yu & Lu 2008) prevent us to draw any firmer conclusion.
A shortcomming of this model – a common feature of any major-merger model
– is instead the underestimation of the number of faint, low-redshift AGNs, as also
recently re-addressed by Shen (2009b). This is principally because the major-merger
events involving massive haloes get too rare at low redshifts to accommodate for
190
the numerous faint, low-z AGNs. We will further discuss below ways to constrain
extensions of our model tuned to reproduce this portion of the AGN luminosity
function.
5.6.2 Quasar clustering
The large scale clustering
Figure 5.10 compares our reference model to the bias measurements (listed in the
caption) from several large and small quasar surveys at different redshifts and lumi-
nosities. The grey stripes encompass a systematic 10% uncertainty in the analytical
fits of the bias (e.g., Shankar et al., 2009c). Overall, our model provides a reasonable
good match to the large scale clustering, although the systematic and statistical er-
rors in the measurements are still large. It is also fair to mention that not all the
clustering measurements have the same statistical weight. For example, the data
based on large scale surveys such as SDSS and 2dF (e.g., Myers et al., 2007; da
Angela et al., 2008; Shen et al., 2009a) should be less affected by cosmic variance
with respect to those obtained from small fields, such as the compilation of data of
Plionis et al. (2008) from X-ray surveys (see also discussions in Gilli et al., 2007;
Marulli et al., 2009). Also, the SDSS and 2dF clustering measurements are based
on thousands of quasars and therefore should be more reliable than data obtained
from much smaller samples (see, e.g., discussion in Coil et al., 2007, 2009, and
references therein).
191
Figure 5.10: Predicted bias as a function of luminosity at different redshifts, as
labeled. Our merger model is consistent with all the available data. Data are
from Padmanabhan et al. (2008) (inverse solid triangles), Mountrichas et al. (2009)
(filled circles), da Angela et al. (2008) (open triangles), Francke et al. (2008) (large
triangle), Shen et al. (2009a) (double circles), Myers et al. (2007) (open squares),
Porciani & Norberg (2006) (filled triangles), Shen et al. (2007) (diamonds and
inverse open triangles), Plionis et al. (2008) (open circles), Coil et al. (2007) (filled
squares). In particular, measurements from small samples (e.g., Plionis et al., 2008)
suffer from cosmic variance and are not as reliable as those from broad surveys like
SDSS and 2dF.
192
Nevertheless, several features are worth mentioning. At high redshifts (z > 3),
the model is only marginally consistent with the data. This is a feature common
to all models based on ‘low’ radiative efficiencies of ǫ . 0.1 (see, e.g., Shen, 2009b).
To boost the predicted bias some modification of the input parameters must be
adopted. As discussed in detail by Shankar et al. (2009c) (see also White et al.,
2008), boosting the bias implies placing the quasars in more massive, but less nu-
merous, haloes. This in turn implies increasing the median radiative efficiency to
at least ǫ & 0.15 (at fixed Eddington ratio λ & 0.15), to allow for the less numerous
hosts to produce the same amount of emissivity to match observations.
Alternatively, some authors (e.g., Wyithe & Loeb, 2008, and references therein)
have put forward the hypothesis that recently merged haloes are characterized by
some ‘excess’ bias with respect to haloes of similar mass. Adopting this extreme
bias would then ease the tension between model predictions and the high clustering
signal measured at z > 3. Detailed studies with cosmological simulations will be
able to test such interesting proposals (e.g., Gao & White, 2007; Bonoli et al., 2009).
Another important point concerns the clustering of low-luminosity sources at
z . 1. Taken at face value, the model predicts a flat bias below log(L/ergs−1) . 45
over several orders of magnitude in luminosity. This is due to the lower limit
in massive haloes which can host quasars, that in our reference model is set to
Mmin = 3 × 1011M⊙. We will further discuss this point, and modifications to this
assumption, in § 5.6.2.
193
A signature of rapid black hole growth
As anticipated at the end of § 5.4.2, the high redshift clustering measured from
SDSS at z > 3 is also powerful in constraining the delay time experienced by
quasars from their triggering epoch until their shining time. The high redshift
quasars are on average very luminous, and therefore they must be produced at
times close to the peak of the light curve, when the luminosity emitted by the BH is
around the maximum. Therefore the delay time tdelay = t(zshi)− t(ztri) is primarily
composed of the time spent by the BH in the ascending phase. In other words,
tdelay ≃ tasc ∝ log(µBH)/λ0 (see equation 5.16).
Figure 5.11 compares the predicted z = 4 bias as a function of luminosity for
models characterized by different delay times, as labeled. Models with longer delays
between the virialization epoch and the shining of the quasar, i.e., with lower λ0
and/or higher µBH = MBH,peak/MBH,seed, tend to generate a lower bias at fixed
luminosity, due to the passive evolution of the bias, especially at high redshifts (see
Section 5.4.2). The high-z quasar clustering measured by Shen et al. (2007) favors
models characterized by massive seeds and high Eddington ratios, conditions which
minimize the delay. We conclude from our analysis that a delay time of . 108 yr is
favored by the high-z data on quasar clustering (and number counts, see § 5.7.1).
This delay time is hard to reconcile with models that adopt small BH seeds of
. 103M⊙ and, especially, Eddington or sub-Eddington rates in the initial growth
phases of the BH (a 109M⊙ BH in these conditions would in fact require a growth
194
Figure 5.11: Comparison among models characterized by different delay times at
two different redshifts, as labeled. Models with longer delays between the virial-
ization epoch and the shining of the quasar, i.e., with lower λ0 and/or higher ratio
between peak and seed masses, tend to generate a lower bias at fixed luminosity,
due to the passive evolution of the bias, especially at high redshifts. The high-z
quasar clustering measured by Shen et al. (2007) favors models characterized by
massive seeds and high Eddington ratios, conditions which minimize the delay.
195
time in the ascending phase of & 5× 108 years).
Observationally, our understanding of BH evolution is richer in their final, op-
tically visible stages than in the earlier (possibly obscured) ones. Therefore, con-
straints like the one presented here derived from clustering, are invaluable for mod-
els that attempt to describe the full evolution of BHs. For example, Granato et
al. (2006) and Lapi et al. (2006) discuss the importance of the relative duration
of the obscured and visible phases within their quasar evolutionary models tuned
to simultaneously reproduced both the dust-enshrouded number counts of SCUBA
galaxies and ULIRGs, and the quasar optical luminosity functions. They conclude
that successful models must be characterized by a rather short tdelay, a condition
they are able to meet by assuming an initial fast, super-Eddington accretion phase
(see also Di Matteo et al. 2008).
Making faint AGNs at z < 1
As anticipated in § 5.6.1, the merging model fails at reproducing the faint-end of
the AGN luminosity function at z < 1 and L . 1045erg s−1. This is an old problem
several times discussed in the literature (e.g, Vittorini et al., 2005; Hopkins et al.,
2006b; Shen, 2009b, and references therein), although the solution to this issue has
not yet found a definite answer.
Faint AGNs at these epochs are predominantly Seyfert-like (see e.g., Hopkins
& Hernquist, 2009), usually associated to disky galaxies (e.g., Kim et al., 2008).
196
However, it is still not clear what the exact masses of the haloes that host them
actually are. In Figure 5.12 we compare our reference model (solid lines), at z = 0.5
(upper panels) and z = 0.1 (lower panels), with the outputs of two extension of our
model. The first is characterized by (1) faint AGNs predominantly hosted by low-
mass haloes with log(Mmin/M⊙) = 10 undergoing major merger events (dot-dashed
lines); while the other one has (2) faint AGNs hosted by the same range of halo
masses considered in the reference model (i.e., log(Mmin/M⊙) = 11.5), but allowing
minor mergers as triggering agents. It is clear that both alternative models can
reproduce the number counts of faint AGNs although, as expected, their clustering
predictions are significantly different.
In the lower right panel of Figure 5.12 we compare with the data on clustering
extracted by us for faint AGNs from SDSS (see Section 5.5). Given the large
number of sources in each luminosity bin (about 8,000 AGNs), we are able to derive
very small error bars which can rule out the model with low-mass haloes at a high
significance level. We conclude that the faint-end of the AGN luminosity function at
low redshifts is mainly produced in haloes more massive than log(Mmin/M⊙h−1) ∼
11.5−12 (see also Lidz et al., 2006). Obviously, our simple model cannot isolate the
true triggering mechanisms causing the shining of the faint, low-z AGNs; although
these are probably in-situ processes, uncorrelated from large scale events as in the
more luminous quasars. As discussed below, having constrained the haloes which
are hosting the faint AGNs also implies a different local BH mass distribution.
197
Figure 5.12: Comparison among the reference model (solid lines) with other two
models, one specified by having fainter AGNs produced in less massive haloes, with
log(Mmin/M⊙) = 10 (dot-dashed lines), and another one where the faint-end of
the AGN luminosity function is produced by black holes in more massive haloes
triggered by minor events (long-dashed lines). These models can be readily tested
by current and future measurements.
198
5.6.3 The predicted black hole mass function
Following § 5.4, we can compute the expected BH mass function at any redshift with
our formalism. The result for our reference model is shown in Figure 5.13, plotted at
different redshifts, as labeled. It is clear that a natural, mild downsizing is generated
during the evolution in the model, with the most massive BHs completing their
growth faster than less massive ones. This general feature is a natural outcome
of all models tied to the observed AGN luminosity (see discussion in SWM and
references therein).
The long-dashed line is the local BH mass function found by convolving the
Sheth (2003) velocity dispersion function with the MBH-σstar relation for early-type
galaxies recently calibrated by Tundo et al. (2007) and (Shankar & Ferrarese, 2009,
and references therein). The grey band is the statistical 1-σ error bar estimated
derived from Monte Carlo simulations (see Shankar & Ferrarese, 2009, for further
details). We find that our merger model predicts a cumulative BH mass function
at z = 0 consistent with the local distribution of BHs in early-type galaxies, when
adopting ǫ . 0.07 − 0.1, consistent with Soltan-type arguments (SWM and ref-
erences therein), with the systematic uncertainty of ∼ 30% caused by the actual
bolometric corrections adopted (see discussion in SWM). Our model therefore nat-
urally predicts that all BHs in early-type galaxies were mainly triggered in major
merging events, as long as ǫ . 0.07− 0.1. Also note that our model grows massive
BHs less than the model adopted by Shen (2009b) because our input light curves
199
Figure 5.13: Predicted black hole mass function at different redshifts, as labeled.
There is sign of a mild downsizing, with the cumulative number the more massive
black holes being in place at higher redshifts. The grey band is the local black
mass function computed from the velocity dispersion function of early-type galaxies,
while the dotted lines are from Shankar et al. (2009c) and bracket the systematic
uncertainties in estimating the total local black hole mass function.
200
for massive BHs have negligible descending phase. This allows a better match with
the local BH mass function at the high-mass end.
Figure 5.14 shows the predicted local BH mass function derived from the three
models discussed in § 5.6.2 in reference to Figure 5.12. The models characterized
by quasars shining only in more massive / less abundant haloes with major and
minor mergers (solid and long-dashed lines) produce, as expected, less numerous
BHs with mass log MBH < 107M⊙ by a factor & 2−3, relative to the model in which
BHs are allowed to be triggered by major mergers even in less massive haloes (dot-
dashed lines). SDSS clustering measurements presented in the lower right panel
of Figure 5.12 and discussed in § 5.6.2, however, support the former massive-host
models. This has profound implications regarding our knowledge of the local BH
mass function. Many of the previous attempts (e.g., Salucci et al., 1999; Shankar
et al., 2004; Marconi et al., 2004) might have overestimated the actual contribution
of late-type galaxies to the local BH mass function. We here find that models
that match the SDSS bias values are consistent with a much flatter local BH mass
function. This in turn implies that the Eddington ratio distribution characterizing
the local population of AGNs is peaked at significantly lower λ with respect to the
more luminous high-z quasars. This is simply due to fact that the characteristic
BH mass is always growing in time while the L∗ in the AGN luminosity function is
decreasing (e.g., Hopkins et al., 2006b; Shankar et al., 2009; Shankar, 2009).
This behavior in the Eddington ratio distribution with time is also confirmed
201
Figure 5.14: Predicted black hole mass functions for the same models discussed in
§ 5.6.2 and also labeled here. The models characterized by quasars shining only in
massive haloes produce less numerous black holes with mass log MBH < 107M⊙ by
a factor & 2 − 3 (and a flatter bias as shown in Figure 5.12). The grey band and
dotted lines are as in Figure 5.13.
202
by continuity equation arguments. Shankar et al. (2004) and Shankar et al. (2009)
(as also reviewed in Shankar 2009), have shown that a continuous decrease of the
characteristic λ with decreasing redshift is the only way to reduce the number
of low-mass BHs predicted by direct integration of the observed AGN luminosity
function.
5.7 Discussion
5.7.1 Further constraint from high-z X-ray counts.
As discussed by several authors (e.g., Richards et al. 2006, Fontanot et al. 2007,
Hopkins et al. 2007, Shankar et al. 2009), the shape of the bolometric z > 3
AGN luminosity function is still quite uncertain, as also directly evident from the
data in Figure 5.8. Brusa et al. (2009) have recently measured the space density
of X-ray selected quasars in the deep and uniform multiwavelength coverage of the
COSMOS survey in the redshift range 3 . z . 4.5. In agreement with previous
findings on optically selected quasars by Richards et al. (2006), Brusa et al. (2009)
confirm an exponential drop in the space density of quasars at z & 3. Recent re-
calibrations of the global hard X-ray luminosity function based on 2 Ms Chandra
Deep Fields and the AEGIS-X 200 ks survey (Aird et al., 2009, open squares in the
upper right panel of Figure 5.15) also point towards a lower number density of faint
quasars at z > 3. These findings set further constraints on the actual shape of the
203
Figure 5.15: Upper panels : Predicted X-ray number counts for different models, as
labeled, compared with the data by Brusa et al. (2009) at z = 3 (left) and z = 4
(right). Lower panels : corresponding AGN luminosity functions predicted by the
same models at the same redshifts. Data at z & 3 favor models with a higher Mmin
and minimal descending phase.
204
overall, high-z bolometric AGN luminosity function at luminosities L & (0.5−1)L∗,
favoring a flat faint-end for the AGN luminosity function at L ≤ L∗. Note in fact
that the minimum luminosity probed by the COSMOS survey is L ∼ 3×1045erg s−1
(LX ∼ 1044), a factor of a few lower than the knee of the AGN luminosity function,
occurring around L ∼ 1046erg s−1 at z & 2 (Figure 5.8).
As already partially discussed above, our reference model showed in Figure 5.8
tends to predict too many faint quasars with respect to the data, although the
mismatch is somewhat dependent on the exact bolometric corrections, completeness
corrections, and several other systematics affecting the data (see, e.g., Fontanot
et al. 2007). The thick dotted lines in Figure 5.15 show the corresponding X-ray
number counts at z = 3 and z = 4 in our ‘reference’ model discussed above. As note
before, the model overproduces the number of faint AGNs at z > 2 and therefore,
as expected, it also overpredicts the X-ray number counts.
Here we plot, in agreement with the data, the number counts of all X-ray sources
with column densities in the range 21 < log NH/cm−2 < 24. The number counts
are computed following, e.g., Gilli et al. (2001, and references therein). We first
determine the relative contribution of X-ray sources of a given column density, red-
shift and luminosity to the overall number counts by first correcting the bolometric
AGN luminosity function by the appropriate redshift- and luminosity-dependent
fraction of sources f(NH , z, LX) (taken from Ueda et al. 2003). We then compute
the integral N(> S) ∝∫ Lmax
Lmin(NH ,z,LX)Φ(LX , z, NH)dzdLX with Lmin(NH , z, LX) dif-
205
ferently K-corrected depending on the type of spectrum considered (each spectrum
is computed from the PEXRAV code of Magdziarz & Zdziarski (1995), and de-
pends on the amount of column density NH considered). We note that our results
do not depend significantly on the exact choice of column density distribution or
X-ray spectrum adopted, but more on the shape of the underlying AGN bolometric
luminosity function.
We find that to decrease the number of sources shining at luminosities L . L∗,
thus simultaneously improving the match to the AGN luminosity function and X-
ray number counts, we could either increase the minimum halo mass Mmin host-
ing quasars, and/or minimize the time spent by BHs in their descending, low-
Eddington ratios, post-peak phases. Figure 5.15 shows examples of such mod-
els characterized by higher minimum halo masses of Mmin= 1012M⊙ (long-dashed
lines), of Mmin= 1012.3M⊙ (dot-dashed lines), and a model with higher minimum
mass Mmin= 1012M⊙ and without descending phase (solid lines). All these models
characterized by a higher Mmin provide better matches to the data, further sup-
porting the need for an evolving minimum halo mass hosting quasars.
Also, these models predict a strong break and a flat faint-end slope in the high-z
AGN luminosity function. In principle, all these improved models are quite degener-
ate with respect to the data (although the model with no descending phase provides
a better match to the bright end of the AGN luminosity function). In contrast with
what has been discussed in reference to the low redshift universe, however, the
206
still scarce data on the faint-end, high-z AGN luminosity function and the still too
loose clustering measurements in the corresponding region in the (L, z)-plane (e.g.,
Francke et al., 2008) prevent us to draw any firmer conclusions.
Nevertheless, we note that those models characterized by BH light curves sharply
peaked with a negligible post-peak phase (e.g., Lapi et al., 2006), better reproduce
the obscured fraction as a function of luminosity within evolutionary models (e.g.,
Lapi et al., 2006; Shankar et al., 2008). In fact, we have checked that a long
post-peak luminous, unobscured phase requires a non-trivial, physically challenging,
fine tuning of the parameters to reproduce an obscured fraction decreasing with
increasing luminosity. We reserve a full study of the obscuration fraction as a
function of luminosity and redshift and its connection with evolutionary/orientation
models and input light curves for future work.
One key result in several studies that attempted to empirically constrain the
average quasar light curve (e.g., Yu & Lu, 2008; Hopkins & Hernquist, 2009; Shen,
2009b), is that most BHs should possess a significant post-peak luminous phase,
most of the times characterized by a more or less pronounced descending phase.
However, most of these studies are sensitive to data at z . 1.5 − 2, and cannot
really rule out alternative models characterized by only an ascending phase at higher
redshifts. In fact, most of the models predict that the bright high-z quasar emissivity
is produced around the peak of the light curve (e.g., Haehnelt et al., 1998; Wyithe &
Loeb, 2003; Lapi et al., 2006). However, we have also verified that adopting a light
207
curve with no descending phase even at very low redshifts implies that only a really
tiny portion of the AGN luminosity function at z . 1.5 is produced by the model,
specifically only the number density of AGNs shining with above L & 1047erg s−1.
We have seen that in this case the natural extensions of our models, as the ones
discussed in § 5.6.2 characterized by either less massive haloes hosting AGNs and/or
lower fraction of the minimum triggering ratio ξmin, do not provide any match
to the data, and some non-trivial, heavy fine-tuning of the models (such as an
evolutionary, mass and redshift-dependent normalization in the Lpeak−M relation)
must be invoked to get better agreement with the data. We therefore feel that a
much cleaner, simpler, and possibly more physical model is to just allow lower-mass
BHs to have more extended light curves (at least at low redshifts).
5.7.2 Scaling relations
It is now of wide interest the understanding of the redshift and, possibly, the mass
dependence of the basic scaling relations between BHs and their host galaxies and
dark matter haloes (e.g., Peng et al., 2006; Shankar et al., 2009b). Although flux-
limited biases might still affect in part some of the observational results (Lauer
et al., 2007), an increasing number of groups are finding a non-negligible increase,
within a factor of ∼ 2 − 3 in the overall normalization of the MBH-Mstar relation
at 0.5 < z < 2 (e.g., Peng et al., 2006; Jahnke et al., 2009; Somerville, 2009). The
actual evolution in the MBH-σstar relation seems instead to be milder, at least for
208
the bulk of the BH population, as concluded both from cumulative, Soltan-type
arguments (e.g., Shankar et al., 2009b) and detailed, direct observations of high-z
AGNs (e.g., Shields et al., 2006; Woo et al., 2008; Gaskell, 2009).
Understanding the true evolution of these scaling relations is, of course, invalu-
able for setting constraints on the actual co-evolution of BHs and galaxies. In this
section we address this issue by making use of our output MBH-M relation. The
feedback-constrained model developed in this work allows us in fact to trace the
underlying relation between BH mass and host halo mass/virial velocity at the peak
of the quasar activity. We can then work out the MBH-σstar relation predicted by
this model by first assuming a mapping between σstar and circular velocity Vc, as
observed in the local universe (e.g., Ferrarese, 2002), and mapping circular velocities
to virial velocities Vvir, which are linked to halo masses by the virial theorem (e.g.,
Barkana & Loeb, 2001).
We perform this exercise in Figure 5.16, where we show the predicted MBH-σstar
relations in four different redshift bins, as labeled. The grey band represents one
of the latest determinations of the local MBH-σstar relation by Tundo et al. (2007).
In each panel we show with a solid line the predicted MBH-σstar relation at the
moment of the shining of the quasar, derived by imposing the circular velocity Vc
equal to Vvir. This proves that if the velocity dispersion σstar is a good tracer of the
potential well, then a tight relation between virial velocity and BH mass, such as the
feedback-type relation used here, naturally translates into a non-evolving MBH-σstar
209
Figure 5.16: Predicted MBH-σstar relation at different redshift and different values
of the ratio between circular velocity Vc and virial velocity Vvir, as labeled. The
reference model is consistent with a nearly constant MBH-σstar relation with time,
as also found from independent empirical studies. Coupled with the information
of a smaller sizes at fixed stellar mass, the non-evolution in the MBH-σstar relation
implies a significant evolution in the Magorrian relation (see text).
210
relation (e.g., Wyithe & Loeb, 2003). Such findings, in turn, are in agreement with
the above mentioned empirical results concerning the MBH-σstar relation observed
in distant quasars, which seem to be consistent with the local one (e.g., Gaskell,
2009) – (see also Shankar et al., 2009b). For completeness, we also show with filled
circles the predicted remnant MBH-σstar relation at z = 0, obtained by assuming the
velocity dispersions do not evolve since the quasar shining epoch. It is clear that
our models predict final BH masses still within the range of the observed scatter in
the local MBH-σstar relation. Models characterized by much more prolonged light-
curves than the ones adopted here, would obviously produce BHs more massive
than allowed by the scatter in the z = 0 relation.
Direct observations of high-z massive, early-type galaxies have shown that, at
fixed stellar mass, galaxies are more compact (e.g., Trujillo et al., 2006, 2007;
Cimatti et al., 2008; Buitrago et al., 2008; Franx et al., 2008; Tacconi et al., 2008;
van Dokkum et al., 2008; Younger et al., 2008; Saracco et al., 2009; van der Wel et
al., 2009; Damjanov et al., 2009; Williams et al., 2009), and have higher velocity
dispersions (e.g., Cenarro & Trujillo, 2009; Cappellari et al., 2009; van Dokkum et
al., 2009). In particular, van Dokkum et al. (2009) measured the velocity dispersion
of a z ∼ 2 massive galaxy to be about around 510 km/s, a factor of ∼ 1.4 higher
than the most rapidly rotating early type galaxies in the local universe. We show
with dot-dashed lines in Figure 5.16 the results of assuming such a scaling relation
between stellar and dark matter velocities. It is clear that if the stellar circular
211
velocity or velocity dispersion is significantly higher than that of the underlying
dark matter, the resulting MBH-σstar relation gets shifted to higher σstar at fixed
BH mass, and just the opposite is true assuming Vc to be lower than Vvir, as in the
long-dashed lines. However, such trends contradict empirical results (e.g., Woo et
al., 2008; Gaskell, 2009; Shankar et al., 2009b).
The only way to reconcile higher velocity dispersions in high-z quasar hosts and,
at the same time, minimizing any (negative) evolution in the MBH-σstar relation,
would be to allow for a lower host stellar bulge hosting the BH at the moment of the
shining. This would automatically decrease the velocity dispersion associated with
a given BH mass, without affecting the underlying Mstar-σstar relation. A direct
one-to-one matching between the stellar mass and halo mass functions at different
redshifts, allows to determine the mean Mstar-M relation at all times (e.g., Vale &
Ostriker, 2004; Shankar et al., 2006; Conroy & Wechsler, 2009). The most recent
estimate of this relation has been performed by Moster et al. (2009), who also
show that the resulting stellar mass-halo relation is consistent with a compilation
of galaxy large-scale clustering at different masses and redshifts. Associating their
Mstar-M relation to our empirical MBH-M relation yields a redshift-dependent MBH-
Mstar relation which we show in Figure 5.17. Although it is difficult to draw firmer
conclusions due to our ignorance in how to exactly parameterize the true stellar
mass fractions associated to a given bin of BH mass at different redshift, there is
nevertheless tentative evidence for a positively evolving MBH-Mstar relation, in a
212
Figure 5.17: Predicted MBH-Mstar relation at different redshift, as labeled, obtained
by matching our reference model MBH-M relation with the empirical Mstar-M rela-
tion obtained from cumulative number matching techniques. The reference model
is consistent with a relation with normalization increasing with increasing redshift,
i.e., higher BH masses at fixed stellar bulge. Coupled with the information of a
smaller sizes at fixed stellar mass, the evolution in the MBH-Mstar relation can be
translated into a non-evolving MBH-σstar relation, as observed (see text).
213
way that higher BH masses correspond to the same stellar bulge at higher redshifts.
We also note that the above discussion relies on the basic assumption that our
adopted MBH-M relation actually holds at the shining epoch. However, as also
discussed in § 5.2 and § 5.7.3, although supported by simulations and observations,
dropping the assumption of no dynamical friction delay between halo mergers and
quasar triggering, significantly reduces the number of mergers per halo at z & 3 (by a
factor of ∼ 2−4; see, e.g. Shen, 2009b). Recovering the match to the observed AGN
luminosity function without worsening the predicted bias then implies increasing
the normalization of the MBH-M relation by a proportionally higher factor. While
this has a minor impact on the overall local BH mass function (as long as the
condition of a higher normalization is limited to z & 3), it has a major impact on
the resulting MBH-σstar relation. In fact, a higher BH mass compensates the lower
MBH-σstar normalization induced by the higher σstar at the quasar shining due to
a more compact host, thus yielding a resulting MBH-σstar at z > 0 closer to the
local one, and in better agreement with direct observations. A drawback of this
model is, however, that due to the larger BH mass for a given host, the resulting
MBH-Mstar relation is much more offset with respect to the local one than the one
showed in Figure 5.17, requiring a major stellar mass growth at late times, difficult
to reconcile with observations and most SAMs.
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5.7.3 Comparison with previous models
The approach described in this work is significantly different from previous ones. Al-
though previous models aimed at describing the AGN luminosity function and clus-
tering properties using similar techniques of convolution between halo merger/creation
rates and light curves, the physical assumptions were, sometimes, substantially dif-
ferent.
As anticipated in § 5.6, the seminal paper by Wyithe & Loeb (2003) showed
that reproducing the AGN luminosity function at all relevant redshifts through
feedback-constrained relations, is not trivial. Possibly the main concern with these
kind of models is the overproduction of bright quasars at z . 2. In order to solve
such a problem, Scannapieco & Oh (2004) proposed that due to quasar preheating,
massive haloes at late times with an entropy floor of ∼ 100 keV cm−2 cannot cool
within a Hubble time inhibiting further star formation and BH fueling (see also
Marulli et al., 2007).
Our model differs from theirs in two ways. First, we here just assume that
quasar activity occurs only within haloes with mass below Mmax, in line with a
number of other groups (e.g., Granato et al., 2004; Mahmood et al., 2005; Lapi et
al., 2006). Although such a cut has a minor effect on the predictions at z > 2, due
to the scarce number density of very massive, cluster-size haloes at high redshifts,
it does have a significant impact on the AGN luminosity function at lower redshifts,
as discussed above. However, cutting out the mergers of very massive haloes as
215
triggers of quasar activity is a reasonable approximation. In fact, as discussed in
detail by, among others, Cavaliere & Vittorini (2000), Menci et al. (2005), and
Vittorini et al. (2005), most of the haloes more massive than Mmax at late times
are massive groups and clusters of galaxies with much longer cooling time-scales
and in which mergers among galaxies are replaced by milder dynamical events such
interactions, rarely capable of triggering very luminous quasars. In other words, the
massive BHs at the center of massive groups and clusters, such as their galaxies,
do not experience many major mergers, especially at late times (e.g., Bouwens et
al., 2009), and thus cannot be accounted for in quasar statistics. The SAMs by
Marulli et al. (2008) and Bonoli et al. (2009) also find, for similar reasons, that
AGN activity is negligible in haloes more massive than ∼ (1 − 2) × 1013M⊙. The
large hydro-simulations of Colberg & di Matteo (2008), which also include recipes
for the build-up of BHs, also show a similar drop in the star formation rate and BH
accretion rate above Mmax, at least at z & 3.
However, simply cutting out haloes more massive than Mmax from quasar num-
ber counts is not enough. As clearly re-discussed recently by Shen (2009b), allowing
for quasars to evolve with the same constant light curve at all times and luminosities
still produces too many bright sources at z . 2 (despite the upper cut at a fixed
Mmax). A further, progressive inhibition of BH gas accretion at late times still needs
to be inserted into the models to make them successful. Mahmood et al. (2005)
and Shen (2009b) mimicked this behavior by inserting a redshift dependence Mmax,
216
and/or a decreased BH growth efficiency at late times. While this behavior in the
BH accretion mode might at least in part be true, here we have shown that one
may alternatively allow for mass-dependent light curves (with no redshift variation
in halo hosts or BH efficiency), and still reproduce the observables. Moreover, the
latter choice can be physically coonnected with the notion of galaxy downsizing, as
discussed above, in the hypothesis that AGN and star formation in the galaxy are
coupled at some level and proceed on similar timescales.
The faint end of the AGN luminosity function at low redshifts seems to be better
fitted by models that allow for a significantly long post-peak phase, as discussed in
§ 5.6. These conclusions agree with the results of more advanced SAMs, such as
those by Marulli et al. (2009) and Bonoli et al. (2009), and also with the statistics of
AGNs as extracted from large hydro-simulations of Di Matteo et al. (2008), Colberg
& di Matteo (2008), and DeGraf et al. (2009). We also note here that the Marulli
et al. (2008), Bonoli et al. (2009), and Di Matteo et al. (2008) models have similar
mass-dependent light curves to the ones used here, with more massive BHs shutting
down faster than low mass ones. In particular, Marulli et al. (2008) and Bonoli et
al. (2009) showed that, in agreement with our findings, a significant post-peak phase
is required to fit the faint-end of the AGN luminosity function at z . 0.5, although
it is not necessary, and possibly disfavored, at higher redshifts (z & 2).
Moreover, we also find evidence from the z < 0.3 SDSS AGN clustering, that
faint sources predominantly live in haloes more massive than 5× 1011M⊙h−1. This
217
information breaks the degeneracies discussed by several groups (e.g., Vittorini et
al., 2005; Marulli et al., 2008; Bonoli et al., 2009), favoring models where the faint
end is dominated by the post-peak phases of BHs shining in haloes more massive
than 5× 1011M⊙h−1(in this context, see also Lidz et al., 2006).13
A key difference, however, is that the model in our work consistent with the
SDSS z < 0.3 clustering predicts a local BH mass function flatter than the one by
Marulli et al. (2008) and Bonoli et al. (2009). The discrepancy can be ascribed by
the fact that still a large fraction of sources in their models still lives in low mass
haloes (see, e.g., Figure 14 of Bonoli et al., 2009). A more precise measurement
of the local BH mass function and AGN duty cycles as a function of redshift and
luminosity will further break these degeneracies.
The fact that the SDSS clustering measurements point to high mass hosts can
favor AGN models which grow their BHs through in situ processes, such as those
by Lagos et al. (2008), where most of the BH growth is triggered by disk instability,
especially at late times. The models put forward by Granato et al. (2004), Cat-
taneo et al. (2005) and Fontanot et al. (2006) link the star formation rate to the
accretion rate on to the central BH, all of them finding good agreement with the
AGN luminosity function at different epochs. In particular, Fontanot et al. (2006)
push their model to very low redshifts, predicting an Eddington ratio distribution
13We cannot directly compare with the results from hydro-simulations here as no clustering
information has still been discussed by Di Matteo et al. (2008) and Colberg & di Matteo (2008)
at z < 1.
218
peaked at very low values, in agreement with observational data (e.g., Ballo et. al.,
2004; Kauffmann & Heckman, 2009). Also, they predict a flatter BH mass function,
consistent with our model where AGNs are preferentially hosted by massive haloes
(long-dashed line in Figure 5.14), although a closer comparison cannot be made
given that no clustering predictions have been discussed by these groups.
Particularly interesting is the Malbon et al. (2007) model that predicts, at vari-
ance with several other SAMs, a late evolution for massive BHs governed by BH
coalescence with negligible gas accretion. They predict all local BHs to be hosted
by haloes more massive than 1011M⊙h−1, only marginally consistent with our SDSS
data, and with a cumulative BH number density higher than the models discussed
above. Their match to the optical luminosity function is good, at least at z . 2,
although a comparison with the full bolometric AGN luminosity function prevents
us to further test and compare with their model.
We showed that the z & 3 SDSS clustering measurements of luminous quasars
allows us to set stringent constraints on the delay time (< 5× 108 yr) between the
triggering and shining of the quasar, mostly due to the rapid decline in time of
the clustering strength of the high-σ peaks hosting quasars. Such findings are in
line with several other groups. Marulli et al. (2008) and Bonoli et al. (2009) use
light curves with a delay time < 1 Gyr. (see also Li et al., 2007; Sijacki et al.,
2009, for comparable time-scales from simulations), and similarly Lapi et al. (2006)
adopt super-Eddington accretion at z & 3 to enhance the initial growth of the most
219
massive BHs. Our values of the median bias and scatter in the Lpeak-M relation
are also consistent with these studies.
Moreover, all these studies agree in showing that a negligible post-peak phase is
better suited to reproduce the faint-end of the high-z AGN luminosity function and,
in the case of Lapi et al. (2006), to reproduce the fraction of obscured sources as
a function of luminosity. We also showed that the exponential drop in the number
density of deep X-ray counts further supports the need for a flatter faint-end slope
of the AGN z > 3 luminosity function, thus further limiting the actual need for
a long post-peak phase. Integrating over the pre-peak obscured phases of quasars,
Granato et al. (2006) also showed consistency with the cumulative SCUBA number
counts at 850µm.
A noticeable difference between the model described here and the one discussed
by Marulli et al. (2008) and Bonoli et al. (2009) is the fact that the latter falls short
in reproducing the bright end of the z & 2 AGN luminosity function. We believe one
of the main reasons for this discrepancy lies on our assumption to neglect any delay
time for the progenitors galaxies to actually merge. Such an assumption naturally
enhances the merger triggering rates of quasars at z & 2 while it has a much more
negligible impact at lower redshifts (e.g., Shen 2009), irrespective of the differences
in dynamical friction delay times (Chandrasekhar, 1943; Zentner & Bullock, 2003;
Taylor & Babul, 2004; Zentner et al., 2005; Boylan-Kolchin et al., 2008; Hopkins et
al., 2008b; Wetzel et al., 2009). As discussed in § 5.2 and § 5.7.2, neglecting any
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delay is a good approximation given that the central BH can be triggered at any
time during the interaction, as suggested by numerical experiments (e.g., Hopkins
et al., 2006a) and observational evidence (e.g, Ballo et. al., 2004; Guainazzi et. al.,
2005).
Overall, we believe that the basic approach undertaken here has revealed several
interesting, and quite general, aspects of BH / AGN evolution that should be taken
seriously into account by more advanced SAMs that attempt to connect BH and
galaxy evolution.
5.7.4 Assumptions, input parameters, and degeneracies
In line with the conclusions of the previous paragraph, we stress here that the aim
of this paper is not to provide the reader with a specific model for BH evolution, but
rather to show how the simultaneous match to a number of independently derived
observables is effective in constraining some of the main ‘characteristics’ that any
model must satisfy. In other words, the model presented here is by no means unique,
in fact several of the parameters are degenerate, and this is the main reason why we
do not present a full χ2 fitting. However, our study shows that some basic ‘trends’
must be satisfied for a basic AGN evolution model to actually work. In this section
we attempt to discuss these issues in some detail.
First of all, we stress that this work is based on a set of very general and widely
accepted model assumptions that cannot be considered as true free parameters. Our
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first assumption is that quasar activity is triggered in major merger events (with
ξmin & 0.25 − 0.3), close to the epoch of virialization of the halo and build up of
the central potential. The number density of these events as a function of mass and
redshift, is well described by the analytical formalism presented in § 5.2.1, which
was tuned to fit the results of large, high-resolution numerical simulations.
The redshift and halo dependence of the Lpeak-M relation have also been fixed
to the values predicted by theory (see § 5.2), and the normalization chosen in way to
match the z = 0 MBH-σstar relation (adopting the observed σstar-Vvir relation). The
model described here therefore, by definition, satisfies the local relations between
BH and its host masses.
Our prediction of the delay time between triggering and quasar shining is not
very well constrained by the AGN luminosity function, at least at z ≤ 2. There-
fore, also the associated parameters are not very well defined, such as the initial
Eddington ratio λ0 which can take any value in the range ∼ 0.1 − 10, or the ra-
diative efficiency, which is degenerate with λ0. Nevertheless, additional constraints
for these parameters come from other observables. For instance, the strong z > 3
clustering signal forces the model to have short delays (see § 5.6.2), which imply
super-Eddington values for λ0, at least at high-z. From arguments linking the the
integrated BH mass function from luminous quasars events matches the local early-
type BH mass function (e.g., Salucci et al., 1999), we get ǫ . 0.1 (see § 5.6.3),
independent of the systematic uncertainties linked to the AGN luminosity function
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and / or bolometric corrections (e.g., Shankar et al. 2009).
The parameter α0 (in equation 5.15) that controls the overall ‘strength’ of the
light curve descending phase, is by no means unique, and it is in fact degenerate
with the normalization of the Lpeak-M relation. Lower values of α0, such as in Shen
(2009b) and Yu & Lu (2008), imply longer decreasing phases and, overall, more
AGN counts at approximately all luminosities, and larger final BH masses. For
example, setting α0 = 1.5, requires a reduction of ∼ 40% in the normalization of
the Lpeak-M relation to recover the match to the AGN luminosity function, which
anyhow provides a MBH-σstar relation still well within the observational constraints.
However, the major variation in this kind of model is that the final BH mass in-
creases by a factor of ∼ 10, with respect to the its mass at the shining, i.e. a much
higher BH mass at fixed σstar. While this effect might help in reconciling high-z
observations of a higher σstar and a non-negative MBH-σstar redshift evolution (see
§ 5.7.2), it is for sure at variance with the very low z > 3 X-ray number counts
(§ 5.7.1). Also, allowing for too much post-peak BH growth produces far too many
massive BHs in the local universe, in clear contradiction with measurements of the
local BHMF. In conclusion, the multiple match to different data sets favors higher
values of α0.
The mass-dependence of the α parameter presented in Eq. 5.15, is obviously not
unique. For example, lowering (increasing) its mass-dependence can be counteracted
by a decrease (increase) in Mmax. Nevertheless, models characterized by a sharp
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cut-off for BH masses above a few times 108M⊙ are better-suited to reproduce the
bright end of the AGN luminosity if no redshift dependence in Mmax is invoked.
As discussed above, the maximum and minimum masses hosting quasars are
input parameters in our model. A natural upper mass cut-off Mmax is expected
because multiple quasars will be hosted by too massive haloes with longer cooling
timescales (e.g., Cavaliere & Vittorini, 2000). However, the exact value of this
parameter is degenerate, again, with the normalization of the Lpeak-M relation.
For example, increasing (decreasing) Mmax by a factor of ∼ 2, implies decreasing
(increasing) the normalization by∼ 40%, which has a minimal effect on the resulting
MBH-σstar relation or other observables. Similar uncertainties affect the minimum
halo mass Mmin, although, as discussed above, larger variations of this parameter
are ruled at high significance by clustering and number counts measurements.
Finally, Mmax and the normalization of the Lpeak-M relation, are degenerate
with the value of the intrinsic scatter Σ. While Σ = 0.28 dex was chosen to be
in agreement with the average value of the scatters in the local relations, its exact
value does not have much impact in the overall results, provided that (rather small)
appropriate changes in Mmax and/or Lpeak-M normalizations are taken into account.
The parameter Σ has a larger impact on the results at z > 3, where smaller values
of Σ are preferred because the boost the clustering associated to a given luminosity
bin (see § 5.6.2 and references therein), although the main conclusions on delay,
number counts, etc.. discussed above, again, do not depend much on Σ.
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5.7.5 Black holes masses at triggering
The purpose of this section is to discuss, for the first time, a conceptual limitation
present in most theoretical models of quasar activity (ours included). Recall that
in our framework, we assume that circumnuclear material is initially accreted at
a mildly super-Eddington rate by the central BH, which in turn grows from some
triggering mass MBH, tri to some mass MBH, peak during some time tasc. One usually
argues that choosing the BH mass at triggering is not trivial, due to our lack of
knowledge in the early obscured phases. For this reason, we simply assume that
the BH mass at triggering is some fixed fraction µBH of its mass at the peak.
Nevertheless, this issue is not merely an observational limitation, but a theoretical
one as well.
In general, models based on halo merger / formation rates (the so-called ‘analytic
models’) and models involving detailed simulations of individual galactic mergers
(‘hydrodynamic simulations’) often ignore the past history of the black holes prior
to the triggering event in question. An interesting question to pose is, how massive
are the incoming BHs in the merging haloes? In particular, did they follow the
same MBH, peak −M relation? Are the BH masses in the incoming haloes compa-
rable with the initial mass MBH, tri of the final halo? Below we perform a simple
analytic estimate to address this problem. As a first step, recall that picking the
BH mass at triggering to be a fixed fraction of the BH mass at the peak of the light
curve is equivalent to fixing the time tasc spend by the BH in the ascending phase.
225
Conversely, one could pick a value of tasc and infer µBH (and MBH, tri). The result
of this exercise is shown in Figure 5.18.
Each column in Figure 5.18 corresponds to a given host halo mass at triggering
(as labeled, note the different vertical mass scales used in each column) and each
row corresponds to different triggering redshifts. In each panel, the lower dotted
line represents MBH, tri, while the upper dotted line represents MBH, peak. Their mass
ratio is given by MBH, peak/MBH, tri = µBH = exp (tasc/tef), where the e-folding time
is tef = 0.43(f/λ0) Gyr. Notice that our values of MBH, tri are compatible or above
those used in hydrodynamical simulations (e.g, Di Matteo et al., 2005; Robertson
et al., 2006; Hopkins et al., 2008a,b). In the Figure we assume that tasc = 0.11 Gyr
and λ0 = 3, which corresponds to an e-folding time of 15.9 Myr, and a BH mass
ratio of log µBH = 3. This is our original reference model, broadly consistent with
the bias at high redshift (see Figure 5.11).
Consider a merger of two haloes with masses m and m′ respectively (with m′ <
m), producing a halo of mass M = m + m′ at redshift z = Ztri as a result. In our
model, we would invoke the MBH, peak −M relation at Ztri to find MBH, peak. Then,
with our assumed values of tasc and λ0, the black hole mass MBH, tri at triggering
could be inferred (the upper and lower dotted lines in Figure 5.18, respectively).
Estimating the masses of the incoming BHs (at the centers of the merging haloes)
is a more difficult task. First of all, recall that only major mergers (with ξ =
m′/m > 1/4) are considered as triggering agents. Denote the mass of the BH in
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Figure 5.18: Relevant black hole masses involved in a halo merger of haloes with
masses m and m′ vs. the mass ratio of the merging haloes (ξ = m/m′). We consider
final halo masses M = 1012M⊙ (left panel) and 1013M⊙ (right panel). The rows refer
to the merger redshifts z = 0.5, 2, 4, 6 (top-to-bottom). See text for a discussion of
the different line styles.
227
the m-halo as mBH, and that in the m′-halo as m′BH. Without loss of generality,
we will focus our discussion in terms of ‘unprimed’ quantities – similar expressions
hold for ‘primed’ quantities. It is safe to assume that the bulk of their masses was
accreted during their respective ascending phases, especially at these early epochs
(see § 5.7.1). In other words, let us assume that mBH ≃ mBH, peak. To estimate the
masses of the incoming BHs at their peak luminosity, we must know their respective
triggering epochs and the masses of their hosts at those epochs. Once these pieces
of information are established, the mBH −m at ztri can be invoked for each of the
merging haloes. A possible way to do this is to use a merger history tree algorithm
(e.g., Moreno et. al. , 2008). In the present work we will use a simpler, but
still reasonable estimate instead. A more rigorous approach with our merger tree
algorithm will be the subject of future work.
Recall that in Chapter 3, we introduced the term ‘halo formation’ to describe
when a dark matter halo is formed (see also, Lacey & Cole, 1993; Harker et al.,
2006; Sheth & Tormen, 2004). Formally, this is defined as the first time a halo had
at least one half its final mass. In our estimate of the incoming BH masses, we will
assume that each host halo (i.e., with ‘final’ masses m at Ztri, etc.) was triggered
at redshifts ztri = zF. At triggering, we denote the host halo mass by mtri = mF.
One caveat is that for a halo at the final epoch Ztri, one does not expect a single
value of formation redshift or formation mass, but a distribution of these formation
values with a certain width. Given that we here interested in average trends, we
228
will simply estimate the formation redshift as zF = zF, where zF is the median
formation redshift. For similar reasons we will also assume that mF = m/2, since
one-half the final mass is the most typical formation mass (Sheth & Tormen, 2004).
Following Giocoli et al. (2007), the median formation redshift (of haloes identified
at final redshift Ztri) is given by
δc(zF) = δc(Ztri) + 1.26√
σ2(m/2)− σ2(m). (5.51)
Here, δc(z) is the overdensity threshold required for spherical collapse at redshift z
and σ(m) is the rms in the initial linear fluctuation field, smoothed with a top-hat
filter of size R = (3m/4πρ)13 (with comoving background density ρ). With this,
we can invoke the mBH −mtri self-regulation condition (Eq. 5.12 with λ0 = 3) with
mtri = mF = m/2 and ztri = zF (and similarly so for the ‘primed’ quantities).
For each halo of mass M and triggering redshift z = Ztri, Figure 5.18 shows
the values of mBH and m′BH versus the mass ratio of their merging hosts (ξ =
m′/m). These are denoted by the short-dash-dotted and long-dash-dotted lines
respectively. Notice that m′BH/mBH approaches unity as ξ → 1, and become greater
than unity for smaller values of ξ. In all panels, the incoming BH masses are
orders of magnitude larger than our BH mass at triggering. One might think that
this is good news, since growing massive BHs takes less time. From the point of
view of clustering, shorter accretion phases implies that the bias (as described by
the Fry formula) does not drop so much, improving the match of our model with
observations. However, this feature is spoiled by a more important issue. Notice
229
that in many cases, the incoming BHs are more massive than the MBH, peak (at
z < Ztri) predicted by our self-regulation condition. This is especially true for
massive hosts and recent epochs. In other words, if either of the two incoming BHs
gets triggered, then the correspoding AGN would extinguish itself immediately,
completely inhibiting the very existence of quasars, which is obviously contrary to
observations.
Several models of BH growth assume that the two incoming BHs coalesce imme-
diately after the merger of their hosts (Kauffmann & Haehnelt, 2000; Menou et al.,
2001; Erickcek et al., 2006; Malbon et al., 2007). In Figure 5.18, the solid line rep-
resents the sum mBH + m′BH. Except for low-mass hosts at high redshifts, the mass
of the BH resulting from the coalescence of the binary is larger than that allowed
by the MBH−M relation. Conserving the total entropy of the system instead (e.g.,
Ciotti & Ostriker, 2001) leads to a final mass given by√
m2BH + m′2
BH < mBH+m′BH,
while the rest leaves the system in the form of gravitational waves. This final mass
is depicted as dashed lines. While this is certainly smaller than the sum of the
incoming BH masses, the effect is small.
From this exercise we conclude that merging models like ours cannot, by con-
struction, be strictly self-consistent. This means that such models require that at
each merging event a “seed” BH must exist at the center of the remnant halo that is
not necessarily related to any previous, self-regulated quasar phase. Indeed, these
findings seem to support the choice of some of the current SAMs that actually take
230
the BH seed as an external free parameter not related to the previous merging his-
tory of the haloes (e.g., Granato et al. 2004, Marulli et al. 2008). For these reasons,
we here caution against the use of predicting, at least in merger-driven models of
this kind, the BH mass evolution at each time during the hierarchy, by simple wild
extrapolations of some self-regulated condition. Nevertheless, despite this concep-
tual complication, simple models like ours still provide very powerful descriptions of
quasar activity. Our formalism remains an excellent approximation. This is because
galactic haloes are not expected to experience more than one major merger during
the epochs studied here (e.g., Angulo & White, 2009). In other words, in our regime
of interest, it is for the most part safe to assume that there is only one triggering
event, allowing us to ignore any previous history of the supermassive black hole.
It is true that a halo might had experience a series of major mergers prior to its
final major merger. Nevertheless, this would only occur at extremely high redshifts,
when the first generation of black holes first assembled. This regime is beyond the
scope of models of quasar activation by major mergers like ours.
5.8 Final remarks
In this work we discuss the predictions of a simple model of quasar activation by
major mergers of dark matter ‘galactic’ haloes in the range 1011.5 − 1013M⊙h−1.
The model consists of two main ingredients: (1) the halo merger rate as the main
triggering agent, and (2) the quasar light curve, which describes the evolution of
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individual quasars. After a phase of exponential growth, the BH reaches a peak
luminosity set by a feedback-constrained relation of the type Lpeak ∝ V 5vir, after
which the AGN luminosity progressively decreases. This model has already been
widely used in the literature and, in its essence, it is the core of all SAMs that tried
to explain the evolution of BHs and their hosts. By carefully revisiting some of
the main issues linked to this approach, we were able to derive several interesting
and physically meaningful constraints regarding BH evolution that will be useful
for updating more advanced SAMs.
Our results can be summarized as follows:
• We find that allowing less massive BHs to possess more extended light curves,
i.e., to be active for a longer time after their peak luminosity, can well repro-
duce the bright end of the AGN luminosity function at all epochs, without the
need to invoke a not very well defined declining AGN efficiency at low red-
shifts. In turn, this recipe is in line with models that postulate AGN feedback
to be the ‘clock’ regulating the star-formation timescales in the host galax-
ies. Chemical evolution models of early-type galaxies in fact require rapid
shut-downs after a burst of intense star formation, the duration of which is
inversely proportional to the host potential well and final stellar mass.
• The integrated BH mass function from the major-merger model is consistent
with the local BH mass function, derived from the early-type galaxy pop-
ulation, if the radiative efficiency is ǫ ≤ 0.1. This result is robust against
232
systematics affecting the bolometric corrections or the exact shape of the
quasar luminosity function.
• We measure the average bias of Type II AGNs in SDSS to be, on average, b =
1.282 ± 0.192, independent of luminosity in the range 42.5 ≤ log L/ergs−1 ≤
45.5. Such a high value of the bias implies that faint AGNs at z < 0.3 are
mainly hosted by haloes more massive than M ∼ 1011.5−12M⊙/h, thus favoring
models in which AGN activity in the local universe is triggered by minor,
low-Eddington ratio events, such as galaxy interactions, minor mergers, and
in-situ processes (e.g., bar instability). We find that if AGNs are preferentially
hosted by more massive haloes, then the model predicts at z = 0 much less
numerous low-mass BHs, i.e., a flat BH mass function at the low-mass end,
which needs to be confirmed by independent measurements of the local BH
mass function. As discussed in § 5.7.3, an independent, secure estimate of the
shape of the low-mass end of the local BH mass function will be invaluable to
break some degeneracies now present in current, different SAMs.
• The high clustering signal measured at z > 3 in SDSS forces successful models
to be characterized by rather short delay times of tdelay . 5× 108 yr. between
the triggering and the shining epochs. If the delay time is too long in fact, the
bias characterizing the high-σ peaks of quasar host haloes will rapidly drop as
b ∝ (1 + z)1.5 for the masses of interest here. Such conditions of short delays
are more easily met if the ‘seed’ BH mass is & 105M⊙, and the initial growth
233
is super-Eddington.
• The exponential drop characterizing the number counts of X-ray AGNs – as
measured in recent deep surveys and the observed obscured AGN fraction
decreasing with increasing luminosity – are better reproduced by models with
a minimal post-peak phase, as expected from a more efficient AGN feedback
at high redshifts and a higher minimum halo mass (of & 1012M⊙h−1) hosting
quasars.
• We are the first to notice, intriguingly, that merger-driven models (including
the one presented here) are, by construction, conceptually limited. In fact,
by applying the self-regulation condition to the incoming, merging haloes at
their respective triggering epochs, would force them to host BHs much more
massive than the final BH in the descendant halo. This indicates that the
BH mass at the triggering in the descendant halo must be close to the true
seed BH mass, or at least it should not have experienced any previous major,
self-regulated growth. Scenarios with a single major merger remain a good
approximation because galactic haloes are not expected to experience several
major mergers in the epochs and range of halo masses of interest.
• Our feedback-constrained relation MBH-Vvir, consistent with the clustering and
AGN luminosity function at all redshifts, implies a non-evolving MBH-σstar re-
lation consistent with the local one, assuming σstar and Vvir scale as observed in
234
the local universe. Observations suggest, however, that the host galaxy might
be more compact and with a higher σstar at the moment of the quasar shining,
implying a lower BH mass at fixed velocity dispersion (i.e., a lower normal-
ization in the MBH-σstar relation), if the BH mass is mostly formed during the
quasar phase. On the other hand, however, direct observations and cumulative
arguments support, if anything, a mild positive evolution in the normalization
of the MBH-σstar relation. We speculate that such apparently contradictory
results might be resolved either by a non-trivial, redshift-dependent mapping
between σstar and Vvir, or either by assuming that the BH host had a smaller
bulge stellar mass associated to the BH at the moment of the shining of the
quasar.
• Cross-correlating the feedback-constrained MBH-M relation, with the redshift-
dependent Mstar-M relation obtained from the cumulative number matching
of the stellar and halo mass functions, we find tentative evidence for a factor
of ∼ 2 larger BH-to-stellar mass ratio at high redshifts, which might sup-
port the above argument and also some direct observations and clustering
measurements.
The wealth of observational information now available is capable of constraining
the main effects describing BH growth in AGNs. Simple models with very few
realistic assumptions, like the one presented here, are flexible enough to easily probe
and study the impact of these different effects (e.g., host halo masses, delay times
235
in the early stages, shut-down in the late stages, etc.). Such models have already
been widely used in the literature and, in essence, they are the core of all SAMs
that tried to explain the evolution of BHs and their hosts. By carefully revisiting
some of the main issues linked to this approach, we were able to derive several
interesting and physically meaningful constraints regarding BH evolution. It is our
hope and intention that simple, yet powerful models like ours, will help guide the
next generation of semi-analytic models of galaxy formation.
236
Appendix A
Diffusion theory
A.1 Diffusion theory
We now present the solutions to the constant and square-root barrier problems
from the point of view of the diffusion equation. This Appendix is particularly
technical, and may be skipped. Nevertheless, we point out the most important
formulas. These are equation (A.38) and equation (A.33) for the exact square-root
unconditional and conditional mass functions, respectively.
A.1.1 Constant barriers
Here we present an alternative derivation of the Press-Schechter formula. The
statistics of random walks have been studied in a completely different context.
Imagine an ensemble of weakly-interacting dust particles in one-dimensional random
237
motion. A random walk then can be assigned to each particle. δ can be interpreted
as the distance walked, and S as a time variable. The question is, what is the
distribution of times these particles take before they are deposited on a barrier
at a distance δc. In our cosmological context, the meaning of these variables is
completely different, but the underlying mathematics is the same. The key idea
here is to acknowledge that Q(S, δ) obeys the one-dimensional diffusion equation:
∂Q(S, δ)
∂S=
1
2
∂2Q(S, δ)
∂δ2. (A.1)
In the absence of a barrier, the solution is simply
Q0(S, δ) =1√2πS
exp(
− δ2
2S
)
. (A.2)
(Notice the ‘0’ index denoting the barrier-free situation.)
Including a barrier amounts to imposing the constraint
Q(S, δ = δc) = 0 (A.3)
at the barrier, known as a Dirichlet boundary condition. Such barriers are called
‘absorbing’ barriers (Chandrasekhar, 1943). Notice that the solution is only valid
for −∞ < δ ≤ δc. An additional condition, which allows Q to be normalizable, is
that Q and its first partial derivatives vanish as S →∞ and δ → −∞. The points
along the red walk in Figure A.1 satisfy equation (A.1) with the boundary condition
given in (A.3). To obtain this solution, we must take the set of all walks beneath
the barrier at S ′, and substract off those forbidden ones that have been absorbed
238
Figure A.1: Diffusion with a constant absorbing barrier. The solution Q (in red)
of the diffusion equation (A.1) with boundary condition (A.3) is obtained by sub-
stracting off forbidden walks (black-gray). One can get the latter by exploting the
symmetry Q0(S, δ)→ Q0(S, 2 δc − δ) about the barrier of height δc.
239
already by the barrier (e.g., the black-gray walk). By symmetry, a walk is equally
likely to go above (black) or beneath (gray) the barrier after the crossing point. To
get the latter from the former, one just needs to replace δ → 2 δc − δ. Thus, to get
the set of walks that have never touched the barrier, we must take all and substract
off the forbidden ones. Explicitly,
Q(S, δ) = Q0(S, δ)−Q0(S, 2 δ − δc) (A.4)
=1√2πS
[
exp(
− δ2
2S
)
− exp(
− (2δc − δ)2
2S
)]
.
(Notice the absence of the ‘0’ index on the left, indicating the presence of a barrier).
This is the solution to the diffusion equation in the presence of an absorbing-barrier
boundary condition. See Zentner (2007) for an alternative method of solving the
diffusion equation, which uses a Fourier transform.
The cumulative fraction of walks that have crossed the barrier at some S < S ′
includes all the walks minus those that have never crossed the barrier (the red
walks), and it is given by
F (< S ′|δc) = 1−∫ δc
−∞
dδ Q(S ′, δ), (A.5)
From this, one can compute
f(S|δc)dS =∂
∂S ′F (< S ′|δc)
∣
∣
∣
S′=SdS = −
∫ δc
−∞
dδ∂Q(S ′, δ)
∂S ′
∣
∣
∣
S′=SdS
= −1
2
∫ δc
−∞
dδ∂2Q(S, δ)
∂δ2dS =
√
δ2c
2πSexp
(
− δ2c
2S
)dS
S, (A.6)
which is precisely the extended Press-Schechter result. Notice that in the second
240
line we have used the diffusion equation and the vanishing condition as δ → −∞.
We now show a derivation of the exact solution to the square-root barrier.
A.1.2 Square root barriers
In this section we solve the diffusion equation in the presence of square-root barrier
(Breiman, 1967). In terms of the ensemble of random-motion dust particles, the
distance to the absorbing barrier is increasing with S (interpreted as a time variable)
– that is, the barrier is moving. Another interpretation is that the barrier is fixed,
and the particles ‘drift’ away because they experience a constant force. Such random
walk models with drift have been studied extensively in the literature (see Grimmett
& Stirzaker, 2001, and references therein).
In this presentation we follow Mahmood & Rajesh (2005). We will work out the
conditional solution because, and the unconditional one will be obtained as a special
case. We are interested in the first crossing distribution of a barrier B(Sm, δc(z))
(equation 2.79) by random walks with origin at (SM , B(Sm, δc(Z)). Close inspection
of Figure A.2 shows that this problem is simplifed by a change of coordinates:
s = SM − S,
a = δ − δc(Z)− β√
SM . (A.7)
Compare with the change of variables with constant barrier (equation 2.68). With
this, the above two-barrier problem reduces to the crossing distribution of one com-
241
Figure A.2: Exact square-root barrier solution. The first crossing distribution can
be obtained from the first crossing distribution of a barrier B(s, ac) (see equa-
tion A.8), shown in magenta. For this it is necessary to change coordinates
δ → a = δ − δc(Z)− β√
SM and S → s = S − SM . Compare to Figure 2.11.
242
plicated barrier,
B(s, ac) = ac + β√
s + SM ,
ac ≡ δc(z)− δc(Z)− β√
SM . (A.8)
We suppress the parameter q here for simplicity. The shape of the effective barrier
B depends explicitly on SM . This barrier does not belong to the family of barriers
in equation (2.78), and cannot be written in self-similar fashion (equation 2.81).
For this reason, the moving barrier prediction of the conditional mass function is
not expected to be self-similar.
The diffusion equation in these coordinates is given by
∂Q(s, a)
∂s=
1
2
∂2Q(s, a)
∂a2. (A.9)
In the presence of an absorbing square-root barrier (equation A.8), the Dirichlet
boundary condition at the barrier is
Q(s, a = B(s, ac)) = 0. (A.10)
The solution and its derivatives vanish as s→∞ and a→ −∞. Finally, at s = 0,
it is certain that a = 0. Mathematically,
Q(s = 0, a) = δD(a = 0). (A.11)
Our calculations are facilitated by using the distance to the barrier as one of the
coordinates: a→ ξ = B − a. In this case, the diffusion equation becomes
∂Q(s, ξ)
∂s+
β
2√
s + SM
∂Q(s, ξ)
∂ξ=
1
2
∂2Q(s, ξ)
∂ξ2. (A.12)
243
This is seemingly more complicated than equation (A.9), but the boundary condi-
tions are simplified considerably. Explicitly, these become
Q(s, ξ = 0) = 0 ,
Q(s = 0, ξ) = δD(ξ − B) . (A.13)
We will make another change of variables, which the will allow the solution to
be separable. By setting ξ → y = ξ/√
s + SM , the diffusion equation becomes
∂Q(s, y)
∂s+
1
2
β − y
s + SM
∂Q(s, y)
∂y− 1
2(s + SM)
∂2Q(s, y)
∂y2= 0 . (A.14)
Setting
Q(s, y) = Θ(s)Y (y) , (A.15)
and substituting in equation (A.14) yields
s + SM
Θ(s)
dΘ(s)
ds=
1
2Y (y)
[
(y − β)dY (y)
dy+
d2Y (y)
dy2
]
= −ζ . (A.16)
The left-hand side only depends on s, while the right-hand side depends only on y.
In reality, both sides are just a constant −ζ. Let’s solve for Θ(s) first. The integral
of the left-hand size of (A.16) is:
Θζ(s) =1
(s + SM)ζ, (A.17)
where the constant of integration is zero because Q = 0 as s→∞.
For the y-factor, let x = y − β, which turns the right-hand side of (A.16) into
Y ′′(x) + xY ′(x) + 2µY (x) = 0 . (A.18)
244
Now define
D(x) = e−x2/4Y (x) . (A.19)
Substituting back into (A.18) and defining λ = 2ζ − 1 we are left with
D′′(x) + (λ +1
2− x2
4)D(x) = 0 . (A.20)
This is known as the Webber differential equation, whose solutions are the Parabolic
Cylinder Functions Dλ(x) (Abramowitz & Stegun, 1972). The general solution is a
linear combination
Q(s, x) =∑
λ
AλΘλ(s)Dλ(x)e−x2/4 . (A.21)
where the Θλs can be thought of as eigenfunctions. The set λ of eigenvalues and
the Aλ coefficients will be fixed by the boundary conditions.
In these variables, the first (Dirichlet) boundary condition is
Q(s, x = −β). (A.22)
This fixes the eigenvalues, which are defined by this condition:
Dλ(−β) = 0 . (A.23)
The second boundary condition (at the origin) is
Q(s = 0, x) =δD(x = ηβ)√
s + SM
, and ηβ ≡ac√SM
. (A.24)
245
Here we have used the fact that δD(κx) = δD(x)/|κ|. With equation (A.23), this
becomes
∑
λ
AλΘλ(0)Dλ(x)e−x2/4√
SM = δD(x = ηβ) . (A.25)
In order to solve for the Aλ coefficients, we use the fact that the Parabolic cylinder
functions are orthogonal:
∫ +∞
−β
Dλ(x)Dλ′(x)dx ≡ δλλ′Iλ(−β) , (A.26)
where δλλ′ is the Kronecker delta. Multiplying equation (A.25) by Dλ′(x)ex2/2 and
integrating, we get
Aλ =S
λ/2M eη2
β/4Dλ(ηβ)
Iλ(−β)(A.27)
and the solution to the diffusion equation is
Q(s, η) =∑
λ
( SM
s + SM
)λ2 eη2
β/4e−(ηβ−η)2/4Dλ(ηβ)Dλ(ηβ − η)
Iλ(−β)√
s + SM
, (A.28)
where we have set
x = ηβ − η, and η ≡ a√s + SM
. (A.29)
The first crossing distribution is
f(s|ac)ds = − ∂
∂s′
∫ B(s,ac)
−∞
daQ(s′, a)∣
∣
∣
s′=sds = −1
2
∂Q(s, a)
∂a
∣
∣
∣
B(s,ac)
−∞
= − 1
2√
s + SM
∂Q(s, η)
∂η
∣
∣
∣
ηβ+β
−∞=(ηβ − η
2+
D′λ(ηβ − η)
Dλ(−β)
) Q(s, η)
2√
s + SM
∣
∣
∣
ηβ+β
−∞. (A.30)
Using the fact that Q and its derivatives vanish as η → −∞, we obtain
f(s|ac)ds =∑
λ
( SM
s + SM
)λ2eη2
β/4Dλ(ηβ)lλ(−β)
ds
s + Sm
, (A.31)
246
where
lλ(−β) ≡ e−β2/4
2
D′λ(−β)
Iλ(−β). (A.32)
This is the exact first-crossing distribution to the square-root barrier we were after.
The conditional crossing distribution is
f(Sm|δc(z), SM , δc(Z))dSm =∑
λ
(SM
Sm
)λ2eη2
β/4Dλ(ηβ)lλ(−β)
dSm
Sm
, (A.33)
where
ηβ =δc(z)− δc(Z)√
SM
− β. (A.34)
For the unconditional case, we need to set (SM , δc(Z)) = (0, 0). That is, we must
take the limit as ηβ →∞. We can use the fact that as x→∞,
Dλ(x)→ xλe−x2/4 . (A.35)
With this
(SM
Sm
)λ2eη2
β/4Dλ(ηβ)→
(SM
Sm
)λ2ηλ
β =(δ2
c (z)
Sm
)λ2, (A.36)
and the unconditional mass function becomes
f(Sm|δc(z))dSm =∑
λ
(δ2c (z)
Sm
)λ2lλ(−β)
dSm
Sm
. (A.37)
As expected, this solution is self-similar
f(ν)dν =∑
λ
νλ2 lλ(−β)
dν
ν. (A.38)
Our square-root barrier limit of ellipsoidal collapse allows us to compare against
a exact solution. Obtaining exact solutions for the crossing distribution of a bar-
rier of general shape remains an open problem (but see Zhang & Hui, 2006, for a
numerical solution).
247
Appendix B
Merger tree algorithm
B.1 The algorithm
The aim of this section is to discuss the implementation of the merger history tree
algorithm (presented in Chapter 3) in more detail. Consider a dark matter halo
of mass M0 at redshift Z0 (Figure B.1). Realizations of its merger history may be
constructed with our tree algorithm. Below we explain how each branch is built
and how these branches are connected.
B.1.1 The branch: from random walks to mass histories
The mass growth history of a halo is contained in a random walk (Section 3.2). First
we discuss the constant-barrier (spherical collapse) model and then the square-root
barrier (ellipsoidal collapse) case.
248
Figure B.1: A sample merger history tree of a halo with final mass M0 at Z0. The
mass history (black branch) is constructed with a random walk. Mass jumps larger
than mdust are identified, and branches are connected there (medium-gray). This
process is repeated for each branch, and new branches are connected (light-gray)
until the tree is complete.
249
A random walk is essentially a collection of steps and heights: s0, . . . , si, . . .
and h0, . . . , hi, . . . (e.g., the jagged line in Figure 3.2). Consider a horizontal
barrier of height δc(z). This line may intersect the walk at several values of S. The
smallest of such of values, Sm, indicates what mass the halo had at redshift z. As
z increases, so does δc – and m decreases accordingly, as expected.
As we increase the height of the barrier, a subset (S0, H0), . . . , (Sj, Hj), . . .
of the walk is chosen. These points contain the mass history (e.g., the dark-filled
circles in Figure 3.2). Every point in this subset has the following property: they
are higher than all their predecessors. This is illustrated by the point at (Sm, δc(z))
Figure 3.2: it has the maximum height in the range SM ≤ S ≤ Sm. In other words,
for any point (si, hi) on the walk to be selected as part of the history, it must satisfy:
hi > hk, ∀ k < i. (B.1)
Now we discuss what modifications are necessary when the square-root barrier
model of ellipsoidal collapse is used (Figure 3.3). Consider a barrier (equation 2.79)
with δ-intercept√
qδc(z). This curve may intersect the random walk at several
places. We pick the smallest of these to find the mass at that redshift. As z
increases, the δ-intercept increases, but the overall shape of the barrier remains
unchanged. This is how the mass history (S0, H0), . . . , (Sj, Hj), . . . is selected.
250
From this subset we can construct a string √qD0, . . . ,√
qDi, . . . of ‘δ-intercepts’:
Di =1√q(Hi − β
√
Si)1 (B.2)
Every point on the history has the following property: its δ-intercept is higher than
that of its predecessors. In other words, for a point (si, hi) on the walk to belong
to the mass history, condition (B.1) is replaced by
di > dk, ∀ k < i, where di =1√q(hi − β
√si). (B.3)
For a given cosmology and initial power spectrum one may map excursion set vari-
ables into physical ones: (S(m), δc(z)) ↔ (m, z). With this prescription, the mass
accretion history is obtained: (M0, Z0), . . . , (Mj, Zj) . . . . The masses in the his-
tory are such that S(Mj) = Sj and the redshifts are given by δc(Zj) = Dj. When
barriers are constant, Dj = Hj.
B.1.2 The tree: connecting the branches
Consider a halo of mass M0 at redshift Z0. We are interested in constructing its
merger history tree. The first step is to draw a random walk with origin at
s0 = S(M0), h0 = B(S(M0), δc(Z0)), (B.4)
where B(S, δc) is given by equation (2.79). The mass history is then collected:
(M0, Z0), . . . , (Mj, Zj), . . . (see section B.1.1). In Figure B.1, this is represented
by the black branch.
1Note that this mapping is ill-defined when γ > 1/2, because barriers cross in that case.
251
The mass history can also be seen as a series of jumps in mass: (M0 ←
M ′0, Z0), . . . , (Mj ← M ′
j, Zj), . . . , with M ′j = Mj+1. Such jumps are interpreted
as binary mergers: M ′j + (Mj −M ′
j) → Mj. In practice, we only care about the
mass jumps above the branching-mass resolution. Denote this subset of jumps as
. . . , (mJ ← m′J , zJ), . . . , where
mJ −m′J > mdust. (B.5)
These jumps are shown in Figure B.1. The next step is to construct the history of
each halo with mass (mJ −m′J) that falls on to the mass history of the M0-halo.
This is done by generating random walks originating from
s0 = S(mJ −m′J), h0 = B(S(mJ −m′
J), δc(zJ)), (B.6)
Such mass histories correspond to the medium-gray branches in Figure B.1. The
above process is repeated for each of these branches: retrieve their mass history,
identify large enough mass jumps, and attach new branches there (light-gray in
Figure B.1). Eventually, all the new branches will only involve mass jumps that do
not satisfy condition (B.5). At this point the tree is complete.
B.2 The main branch
In a merger history tree, the ‘main’ branch of a halo is obtained by following the
most massive progenitor at each mass split. In this section, we show that the mass
history algorithm described in section B.1.1 can be easily modified to construct
252
Figure B.2: Main branch algorithm. The symbols and styles are the same as in
Figure 3.3. The dotted vertical line denotes S = Sm/2. The main branch is built by
following the most massive piece at each split: Sm′ > Sm/2 indicates that m′ < m/2,
implying that m−m′ > m′, whose mass history we must follow.
253
the main branch. This process requires continuous monitoring of the walk and its
associated history, and is illustrated in Figure C.7 (compare to Figure 3.3).
Recall that mass decreases with increasing S. For the portions in the mass
history consisting of jumps where the mass loss is less than half, the algorithm is
unchanged (e.g., the portion with SM ≤ S ≤ Sm in Figure C.7). Occasionally,
there are jumps where more than half the mass is lost. Such is the case for the
Sm → Sm′ jump illustrated in Figure C.7, with Sm′ > Sm/2 (i.e., m′ < m/2 and
(m − m′) > m/2). To construct the main branch in that situation, one must
simply follows (m−m′), not m′. In other words, instead of continuing the walk at
(Sm′ ,√
qδc(z)+β√
Sm′), one must continue from (Sm−m′ ,√
qδc(z)+β√
Sm−m′) (i.e.,
the dark filled circles in Figure C.7).
254
Appendix C
Coagulation and fragmentation
In Chapter 4, we used information from the Smoluchowski coagulation formalism
to infer the functional form of the halo creation rate (equation 4.34). In this Ap-
pendix, we discuss coagulation theory (the discrete Smoluchowski equation) in more
detail. In particular we address varios kernels, their associated solutions, and their
interpretation in the context of polymers. Coagulation theory assumes that clumps
never fragment, in contrast to what is actually observed in N-body simulations of
dark matter haloes (Tormen, 1998; Fakhouri & Ma, 2008a). Fragmentation can
change the shape of the mass function significantly. In this Appendix, we present a
special case known as ‘reversible’ fragmentation, in which the mass function is left
intact.
255
C.1 Introduction
Whereas coagulation models (Smoluchowski, 1916, 1917; Ziff, 1980) and fragmenta-
tion models (Montroll & Simha, 1940; Ziff & McGrady, 1985) have been reasonably
well-studied, models which combine the two processes are less well-developed.
Barrow & Silk (1981) studied perhaps the simplest merger-fragmentation model:
one in which the merger and fragmentation rates were independent of both time
and of the masses of the objects which were merging and fragmenting. That model
admitted a solution in which the mass spectrum did not evolve, and could therefore
be interpreted as the equilibrium state. They noted that it would be interesting if
equilibrium states also existed when the merger and fragmentation rules were more
complex. Below we show that this is indeed the case.
C.2 The binary merger-fragmentation model
The binary merger-fragmentation model is
dn(m, t)
dt=
1
2
m−1∑
m′=1
K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑
m′=1
K(m,m′)n(m, t)n(m′, t)
+∞∑
M=m+1
F (M |m)n(M, t)−m−1∑
m′=1
F (m|m′)n(m, t),(C.1)
where K and F denote the coagulation and fragmentation rates which are assumed
to be independent of time t. In what follows, we will derive expressions for n(m, t)
which solve this model for specific choices of K and F . We will be particularly
256
interested in solutions which are independent of time, which we will interpret as the
equilibrium state.
C.2.1 Linear polymers
If, in addition to being independent of time, K and F are both also independent of
m, then
dn(m, t)
dt=
K0
2
m−1∑
m′=1
n(m′, t)n(m−m′, t)−K0n(m, t)∞∑
m′=1
n(m′, t)
−F0(m− 1)n(m, t) + 2F0
∞∑
M=m+1
n(M, t), (C.2)
where we have used K0 and F0 to denote the constant coagulation and fragmentation
rates, respectively. The coagulation terms are the same as those which describe the
growth of linear polymers. The fragmentation terms can be understood as follows.
In the model, all bonds in the system are equally likely to be broken. Since an
m−mer contains (m − 1) bonds which may be broken, the rate at which m−mers
fragment is the fragmentation rate, times the number of bonds associated with
an m−mer, times the number density of m−mers: F0 (m − 1) n(m, t). Each such
fragmentation decreases the number of m−mers. The number of m−mers increases
if an M−mer with M > m fragments into an m−mer and an (M − m)−mer.
For linear polymers, of the (M − 1) bonds in a massive M−mer, only a fraction
p = 2/(M − 1) will lead to the creation of an m−mer (recall m < M). The rate at
257
which this happens is F (M |m) = F0 (M − 1)p = 2F0. If
n(m, t) = (1− b)2 bm−1 for some b(t), (C.3)
then (C.2) requires that
dn(m, t)
dt= n(m, t)
(
m−1b− 2
1−b
)
dbdt
= n(m, t)(
m−1b− 2
1−b
)
[
K0(1−b)2
2− F0b
]
, (C.4)
where the first equality comes from evaluating dn/db and the second from inserting
(C.3) in the right hand side of (C.2). Thus, the term in square brackets shows what
is required of db/dt if (C.3) is to solve (C.2). Notice that this solution corresponds
to monomer initial conditions. We must point out that in this Appendix, the term
‘initial conditions’ refers to the size-distribution of clumps, not the distribution of
particles (the power spectrum). In the continuum limit at late times, the solution
has no memory of the initial mass function. Mergers erase this information, making
the mass spectrum approach a universal shape.
Before solving for b(t), notice that when
F0
K0
=(1− b)2
2b, (C.5)
then dn/dt = 0 for all m. This shows that the merger-fragmentation process leads
to an equilibrium mass spectrum given by (C.3), with the parameter b related to
the ratio of the coagulation and fragmentation rates. Notice that an equilibrium
state exists even if the coagulation and fragmentation rates are vastly different! If
258
fragmentation is large, F0/K0 ≫ 1, then b≪ 1 and there are very few objects with
large m (in the limit, almost all polymers are monomers). If mergers dominate,
then b→ 1 and the mass spectrum tends to
(1− b)2
bbm → 2F0
K0
exp(
−m(1− b))
≈ 2F0
K0
exp
(
−m
√
2F0
K0
)
.
Our expression for the equilibrium mass spectrum represents the discrete version of
the calculation presented in Barrow & Silk (1981). As Barrow notes, the interesting
thing about this equilibrium mass spectrum is that an exponential form is also the
solution to the case in which there is no fragmentation. This remains true in the
discrete calculation here.
However, our calculation demonstrates that it is not just the equilibrium mass
spectrum which has the same form as when fragmentation is absent. In particular, in
our merger-fragmentation model, dn/db does not depend on F and K individually,
so the functional form of the mass spectrum is the same as when fragmentation
is absent—the net result of allowing for fragmentation is to rescale the relation
between b and t. For instance, if the initial distribution at t = t0 is given by (C.3)
with b0, then (C.3) with
b(t) = b0 +K0(t− t0)(1− b0)
2
K0(1− b0)(t− t0) + 2
solves (C.2) when F0 = 0.
When both coagulation and fragmentation are present
b(t) =F0
K0
r+ + κ(t)r−1− κ(t)
,
259
where
r± =√
1 + 2K0/F0 ± (1 + K0/F0), κ(t) = κ0 exp[−√
1 + 2K0/F0(t− t0)],
and
κ0 =b0K0/F0 − r+
b0K0/F0 + r−.
Notice that b(t) depends on the ratio K0/F0 and not on K0 or F0 individually. The
pure coagulation solution is recovered in the limit as F0 → 0. The number density
n(m, t) depends on K0/F0 implicitly through b(t). Thus, fragmentation modifies
the rate at which n(m, t) evolves, but not the functional form of the mass spectrum
itself. sequence of states it takes. The system reaches equilibrium when
b→(
1 +F0
K0
)
[
1−√
1− 1
(1 + F0/K0)2
]
; (C.6)
equilibrium is reached earlier as F0/K0 is increased.
We are not aware of any astrophysical applications of this kernel (or any others
besides the additive one discussed below). But, in the presence of a constant kernel,
the growth of clumps is associated with percolation – similar to that behind the
‘friends-of-friends’ algorithm for indentifying haloes in N-body simulations.
C.2.2 Branched polymers
A simple model of the evolution of branched polymers follows from setting K(m,m′) =
K0(m + m′) (see Chapter 4). In this case, an m-mer is best visualized not as being
linear, but as being a ‘tree’: a singly connected planar graph containing (m − 1)
260
bonds which connect m vertices with no loops. We must emphasize that on this
context, the word ‘tree’ refers to spatial structure of the clumps, not to their as-
sembly history, as in Chapter 3. Nevertheless, the two are related by following
the creation of the individual bonds as subclumps merge (Sheth & Pitman, 1997).
If fragmentation follows from the same simple process as before, i.e., the deletion
of a randomly chosen bond, then F (M |m) 6= 2F0 as it was for the linear graphs,
since more than two possible choices of the bond to cut may give rise to the pair of
subtrees (m,M −m). Rather, the probability that deletion of a random edge of an
M−tree leads to an m−tree is
pm|M =1
2(M − 1)
(
M
m
)
(m
M
)m−1 (
1− m
M
)M−m−1
(C.7)
where 1 ≤ m ≤M − 1 (see equation 10 in Sheth & Pitman, 1997) – see Figure C.1.
Note that although this fragmentation term is considerably more complicated than
those usually studied in the literature, it is, in fact, the natural choice.
With this choice, F (M |m) = F0 (M − 1) pm|M , so (C.1) becomes
dn(m, t)
dt=
K0m
2
m−1∑
m′=1
n(m′, t)n(m−m′, t)−K0n(m, t)∞∑
m′=1
(m + m′) n(m′, t)
−F0(m− 1)n(m, t) + F0
∞∑
M=m+1
n(M, t) (M − 1)(
pm|M + pM−m|M
)
.(C.8)
A little algebra shows that if
n(m, t) = (1− b)(mb)m−1
m!e−mb, (C.9)
261
Figure C.1: Haloes as branched polymers. An m-clump and and m′-clump are
obtain by deleting a random bond in an (m + m′) clump.
262
then
dn(m, t)
dt= n(m, t)
(
m− 1
b−m− 1
1− b
)
db
dt
db
dt= K0(1− b)− F0b. (C.10)
Note that when
F0
K0
=1− b
b(C.11)
then dn/dt = 0 for all m. If fragmentation dominates, then b → 0, and almost all
clusters are monomers. If mergers dominate, b→ 1, at t =∞.
As in the linear polymer (constant kernel) model, the equilibrium mass spectrum
(Borel, 1942) has the same form as the solution to (C.8) when F0 = 0. The only
difference is that here, if b = b0 at t0, then
b(t)− b0 =(
1− exp[
−K0(t− t0)(1 + F0/K0)])
(
1
1 + F0/K0
− b0
)
. (C.12)
From (C.11), equilibrium is reached when
b→(
1 + F0/K0
)−1
; (C.13)
as before, larger values of F0/K0 lead to earlier equilibrium.
Ziff & McGrady (1985) discuss the pure fragmentation model (K = 0) for linear
polymers with random bonds, so, for completeness, we now discuss the correspond-
ing solution for branched polymers. Let N0(t) ≡∑
n(m, t) denote the total number
density of polymers present at time t. If all polymers initially have the same num-
ber of particles, say M (note that branched polymers are not indistinguishable from
263
each other, as there are MM−2 different tree configurations; in contrast, linear poly-
mers of mass M are indistinguishable), and the system then fragments according
to (C.8) with K0 = 0 and F0 = 1, then the number density of polymers increases
with time: N0(t) = M + (1−M) e−t. In this case, the mass spectrum evolves as
n(m, t|M) = (1−B)
(
M
m
)(
mB
M
)m−1(
1− mB
M
)M−m−1
, (C.14)
where B(t) = exp(−t) and 1 ≤ m ≤M . This can be verified by direct substitution
(tedious), or by noting that this expression for n(m, t) is the same as that in Sheth
(1996) and Sheth & Pitman (1997) for the average number of m−subtrees within
an M−tree, given that a fraction (1 − B), chosen at random from among the
available M−1 branches, has been cut. If there is a distribution of initial polymers,
then the solution is got by simple superposition of the one derived here. If the
initial distribution is given by (C.9) with parameter b0, then this superposition
can be solved analytically: the distribution at some later time is given by (C.9)
with parameter 0 ≤ b1 ≤ b0. To see this, simply set B = b1/b0 and compute
∑
n(M, b0)n(m, t|M).
Barrow & Silk (1981) asked if a coagulation–fragmentation process could give
rise to a mass spectrum of the form n(m) ∝ m−3/2 exp(−m/m∗(t)). If K0 ≫ F0,
then b→ 1−F0/K0, and the mass spectrum in our (C.9) tends to this form. Thus,
our model shows that the answer to his question is ‘yes’, provided coagulation is
proportional to the sum of the masses, and fragmentation is described by (C.7). Al-
though our fragmentation term is considerably more complicated than those usually
264
studied in the literature, we argued that it is, in fact, a natural choice: it corre-
sponds to a model in which a system of branched (rather than linear) polymers
evolves by the addition or deletion of randomly chosen bonds.
C.3 Generalization
Our model of reversible coagulation and fragmentation can be generalized by setting
F (M |m) = F0 (M − 1)n(m, t)n(M −m, t) K(m,M −m)
∑M−1m=1 n(m, t)n(M −m, t) K(m,M −m)/2
and
F (m|m′) = F0 (C.15)
in the binary merger-fragmentation model of (C.1). Notice that the denominator in
F (M |m) is simply the rate of creation of objects of mass M from pure coagulation.
So F (M |m), the rate at which objects of mass M fragment to produce objects of
mass m is the product of a fragmentation rate F0 times the number of bonds (M−1)
times twice the probability that coagulation produced an object with M − 1 bonds
by the merger of objects with mass m and M −m. [Twice, because fragmentation
can produce m from M either by reversing the merger of m with M − m, or of
M −m with m, and, by symmetry F (M |m) = F (M |M −m).]
If N0(t) ≡∑∞
m=1 n(m, t) and N1(t) ≡∑∞
m=1 mn(m, t). then mass conservation
265
means that N1 is constant. This can be checked explicitly, since
dN1(t)
dt=
1
2
∑
m>0
m∑
m′<m
K(m,m′)n(m′, t)n(m−m′, t)
−∑
m>0
m∑
M>0
K(m,M)n(M, t)n(m, t)
+∑
m>0
m∑
M>m
F (M |m)n(M, t)−∑
m>0
m∑
m′<m
F (m|m′)n(m, t)
=1
2
∑
m′>0
∑
m>m′
(m′ + m−m′)K(m,m′)n(m′, t)n(m−m′, t)
−∑
m>0
m∑
M>0
K(m,M)n(M, t)n(m, t)
+∑
M>0
n(M, t)∑
m<M
mF (M |m)−∑
m>0
m(m− 1)F0n(m, t). (C.16)
Notice that the first two terms, which arise from the coagulation part of the process,
cancel each other. The second two terms, arising from the fragmentation process,
also cancel, as can be seen by inserting (C.15) for F (M |m) and using the fact that
M−1∑
m=1
m n(m, t)n(M −m, t) K(m,M −m)
=M−1∑
m=1
(m + M −m)
2n(m, t)n(M −m, t) K(m,M −m)
=M
2
M−1∑
m=1
n(m, t)n(M −m, t) K(m,M −m). (C.17)
Thus, dN1(t)/dt = 0, and it is usual to set this constant to unity: N1 = 1. Similarly,
(C.1) and (C.15) imply that
dN0(t)
dt= −
∞∑
m=1
n(m, t)∞∑
m′=1
n(m′, t)K(m,m′)
2− F0
[
N0(t)−N1
]
. (C.18)
A pure coagulation model does not have the extra term involving F0. Equilibrium
is reached if dN0(t)/dt = 0.
266
Writing out dn(m, t)/dt explicitly yields
dn(m, t)
dt=
1
2
m−1∑
m′=1
K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑
m′=1
K(m,m′)n(m, t)n(m′, t)
+∞∑
M=m+1
F (M |m)n(M, t)−m−1∑
m′=1
F (m|m′)n(m, t),
=1
2
m−1∑
m′=1
K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑
m′=1
K(m,m′)n(m, t)n(m′, t)
+2F0
∞∑
M=m+1
(M − 1)n(M, t)n(m, t)n(M −m, t)K(m,M −m)
∑M−1m=1 n(m, t)n(M −m, t)K(m,M −m)
−F0
m−1∑
m′=1
n(m, t),
=1
2
m−1∑
m′=1
K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑
m′=1
K(m,m′)n(m, t)n(m′, t)
+2F0n(m, t)∞∑
M=m+1
(M − 1)n(M, t)n(M −m, t)K(m,M −m)∑M−1
m=1 n(m, t)n(M −m, t)K(m,M −m)
−F0(m− 1)n(m, t),
=1
2
m−1∑
m′=1
K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑
m′=1
K(m,m′)n(m, t)n(m′, t)
+2F0n(m, t)∞∑
m′=1
(m′ + m− 1)n(m′ + m, t)n(m′, t)K(m,m′)∑m′+m−1
m′′=1 n(m′′, t)n(m′ + m−m′′, t)K(m′′,m′ + m−m′′)
−F0(m− 1)n(m, t),
=1
2
m−1∑
m′=1
n(m′, t)n(m−m′, t)[
K(m′,m−m′)−F(m′,m−m′, t)]
−n(m, t)∞∑
m′=1
n(m′, t)[
K(m,m′)−F(m,m′, t)]
(C.19)
267
where we have defined
F(m,m′, t) ≡ 2F0 (m′ + m− 1)n(m′ + m, t)K(m,m′)∑m′+m−1
m′′=1 n(m′′, t)n(m′ + m−m′′, t) K(m′′,m′ + m−m′′), (C.20)
and used the fact that
1
2
m−1∑
m′=1
F(m′,m−m′, t) n(m′, t) n(m−m′, t)
= F0 (m− 1) n(m, t)m−1∑
m′=1
n(m′, t)n(m−m′, t) K(m′,m−m′)∑m−1
m′′=1 n(m′′, t)n(m−m′′, t) K(m′′,m−m′′)
= F0 (m− 1) n(m, t). (C.21)
Thus,
dn(m, t)
dt=
1
2
m−1∑
m′=1
n(m′, t)n(m−m′, t)[
K(m′,m−m′)−F(m′,m−m′, t)]
−n(m, t)∞∑
m′=1
n(m′, t)[
K(m,m′)−F(m,m′, t)]
(C.22)
(Note that F is symmetric in m and m′.) This illustrates that the system with co-
agulation and reversible fragmentation has the form of a pure coagulation problem,
but with time-dependent coagulation kernel
K(m,m′, t) ≡ K(m,m′)−F(m,m′, t). (C.23)
Because of this time dependence, it is possible for the system to evolve to an equi-
librium state. This corresponds to the time when K = F(t).
If F(m,m′, t) separates into the product of a mass-dependent part and a time-
dependent part, and the mass-dependent part is similar to the mass-dependence of
K(m,m′), then the mass spectrum associated with the coagulation-fragmentation
268
process is the same as that associated with pure coagulation, except for a rescaling of
the time variable. Equation (C.20) shows that this will happen for all systems where
the creation term associated with coagulation (notice that it is the denominator
in the expression which defines F) can be written as (m + m′ − 1)n(m + m′, t)
times a piece which depends on time but not mass. (This is not an unreasonable
requirement, since this simply says the creation rate can be written as the product
of the number of bonds in a system with M = m + m′ particles, times the number
density of such systems, times a function of time.) When this is true, the ensemble
averaged coagulation-fragmentation system evolves through the same sequence of
states as when fragmentation is absent, only at a different rate. Of course, the
system associated with both coagulation and fragmentation evolves only until the
equilibrium state is reached.
C.3.1 General linear kernel
To illustrate, consider the general linear kernel
K(m,m′) = A + B(m + m′). (C.24)
In the pure coagulation case and monomer initial conditions,
n(m, t) = (1− b) (1− A∗b)m/A∗−m+1 (A∗b)
m−1
m!
Γ(m/A∗ + 1)
Γ(m/A∗ −m + 2)(C.25)
with b =1− exp(−Bt)
1− A∗ exp(−Bt)and A∗ =
A
A + 2B(C.26)
269
(Lu , 1987; Treat, 1990), so
dn(m, t)
dt= n(m, t)
[
m− 1
b− 1
1− b−(
m
A∗
−m + 1
)
A∗
1− A∗b
]
db
dt, (C.27)
where
db
dt= B (1− b)
1− A∗b
1− A∗
. (C.28)
Computing F(m,m′, t) involves finding the creation rate in the pure coagulation
equation (C.27). However, the required sum involves Pochhammer symbol iden-
tities, making the calculation cumbersome. To circumvent this, one can instead
compute dn(m, t)/dt and then subtract-off the destruction rate (which only involves
moments of n(m, t)). This shows that the creation rate is
m−1∑
m′=1
n(m′, t) n(m−m′, t)K(m′,m−m′)
2= (m− 1) n(m, b)
d ln b
dt
∣
∣
∣
F0=0, (C.29)
so
F(m,m′, t) = F0 K(m,m′)
(
d ln b
dt
∣
∣
∣
F0=0
)−1
, (C.30)
and the net coagulation kernel is
K(m,m′, t) = K(m,m′)
[
1− F0
(
d ln b
dt
∣
∣
∣
F0=0
)−1]
. (C.31)
This is equivalent to replacing
db
dt→ db
dt
∣
∣
∣
F0=0− F0b. (C.32)
(e.g. C.22). These four equations hold for all the kernels considered so far; only the
evolution in b(t) differs.
270
In particular, for the general linear kernel, (C.27) still holds in the presence of
fragmentation, but now
db
dt=
(1− b)(1− A∗b)
1− A∗
B − F0b. (C.33)
Solving for b(t) here is analogous to the constant kernel case, so we do not show it
here. The condition for equilibrium is
F0
A + 2B= (1− b)
1− A∗b
2b. (C.34)
It is a simple matter to check that when (A,B) = (1, 0), then equilibrium is given
by (C.5). Likewise, when (A,B) = (0, 1), equilibrium is given by (C.11). The
larger the value of F0, the sooner the equilibrium state is reached. In the limit
when F0/A≫ 1 and F0/B ≫ 1, then b→ 0 and one finds mostly monomers.
Moreover, because the creation rate has the form given by (C.29), the net coag-
ulation rate K(m,m′, t) separates into the product of a mass-dependent part and a
time-dependent part. Therefore, the mass-dependence of K is the same as that of
the original coagulation piece K.
C.3.2 Multiplicative kernel
There is another exactly solvable case for which our merger-fragmentation model
shows similar behavior: the multiplicative kernel
K(m,m′) = K0 mm′. (C.35)
271
In contrast to the generalized linear case (and its two special instances), this model
evolves towards gelation—an infinite-size cluster is created in finite time—if there
is no fragmentation. Prior to gelation, the mass spectrum is
n(m, t) =(2mb)m−1
mm!e−2mb where 0 ≤ b(t) = t/2 < 1. (C.36)
The explicit factor of 2 is chosen so that N0(t) ≡∑
m n(m, t) = 1− b(t), as it is for
the constant, additive and general linear kernels, and the upper limit on b derives
from requiring that N0 be positive. In fact, one could argue that N0(t) is a more
natural choice for the time parameter than is b(t).
In the pure coagulation case, this kernel results in
dn(m, t)
dt= n(m, t)
(
m−1b− 2m
)
dbdt
= n(m, t)(
m−1b− 2
1−b
)
K0
2. (C.37)
Here
b(t) = b0 + (K0/2) (t− t0) (C.38)
and the system evolves towards gelation in finite time:
tgel ≡ t0 + (2/K0) (1− b0). (C.39)
Can fragmentation prevent the onset of gelation? The creation term in this case
has the form given by (C.29), so the three equations which follow it also apply.
Therefore, in the presence of fragmentation, b(t) obeys
db
dt=
C0
2− F0b, (C.40)
272
and the condition for equilibrium is simply
F0
K0
=1
2b. (C.41)
The evolution is given by
b(t) =K0
2F0
−(
K0
2F0
− b0
)
exp[
− F0(t− t0)]
. (C.42)
Notice that equilibrium is only reached if F0 > 2K0; if F0 is smaller, then gelation
will still occur.
As in all the other cases studied in this Appendix, if fragmentation dominates
over coagulation, b→ 0 and one finds mostly monomers. In this case too, because
F(m,m′, t) is the product of a time dependent piece and a mass dependent piece
which scales as K(m,m′), the evolution to equilibrium is through the same sequence
of mass spectra as when fragmentation was absent, only at a different rate.
C.4 r−mer initial conditions
The above examples assumed that the initial conditions were monodisperse. For
arbitrary initial conditions, the solution to the coagulation equation with kernel
K(m,m′) = A + B(m + m′) is
n(m, t) = (1− b) (1− A ∗ b)(1+2Bm/A)
m−1∑
n=0
(A∗b)n
(n + 1)!(2 + 2Bm/A)nc(n+1)
m (C.43)
(Treat, 1990). Here
b =1− exp(−Bµt)
1− A∗ exp(−Bµt), A∗ =
A
A + 2µB, (C.44)
273
µ ≡ N1(0), (a)n ≡ Γ(a + n)/Γ(a) denotes the Pochhammer symbol, and
∞∑
m=1
c(n+1)m sm =
(
∞∑
i=1
n(i, 0)si)n+1
. (C.45)
The case of monomers considered before corresponds to n(m, 0) = δm,1 and
therefore c(n+1)m = δm,n+1. It is straightforward to generalize to r-mer initial condi-
tions:
n(m, 0) = δm,r, where c(n+1)m = δm,r(n+1).
In this case
dn(m, t)
dt= n(m, t)
(
m/r − 1
b− 1
1− b− (
m/r
A∗
−m/r + 1)A∗
1− A∗b
)
db
dt(C.46)
if there is no fragmentation. This differs from the solution (C.25) for monomers
only by the replacements m → m/r and µ = 1 → r, reflecting the fact that, with
pure coagulation, only polymers of size being a multiple of r will ever form. The
evolution of b is given by
db
dt= rB
(1− b)(1− A∗b)
(1− A∗). (C.47)
Our prescription for fragmentation yields
F(m,m′, t) = rF0
(
m + m′ − 1
m + m′ − r
)
K(m,m′)
d ln b/dt|F0=0
; (C.48)
this does not have the same mass dependence as K(m,m′) unless r = 1. Therefore,
in this case, fragmentation not only changes the rate with which the system evolves,
but also the mass spectrum at any give time. This is easily understood: absent
274
fragmentation, the only allowed states were multiples of r−mers; our fragmentation
treats all bonds equally—the primordial bonds between the initial r-mers as well
as those formed by coagulation—so fragmentation results in a system in which
polymers whose sizes are no longer integer multiples of r. Therefore, in the presence
of fragmentation, (C.43) for the mass spectrum no longer applies. The only way
to preserve the same states as in the pure coagulation case is to have a different
fragmentation rule in which the bonds in the initial r−mers where indestructible.
C.5 Final remarks
We have studied a number of systems in which fragmentation is the time-reverse
process of coagulation. The particular examples presented here for which exact
analytic solutions were obtained suggest a general conclusion: When the fragmen-
tation is the time-reverse process of coagulation, then the mass spectrum evolves
to an equilibrium state provided that the pure-coagulation process would not have
formed an infinite cluster. In this case, the shape of the equilibrium spectrum is
described by the same functional form as when fragmentation was absent. More-
over, until it reaches this equilibrium state, the mass spectrum evolves similarly to
how it does when there is no fragmentation; the only difference is that the rate of
evolution is different. The constant, additive and generalized linear polymers are
examples of such systems.
On the other hand, if the pure-coagulation process would have led to forma-
275
tion of an infinite cluster, then an equilibrium state exists only if fragmentation is
sufficiently efficient. The results presented here allow one to estimate this critical
efficiency. They also show that the evolution to equilibrium is through a similar
series of mass spectra as when fragmentation was absent, only the rate of evolution
is slower, and the mass spectrum reaches a steady state before infinite clusters form.
The multiplicative polymer is a particular example of this case, in which F0 > 2K0
results in equilibrium before gelation.
These conclusions depend on the initial spectrum of clusters in the system. In
general, if the initial conditions contain r-mers, rather than monomers, the spectrum
of states is changed in the presence of fragmentation. This is because in pure
coagulation, the sizes of polymers are always multiples of r; this is no longer true if
fragmentation is able to break the primordial bonds between the initial r-mers.
276
Bibliography
Abazajian K. N., et al., The Seventh Data Release of the Sloan Digital Sky Survey,
2009, ApJS, 182, 543
Abramowitz M., Stegun I. A., Handbook of mathematical functions, 1972, New York:
Dover
Acquaviva V., Bartolo N., Matarrese S., Riotto A., Second-order cosmological per-
turbations from inflation, 2003, Nucl. Phys. B, 667, 119
Adelberger K. L., Steidel C. C., Kollmeier J. A., Reddy N. A., 2006, ApJ, Possible
Detection of Ly-alpha Fluorescence from a Damped Lyalpha System at Redshift z
2.8,, 2006, 637, 74
Afshordi N., Tolley A. J., Primordial non-Gaussianity, statistics of collapsed objects,
and the integrated Sachs-Wolfe effect, 2008, PRD, 78, 123507
Aird J., et al., The evolution of the hard X-ray luminosity function of AGN, 2009,
preprint (astro-ph/0910.1141)
277
Allen D. A., Roche P. F., Norris R. P., Faint IRAS galaxies - A new species in the
extragalactic zoo, 1985, MNRAS, 213, 67P
Aller M. C., Richstone D., The cosmic density of massive black holes from galaxy
velocity dispersions, 2002, AJ, 124, 3035 h
Almeida C., Baugh C. M., Wake D. A., Lacey C. G., Benson A. J., Bower R. G.,
Pimbblet K., Luminous red galaxies in hierarchical cosmologies, 2008, MNRAS,
466
Angulo R. E., Baugh C. M., Lacey C. G., The assembly bias of dark matter haloes
to higher orders, 2008, MNRAS, 387, 921
Angulo R. E., White S. D. M., The birth and growth of neutralino haloes, 2009,
preprint (astro-ph/0906.1730)
Armitage P. J., Natarajan P., Accretion during the merger of supermassive black
holes, 2002, ApJL, 567, L9
Armus L., Heckman T., Miley G., Multicolor optical imaging of powerful far-infrared
galaxies - More evidence for a link between galaxy mergers and far-infrared emis-
sion, 1987, AJ, 94, 831
Audit E., Alimi J.-M., Gravitational Lagrangian dynamics of cold matter using the
deformation tensor, 1996, AAP, 315, 11
278
Audit E., Teyssier R., Alimi J.-M., Non-linear dynamics and mass function of cos-
mic structures. I. Analytical results, 1997, AAP, 325, 439
Avelino P. P., Viana P. T. P., The cloud-in-cloud problem for non-Gaussian density
fields, 2000, MNRAS, 314, 354
Avila-Reese V., Understanding galaxy formation and evolution, 2006, preprint
(astro-ph/0605212)
Avila-Reese V., Firmani C., Hernandez X., On the formation and evolution of disk
galaxies: cosmological initial conditions and the gravitational collapse, 1998, ApJ,
505, 37
Bachelier L., Theorie de la speculation, 1900, Annales Scientifiques de l’E.N.S. 3eme
serie, Tome 17.
Baes M., Buyle P., Hau G. K. T., Dejonghe H., Observational evidence for a con-
nection between supermassive black holes and dark matter haloes, 2003, MNRAS,
341, L44
Ballo L., et al., Arp 299: a second merging system with two active nuclei? 2004,
ApJ, 600, 634
Bandara K., Crampton D., Simard L., A Relationship between Supermassive Black
Hole Mass and the Total Gravitational Mass of the Host Galaxy, 2009, preprint
(astro-ph/0909.0269)
279
Bardeen J. M., Kerr metric black holes, 1970, Nat, 226, 64
Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., The statistics of peaks of
Gaussian random fields, 1986, ApJ, 304, 15
Barger A. J., Cowie L. L., Mushotzky R. F., Yang Y., Wang W.-H., Steffen A. T.,
Capak P., The cosmic evolution of hard X-ray-selected active galactic nuclei, 2005,
AJ, 129, 578
Barkana R., On correlated random walks and 21-cm fluctuations during cosmic
reionization, 2007, MNRAS, 376, 1784
Barkana R., Loeb A., In the beginning: the first sources of light and the reionization
of the universe, 2001, Phys. rep., 349, 125
Barnes J. E. 1988, Encounters of disk/halo galaxies, ApJ, 331, 699
Barnes J. E., Transformations of galaxies. I - Mergers of equal-mass stellar disks ,
1992, ApJ, 393, 484
Barnes J. E., Hernquist L. E. Fueling starburst galaxies with gas-rich mergers, 1991,
ApJL, 370, L65
Barnes J. E., Hernquist L., Transformations of galaxies. II. Gasdynamics in merging
disk galaxies, 1996, APJ, 471, 115
Barthel P. D., Star-forming QSO host galaxies, 2006, A&A, 458, 107
280
Barrow J. D., Coagulation with fragmentation, 1981, JPA, 14, 729
Barrow J. D., Saich P., Growth of large-scale structure with a cosmological constant,
1993, MNRAS, 262, 717
Barrow, J. D., Stein-Schabes, J., Inhomogeneous cosmologies with cosmological con-
stant, 1984, PLA, 103, 315
Barrow J. D., Silk J., The growth of anisotropic structures in a Friedmann universe,
1981, ApJ, 250, 432
Barrow J. D., Saich P., Growth of large-scale structure with a cosmological constant,
1993, MNRAS, 262, 717
Barthel P. D., Star-forming QSO host galaxies, 2006, A&A, 458, 107
Bartolo N., Komatsu E., Matarrese S., Riotto A., Non-Gaussianity from inflation:
theory and observations, 2004, Phys. Rep., 402, 103
Basilakos S., Cluster formation rate in models with dark energy, 2003, ApJ, 590,
636
Baugh C. M., A primer on hierarchical galaxy formation: the semi-analytical ap-
proach, 2006, Rep. Prog. Phys., 69, 3101
Begelman M. C. Black holes in radiation-dominated gas - an analogue of the Bondi
accretion problem, 1978, MNRAS, 184, 53
281
Begelman M. C., Blandford R. D., Rees M. J., Massive black hole binaries in active
galactic nuclei, 1980, Nat, 287, 307
Begelman M. C., Super-Eddington fluxes from thin accretion disks?, 2002, ApJL,
568, L97
Begelman M. C., Rossi E. M., Armitage P. J., Quasistars: Accreting black holes
inside massive envelopes, 2008, MNRAS, 387, 1649
Begelman M. C., AGN feedback mechanisms, 2004, Coevolution of Black Holes and
Galaxies, 374
Bekenstein J. D., Relativistic gravitation theory for the modified Newtonian dynam-
ics paradigm, 2004, PRD, 70, 083509
Bell E. F., Zheng X. Z., Papovich C., Borch A., Wolf C., Meisenheimer K., Star
formation and the growth of stellar mass, 2007, ApJ, 663, 834
Bennert N., Canalizo G., Jungwiert B., Stockton A., Schweizer F., Peng C. Y.,
Lacy M., Evidence for merger remnants in early-type host galaxies of low-redshift
QSOs, 2008, ApJ, 677, 846
Benson A. J., Bower R. G., Frenk C. S., Lacey C. G., Baugh C. M., Cole S., What
shapes the luminosity function of galaxies? 2003, ApJ, 599, 38
Benson A. J., Kamionkowski M., Hassani S. H., Self-consistent theory of halo merg-
ers, 2005, MNRAS, 357, 847
282
Benson A. J., Constraining cold dark matter halo merger rates using the coagulation
equations, 2008, MNRAS, 388, 1361
Bentz M. C., Peterson B. M., Pogge R. W., Vestergaard M., The black hole mass-
bulge luminosity relationship for active galactic nuclei From reverberation map-
ping and Hubble Space Telescope imaging, 2009, ApJL, 694, L166
Bernardi M., Renzini A., da Costa L. N., Wegner G., Alonso M. V., Pellegrini P. S.,
Rite C., Willmer C. N. A., Cluster versus Field Elliptical Galaxies and Clues on
Their Formation, 1998, ApJL, 508, L143
Bernardi M., Sheth R. K., Tundo E., Hyde J. B., Selection bias in the M-σ and
M-L correlations and its consequences, 2007, ApJ, 660, 267
Bertacca D., Bartolo N., Diaferio A., Matarrese S., How the scalar field of unified
dark matter models can cluster, 2008, JCA-P Phys., 10, 23
Bertschinger E., Jain B., Gravitational clustering from scale-free initial conditions,
1994, ApJ, 431, 486
Bettoni D., Falomo R., Fasano G., Govoni F., The black hole mass of low redshift
radiogalaxies, 2003, AAP, 399, 869
Blandford R. D., Znajek R. L., Electromagnetic extraction of energy from Kerr black
holes, 1977, MNRAS, 179, 433
283
Blanton M. R., et al., New York University Value-Added Galaxy Catalog: A Galaxy
Catalog Based on New Public Surveys, 2005, AJ, 129, 2562
Blumenthal G. R., Faber S. M., Primack J. R., Rees M. J., Formation of galaxies
and large-scale structure with cold dark matter, 1984, Nat, 311, 517
Binglin, L. The exact solution of the coagulation equation with kernel Kij = A(i +
j) + B, 1987, J. Phys. A., 20 2347
Blain A. W., Longair M. S., Submillimetre cosmology, 1993, MNRAS, 264, 509
Blain A. W., Longair M. S., Millimetre background radiation and galaxy formation,
1993, MNRAS, 265, L21
Bond J. R., Cole S., Efstathiou G., Kaiser N., Excursion set mass functions for
hierarchical Gaussian fluctuations, 1991, ApJ, 379, 440
Bond J. R., Myers S. T., The peak-patch picture of cosmic catalogs. I. Algorithms,
1996, ApJS, 103, 1
Bondi H., Hoyle F., On the mechanism of accretion by stars, 1944, MNRAS, 104,
273
Bondi H., On spherically symmetrical accretion, 1952, MNRAS, 112, 195
Bonoli S., Marulli F., Springel V., White S. D. M., Branchini E., Moscardini L.,
Modeling the cosmological co-evolution of supermassive black holes and galaxies:
284
II. The clustering of quasars and their dark environment, 2008, preprint (astro-
ph/0812.0003)
Bonoli S., Shankar F., White S., Springel V., Wyithe S., On merger bias and the
clustering of quasars, 2009, preprint (astro-ph/0909.0003)
Booth C. M., Schaye J., Cosmological simulations of the growth of supermassive
black holes and feedback from active galactic nuclei: method and tests, 2009,
preprint (astro-ph/0904.2572)
Borch A., et al., The stellar masses of 25 000 galaxies at 0.2 1.0 estimated by the
COMBO-17 survey, 2006, A&A, 453, 869
Borel E., Sur l’emploi du thorme de Bernouilli pour faciliter le calcul d’une infinit
de coefficients. Application au probleme de l’attente a un guichet, 1942, Comptes
Rendus de l’Academie des Sciences, 214, 452
Borne K. D., et al., A morphological classification scheme for ULIRGs, 1999,
Ap&SS, 266, 137
Bournaud F., Jog C. J., Combes F., Galaxy mergers with various mass ratios:
Properties of remnants, 2005, A&P, 437, 69
Bouwens R. J., et al., z 7-10 Galaxies Behind Lensing Clusters: Contrast with
Field Search Results, 2009, ApJ, 690, 1764
285
Bower R. G., The evolution of groups of galaxies in the Press-Schechter formalism,
1991, MNRAS, 248, 332
Bower R. G., Benson A. J., Malbon R., Helly J. C., Frenk C. S., Baugh C. M., Cole
S., Lacey C. G., Breaking the hierarchy of galaxy formation, 2006, MNRAS, 370,
645
Breiman L., First exit times from a square root boundary, 1967, in Cam. L. L.,
ed. Proc. 5th Berk. Symp. Math. Statist. Prob. Vol 2, Univ. California Press,
Berkeley, p. 9
Boylan-Kolchin M., Ma C.-P., Quataert E., Dynamical friction and galaxy merging
time-scales, 2008, MNRAS, 383, 93
Boyle B. J., Terlevich R. J., The cosmological evolution of the QSO luminosity
density and of the star formation rate, 1998, MNRAS, 293, L49
Bromley J. M., Somerville R. S., Fabian A. C., High-redshift quasars and the super-
massive black hole mass budget: constraints on quasar formation models, 2004,
MNRAS, 350, 456
Bromm V., Loeb A., Formation of the first supermassive black holes, 2003, ApJ,
596, 34
Brotherton M. S., et al., A Spectacular poststarburst quasar, 1999, ApJL, 520, L87
286
Brueggen M., Scannapieco E., Self-regulation of AGN in galaxy clusters, 2009,
preprint (astro-ph/0905.4726)
Buitrago F., Trujillo I., Conselice C. J., Bouwens R. J., Dickinson M., Yan H., Size
Evolution of the Most Massive Galaxies at 1.7 < z < 3 from GOODS NICMOS
Survey Imaging, 2008, ApJL, 687, L61
Brusa M., et al., High-Redshift Quasars in the COSMOS Survey: The Space Density
of z > 3 X-Ray Selected QSOs, 2009, ApJ, 693, 8
Bryan G. L., Norman M. L., Statistical properties of X-ray clusters: Analytic and
numerical comparisons, 1998, ApJ, 495, 80
Bundy K., et al., The mass assembly history of field galaxies: detection of an
evolving mass Limit for star-forming galaxies, 2006, ApJ, 651, 120
Bundy K., et al., AEGIS: New evidence linking active galactic nuclei to the quench-
ing of star formation, 2008, ApJ, 681, 931
Bundy K., Fukugita M., Ellis R. S., Targett T. A., Belli S., Kodama T., The
Greater Impact of Mergers on the Growth of Massive Galaxies: Implications for
Mass Assembly and Evolution since z sime 1, 2009, ApJ, 697, 1369
Busha M. T., Adams F. C., Wechsler R. H., Evrard A. E., Future evolution of
cosmic structure in an accelerating universe, 2003, ApJ, 596, 713
287
Busha M. T., Evrard A. E., Adams F. C., Wechsler R. H., The ultimate halo mass
in a ΛCDM universe, 2005, MNRAS, 363, L11
Busha M. T., Evrard A. E., Adams F. C., The asymptotic form of cosmic structure:
small-scale power and accretion history, 2007, ApJ, 665, 1
Caditz D., Petrosian V., Statistical and physical evolution of quasi-stellar objects,
1990, ApJ, 357, 326
Caditz D. M., Petrosian V., Wandel A., Evolution of the luminosity function of
quasar accretion disks, 1991, ApJL, 372, L63
Caldwell N., Rose J. A., Concannon K. D., Star formation histories of early-type
galaxies. I. Higher order Balmer lines as age indicators, 2003, AJ, 125, 2891
Canalizo G., Stockton A., Spectroscopy of Close Companions to Quasi-Stellar Ob-
jects and the Ages of Interaction-Induced Starbursts, 1997, ApJL, 480, L5
Canalizo G., Stockton A., Quasi-stellar objects, ultraluminous infrared galaxies, and
Mmergers, 2001, ApJ, 555, 719
Canalizo G., Bennert N., Jungwiert B., Stockton A., Schweizer F., Lacy M., Peng
C., Spectacular shells in the host galaxy of the QSO MC2 1635+119, 2007, ApJ,
669, 801
Cao X., Li F., Rapidly spinning massive black holes in active galactic nuclei: evi-
dence from the black hole mass function, 2008, MNRAS, 390, 561
288
Cappellari M., et al., Dynamical masses of early-type galaxies at z ∼ 2: Are they
truly superdense? 2009, preprint (astro-ph/0906.3648)
Carlberg R. G., Quasar evolution via galaxy mergers, 1990, ApJ, 350, 505
Carollo C. M., Danziger I. J., Buson L., Metallicity Gradients in Early Type Galax-
ies, 1993, MNRAS, 265, 553
Carroll S. M., Spacetime and geometry. An introduction to general relativity, 2004,
San Francisco, CA, USA: Addison Wesley, ISBN 0-8053-8732-3, XIV + 513 pp.
Carroll S. M., Press W. H., Turner E. L. The cosmological constant, 1992, ARAA,
30, 499
Cattaneo A., Quasars and galaxy formation, 2001, MNRAS, 324, 128
Cattaneo A., Haehnelt M. G., Rees M. J., The distribution of supermassive black
holes in the nuclei of nearby galaxies, 1999, MNRAS, 308, 77
Cattaneo A., Blaizot J., Devriendt J., Guiderdoni B., Active Galactic Nuclei In
Cosmological Simulations - I. Formation of black holes and spheroids through
mergers 2005, MNRAS, 364, 407
Cattaneo A., et al., Accretion, feedback and galaxy bimodality: a comparison of the
GalICS semi-analytic model and cosmological SPH simulations, 2007, MNRAS,
377, 63
Cavaliere A., Morrison P., Wood K., On quasar evolution, 1971, ApJ, 170, 223
289
Cavaliere A., Danese L., de Zotti G., Franceschini A., Constraints to the QSO
contribution to the X-ray background, 1981, AAP, 97, 269
Cavaliere A., Giallongo E., Vagnetti F., Messina A., Quasar evolution and gravita-
tional collapse, 1983, ApJ, 269, 57
Cavaliere A., Szalay A. S., Primeval QSOs, 1986, ApJ, 311, 589
Cavaliere A., Giallongo E., Padovani P., Vagnetti F., Activity patterns and evolution
of AGNs, 1988, Optical Surveys for Quasars, 2, 335
Cavaliere A., Padovani P., The connection between active and normal galaxies, 1989,
ApJ, 340, L5
Cavaliere A., Colafrancesco S., Menci N., The merging runway, 1991, ApJL, 376,
L37
Cavaliere A., Colafrancesco S., Scaramela R., The mass distribution of groups and
clusters of galaxies, 1991, ApJ, 380, 15
Cavaliere A., Colafrancesco S., Menci N., Merging in cosmic structures, 1992, ApJ,
392, 41
Cavaliere A., Menci N., Binary merging: the other mode of galaxy evolution, 1993,
ApJL, 407, L9
290
Cavaliere A., Vittorini V., The rise and fall of the quasars, 1998, The Young Uni-
verse: Galaxy Formation and Evolution at Intermediate and High Redshift, 146,
26
Cavaliere A., Vittorini V., The fall of the quasar population, 2000, ApJ, 543, 599
Cavaliere A., Vittorini V., Supermassive black holes in galactic Nuclei, 2002, ApJ,
570, 114
Chandrasekhar S., Stochastic problems in physics and astronomy, 1943, Rev. Mod.
Phys., 15, 1
Chiueh T., Lee J.,On the nonspherical nature of halo formation, 2001, ApJ, 555,
83
Chornock R., et al., The quasar SDSS J1536+0441: An unusual double-peaked
emitter, 2009, preprint (astro-ph/0906.0849)
Cenarro A. J., Trujillo I., Mild Velocity Dispersion Evolution of Spheroid-Like Mas-
sive Galaxies Since z 2, 2009, ApJL, 696, L43
Cimatti A., Daddi E., Renzini A., Mass downsizing and “top-down” assembly of
early-type galaxies, 2006, AAP, 453, L29
Cimatti A., et al., GMASS ultradeep spectroscopy of galaxies at z ∼ 2. II. Superdense
passive galaxies: how did they form and evolve? 2008, A&P, 482, 21
291
Ciotti L., Ostriker J. P., Cooling flows and quasars: Different aspects of the same
phenomenon? I. Concepts, 1997, ApJL, 487, L105
Ciotti L., Ostriker J. P., Cooling flows and quasars. II. Detailed models of feedback-
modulated accretion flows, 2001, ApJ, 551, 131
Ciotti L., Lanzoni B., Volonteri M., The importance of dry and wet merging on the
formation and evolution of elliptical galaxies, 2007, ApJ, 658, 65
Ciotti L., Ostriker J. P., Proga D., Feedback from central black holes in elliptical
galaxies. I. Models with either radiative or mechanical feedback but not both, 2009,
ApJ, 699, 89
Coil A. L., Hennawi J. F., Newman J. A., Cooper M. C., Davis M., The DEEP2
Galaxy Redshift Survey: Clustering of Quasars and Galaxies at z = 1, 2007, ApJ,
654, 115
Coil, A. L., et al., AEGIS: The Clustering of X-Ray Active Galactic Nucleus Relative
to Galaxies at z ∼ 1, 2009, ApJ, 701, 1484
Colberg J. M., di Matteo T., Supermassive black holes and their environments, 2008,
MNRAS, 387, 1163
Cole S., Lacey C. G., Baugh C. M., Frenk C. S., Hierarchical galaxy formation,
2000, MNRAS, 319, 168
292
Colpi M., Mayer L., Governato F., Dynamical friction and the evolution of satellites
in virialized Halos: The theory of linear response, 1999, ApJ, 525, 720
Colpi M., Dotti M., Massive binary black holes in the cosmic landscape, 2009,
preprint (astro-ph/0906.4339)
Comerford J. M., Griffith R. L., Gerke B. F., Cooper M. C., Newman J. A., Davis
M., Stern D., A 1.75 kpc/h Separation Dual AGN at z=0.36 in the COSMOS
Field, 2009, preprint (astro-ph/0906.3517)
Conroy C., Wechsler R. H., Kravtsov A. V., Modeling luminosity-dependent galaxy
clustering through cosmic time, 2006, ApJ, 647, 201
Conroy C., Ho S., White M., Constraints on the merging time-scale of luminous red
galaxies, or, where do all the haloes go?, 2007, MNRAS, 379, 1491
Conroy C., Wechsler R. H., Kravtsov A. V., The hierarchical build-up of massive
galaxies and the intracluster light since z = 1, 2007, ApJ, 668, 826
Conroy C., Wechsler R. H., Connecting Galaxies, Halos, and Star Formation Rates
Across Cosmic Time, 2009, ApJ, 696, 620
Constantin A., Vogeley M. S., The Clustering of Low-Luminosity Active Galactic
Nuclei, 2006, ApJ, 650, 727
Cowie L. L., Songaila A., Hu E. M., Cohen J. G., New insight on galaxy formation
293
and evolution from Keck Spectroscopy of the Hawaii Deep Fields, 1996, AJ, 112,
839
Cowie L. L., Barger A. J., An integrated picture of star formation, metallicity evo-
lution, and galactic stellar mass assembly, 2008, ApJ, 686, 72
Cox T. J., Star formation and feedback in simulations of interacting galaxies, 2004,
Ph.D. Thesis,
Cox T. J., Dutta S. N., Di Matteo T., Hernquist L., Hopkins P. F., Robertson B.,
Springel V., The kinematic structure of merger remnants, 2006, ApJ, 650, 791
Cox T. J., Jonsson P., Somerville R. S., Primack J. R., Dekel A., The effect of
galaxy mass ratio on merger-driven starbursts, 2008, MNRAS, 384, 386
Creminelli P.,On non-Gaussianities in single-field inflation, 2003, JCA-PPhys, 10,
3
Croom S. M., Shanks T. QSO clustering - III. Clustering in the Large Bright Quasar
Survey and evolution of the QSO correlation function, 1996, MNRAS, 281, 893
Croom S. M., et al., The 2dF QSO Redshift Survey - XIV. Structure and evolution
from the two-point correlation function, 2005, MNRAS, 356, 415
Croton D. J., et al., The many lives of active galactic nuclei: cooling flows, black
holes and the luminosities and colours of galaxies, 2006, MNRAS, 365, 11
294
Csaki E., Foldes A., Salmien P., On the joint distribution of the maximum and its
location for a linear diffusion, Annales de lI. H. P., B, 23-2, 179
da Angela J., et al., The 2dF-SDSS LRG and QSO survey: QSO clustering and the
L-z degeneracy, 2008, MNRAS, 383, 565
Dalal N., Dore O., Huterer D., Shirokov A., Imprints of primordial non-
Gaussianities on large-scale structure: Scale-dependent bias and abundance of
virialized objects, 2008, PRD, 77, 123514
Damjanov I., et al., Red Nuggets at z ∼ 1.5: Compact Passive Galaxies and the
Formation of the Kormendy Relation, 2009, ApJ, 695, 101
Dasyra K. M., et al., Dynamical properties of ultraluminous infrared alaxies. I. Mass
ratio conditions for ULIRG activity in interacting pairs, 2006, ApJ, 638, 745
Dasyra K. M., et al., Dynamical properties of ultraluminous infrared galaxies. II.
Traces of dynamical evolution and end products of local ultraluminous mergers,
2006, ApJ, 651, 835
DeBuhr J., Quataert E., Ma C.-P., Hopkins P., Self-regulated black hole growth
via momentum deposition in galaxy merger simulations, 2009, preprint (astro-
ph/0909.2872)
DeGraf C., Di Matteo T., Springel V., 2009, preprint (astro-ph/0910.1843)
295
De Lucia G., Blaizot J., The hierarchical formation of the brightest cluster galaxies,
2007, MNRAS, 375, 2
De Robertis M., QSO evolution in the interaction model, 1985, AJ, 90, 998
Dekel A., Silk J., The origin of dwarf galaxies, cold dark matter, and biased galaxy
formation, 1986, ApJ, 303, 39
Del Popolo A., Gambera M., Tidal torques and the clusters of galaxies evolution
1998, AAP, 337, 96
Desjacques V. Environmental dependence in the ellipsoidal collapse model, 2008,
MNRAS, 727
Diaferio A., Ostorero L., X-ray clusters of galaxies in conformal gravity, 2008,
preprint (astro-ph/0808.3707)
Diemand J., Kuhlen M., Madau P., 2007, ApJ, 667, 859
Di Matteo T., Springel V., Hernquist L., Energy input from quasars regulates the
growth and activity of black holes and their host galaxies, 2005, Nat, 433, 604
Di Matteo T., Colberg J., Springel V., Hernquist L., Sijacki D., Direct cosmological
simulations of the growth of black holes and galaxies, 2008, ApJ, 676, 33
Dodelson S. 2003, Modern cosmology. Amsterdam (Netherlands): Academic
Press. ISBN 0-12-219141-2 2003, XIII + 440 p.,
296
Doroshkevich, A. G., The space structure of perturbations and the origin of rotation
of galaxies in the theory of fluctuation, 1970, Astrofizika, 3, 175
Dubinski, J., Cosmological tidal shear, 1992, ApJ, 401, 441
Dunlop J., Cosmic star-formation history, as traced by radio source evolution, 1997,
preprint (astro-ph/9704294)
Dunlop J. S., McLure R. J., Kukula M. J., Baum S. A., O’Dea C. P., Hughes D. H.,
Quasars, their host galaxies and their central black holes, 2003, MNRAS, 340,
1095
Dvali G., Gabadadze G., Porrati M., 4D gravity on a brane in 5D Minkowski space,
2000, Phys. Let. B, 485, 208
Enoki M., Nagashima M., Gouda N., Relations between galaxy formation and the
environments of quasar, 2003, PASJ, 55, 133
Efstathiou G., Rees M. J., High-redshift quasars in the cold dark matter cosmogony,
1988, MNRAS, 230, 5P
Efstathiou G., Frenk C. S., White S. D. M., Davis M., Gravitational clustering from
scale-free initial conditions, 1988, MNRAS, 235, 715
Efstathiou G., Bond J. R., White S. D. M., Cosmic background anisotropies in cold
dark matter cosmology, 1992, MNRAS, 258, 1P
297
Eisenstein, D. J., & Hu, W., Power spectra for cold dark matter and its variants,
1999, ApJ, 511, 5
Eke V. R., Cole S., Frenk C. S., Cluster evolution as a diagnostic for Ω, 1996,
MNRAS, 282, 263
Enoki M., Nagashima M., Gouda N., Relations between galaxy formation and the
environments of quasars 2003, PASJ, 55, 133
Epstein R. I. Proto-galactic perturbations, 1983, MNRAS, 205, 207
Erickcek A. L., Kamionkowski M., Benson A. J., Supermassive black hole merger
rates: uncertainties from halo merger theory, 2006, MNRAS, 371, 1992
Escala A., Toward a comprehensive fueling-controlled theory of the growth of massive
black holes and host spheroids, 2007, ApJ, 671, 1264
Faber S. M., Worthey G., Gonzales J. J., Absorption-line spectra of elliptical galax-
ies and their relation to elliptical formation, 1992, The Stellar Populations of
Galaxies, 149, 255
Fabian A. C., The obscured growth of massive black holes, 1999, MNRAS, 308, L39
Fakhouri O., Ma C.-P., The nearly universal merger rate of dark matter haloes in
ΛCDM cosmology, 2008, MNRAS, 386, 577
Fakhouri O., & C.-P., Environmental dependence of dark matter halo growth I: Halo
merger rates, 2008, MNRAS, 394, 1825
298
Feoli A., Mele D., Is there a relationship between the mass of a Smbh and the kinetic
energy of its host elliptical galaxy?, 2005, IJMP-D, 14, 1861
Feoli A., Mele D., Improved tests on the relationship between the kinetic energy of
galaxies and the mass of their central black holes, 2007, IJMD-D, 16, 1261
Ferrarese L., Beyond the bulge: A fundamental relation between supermassive black
holes and dark matter halos, 2002, ApJ, 578, 90
Ferrarese L., Merritt D., A fundamental relation between supermassive black holes
and their host galaxies, 2000, ApJL, 539, L9
Ferrarese L., Ford H., Supermassive black holes in galactic nuclei: past, present and
future research, 2005, SSR, 116, 523
Flory P. J., Molecular size distribution in three dimensional polymers. I. Gelation,
1941, J. Am. Chem. Soc., 63, 3083
Flory P. J., Molecular size distribution in three dimensional polymers. II. Trifunc-
tional branching units, 1941, J. Am. Chem. Soc., 63, 3091
Flory P. J., Molecular size distribution in three dimensional polymers. III. Tetra-
functional branching units, 1941, J. Am. Chem. Soc., 63, 3096
Fontanot F., Monaco P., Cristiani S., Tozzi P., The effect of stellar feedback and
quasar winds on the AGN population, 2006, MNRAS, 373, 1173
299
Fontanot F., Monaco P., Silva L., Grazian A., Reproducing the assembly of massive
galaxies within the hierarchical cosmogony, 2007, MNRAS, 382, 903
Ford H. C., Tsvetanov Z. I., Ferrarese L., Jaffe W. 1998, HST detections of massive
black holes in the centers of galaxies, The Central Regions of the Galaxy and
Galaxies, 184, 377
Foreman G., Volonteri M., Dotti M., Double quasars: Probes of black hole scaling
relationships and merger scenarios, 2009, ApJ, 693, 1554
Francke H., et al., , Clustering of Intermediate-Luminosity X-Ray-Selected Active
Galactic Nuclei at z ∼ 3, 2008, ApJL, 673, L13
Franx M., van Dokkum P. G., Schreiber N. M. F., Wuyts S., Labbe I., Toft S.,
Structure and Star Formation in Galaxies out to z = 3: Evidence for Surface
Density Dependent Evolution and Upsizing, 2008, ApJ, 688, 770
Fry J. N., The evolution of bias, 1996, ApJL, 461, L65
Fujimoto M., Gravitational collapse of rotating gaseous ellipsoids, 1968, ApJ, 152,
523
Furlanetto S. R., Zaldarriaga M., Hernquist L., The growth of H II regions during
reionization, 2004, ApJ, 613, 1
Gammie C. F., Nonlinear outcome of gravitational instability in cooling, gaseous
disks 2001, ApJ, 553, 174
300
Gao L., Loeb A., Peebles P. J. E., White S. D. M., Jenkins A., Early formation and
late merging of the giant galaxies, 2004, ApJ, 614, 17
Gao L., White S. D. M., Jenkins A., Stoehr F., Springel V., The subhalo population
of ΛCDM dark haloes, 2004, MNRAS, 355, 819
Gao L., White S. D. M., Jenkins A., Frenk C. S., Springel V., Early structure in
ΛCDM, 2005, MNRAS, 363, 379
Gao L., White S. D. M., Assembly bias in the clustering of dark matter haloes, 2007,
MNRAS, 377, L5
Gaskell C. M., An Improved [O III] Line Width to Stellar Velocity Dispersion Cali-
bration: Curvature, Scatter, and Lack of Evolution in the Black-Hole Mass Versus
Stellar Velocity Dispersion Relationship, 2009, preprint (astro-ph/0908.0328)
Gebhardt K., et al., A relationship between nuclear black hole mass and galaxy
velocity dispersion, 2000, ApJL, 539, L13
Gebhardt K., et al., The black hole mass and extreme orbital structure in NGC 1399
, 2007, ApJ, 671, 1321
Genel S., Genzel R., Bouche N., Naab T., Sternberg A., The Halo Merger Rate in the
Millennium Simulation and Implications for Observed Galaxy Merger Fractions,
2009, ApJ, 701, 2002
301
Genzel R., et al., What Powers Ultraluminous IRAS Galaxies?, 1998, ApJ, 498,
579
Georgakakis A., Gerke B. F., Nandra K., Laird E. S., Coil A. L., Cooper M. C.,
Newman J. A., X-ray selected AGN in groups at redshifts z ∼ 1, 2008, MNRAS,
391, 183
Georgakakis A., et al., The role of AGN in the colour transformation of galaxies at
redshifts z ∼ 1, 2008, MNRAS, 385, 2049
Gelb J. M., Bertschinger E.,Cold dark matter. 1: The formation of dark halos, 1994,
ApJ, 436, 467
Gilli R., Comastri A., Hasinger G., The synthesis of the cosmic X-ray background
in the Chandra and XMM-Newton era, 2007, A&A, 463, 79
Giocoli C., Moreno J., Sheth R. K., Tormen G., An improved model for the forma-
tion times of dark matter haloes 2007, MNRAS, 376, 977
Giocoli C., Pieri L., Tormen G., Analytical approach to subhalo populations in dark
matter haloes 2008, MNRAS, 387, 689
Giocoli C., Tormen G., van den Bosch F. C., The population of dark matter sub-
haloes: mass functions and average mass-loss rates, 2008, MNRAS, 468
Graham A. W., Erwin P., Caon N., Trujillo I., A correlation between galaxy light
concentration and supermassive black hole mass, 2001, ApJL, 563, L11
302
Graham A. W., Driver S. P., A log-quadratic relation for predicting supermassive
black hole masses from the host bulge Srsic index, 2007, ApJ, 655, 77
Graham A. W., The black hole mass - spheroid luminosity relation, 2007, MNRAS,
379, 711
Granato G. L., De Zotti G., Silva L., Bressan A., Danese L., A physical model for
the coevolution of QSOs and their spheroidal hosts, 2004, ApJ, 600, 580
Granato G. L., Silva L., Lapi A., Shankar F., De Zotti G., Danese L., The growth
of the nuclear black holes in submillimeter galaxies, 2006, MNRAS, 368, L72
Greene J. E., Barth A. J., Ho L. C. The smallest AGN host galaxies, 2006, NAR,
50, 739
Grimmett G. R., Stirzaker D. R., Probability and random processes, 2001, Oxford,
UK: Oxford University Press, ISBN 0-19-857223, XII + 596 pp.
Groeneboem P., Brownian motion with a parabolic drift and Airy functions, 1989,
Prob. Theory and Related Fields, 81, 79-109
Grossi M., Dolag K., Branchini E., Matarrese S., Moscardini L., Evolution of mas-
sive haloes in non-Gaussian scenarios, 2007, MNRAS, 382, 1261
Guainazzi M., Piconcelli E., Jimenez-Bailon E., Matt G., 2005, The early stage of
a cosmic collision? XMM-Newton unveils two obscured AGN in the galaxy pair
ESO509-IG066, A&A, 429, L9
303
Gultekin K., et al., The M-σ and M-L relations in galactic bulges, and determina-
tions of their intrinsic scatter, 2009, ApJ, 698, 198
Gunn J. E., Gott J. R. I., On the infall of matter into clusters of galaxies and some
effects on their evolution, 1972, ApJ, 176, 1
Gunn J. E., Feeding the monster: gas discs in elliptical galaxies, 1979, Active Galac-
tic Nuclei, 213
Guth A. H., Inflationary universe: a possible solution to the horizon and flatness
problems, 1981, PRD, 23, 347
Guth A. H., Pi S.-Y., Fluctuations in the new inflationary, 1982, PRL, 49, 1110
Haehnelt M. G., Rees M. J., The formation of nuclei in newly formed galaxies and
the evolution of the quasar population, 1993, MNRAS, 263, 168
Haehnelt M. G., Natarajan P., Rees M. J., High-redshift galaxies, their active nuclei
and central black holes, 1998, MNRAS, 300, 817
Haiman Z., Loeb A., Observational signatures of the first quasars, 1998, ApJ, 503,
505
Haiman Z., Menou K., On the cosmological evolution of the luminosity function and
the accretion rate of quasars, 2000, ApJ, 531, 42
Haiman Z., Hui L., Constraining the Lifetime of Quasars from Their Spatial Clus-
tering, 2001, ApJ, 547, 27
304
Haiman Z., Constraints from gravitational recoil on the growth of supermassive black
holes at high redshift, 2004, ApJ, 613, 36
Haring N., Rix H.-W., On the black hole mass-bulge mass relation, 2004, ApJL,
604, L89
Harker G., Cole S., Helly J., Frenk C., Jenkins A., A marked correlation function
analysis of halo formation times in the Millennium Simulation, 2006, MNRAS,
367, 1039
Harrison E. R., Fluctuations at the threshold of classical cosmology, 1970, PRD, 1,
2726
Hatziminaoglou E., Siemiginowska A., Elvis M., Accretion disk instabilities, cold
dark matter models, and their role in quasar evolution, 2001, ApJ, 547, 90
Hatziminaoglou E., Mathez G., Solanes J.-M., Manrique A., Salvador-Sole E., Major
mergers of haloes, the growth of massive black holes and the evolving luminosity
function of quasars, 2003, MNRAS, 343, 692
Hawkins E., et al., The 2dF Galaxy Redshift Survey: correlation functions, peculiar
velocities and the matter density of the Universe, 2003, MNRAS, 346, 78
Hazard C., Mackey M. B., Shimmins A. J., Investigation of the radio source 3C 273
by the method of lunar occultations, 1963, Nat, 197, 1037
305
Heath D. J., The growth of density perturbations in zero pressure Friedmann-
Lemaitre universes, 1977, MNRAS, 179, 351
Heckman T. M., Bothun G. D., Balick B., Smith E. P., Low-redshift quasars as
the active nuclei of cosmologically distant interacting galaxies - A spectroscopic
investigation, 1984, AJ, 89, 958
Heger A., Woosley S. E., The nucleosynthetic signature of population III, 2002, ApJ,
567, 532
Heinz S., Brueggen M., Morsony B., Prospects of high-resolution X-ray spectroscopy
for AGN feedback in galaxy clusters, 2009, preprint (astro-ph/0906.2787)
Hennawi J. F., et al., Binary quasars in the Sloan Digital Sky Survey: Evidence for
excess clustering on small scales, 2006, AJ, 131, 1
Hennawi J. F., et al., Binary Quasars at High Redshift I: 24 New Quasar Pairs at
z ∼ 3-4, 2009, arXiv:0908.3907
Hernquist L., Tidal triggering of starbursts and nuclear activity in galaxies, 1989,
Nat, 340, 687
Hernquist, L. 1992, Structure of merger remnants. I - Bulgeless progenitors, ApJ,
400, 460
Hernquist L., Structure of merger remnants. II - Progenitors with rotating bulges,
1993, ApJ, 409, 548
306
Hernquist L., Mihos J. C., Excitation of activity in galaxies by minor mergers, 1995,
ApJ, 448, 41
Hiotelis N., Del Popolo A., On the reliability of merger-trees and the mass-growth
histories of dark matter haloes, 2006, APSS, 301, 167
Holder G. P., McCarthy I. G., Babul A., The Sunyaev-Zeldovich background, 2007,
MNRAS, 382, 1697
Hoffman Y., The dynamics of superclusters - The effect of shear, 1986, ApJ, 308,
493
Hoffman Y., Angular momentum, hierarchical clustering, and local density maxima,
1988, ApJ, 329, 8
Ho L. C., AGNs and starbursts: What is the real connection?, 2005, preprint (astro-
ph/0511157)
Hopkins P. F., Hernquist L., Cox T. J., Di Matteo T., Martini P., Robertson B.,
Springel V., Black holes in galaxy mergers: Evolution of quasars, 2005, ApJ, 630,
705
Hopkins P. F., Hernquist L., Cox T. J., Di Matteo T., Robertson B., Springel V.,
A unified, merger-driven model of the origin of starbursts, quasars, the cosmic
X-ray background, supermassive black Holes, and galaxy spheroids, 2006, ApJs,
163, 1
307
Hopkins P. F., Hernquist L., Cox T. J., Robertson B., Di Matteo T., Springel V.,
The evolution in the faint-end slope of the quasar luminosity function, 2006, ApJ,
639, 700
Hopkins P. F., Robertson B., Krause E., Hernquist L., Cox T. J., An upper limit to
the degree of evolution between supermassive black holes and their host galaxies,
2006, ApJ, 652, 107
Hopkins P. F., Somerville R. S., Hernquist L., Cox T. J., Robertson B., Li Y., The
relation between quasar and merging galaxy luminosity functions and the merger-
driven star formation history of the universe, 2006, ApJ, 652, 864
Hopkins P. F., Hernquist L., Fueling low-level AGN activity through stochastic ac-
cretion of cold gas, 2006, APJS, 166, 1
Hopkins P. F., Hernquist L., Cox T. J., Keres D., A cosmological framework for
the co-evolution of quasars, supermassive black holes, and elliptical galaxies. I.
Galaxy mergers and quasar activity, 2008, ApJS, 175, 356
Hopkins P. F., Cox T. J., Keres D., Hernquist L., A cosmological rramework for
the co-evolution of quasars, supermassive black holes, and elliptical galaxies. II.
Formation of red ellipticals, 2008, ApJS, 175, 390
Hopkins P. F., Hernquist L., Quasars are not light-bulbs: Testing models of quasar
lifetimes with the observed Eddington ratio distribution, 2009, ApJ, 698, 1550
308
Hopkins P. F., Hernquist L., A Characteristic Division Between the Fueling of
Quasars and Seyferts: Five Simple Tests, 2009, ApJ, 694, 599
Hopkins P. F., et al. Mergers and bulge formation in Lambda-CDM: Which mergers
matter? , 2009, preprint (astro-ph/0906.5357)
Hosokawa T., Constraining the lifetime of quasars with the present-day mass func-
tion of supermassive black holes 2002, ApJ, 576, 75
Houghton R. C. W., Magorrian J., Sarzi M., Thatte N., Davies R. L., Krajnovic
D., The central kinematics of NGC 1399 measured with 14 pc resolution, 2006,
MNRAS, 367, 2
Hoyle F., Lyttleton R. A., The effect of interstellar matter on climatic variation,
1939, Mathematical Proceedings of the Cambridge Philosophical Society, 35, 405
Hui L., Bertschinger E., Local approximations to the gravitational collapse of cold
matter 1996, ApJ, 471, 1
Hutchings J. B., Johnson I., Pyke R., Optical images of quasars and radio galaxies,
1988, ApJS, 66, 361
Hutchings J. B., Host galaxies of z∼4.7 quasars, 2003, AJ, 125, 1053
Hutchings J. B., Maddox N., Cutri R. M., Nelson B. O., Host galaxies of 2MASS-
selected QSOs to redshift 0.3, 2003, AJ, 126, 63
309
Hutchings J. B., Cherniawsky A., Cutri R. M., Nelson B. O., Host galaxies of Two
Micron All Sky Survey-selected QSOs at redshift over 0.3, 2006, AJ, 131, 680
Icke V., Formation of galaxies inside clusters, 1973, AAP, 27, 1
Jahnke K., et al., Ultraviolet light from young Stars in GEMS quasar host galaxies
at 1.8¡z¡2.75, 2004, ApJ, 614, 568
Jahnke K., Kuhlbrodt B., Wisotzki L., Quasar host galaxy star formation activity
from multicolour data, 2004, MNRAS, 352, 399
Jahnke K., et al., Galaxies in COSMOS: Evolution of Black hole vs. bulge mass but
not vs. total stellar mass over the last 9 Gyrs? 2009, preprint (astro-ph/0907.5199)
Jedamzik K., The Cloud-in-Cloud Problem in the Press-Schechter Formalism of
Hierarchical Structure Formation, 1995, ApJ, 448, 1
Jenkins A., Frenk C. S., White S. D. M., Colberg J. M., Cole S., Evrard A. E.,
Couchman H. M. P., Yoshida N., The mass function of dark matter haloes, 2001,
MNRAS, 321, 372
Jesseit R., Naab T., Peletier R. F., Burkert A., 2D kinematics of simulated disc
merger remnants, 2007, MNRAS, 376, 997
Jing Y. P., Accurate Determination of the Lagrangian Bias for the Dark Matter
Halos, 1999, ApJL, 515, L45
310
Jing Y. P., Suto Y., Triaxial modeling of halo density profiles with high-resolution
N-Body simulations, 2002, ApJ, 574, 538
Jogee S., The fueling and evolution of AGN: Internal and external triggers, 2006,
Physics of Active Galactic Nuclei at all Scales, 693, 143
Jørgensen I., Franx M., Hjorth J., van Dokkum P. G., The evolution of cluster E
and S0 galaxies measured from the fundamental plane, 1999, MNRAS, 308, 833
Johansson P. H., Naab T., Burkert A., Equal- and unequal-mass mergers of disk
and elliptical galaxies with black holes, 2009, ApJ, 690, 802
Joseph R. D., Wright G. S., Recent star formation in interacting galaxies. II - Super
starburst in merging galaxies, 1985, MNRAS, 214, 87
Juneau S., et al., Cosmic star formation history and its dependence on galaxy stellar
mass, 2005, ApJL, 619, L135
Kaiser N., On the spatial correlations of Abell clusters, 1984, ApJL, 284, L9
Kamenshchik A., Moschella U., Pasquier V., An alternative to quintessence, 2001,
Phys. Let. B, 511, 265
Karlin S., Taylor H.M., 1975, A first course in stochastic processes, 2nd ed. London
Academic Press
311
Kauffmann G., Charlot S., Chemical enrichment and the origin of the colour-
magnitude relation of elliptical galaxies in a hierarchical merger model, 1998,
MNRAS, 294, 705
Kauffmann G., Haehnelt M., A unified model for the evolution of galaxies and
quasars, 2000, MNRAS, 311, 576
Kauffmann G., White S. D. M., The merging history of dark matter haloes in a
hierarchical universe, 1993, MNRAS, 261, 921
Kauffmann G., et al., The host galaxies of active galactic nuclei, 2003, MNRAS,
346, 1055
Kauffmann G., Heckman T. M., Feast and Famine: regulation of black hole growth
in low-redshift galaxies, 2009, MNRAS, 397, 135
Kawakatu N., Wada K., Coevolution of supermassive black holes and circumnuclear
disks, 2008, ApJ, 681, 73
Khalatyan A., Cattaneo A., Schramm M., Gottlober S., Steinmetz M., Wisotzki L.,
Is AGN feedback necessary to form red elliptical galaxies?, 2008, MNRAS, 387,
13
Kim M., Ho L. C., Peng C. Y., Barth A. J., Im M., Martini P., Nelson C. H.,
The Origin of the Intrinsic Scatter in the Relation Between Black Hole Mass and
Bulge Luminosity for Nearby Active Galaxies, 2008, ApJ, 687, 767
312
King A. R., The brightest black holes, 2002, MNRAS, 335, L13
King A., Black Holes, Galaxy Formation, and the MBH-sigma Relation, 2003, ApJL,
596, L27
Kitayama T., Suto Y., Formation rate of gravitational structures and the cosmic
X-ray background radiation, 1996, MNRAS, 280, 638
Kleinmann S. G., Hamilton D., Keel W. C., Wynn-Williams C. G., Eales S. A.,
Becklin E. E., Kuntz K. D., The properties and environment of the giant, infrared-
luminous galaxy IRAS 09104 + 4109, 1988, ApJ, 328, 161
Khochfar S., Burkert A., The importance of spheroidal and mixed mergers for early-
type galaxy formation, 2003, ApJL, 597, L117
Kollmeier J. A., et al., Black hole masses and Eddington ratios at 0.3 < z < 4,
2006, ApJ, 648, 128
Kormendy J., Richstone D., Inward bound—The search for supermassive black holes
In galactic nuclei, 1995, ARAA, 33, 581
Kormendy J., Gebhardt K. 2001, Supermassive black holes in galactic nuclei, 20th
Texas Symposium on relativistic astrophysics, 586, 363
Koushiappas S. M., Bullock J. S., Dekel A., Massive black hole seeds from low
angular momentum material, 2004, MNRAS, 354, 292
313
Kravtsov A. V., Berlind A. A., Wechsler R. H., Klypin A. A., Gottlober S., Allgood
B., Primack J. R., The dark side of the halo occupation distribution, 2004, ApJ,
609, 35
Krolik J. H., AGN Obscuring Tori Supported by Infrared Radiation Pressure, 2007,
ApJ, 661, 52
Kuntschner H., The stellar populations of early-type galaxies in the Fornax cluster,
2000, MNRAS, 315, 184
Kuntschner H., Lucey J. R., Smith R. J., Hudson M. J., Davies R. L., On the
dependence of spectroscopic indices of early-type galaxies on age, metallicity and
velocity dispersion, 2001, MNRAS, 323, 615
Lacey C., Cole S., Merger rates in hierarchical models of galaxy formation, 1993,
MNRAS, 262, 627
Lacey C., Cole S., Merger Rates in Hierarchical Models of Galaxy Formation - Part
Two - Comparison with N-Body Simulations 1994, MNRAS, 271, 676
Lagos C. D. P., Cora S. A., Padilla N. D., Effects of AGN feedback on LCDM
galaxies, 2008, MNRAS, 388, 587
Lahav O., Lilje P. B., Primack J. R., Rees M. J. Dynamical effects of the cosmological
constant, 1991, MNRAS, 251, 128
314
Lapi A., Shankar F., Mao J., Granato G. L., Silva L., De Zotti G., Danese L., Quasar
luminosity functions form joint evolution of black holes and host galaxies, 2006,
ApJ, 650, 42
Laszlo I., Bean R., Nonlinear growth in modified gravity theories of dark energy,
2008, PRD, 77, 024048
Lauer T. R., Tremaine S., Richstone D., Faber S. M., Selection Bias in Observing
the Cosmological Evolution of the M-σ and M-L relationships, 2007, ApJ, 670,
249
Lee J., Shandarin S. F., The cosmological mass distribution function in the Zeldovich
approximation, 1998, ApJ, 500, 14
Lemson G., Dynamical effects of the cosmological constant - the evolution of as-
pherical structures, 1993, MNRAS, 263, 913
Letawe G., Magain P., Courbin F., Jablonka P., Jahnke K., Meylan G., Wisotzki
L., On-axis spectroscopy of the host galaxies of 20 optically luminous quasars at
z 0.3, 2007, MNRAS, 378, 83
Levine R., Gnedin N. Y., Hamilton A. J. S., Kravtsov A. V., Resolving gas dynamics
in the circumnuclear region of a disk galaxy in a cosmological simulation, 2008,
ApJ, 678, 154
Li C., Jing Y. P., Kauffmann G., Borner G., White S. D. M., Cheng F. Z., The de-
315
pendence of the pairwise velocity dispersion on galaxy properties, 2006, MNRAS,
368, 37
Li C., Kauffmann G., Wang L., White S. D. M., Heckman T. M., Jing Y. P., The
clustering of narrow-line AGN in the local Universe, 2006, MNRAS, 373, 457
Li Y., Mo H. J., van den Bosch F. C., Lin W. P., On the assembly history of dark
matter haloes, 2007, MNRAS, 379, 689
Li Y., et al., Formation of z∼6 Quasars from hierarchical galaxy mergers, 2007,
ApJ, 665, 187
Lidz A., Hopkins P. F., Cox T. J., Hernquist L., Robertson B., The luminosity
dependence of quasar clustering, 2006, ApJ, 641, 41
Lightman A. P., Schechter P. L., The Ω dependence of peculiar velocities induced by
spherical density perturbations, 1990, ApJS, 74, 831
Lin C. C., Mestel L., Shu F. H., The gravitational collapse of a uniform spheroid,
1965, ApJ, 142, 1431
Lin W. P., Jing Y. P., Lin L., Formation time-distribution of dark matter haloes:
theories versus N-body simulations, 2003, MNRAS, 344, 1327
Lin L., et al., The redshift evolution of wet, dry, and mixed galaxy mergers from
close galaxy pairs in the DEEP2 galaxy redshift survey, 2008, ApJ, 681, 232
316
Lodato G., Rice W. K. M., Testing the locality of transport in self-gravitating ac-
cretion discs, 2004, MNRAS, 351, 630
Lodato G., Rice W. K. M., Testing the locality of transport in self-gravitating ac-
cretion discs - II. The massive disc case, 2005, MNRAS, 358, 1489
Lo Verde M., Miller A., Shandera S., Verde L., Effects of scale-dependent non-
Gaussianity on cosmological structures 2008, JCAPP, 4, 14
Lu B., The exact solution of the coagulation equation with kernel Kij = A(i+j)+B,
1987, JPA, 20, 2347
Lue A., Scoccimarro R., Starkman G. D., Probing Newton’s constant on vast scales:
Dvali-Gabadadze-Porrati gravity, cosmic acceleration, and large scale structure,
2004, PRD, 69, 124015
Lukic Z., Heitmann K., Habib S., Bashinsky S., Ricker P. M., The halo mass func-
tion: high-redshift evolution and universality, 2007, ApJ, 671, 1160
Lushnikov A. A., Some new aspects of coagulation theory, 1978, Izv. Atmos. Ocean.
Phys., 14, 738
Lushnikov A. A., Coagulation in finite systems, 1978, JCIS, 65, 276
Lynden-Bell D., On large-scale instabilities during gravitational collapse and the
evolution of shrinking Maclaurin spheroids, 1964, ApJ, 139, 1195
Lynden-Bell D., Galactic nuclei as collapsed old quasars, 1969, Nat, 223, 690
317
Lyth D. H., Rodrıguez Y., Inflationary prediction for primordial non-Gaussianity,
2005, Phys. Rev. Let., 95, 121302
Madau P., Rees M. J., Massive black holes as population III remnants, 2001, ApJL,
551, L27
Magdziarz P., Zdziarski A. A., Angle-dependent Compton reflection of X-rays and
gamma-rays, 1995, MNRAS, 273, 837
Magorrian J., et al., The demography of massive dark objects in galaxy centers,
1998, AJ, 115, 2285
Makler M., de Oliveira S. Q., Waga I., Constraints on the generalized Chaplygin gas
from supernovae observations, 2003, Phys. Let. B, 555, 1
Mahmood A., Devriendt J. E. G., Silk J., A simple model for the evolution of
supermassive black holes and the quasar population 2005, MNRAS, 359, 1363
Mahmood A., Rajesh R., Cosmological mass functions and moving barrier models,
2005, preprint (astro-ph/0502513)
Malbon R. K., Baugh C. M., Frenk C. S., Lacey C. G., Black hole growth in hier-
archical galaxy formation, 2007, MNRAS, 382, 1394
Maldacena J., Non-gaussian features of primordial fluctuations in single field infla-
tionary models, 2003, JHEP, 5, 13
318
Maller A. H., Katz N., Keres D., Dave R., Weinberg D. H., Galaxy merger statistics
and inferred bulge-to-disk ratios in cosmological SPH simulations, 2006, ApJ,
647, 763
Mannheim P. D., Conformal cosmology with no cosmological constant, 1990, GRG,
22, 289
Mao S., Mo H. J., White S. D. M., The evolution of galactic discs 1998, MNRAS,
297, L71
Marconi A., Hunt L. K.,The relation between black hole mass, bulge mass, and
near-infrared luminosity, 2003, ApJL, 589, L21
Marconi A., Risaliti G., Gilli R., Hunt L. K., Maiolino R., Salvati M., Local su-
permassive black holes, relics of active galactic nuclei and the X-ray background
2004, MNRAS, 351, 169
Martini P., QSO lifetimes, 2003,
Martini P. 2004, Coevolution of Black Holes and Galaxies, 169
Martini P., Why does low-luminosity AGN fueling remain an unsolved problem?,
2004, The Interplay Among Black Holes, Stars and ISM in Galactic Nuclei, 222,
235
Martini P., Sivakoff G. R., Mulchaey J. S., The evolution of active galactic nuclei
in clusters of galaxies to redshift 1.3, 2009, preprint (astro-ph/0906.1843)
319
Martino M. C., Stabenau H. F., Sheth R. K., Spherical collapse and modified gravity,
2008, preprint (astro-ph/0812.0200)
Martini P., Weinberg D. H., Quasar clustering and the lifetime of quasars, 2001,
ApJ, 547, 12
Marulli F., Crociani D., Volonteri M., Branchini E., Moscardini L., Modelling the
quasi-stellar object luminosity and spatial clustering at low redshifts, 2006, MN-
RAS, 368, 1269
Marulli F., Branchini E., Moscardini L., Volonteri M., Modelling active galactic
nuclei: ongoing problems for the faint-end of the luminosity function, 2007, MN-
RAS, 375, 649 http://adsabs.harvard.edu/abs/2007MNRAS.375..649M
Marulli F., Bonoli S., Branchini E., Moscardini L., Springel V., Modelling the cos-
mological co-evolution of supermassive black holes and galaxies - I. BH scaling
relations and the AGN luminosity function. 2008, MNRAS, 385, 1846
Marulli F., Bonoli S., Branchini E., Moscardini L., Springel V., Modelling the cos-
mological co-evolution of supermassive black holes and galaxies - I. BH scaling
relations and the AGN luminosity function, 2008, MNRAS, 385, 1846
Marulli F., Bonoli S., Branchini E., Gilli R., Moscardini L., Springel V., The spa-
tial distribution of X-ray selected AGN in the Chandra deep fields: a theoretical
perspective, 2009, MNRAS, 396, 1404
320
Masjedi M., Hogg D. W., Blanton M. R., The growth of luminous red galaxies by
merging, 2007, ApJ, 679, 260
Matarrese S., Coles P., Lucchin F., Moscardini L., Redshift evolution of clustering,
1997, MNRAS, 286, 115
Matteucci F., The chemical evolution of the Galaxy, 2001, Astrophysics and Space
Science Library, 253
Maulbetsch C., Avila-Reese V., Colın P., Gottlober S., Khalatyan A., Steinmetz
M., The dependence of the mass assembly history of cold dark matter halos on
environment, 2007, ApJ, 654, 53
Mayer L., Kazantzidis S., Escala A., Formation of nuclear disks and supermassive
black hole binaries in multi-scale hydrodynamical galaxy mergers, 2008, preprint
(astro-ph/0807.3329)
McLeod J. B., On an infinite set of non-linear differential equations, 1962, Q. J.
Math, 13, 119
McLeod J. B., On an infinite set of non-linear differential equations (II), 1962, Q.
J. Math, 13, 192
McLeod J. B., On a recurrence formula in differential equations, 1962, Q. J. Math,
13, 283
321
McLure R. J., Dunlop J. S., On the black hole-bulge mass relation in active and
inactive galaxies, 2002, MNRAS, 331, 795
McLure R. J., Dunlop J. S., The cosmological evolution of quasar black hole masses,
2004, MNRAS, 352, 1390
Menci N., Cavaliere A., Fontana A., Giallongo E., Poli F., Binary aggregations in
hierarchical galaxy formation: The evolution of the galaxy luminosity function,
2002, ApJ, 575, 18
Menci N., Fontana A., Giallongo E., Salimbeni S., Bimodal Color Distribution in
Hierarchical Galaxy Formation, 2005, ApJ, 632, 49
Menou K., Haiman Z., Narayanan V. K., The merger history of supermassive black
holes in galaxies, 2001, ApJ, 558, 535
Merloni A., The anti-hierarchical growth of supermassive black holes, 2004, MNRAS,
353, 1035
Mesinger A., Furlanetto S., Efficient simulations of early structure formation and
reionization, 2007, ApJ, 669, 663
Mihos J. C., Hernquist L., Ultraluminous starbursts in major mergers, 1994, ApJL,
431, L9
Milgrom M., A modification of the Newtonian dynamics as a possible alternative to
the hidden mass hypothesis, 1983, ApJ, 270, 365
322
Miller L., Percival W. J., Croom S. M., Babic A., The cosmological history of
accretion onto dark halos and supermassive black holes, 2006, AAP, 459, 43
Milosavljevic M., Merritt D., Formation of galactic nuclei, 2001, ApJ, 563, 34
Milosavljevic M., Merritt D., Long-term evolution of massive black hole binaries,
2003, ApJ, 596, 860
Milosavljevic M., Merritt D., Ho L. C., Contribution of stellar tidal disruptions to
the X-Ray luminosity function of active galaxies, 2006, ApJ, 652, 120
Minezaki T., Yoshii Y., Kobayashi Y., Enya K., Suganuma M., Tomita H., Aoki T.,
Peterson B. A., Inner Size of a Dust Torus in the Seyfert 1 Galaxy NGC 4151,
2004, ApJL, 600, L35
Miralda-Escude J., Kollmeier J. A., Star captures by quasar accretion disks: A
possible explanation of the M- relation, 2005, ApJ, 619, 30
Mo H. J., White S. D. M., An analytic model for the spatial clustering of dark
matter haloes, 1996, MNRAS, 282, 347
Mo H. J., Mao S., White, S. D. M., The formation of galactic discs, 1998, MNRAS,
295, 319
Mo H. J., Mao S., Galaxy formation in pre-processed dark haloes, 2004, MNRAS,
353, 829
323
Mo H. J., Yang X., van den Bosch F. C., Katz N., Pre-heating by pre-virialization
and its impact on galaxy formation, 2005, MNRAS, 363, 1155
Monaco P., The mass function of cosmic structures with nonspherical collapse, 1995,
ApJ, 447, 23
Monaco P., A Lagrangian dynamical theory for the mass function of cosmic struc-
tures – I. Dynamics, 1997, MNRAS, 287, 753
Monaco P., A Lagrangian dynamical theory for the mass function of cosmic struc-
tures – II. Statistics, 1997, MNRAS, 290, 439
Monaco P., Salucci P., Danese L., Joint cosmological formation of QSOs and bulge-
dominated galaxies, 2000, MNRAS, 311, 279
Monaco P., Theuns T., Taffoni G., Governato F., Quinn T., Stadel J., Predicting
the Number, Spatial Distribution, and Merging History of Dark Matter Halos,
2002, ApJ, 564, 8
Monaco P., Theuns T., Taffoni G., The pinocchio algorithm: pinpointing orbit-
crossing collapsed hierarchical objects in a linear density field, 2002, MNRAS,
331, 587
Monaco P., Theuns T., Taffoni G., Governato F., Quinn T., Stadel J. Predicting the
Number, Spatial Distribution, and Merging History of Dark Matter Halos, 2002,
ApJ, 564, 8
324
Monaco P., Fontanot F., Taffoni G., The MORGANA model for the rise of galaxies
and active nuclei, 2007, MNRAS, 375, 1189
Montroll E W, Simha R, Theory of depolymerization of long chain molecules, 1940,
J. Chem. Phys., 8, 721
Moreno J., Giocoli C., Sheth R. K., Merger history trees of dark matter haloes in
moving barrier models, 2008, MNRAS, 391, 1729
Moreno J., Giocoli C., Sheth R. K., Dark matter halo creation in moving barrier
models, 2007, MNRAS, 397, 299
Moster B. P., Somerville R. S., Maulbetsch C., van den Bosch F. C., Maccio’ A. V.,
Naab T., Oser L., Constraints on the relationship between stellar mass and halo
mass at low and high redshift, 2009, preprint (astro-ph/0903.4682)
Mountrichas G., Sawangwit U., Shanks T., Croom S. M., Schneider D. P., Myers
A. D., Pimbblet K. QSO-LRG two-point cross-correlation function and redshift-
space distortions, 2009, MNRAS, 394, 2050
Myers A. D., Brunner R. J., Richards G. T., Nichol R. C., Schneider D. P., Bahcall
N. A., Clustering Analyses of 300,000 Photometrically Classified Quasars. II. The
Excess on Very Small Scales, 2007, ApJ, 658, 99
Myers A. D., Richards G. T., Brunner R. J., Schneider D. P., Strand N. E., Hall
325
P. B., Blomquist J. A., York D. G., Quasar clustering at 25 h−1 kpc from a
complete sample of binaries, 2008, ApJ, 678, 635
Naab T., Burkert A., Hernquist L., On the formation of boxy and disky elliptical
galaxies, 1999, ApJL, 523, L133
Naab T., Burkert A., Statistical properties of collisionless equal- and unequal-mass
merger remnants of disk galaxies, 2003, ApJ, 597, 893
Naab T., Khochfar S., Burkert A., Properties of early-type, dry galaxy mergers and
the origin of massive elliptical galaxies, 2006, ApJL, 636, L81
Naab T., Trujillo I., Surface density profiles of collisionless disc merger remnants,
2006, MNRAS, 369, 625
Naab T., Jesseit R., Burkert A., The influence of gas on the structure of merger
remnants, 2006, MNRAS, 372, 839
Narayan R., Quataert E., Black hole accretion, 2005, Science, 307, 77
Natarajan P., De Lucia G., Springel V., Substructure in lensing clusters and simu-
lations, 2007, MNRAS, 376, 180
Natarajan P., Treister E., Is there an upper limit to black hole masses?, 2008,
preprint (astro-ph/0808.2813)
Navarro J. F., Steinmetz M. The effects of a photoionizing ultraviolet background
on the formation of disk galaxies, 1997, ApJ, 478, 13
326
Navarro J. F., Frenk C. S., White S. D. M., A universal density profile from hier-
archical clustering, 1997, ApJ, 490, 493
Nelan J. E., Smith R. J., Hudson M. J., Wegner G. A., Lucey J. R., Moore S. A. W.,
Quinney S. J., Suntzeff N. B., NOAO Fundamental Plane Survey. II. Age and
metallicity along the red sequence from line-strength data, 2005, ApJ, 632, 137
Neistein E., van den Bosch F. C., Dekel, A., Natural downsizing in hierarchical
galaxy formation, 2006, MNRAS, 372, 933
Neistein E., Dekel A., Construction merger trees that mimic N-body simulations,
2008, MNRAS, 383, 615
Neistein E., Dekel A., Merger rates of dark-matter haloes, 2008, MNRAS, 388, 1792
Netzer H., Trakhtenbrot B., Cosmic evolution of mass accretion rate and metallicity
in Active Galactic Nuclei, 2007, ApJ, 654, 754
Novak G. S., Faber S. M., Dekel A., On the correlations of massive black holes with
their host galaxies, 2006, ApJ, 637, 96
Nusser A., Sheth R. K., Mass growth and density profiles of dark matter haloes in
hierarchical clustering, 1999, MNRAS, 303, 685
Ohsuga K., Mori M., Nakamoto T., Mineshige S., Supercritical accretion flows
around black holes: Two-dimensional, radiation pressure-dominated disks with
photon trapping, 2005, ApJ, 628, 368
327
Ohsuga K., Mineshige S., Why is supercritical disk accretion feasible?, 2007, ApJ,
670, 1283
Okamoto T., Nemmen R. S., Bower R. G., The impact of radio feedback from ac-
tive galactic nuclei in cosmological simulations: formation of disc galaxies, 2008,
MNRAS, 385, 161
Omukai K., Nishi R., Formation of primordial protostars, 1998, ApJ, 508, 141
Osmer P. S., Evidence for a decrease in the space density of quasars at Z more than
about 3.5, 1982, ApJ, 253, 28
Padmanabhan N., White M., Norberg P., Porciani C., The real-space clustering of
luminous red galaxies around z¡0.6 quasars in the Sloan Digital Sky Survey, 2008,
preprint (astro-ph/0802.2105)
Pan J., Wang Y., Chen X., Teodoro L., Effects of correlation between merging steps
on the global halo formation, 2008, MNRAS, 389, 461
Parkinson H., Cole S., Helly J., Generating dark matter halo merger trees, 2008,
MNRAS, 383, 557
Peacock J. A., Cosmological Physics. pp. 704. ISBN 052141072X. Cambridge, UK:
Cambridge University Press, January 1999
Peacock J. A., Heavens A. F., Alternatives to the Press-Schechter cosmological mass
function, 1990, MNRAS, 243, 133
328
Peebles P. J. E., The large-scale structure of the universe. Princeton, N.J., Prince-
ton University Press, 1980. 435 p.
Pelupessy F. I., Di Matteo T., Ciardi B., How rapidly do supermassive black hole
‘seeds’ grow at early times?, 2007, ApJ, 665, 107
Peng C. Y., Impey C. D., Rix H.-W., Kochanek C. S., Keeton C. R., Falco E. E.,
Lehar J., McLeod B. A., Probing the coevolution of supermassive black holes and
galaxies using gravitationally lensed quasar hosts, 2006, ApJ, 649, 616
Percival W. J., Cosmological structure formation in a homogeneous dark energy
background, 2005, AAP, 443, 819
Percival W., Miller L., Cosmological evolution and hierarchical galaxy formation,
1999, MNRAS, 309, 823
Percival W. J., Miller L., Peacock, J. A., An analytic model for the epoch of halo
creation, 2000, MNRAS, 318, 273
Peterson B. M., An introduction to active galactic nuclei. pp. 254. ISBN
0521479118. Cambridge, UK: Cambridge University Press, 1997
Phinney E. S., 1983, Ph.D. Thesis,
Pillepich A., Porciani C., Hahn O., Universal halo mass function and scale-
dependent bias from N-body simulations with non-Gaussian initial conditions,
2008, preprint (astro-ph/0811.4176)
329
Pitman J., Coalescent random forests, Berk. Stat. Tech. Rep., 457
Pizzella A., Corsini E. M., Dalla Bonta E., Sarzi M., Coccato L., Bertola F., On
the relation between circular velocity and central velocity dispersion in high and
low surface brightness galaxies, 2005, ApJ, 631, 785
Plionis M., Rovilos M., Basilakos S., Georgantopoulos I., Bauer F., Luminosity-
dependent X-Ray Active Galactic Nucleus Clustering? 2008, ApJL, 674, L5
Porciani C., Norberg P., Luminosity- and redshift-dependent quasar clustering, 2006,
MNRAS, 371, 1824
Press W. H., Schechter P., Formation of galaxies and clusters of galaxies by self-
similar gravitational condensation, 1974, ApJ, 187, 425
Quinlan G. D., The dynamical evolution of massive black hole binaries I. Hardening
in a fixed stellar background, 1996, New Astronomy, 1, 35
Raimundo S. I., Fabian A. C., Eddington ratio and accretion efficiency in active
galactic nuclei evolution, 2009, MNRAS, 396, 1217
Reed D. S., Bower R., Frenk C. S., Jenkins A., Theuns T., The halo mass function
from the dark ages through the present day, 2007, MNRAS, 374, 2
Rees M. J., Black hole models for active galactic nuclei, 1984, ARAA, 22, 471
Rice W. K. M., Lodato G., Armitage P. J., Investigating fragmentation conditions
in self-gravitating accretion discs, 2005, MNRAS, 364, L56
330
Richards G. T., et al., The Sloan Digital Sky Survey quasar survey: Quasar lumi-
nosity function from data release 3, 2006, AJ, 131, 2766
Richstone D., et al.. Supermassive black holes and the evolution of galaxies, 1998,
Nat, 395, A14
Ripamonti E., Haardt F., Ferrara A., Colpi M., Radiation from the first forming
stars, 2002, MNRAS, 334, 401
Rhook K. J., Haehnelt M. G., Probing the growth of supermassive black holes at z
> 6 with LOFAR, 2006, MNRAS, 373, 623
Riordan J, 1979, Combinatorial Identities. John Wiley & Sons, Inc., New York
Robertson B., Hernquist L., Cox T. J., Di Matteo T., Hopkins P. F., Martini P.,
Springel V., The evolution of the MBH-σ relation, 2006, ApJ, 641, 90
Roos N., Galaxy mergers and active galactic nuclei, 1981, AAP, 104, 218
Roos N, Galaxy mergers and active nuclei - Part Two - Cosmological evolution,
1985, ApJ, 294, 486
Roos N., Galaxy mergers and active nuclei. I - The luminosity function. II - Cos-
mological evolution, 1985, ApJ, 294, 479
Sadeh S., Rephaeli Y., Silk J., Cluster abundances and Sunyaev-Zel’dovich power
spectra: effects of non-Gaussianity and early dark energy, 2007, MNRAS, 380,
637
331
Safronov V. S., A restricted solution of the accretion equation, 1963, Soviet Physics
– Doklay, 7, 11
Sajina A., Yan L., Armus L., Choi P., Fadda D., Helou G., Spoon H., Spitzer mid-
infrared spectroscopy of infrared luminous galaxies at z∼2 II: Diagnostics, 2007,
ApJ accepted, pre-print (astro-ph/0704.1765)
Salpeter E. E., Accretion of interstellar matter by massive objects, 1964, ApJ, 140,
796
Salucci P., Szuszkiewicz E., Monaco P., Danese L., Mass function of dormant black
holes and the evolution of active galactic nuclei 1999, MNRAS, 307, 637
Salviander S., Shields G. A., Gebhardt K., Bonning E. W., The black hole mass
galaxy bulge relationship for QSOs in the SDSS DR3, 2006, New Astronomy
Review, 50, 803
Sanchez S. F., et al., Colors of active galactic nucleus host galaxies at 0.5<z<1.1
from the GEMS survey, 2004, ApJ, 614, 586
Sandage A., Visvanathan N., The color-absolute magnitude relation for E and S0
galaxies. II - New colors, magnitudes, and types for 405 galaxies, 1978, ApJ, 223,
707
Sanders D. B., Soifer B. T., Elias J. H., Madore B. F., Matthews K., Neugebauer
332
G., Scoville N. Z., Ultraluminous infrared galaxies and the origin of quasars, 1988,
ApJ, 325, 74
Sanders D. B., Soifer B. T., Elias J. H., Neugebauer G., Matthews K., Warm ul-
traluminous galaxies in the IRAS survey - The transition from galaxy to quasar?,
1988, ApJL, 328, L35
Sanders D. B., Mirabel I. F., Luminous infrared galaxies, 1996, ARA&A, 34, 749
Sandvik H. B., Moller O., Lee J., White S. D. M., Why does the clustering of haloes
depend on their formation history?, 2007, MNRAS, 377, 234
Saracco P., Longhetti M., Andreon S., The population of early-type galaxies at 1 <
z < 2 - new clues on their formation and evolution, 2009, MNRAS, 392, 718
Sargent A. I., Sanders D. B., Scoville N. Z., Soifer B. T., Compact molecular gas
structure in the interacting galaxy pair ARP 299 (IC 694-NGC 3690), 1987,
ApJL, 312, L35
Sargent A. I., Sanders D. B., Phillips T. G., CO(2-1) emission from the interacting
galaxy pair NGC 3256, 1989, ApJL, 346, L9
Sasaki S., Formation rate of bound objects in the hierarchical clustering model, 1994,
PASJ, 46, 427
Saslaw W. C., Valtonen M. J., Aarseth S. J., The gravitational slingshot and the
structure of extragalactic radio sources, 1974, ApJ, 190, 253
333
Scannapieco E., Oh S. P., Quasar feedback: The missing link in structure formation,
2004, ApJ, 608, 62
Scannapieco E., Bruggen M., Subgrid modeling of AGN-driven turbulence in galaxy
clusters, 2008, ApJ, 686, 927
Schafer B. M., Koyama K., Spherical collapse in modified gravity with the Birkhoff
theorem, 2008, MNRAS, 385, 411
Schawinski K., Thomas D., Sarzi M., Maraston C., Kaviraj S., Joo S.-J., Yi S. K.,
Silk J., 2007, MNRAS, 382, 1415
Scherrer R. J., Sen A. A., Thawing quintessence with a nearly flat potential, 2008,
PRD, 77, 083515
Schmidt M., 3C 273 : A star-like object with large red-shift , 1963, Nat, 197, 1040
Schmidt M., Green, R. F., 1983, Quasar evolution derived from the Palomar bright
quasar survey and other complete quasar surveys, ApJ, 269, 352
Schmidt M., Schneider D. P., Gunn J. E., Spectroscopic CCD surveys for quasars
at large redshift.IV.Evolution of the luminosity function from quasars detected by
their Lyman-alpha emission, 1995, AJ, 110, 68
Schweizer F., Mergers, 1992, Physics of Nearby Galaxies: Nature or Nurture?, 283
Schroedinger E., Zur theory der fall un steiguersuch an teilchen mit Brownschen
bewegung, 1915, Phys. Z. 16, 289-295
334
Scott, W. T., 1968, J. Atmos. Sci., 25, 54
Scoville N. Z., Sanders D. B., Sargent A. I., Soifer B. T., Scott S. L., Lo K. Y.,
Millimeter interferometry of the molecular gas in ARP 20, 1986, ApJL, 311, L47
Seery D., Lidsey J. E., Primordial non-Gaussianities in single-field inflation, 2005,
JCA-P Phys., 6, 3
Seljak U., Zaldarriaga M., A line-of-sight integration approach to the cosmic mi-
crowave background anisotropies, 1996, ApJ, 469, 437
Seljak U., Constraints on galaxy halo profiles from galaxy-galaxy lensing and Tully-
Fisher/Fundamental Plane relations, 2002, MNRAS, 334, 797
Sen A. A., Scherrer R. J., Generalizing the generalized Chaplygin gas, 2005, PRD,
72, 063511
Shakura N. I., Sunyaev R. A., Black holes in binary systems. Observational appear-
ance, 1973, A&A, 24, 337
Shankar F., Salucci P., Granato G. L., De Zotti G., Danese L., Supermassive black
hole demography: the match between the local and accreted mass functions, 2004,
MNRAS, 354, 1020
Shankar F., Lapi A., Salucci P., De Zotti G., Danese L., New relationships between
galaxy properties and host alo mass, and the role of feedbacks in galaxy formation,
2006, ApJ, 643, 14
335
Shankar F., Mathur S., On the Faint End of the High-Redshift Active Galactic
Nucleus Luminosity Function, 2007, ApJ, 660, 1051
Shankar F., Crocce M., Miralda-Escude’ J., Fosalba P., Weinberg D. H., On the
radiative efficiencies, Eddington ratios, and duty cycles of luminous high-redshift
quasars, 2008, ApJ submitted, preprint (astro-ph/0810.4919)
Shankar F., Ferrarese L., 2009, ApJ, submitted
Shankar F., Dai X., Sivakoff G. R., Dependence of the Broad Absorption Line Quasar
Fraction on Radio Luminosity, 2008, Apj, 687, 859
Shankar F., Weinberg D. H., Miralda-Escude’ J., Self-consistent models of the AGN
and black hole populations: Duty cycles, accretion rates, and the mean radiative
efficiency, 2007, ApJ, 690, 20
Shankar F., The Demography of Super-Massive Black Holes: Growing Monsters at
the Heart of Galaxies, 2009, preprint (astro-ph/0907.5213)
Shankar F., Bernardi M., Haiman Z., The evolution of the M BH-Sigma relation
inferred from the age distribution of local early-type galaxies and active galactic
nuclei rvolution, 2009, ApJ, 694, 867
Shankar F., Crocce M., Miralda-Escude’ J., Fosalba P., Weinberg D. H., 2008,
preprint (astro-ph/0810.4919)
Shankar F., Ferrarese L., 2009, in preparation
336
Shankar F., The Demography of Super-Massive Black Holes: Growing Monsters at
the Heart of Galaxies, 2009, preprint (astro-ph/0907.5213)
Shapiro S. L., Spin, accretion, and the cosmological growth of supermassive black
holes, 2005, ApJ, 620, 59
Shaver P. A., Wall J. V., Kellermann K. I., Jackson C. A., Hawkins M. R. S.
Decrease in the space density of quasars at high redshift, 1996, Nat, 384, 439
Shen J., Abel T., Mo H. J., Sheth R. K., An excursion set model of the cosmic Web:
The abundance of sheets, filaments, and halos, 2006, ApJ, 645, 783
Shen Y., et al., Clustering of High-Redshift (z > 2.9) Quasars from the Sloan Digital
Sky Survey, 2007, AJ, 133, 2222
Shen Y., et al., Quasar Clustering from SDSS DR5: Dependences on Physical Prop-
erties, 2009, ApJ, 697, 1656
Shen Y., Supermassive black holes in the hierarchical universe: A general framework
and observational tests, 2009, preprint (astro-ph/0903.4492)
Shen,Y., et al. Binary Quasars at High Redshift II: Sub-Mpc Clustering at z 3-4,
2009, preprint (astro-ph/0908.3908)
Sheth R. K., Merging and hierarchical clustering from an initially Poisson distribu-
tion, 1995, MNRAS, 276, 796
337
Sheth R. K., Galton-Watson branching processes and the growth of gravitational
clustering, 1996, MNRAS, 281, 1277
Sheth R. K., An excursion set model for the distribution of dark matter and dark
matter haloes, 1998, MNRAS, 300, 1057
Sheth R. K., Substructure in dark matter haloes: towards a model of the abundance
and spatial distribution of subclumps, 2003, MNRAS, 345, 1200
Sheth, R. K., et al., The Velocity Dispersion Function of Early-Type Galaxies, 2003,
ApJ, 594, 225
Sheth R. K., Lemson G., The forest of merger history trees associated with the
formation of dark matter haloes, 1999, MNRAS, 305, 946
Sheth R. K., Moreno J., Models of reversible coagulation and fragmentation, 2009,
J. Phys. A., submitted
Sheth R. K., Pitman J., Coagulation and branching process models of gravitational
clustering 1997, MNRAS, 289, 66
Sheth R. K., Tormen G., Large-scale bias and the peak background split, 1999, MN-
RAS, 308, 119
Sheth R. K., Tormen G., An excursion set model of hierarchical clustering: ellip-
soidal collapse and the moving barrier, 2002, MNRAS, 329, 61
338
Sheth R. K., Tormen G., Formation times and masses of dark matter haloes, 2004,
MNRAS, 349, 1464
Sheth R. K., van de Weygaert R., A hierarchy of voids: much ado about nothing,
2004, MNRAS, 350, 517
Sheth R. K., Mo H. J., Tormen G., Ellipsoidal collapse and an improved model for
the number and spatial distribution of dark matter haloes, 2001, MNRAS, 323, 1
Shi J., Krolik J. H., Radiation Pressure-supported Active Galactic Nucleus Tori with
Hard X-Ray and Stellar Heating, 2008, ApJ, 679, 1018
Shields G. A., Gebhardt K., Salviander S., Wills B. J., Xie B., Brotherton M. S.,
Yuan J., Dietrich M., The black hole-bulge relationship in quasars, 2003, ApJ,
583, 124
Shields G. A., Menezes K. L., Massart C. A., Vanden Bout P., The black hole-bulge
relationship for QSOs at high redshift, 2006, ApJ, 641, 683
Shirata A., Shiromizu T., Yoshida N., Suto Y., Galaxy clustering constraints on
deviations from Newtonian gravity at cosmological scales, 2005, PRD, 71, 064030
Sijacki D., Springel V., di Matteo T., Hernquist L., A unified model for AGN feed-
back in cosmological simulations of structure formation, 2007, MNRAS, 380, 877
Sijacki D., Springel V., Haehnelt M. G., Growing the first bright quasars in cosmo-
logical simulations of structure formation, 2009, MNRAS, 400, 100
339
Silk J., Rees M. J., Quasars and galaxy formation, 1998, AAP, 331, L1
Silk J., White S. D., The development of structure in the expanding universe, 1978,
ApJL, 223, L59
Skibba R. A., Sheth R. K., Martino M. C., Satellite luminosities in galaxy groups,
2007, MNRAS, 382, 1940
Silveira V., Waga I., Decaying Λ cosmologies and power spectrum, 1994, PRD, 50,
4890
Small T. A., Blandford R. D., Quasar evolution and the growth of black holes, 1992,
MNRAS, 259, 725
Smith R. E., Hernandez-Monteagudo C., Seljak U., Impact of scale dependent bias
and nonlinear structure growth on the integrated Sachs-Wolfe effect: Angular
power spectra, 2009, preprint (astro-ph/0905.2408)
von Smoluchowski M., Drie vortrage uber diffusion, Brownsche molekularbewegung
und koagulation von kolloidteilchen, 1916, Phys. Z. 17, 577-571, 585-599
von Smoluchowski M., Versuch einer mathematischen theorie der koagulationsk-
inetik kolloider lusungen, 1917, Z. Phys. Chem. 92, 129-168
Soifer B. T., et. al., The IRAS bright galaxy sample. II - The sample and luminosity
function, 1987, AJ, 320, 238
Soltan A., Masses of quasars, 1982, MNRAS, 200, 115
340
Somerville R. S., Kolatt T. S., How to plant a merger tree, 1999, MNRAS, 305, 1
Somerville R. S., Hopkins P. F., Cox T. J., Robertson B. E., Hernquist L., A semi-
analytic model for the co-evolution of galaxies, black holes and Active Galactic
Nuclei, 2008, MNRAS, 391, 481
Somerville R. S., Empirical Constraints on the Evolution of the Relationship be-
tween Black Hole and Galaxy Mass: Scatter Matters, 2009, preprint (astro-
ph/0908.0927)
Spergel D. N., et al., Three-year Wilkinson Microwave Anisotropy Probe (WMAP)
observations: Implications for cosmology, 2007, ApJS, 170, 377
Spouge J. L., Solutions and critical times for the polydisperse coagulation equation
when α(x, y) = A + B(x + y) + Cxy, 1983, J. Phys. A. 16, 3127
Springel V., Hernquist L., Formation of a spiral galaxy in a major merger, 2005,
ApJl, 622, L9
Springel V., et al., Formation of a spiral galaxy in a major merger, 2005, Nat, 435,
629
Springel V., Di Matteo T., Hernquist L., Modelling feedback from stars and black
holes in galaxy mergers, 2005, MNRAS, 361, 776
Steffen A. T., Barger A. J., Cowie L. L., Mushotzky R. F., Yang Y., The changing
active galactic nucleus population, 2003, ApJL, 596, L23
341
Stewart K. R., Bullock J. S., Wechsler R. H., Maller A. H., Zentner A. R., Merger
histories of galaxy haloes and implications for disk survival, 2007, ApJ, 683, 597
Stewart K. R., Bullock J. S., Wechsler R. H., Maller A. H., Gas-rich mergers in
LCDM: Disk survivability and the baryonic assembly of galaxies, 2009, preprint
(astro-ph/0901.4336)
Stockmayer, W. H., 1943, J. Chem. Phys, 11, 45
Stockton A., Compact companions to QSOs, 1982, ApJ, 257, 33
Stockton A., Canalizo G., Close L. M., PG 1700+518 revisited: Adaptive-optics
imaging and a revised starburst age for the companion, 1998, ApJL, 500, L121
Sugiyama N., Cosmic background anisotropies in cold dark matter cosmology, 1995,
ApJS, 100, 281
Tacconi L. J., et al., Submillimeter Galaxies at z ∼ 2: Evidence for Major Mergers
and Constraints on Lifetimes, IMF, and CO-H2 Conversion Factor, 2008, ApJ,
680, 246
Taffoni G., Monaco P., Theuns T., PINOCCHIO and the hierarchical build-up of
dark matter haloes, 2002, MNRAS, 333, 623
Tanaka T., Haiman Z., The assembly of supermassive black holes at high redshifts
2009, ApJ, 696, 1798
342
Taruya A., Suto Y., Nonlinear stochastic biasing from the formation epoch distri-
bution of dark haloes, 2000, ApJ, 542, 559
Taylor J. E., Babul A., The evolution of substructure in galaxy, group and cluster
haloes - I. Basic dynamics, 2004, MNRAS, 348, 811
Tegmark M., Peebles P. J. E., The Time Evolution of Bias, 1998, ApJL, 500, L79
Terlevich A. I., Forbes D. A., A catalogue and analysis of local galaxy ages and
metallicities, 2002, MNRAS, 330, 547
Thacker R. J., Scannapieco E., Couchman H. M. P., Quasars: What turns them
off?, 2006, ApJ, 653, 86
Thomas D., Maraston C., Bender R., Mendes de Oliveira C., The epochs of early-
type galaxy formation as a function of environment, 2005, ApJ, 621, 673
Tinker J. L, Kravtsov A. V, Klypin A., Abazajian K., Warren M. S, Yepes G.,
Gottlober S., Holz D. E, Toward a halo mass function for precision cosmology:
the limits of universality, 2008, preprint (astro-ph/0803.2706)
Toomre A., Toomre J., Galactic bridges and tails, 1972, ApJ, 178, 623
Toomre A., Mergers and some consequences, 1977, Evolution of Galaxies and Stellar
Populations, 401
Tormen G., The assembly of matter in galaxy clusters, 1998, MNRAS, 297, 648
343
Trager S. C., Faber S. M., Worthey G., Gonzalez J. J., The stellar population
histories of local early-type galaxies. I. Population parameters, 2000, AJ, 119,
1645
Tran H. D., Brotherton M. S., Stanford S. A., van Breugel W., Dey A., Stern D.,
Antonucci R., A polarimetric search for hidden quasars in three radio-selected
ultraluminous infrared galaxies, 1999, ApJ, 516, 85
Treat R. P., An exact solution of the discrete Smoluchowski equation and its corre-
spondence to the solution of the continuous equation, 1990, JPA, 23, 3003
Tremaine S., et al., The slope of the black hole mass versus velocity dispersion
correlation, 2002, ApJ, 574, 740
Treu T., Ellis R. S., Liao T. X., van Dokkum P. G., Keck spectroscopy of distant
GOODS spheroidal galaxies: Downsizing in a hierarchical universe, 2005, ApJL,
622, L5
Trubnikov B. A., Solution of the coagulation equations in the case of a bilinear
coefficient of adhesion of particles, 1971, Soviet Physics - Doklady, 16:2
Trujillo I., et al., Extremely compact massive galaxies at z 1.4, 2006, MNRAS,
373, L36
Trujillo I., Conselice C. J., Bundy K., Cooper M. C., Eisenhardt P., Ellis R. S.,
344
Strong size evolution of the most massive galaxies since z 2 2007, MNRAS, 382,
109
Tundo E., Bernardi M., Hyde J. B., Sheth R. K., Pizzella A., On the inconsistency
between the black hole mass function inferred from M•−σ and M•−L correlations
2007, ApJ, 663, 53
Ueda Y., Akiyama M., Ohta K., Miyaji T., Cosmological evolution of the hard X-
ray active galactic nucleus luminosity function and the origin of the hard X-ray
background, 2003, ApJ, 598, 886
Urrutia T., Lacy M., Becker R. H., Evidence for quasar activity triggered by galaxy
mergers in HST observations of dust-reddened quasars, 2008, ApJ, 674, 80
Vale A., Ostriker J. P., Linking halo mass to galaxy luminosity, 2004, MNRAS, 353,
189
Vale A., Ostriker J. P., The non-parametric model for linking galaxy luminosity with
halo/subhalo mass, 2006, MNRAS, 371, 1173
Vanden Berk D. E., et al., Spectral decomposition of broad-line AGNs and host
galaxies, 2006, AJ, 131, 84
van den Bosch F. C., The universal mass accretion history of cold dark matter
haloes, 2002, MNRAS, 331, 98
345
van den Bosch F. C., Tormen G., Giocoli C., The mass function and average mass-
loss rate of dark matter subhaloes, 2005, MNRAS, 359, 1029
van der Marel R. P., Relics of nuclear activity: Do all galaxies have massive black
holes? 1999, Galaxy Interactions at Low and High Redshift, 186, 333
van der Marel R. P. The black hole mass distribution in early-type galaxies: cusps
in Hubble Space Telescope photometry interpreted through adiabatic black hole
growth, 1999, AJ, 117, 744
van der Wel A., Bell E. F., van den Bosch F. C., Gallazzi A., Rix H.-W., On the
Size and Comoving Mass Density Evolution of Early-Type Galaxies, 2009, ApJ,
698, 1232
van Dokkum P. G., et al., Confirmation of the remarkable compactness of mas-
sive quiescent galaxies at z ∼ 2.3: Early-type galaxies did not form in a simple
monolithic collapse, 2008, ApJL, 677, L5
van Dokkum P. G., Kriek M., Franx M., A high stellar velocity dispersion for a
compact massive galaxy at redshift z = 2.186, 2009, Nat, 460, 717
van Dongen P. G. J., Fluctuations in coagulating systems II, 1987, J. Stat. Phys.,
49, 927
Vass I. M., Valluri M., Kravtsov A. V., Kazantzidis S., Evolution of the dark matter
phase-space density distributions of LCDM halos, 2009, MNRAS, 395, 1225
346
Verde L., Kamionkowski M., Mohr J. J., Benson A. J. On Galaxy-Cluster sizes and
temperatures, 2001, MNRAS, 321, L7
Veilleux S., Kim D.-C., Sanders D. B., Optical and near-infrared imaging of the
IRAS 1 Jy sample of ultraluminous infrared galaxies. II. The analysis, 2002,
ApJS, 143, 315
Vittorini V., Shankar F., Cavaliere A., The impact of energy feedback on quasar
evolution and black hole demographics, 2005, MNRAS, 363, 1376
Volonteri M., Haardt F., Madau P., The assembly and merging history of super-
massive black holes in hierarchical models of galaxy formation, 2003, ApJ, 582,
559
Volonteri M., Rees M. J., Rapid growth of high-redshift black holes, 2005, ApJ, 633,
624
Volonteri M., Salvaterra R., Haardt F., Constraints on the accretion history of
massive black holes from faint X-ray counts, 2006, MNRAS, 373, 121
Volonteri M., Haardt F., Gultekin K., Compact massive objects in Virgo galaxies:
the black hole population, 2008, MNRAS, 384, 1387
Volonteri M., Miller J. M., Dotti M., Supermassive binary black holes: con-
straints and expectations from the Sloan Digital Sky Survey, 2009, preprint (astro-
ph/0903.3947)
347
Wake D. A., Sheth R. K., Nichol R. C., Lam T. Y., et al., The 2dF-SDSS LRG and
QSO Survey: evolution of the clustering of luminous red galaxies since z = 0.6,
2008, MNRAS, 387, 1045
Walter F., Carilli C., Bertoldi F., Menten K., Cox P., Lo K. Y., Fan X., Strauss
M. A., Resolved Molecular Gas in a Quasar Host Galaxy at Redshift z=6.42, 2004,
ApJL, 615, L17
Wang L., Steinhardt P. J., Cluster abundance constraints for cosmological mod-
els with a time-varying, spatially inhomogeneous energy component with negative
pressure, 1998, ApJ, 508, 483
Wang T. G., Dong X. B., Zhou H. Y., Wang J. X., Strong Ca II absorption lines in
the reddened quasar SDSS J2339-0912: Evidence of the collision/merger in the
host galaxy?, 2005, ApJL, 622, L101
Wang L., Mao J., Xiang S., Yuan Y.-F., Negative feedback effects on star formation
history and cosmic reionization, 2009, preprint (astro-ph/0812.4085)
Warren S. J., Hewett P. C., Osmer P. S., A wide-field multicolor survey for high-
redshift quasars, Z greater than or equal to 2.2. 3: The luminosity function, 1994,
ApJ, 421, 412
Warren M. S., Abazajian K., Holz D. E., Teodoro L., Precision determination of
the mass function of dark matter haloes, 2006, ApJ, 646, 881
348
Wechsler R. H., Bullock J. S., Primack J. R., Kravtsov A. V., Dekel A., Concentra-
tions of dark haloes from their assembly histories, 2002, ApJ, 568, 52
Wechsler R. H., Zentner A. R., Bullock J. S., Kravtsov A. V., Allgood B., The
dependence of halo clustering on halo formation history, concentration, and oc-
cupation, 2006, ApJ, 652, 71
Weiss A., Peletier R. F., Matteucci F., Synthetic metal line indices for elliptical
galaxies from super metal-rich α-enhanced stellar models, 1995, AAP, 296, 73
Wetzel A. R., Cohn J. D., White M., Holz D. E., Warren M. S., The clustering of
massive halos, 2007, ApJ, 656, 139
Wetzel A. R., Cohn J. D., White M., Simulating subhaloes at high redshift: merger
rates, counts and types, 2009, MNRAS, 395, 1376
White S. D. M., Efstathiou G., Frenk C. S. The amplitude of mass fluctuations in
the universe, 1993, MNRAS, 262, 1023
White S. D. M., Frenk C. S., Galaxy formation through hierarchical clustering, 1991,
ApJ, 379, 52
White S. D. M., Rees M. J., Core condensation in heavy halos: a two-stage theory
for galaxy formation and clustering, 1978, MNRAS, 183, 341
White S. D. M., Silk J., The growth of aspherical structure in the universe - Is the
Local Supercluster an unusual system?, 1979, ApJ, 231, 1
349
White M., Martini P., Cohn J. D., Constraints on the correlation between QSO lu-
minosity and host halo mass from high-redshift quasar clustering, 2008, MNRAS,
390, 1179
Williams R. J., Quadri R. F., Franx M., van Dokkum P., Toft S., Kriek M., Labbe
I., The evolving relations between size, mass, surface density, and star formation
in 3× 104 galaxies since z = 2, 2009, preprint (astro-ph/0906.4786)
Willott C. J., et al., Dust and Gas Obscuration in ELAIS Deep X-Ray Survey
Reddened Quasars, 2004, ApJ, 610, 140
Woods D. F., Geller M. J., Barton E. J., Tidally triggered star formation in close
pairs of galaxies: Major and minor interactions, 2006, AJ, 132, 197
Woo J.-H., Treu T., Malkan M. A., Blandford R. D., Cosmic evolution of black
holes and spheroids. I. The MBH-sigma relation at z = 0.36 , 2006, ApJ, 645,
900
Woo J.-H., Treu T., Malkan M. A., Blandford R. D., Cosmic Evolution of Black
Holes and Spheroids. III. The MBH-sigma Relation in the Last Six Billion Years,
2008, Apj, 681, 925
Worthey G., Faber S. M., Gonzalez J. J., MG and Fe absorption features in elliptical
galaxies, 1992, ApJ, 398, 69
350
Wyithe J. S. B., Loeb A., A physical model for the luminosity function of high-
redshift quasars, 2002, ApJ, 581, 886
Wyithe J. S. B., Loeb A., Self-regulated growth of supermassive black holes in galax-
ies as the origin of the optical and X-ray luminosity functions of quasars, 2003,
ApJ, 595, 614
Wyithe J. S. B., Loeb A., Calibrating the galaxy halo-black hole relation based on
the clustering of quasars, 2005, ApJ, 621, 95
Wyithe S., Loeb A., Evidence for Merger-Driven Activity in the Clustering of High
Redshift Quasars, 2008, preprint (astro-ph/0810.3455)
Yan R., Madgwick D. S., White M., Constraining evolution in the halo model using
galaxy redshift surveys, 2003, ApJ, 598, 848
Yang X., Mo H. J., van den Bosch F. C., Constraining galaxy formation and cos-
mology with the conditional luminosity function of galaxies, 2003, MNRAS, 339,
1057
Yip C. W., et al., Spectral classification of quasars in the Sloan Digital Sky Survey:
Eigenspectra, redshift, and luminosity effects, 2004, AJ, 128, 2603
Yoo J., Miralda-Escude J., Formation of the black holes in the highest redshift
quasars, 2004, ApJL, 614, L25
351
Yoo J., Miralda-Escude J., Weinberg D. H., Zheng Z., Morgan C. W., The most
massive black holes in the universe: effets of mergers in massive galaxy clusters,
2007, ApJ, 667, 813
Yu Q., Tremaine S., Observational constraints on growth of massive black holes,
2002, MNRAS, 335, 965
Yu Q., Lu Y., Constraints on QSO models from a relation between the QSO lumi-
nosity function and the local black hole mass function, 2004, ApJ, 602, 603
Yu Q., Lu Y., The black hole mass versus velocity dispersion relation in
QSOs/Active Galactic Nuclei: Observational appearance and black hole growth,
2004, ApJ, 610, 93
Yu Q., Lu Y., Kauffmann G., Evolution of accretion disks around massive black
holes: Constraints from the demography of active galactic nuclei, 2005, ApJ,
634, 901
Yu Q., Lu Y., Toward precise constraints on the growth of massive black holes, 2008,
ApJ, 689, 732
Younger J. D., Hopkins P. F., Cox T. J., Hernquist L., The self-regulated growth of
supermassive black holes, 2008, ApJ, 686, 815
Zakamska N. L., et al., Type II quasars from the Sloan Digital Sky Survey. V.
Imaging host galaxies with the Hubble Space Telescope, 2006, AJ, 132, 1496
352
Zel’dovich Y., Novikov I. D., 1964, Soviet Phys. Dokl, 158, 811
Zeldovich Y. B., Gravitational instability: An approximate theory for large density
perturbations, 1970, AAP, 5, 84
Zeldovich Y. B., A hypothesis, unifying the structure and the entropy of the universe,
1972, MNRAS, 160, 1P
Zeldovich Y. B., Novikov I. D., The mass of quasi-stellar objects, 1964, Soviet
Physics – Doklady, 158, 811
Zentner A. R., Bullock J. S., Halo substructure and the power spectrum, 2003, ApJ,
598, 49
Zentner A. R., Berlind A. A., Bullock J. S., Kravtsov A. V., Wechsler R. H., The
physics of galaxy clustering. I. A model for subhalo populations, 2005, ApJ, 624,
505
Zentner A. R., The excursion set theory of halo mass functions, halo clustering, and
halo growth 2007, IJMP-D, 16, 763
Zhang J., Hui L., On random walks with a general moving barrier, 2006, ApJ, 641,
641
Zhang J., Ma C.-P., Fakhouri O., Conditional mass functions and merger rates of
dark matter haloes in the ellipsoidal collapse model, 2008, MNRAS, L54
353
Zhang J., Fakhouri O., Ma C.-P., How to grow a healthy merger tree, 2008, MNRAS,
389, 1521
Zhao D. H., Mo H. J., Jing Y. P., Borner G., The growth and structure of dark
matter haloes, 2003, MNRAS, 339, 12
Zheng Z., et al., Theoretical models of the halo occupation distribution: Separating
central and satellite galaxies, 2005, ApJ, 633, 791
Ziff R. M., Kinetics of Polymerization, 1980, J. Stat. Phys., 23, 241
Ziff R. M., McGrady E. D., The kinetics of cluster fragmentation and depolymeri-
sation, 1985, JPA, 18, 3027
Ziff R. M., Ernst M. H., Hendriks E. M., Kinetics of gelation and universality,
1983, JPA, 16, 2293
354
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