Vereshchagin

Theoretical Astroparticle Physics

Contents

1. Topics 7

2. Participants 92.1. ICRANet participants . . . . . . . . . . . . . . . . . . . . . . . . 92.2. Past collaborators . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Ongoing collaborations . . . . . . . . . . . . . . . . . . . . . . . 102.4. Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. Brief description 133.1. Electron-positron plasma . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1. Bose enhancement and Pauli blocking in the pair plasma 133.1.2. Dynamics and emission from mildly relativistic plasma 143.1.3. Evolution of the pair plasma generated by a strong elec-

tric field . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.4. Electron-positron plasma in GRBs and in cosmology . 15

3.2. Photospheric emission from ultrarelativistic outflows . . . . . 163.3. Correlation dynamics in cosmology . . . . . . . . . . . . . . . 163.4. Neutrinos in cosmology . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1. Massive neutrino and structure formation . . . . . . . . 183.4.2. Cellular structure of the Universe . . . . . . . . . . . . 183.4.3. Lepton asymmetry of the Universe . . . . . . . . . . . . 19

3.5. Semidegenerate self-gravitating system of fermions as a modelfor dark matter halos and universality laws . . . . . . . . . . . 20

3.6. Constraining cosmological models with CMB observations . . 21

4. Publications 234.1. Publications before 2005 . . . . . . . . . . . . . . . . . . . . . . 234.2. Publications (2005 – 2010) . . . . . . . . . . . . . . . . . . . . . 264.3. Publications (2011) . . . . . . . . . . . . . . . . . . . . . . . . . 414.4. Invited talks at international conferences . . . . . . . . . . . . 454.5. Posters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6. Lecture courses . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5. APPENDICES 55

A. Bose enhancement and Pauli blocking in the pair plasma 57A.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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Contents

A.2. Boltzmann equation with QED collisional integrals . . . . . . 59A.2.1. Two-particle interactions . . . . . . . . . . . . . . . . . . 59A.2.2. Three-particle interactions . . . . . . . . . . . . . . . . . 61

A.3. The numerical scheme . . . . . . . . . . . . . . . . . . . . . . . 62A.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B. Dynamics and emission from mildly relativistic plasma 69B.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B.2. Formulation of the problem . . . . . . . . . . . . . . . . . . . . 70B.3. Computational method . . . . . . . . . . . . . . . . . . . . . . . 71B.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72B.5. Conclusions and future perspectives . . . . . . . . . . . . . . . 76

C. Evolution of the pair plasma generated by a strong electric field 77C.1. Cylindrical coordinates in the momentum space . . . . . . . . 78C.2. The Distribution Function . . . . . . . . . . . . . . . . . . . . . 79C.3. Two temperature DF . . . . . . . . . . . . . . . . . . . . . . . . 80C.4. Boltzmann equations . . . . . . . . . . . . . . . . . . . . . . . . 81C.5. Computational scheme . . . . . . . . . . . . . . . . . . . . . . . 82

C.5.1. Acceleration and electric field evolution . . . . . . . . . 82C.5.2. Emission and absorption coefficients . . . . . . . . . . . 82C.5.3. Two particle kinematics . . . . . . . . . . . . . . . . . . 84

C.6. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86C.6.1. Non interacting systems . . . . . . . . . . . . . . . . . . 86C.6.2. Interacting systems . . . . . . . . . . . . . . . . . . . . . 90

C.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D. Electron-positron plasma in GRBs and in cosmology 97D.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97D.2. General equations . . . . . . . . . . . . . . . . . . . . . . . . . . 97

D.2.1. Early Universe . . . . . . . . . . . . . . . . . . . . . . . 99D.2.2. GRBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

D.3. Heavy elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 103D.4. Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104D.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

E. Photospheric emission from relativistic outflows 109E.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109E.2. Optically thick relativistic outflows: wind vs. explosion . . . . 110E.3. Optical depth along the line of sight . . . . . . . . . . . . . . . 112

E.3.1. Pure electron-positron plasma . . . . . . . . . . . . . . 113E.3.2. Acceleration phase . . . . . . . . . . . . . . . . . . . . . 114E.3.3. Coasting phase . . . . . . . . . . . . . . . . . . . . . . . 116

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Contents

E.3.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 117E.4. Geometry and dynamics of the photosphere . . . . . . . . . . 119

E.4.1. Coasting photon thin case . . . . . . . . . . . . . . . . . 120E.4.2. Accelerating and coasting photon thick cases . . . . . . 120

E.5. Observed flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122E.5.1. Adiabatic approximation for evaluation of observed flux 122E.5.2. Acceleration phase . . . . . . . . . . . . . . . . . . . . . 124E.5.3. Coasting phase . . . . . . . . . . . . . . . . . . . . . . . 124E.5.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 130

E.6. Instantaneous and time-integrated spectra from the photosphere133E.6.1. Photon thin outflow . . . . . . . . . . . . . . . . . . . . 134E.6.2. Photon thick outflow . . . . . . . . . . . . . . . . . . . . 136E.6.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 136

E.7. Application to GRBs . . . . . . . . . . . . . . . . . . . . . . . . 137E.7.1. Shell model . . . . . . . . . . . . . . . . . . . . . . . . . 138E.7.2. Wind model . . . . . . . . . . . . . . . . . . . . . . . . . 140

E.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

F. Correlation dynamics in cosmology 149

G. Semidegenerate self-gravitating system of fermions as a model fordark matter halos and universality laws 157G.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157G.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

G.2.1. Properties of the equilibrium configurations . . . . . . 162G.3. Comparison with other DM profiles . . . . . . . . . . . . . . . 163G.4. Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

G.4.1. Solving the scaling laws system of equations . . . . . . 167G.4.2. Application to spiral, elliptical and group of galaxies . 167

H. Features in the primordial spectrum: new constraints fromWMAP7+ACT data and prospects for Planck 171H.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171H.2. Inflationary perturbations in models with interrupted slow roll 172H.3. Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . 174H.4. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 178H.5. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 181

Bibliography 187

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Contents

6

1. Topics

• Electron-positron plasma

– Bose enhancement and Pauli blocking in the pair plasma

– Dynamics and emission from mildly relativistic plasma

– Evolution of the pair plasma generated by a strong electric field

– Electron-positron plasma in GRBs and in cosmology

• Photospheric emission from ultrarelativistic outflows

• Correlation dynamics in cosmology

• Neutrinos in cosmology

• Semidegenerate self-gravitating system of fermions as a model for darkmatter halos and universality laws

• Constraining cosmological models with CMB observations

– Constraints on the power spectrum of primordial density fluctua-tions

– Constraints on the reionization history of the Universe

– Constraints on the variation of fundamental constants

7

1. Topics

8

2. Participants

2.1. ICRANet participants

• Carlo Luciano Bianco

• Massimiliano Lattanzi

• Remo Ruffini

• Gregory Vereshchagin

• She-Sheng Xue

2.2. Past collaborators

• Marco Valerio Arbolino (DUNE s.r.l., Italy)

• Andrea Bianconi (INFN Pavia, Italy)

• Neta A. Bahcall (Princeton University, USA)

• Daniella Calzetti (University of Massachusets, USA)

• Jaan Einasto (Tartu Observatory, Estonia)

• Roberto Fabbri (University of Firenze, Italy)

• Long-Long Feng (University of Science and Technology of China,China)

• Jiang Gong Gao (Xinjiang Institute of Technology, China)

• Mauro Giavalisco (University of Massachusets, USA)

• Gabriele Ingrosso (INFN, University of Lecce, Italy)

• Yi-peng Jing (Shanghai Astronomical Observatory, China)

• Hyung-Won Lee (Inje University, South Korea)

• Marco Merafina (University of Rome “Sapienza”, Italy)

9

2. Participants

• Houjun Mo (University of Massachusetts, USA)

• Enn Saar (Tartu Observatory, Estonia)

• Jay D. Salmonson (Livermore Lab, USA)

• Luis Alberto Sanchez (National University Medellin, Colombia)

• Costantino Sigismondi (ICRA and University of Rome ”La Sapienza”,Italy)

• Doo Jong Song (Korea Astronomy Observatory, South Korea)

• Luigi Stella (Astronomical Observatory of Rome, Italy)

• William Stoeger (Vatican Observatory, University of Arizona USA)

• Sergio Taraglio (ENEA, Italy)

• Gerda Wiedenmann (MPE Garching, Germany)

• Jim Wilson (Livermore Lab, USA)

• Urbano Franca (Instituto de Fısica Corpuscular, Valencia, Spain)

• Julien Lesgourgues (CERN, Theory Division, Geneva, Switzerland)

• Lidia Pieri (Institute d’Astrophysique, Paris, France)

• Sergio Pastor (Instituto de Fısica Corpuscolar, Valencia, Spain)

• Joseph Silk (Oxford University, UK)

• Gustavo de Barros (former IRAP PhD, Brazil)

• Wien Biao Han (former IRAP PhD, now at Shanghai Astronomical Ob-servatory, Chinese Academy of Science, China)

2.3. Ongoing collaborations

• Alexey Aksenov (ICAD, RAS, Russia)

• Roustam Zalaletdinov (Tashkent University, Uzbekistan)

• Valeri Chechetkin (Keldysh Institute, Russia)

• Alessandro Melchiorri (Univ. “Sapienza” di Roma, Italy)

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2.4. Students

2.4. Students

• Alberto Benedetti (Erasmus Mundus IRAP PhD, Italy)

• Ivan Siutsou (IRAP PhD, Belarus)

• Damien Begue (Erasmus Mundus IRAP PhD, France)

• Micol Benetti (IRAP PhD, Italy)

• Eloisa Menegoni (IRAP PhD, Italy)

• Stefania Pandolfi (IRAP PhD, Italy)

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2. Participants

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3. Brief description

Astroparticle physics is a new field of research emerging at the intersectionof particle physics, astrophysics and cosmology. Theoretical development inthese fields is mainly triggered by the growing amount of experimental dataof unprecedented accuracy, coming both from the ground based laboratoriesand from the dedicated space missions.

3.1. Electron-positron plasma

Electron-positron plasma is of interest in many fields of astrophysics, e.g.in the early universe, active galactic nuclei, the center of our Galaxy, com-pact astrophysical objects such as hypothetical quark stars, neutron stars andgamma-ray bursts sources. It is also relevant for the physics of ultraintenselasers and thermonuclear reactions. We study physical properties of denseand hot electron-positron plasmas. In particular, we are interested in the is-sues of its creation and relaxation, its kinetic properties and hydrodynamicdescription, baryon loading and radiation from such plasmas.

Two completely different states exist for electron-positron plasma: opti-cally thin and optically thick. Optically thin pair plasma may exist in activegalactic nuclei and in X-ray binaries. The theory of relativistic optically thinnonmagnetic plasma and especially its equilibrium configurations was es-tablished in the 80s by Svensson, Lightman, Gould and others. It was shownthat relaxation of the plasma to some equilibrium state is determined by adominant reaction, e.g. Compton scattering or bremsstrahlung.

Developments in the theory of gamma ray bursts from one side, and ob-servational data from the other side, unambiguously point out on existenceof optically thick pair dominated non-steady phase in the beginning of for-mation of GRBs. The spectrum of radiation from optically thick plasma isusually assumed to be thermal.

3.1.1. Bose enhancement and Pauli blocking in the pairplasma

We consider relaxation of nonequilibrium optically thick pair plasma to com-plete thermal equilibrium by integrating numerically relativistic Boltzmannequations with exact QED two-particle and three-particle collisional inte-

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grals. Quantum nature of particle statistics is accounted for in collisionalintegrals by the corresponding Bose enhancement and Pauli blocking factors.

We point out that unlike classical Boltzmann equation for binary interac-tions such as scattering, more general interactions are typically described byfour collisional integrals for each particle that appears both among incomingand outgoing particles.

Our numerical results indicate that the rates of three-particle interactionsbecome comparable to those of two-particle ones for temperatures exceedingthe electron rest-mass energy. Thus three particle interactions such as rela-tivistic bremsstrahlung, double Compton scattering and radiative pair cre-ation become essential not only for establishment of thermal equilibrium, butalso for correct estimation of interaction rates, energy losses etc. Our resultson this topic are reported in Appendix A.

3.1.2. Dynamics and emission from mildly relativistic plasma

Interactions and emission in a spherical region with optically thick relativisticplasma is studied using kinetic Boltzmann equations, see Appendix B. Beinglimited by the computational requirements, we selected initial optical depthof the order of τ = 107, and initial temperature of the order of electron restmass energy. Such initial conditions allow as first of all to study opticallythick pair plasma. Secondly, we may follow the process of self accelerationand formation of the shell which reaches mildly relativistic bulk velocity ofexpansion before it becomes transparent for radiation, similarly to electron-positron plasma in GRB sources. Such initial value of the optical depth is toosmall for GRBs, as we show in Appendix E, and consequently large Lorentzfactors are not attained, but important relativistic effect may be studied. Atthe same time such initial optical depth is too large for existing laboratoryexperiments. However we believe that this simulation, exploring intermedi-ate region between laboratory and astrophysical conditions, allows to obtainsome important physical insights into kinetic evolution of electron-positronplasma. We follow dynamical evolution of particle number density, opticaldepth, hydrodynamic velocity, luminosity and spectra. Most important wefind unexpectedly that the spectrum of emission near its peak is differentfrom pure thermal one.

3.1.3. Evolution of the pair plasma generated by a strongelectric field

We investigate the behavior of the electron-positron pairs created by a strongelectric field, see Appendix C. This problem has been studied analytically inour previous work using simple formalism based on continuity and energy-momentum conservation equations. Now we extend that work using the

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3.1. Electron-positron plasma

more general kinetic approach.We consider a system which is uniform and homogeneous in the physical

space and axially symmetric in the momentum space. The axis of symme-try is given by the direction of the initial external electric field. With thesehypotheses, the relativistic Boltzmann equation is solved numerically for dif-ferent starting values of the field. In this framework we can describe theDistribution Function for each kind of particle in a two dimensional momen-tum space. Our numerical code allows us to take into account the interactionsbetween particles as well. We study the non-interacting and the interactingcases separately, then we compare the two runs and the role of the interac-tions can be analyzed; in particular we are interested in the dynamical ap-proach to the thermal equilibrium configuration.

There are many analogies between the results we obtain using the twomethods methods, namely the time dependence during the first half oscil-lation of all the quantities involved. However, after this short period, the twomethods give substantially different results.

We analyzed in details the case when interactions are not taken into ac-count. For all the considered parameter sets we find that after several oscilla-tions the magnitude of the electric field becomes much less than the startingvalue; as a consequence, acceleration and pair production are strongly sup-pressed. The bulk momentum parallel to the external field becomes verysmall, while the number density of the pairs saturates to a small fraction ofthe maximum achievable one, well below 10 percent for all cases considered.This result comes out only when kinetic treatment is adopted, and conse-quently distribution of particles in momentum space is accounted for. Wefind that production of pairs and their acceleration in the same electric fieldsimultaneously produce a peculiar distribution function of particles whichquickly settles down in a sort of equilibrium. This equilibrium is well de-scribed using a relativistic two-temperature distribution function, with thetemperature along the electric field being much larger than the one in or-thogonal direction. Therefore we find that substantial part of total energy,initially stored in the electric field is converted after few oscillations into in-ternal energy. This effect could not be obtained within the simple treatmentwe used before, since all particles were assumed to have single momentum(delta-function distribution in momentum space).

When particle interactions are taken into account photon interactions areexpected to bring this two-temperature system into thermal equilibrium.

3.1.4. Electron-positron plasma in GRBs and in cosmology

Analogy and difference between electron-positron plasma in the early Uni-verse and in sources of GRBs are discussed in Appendix D. We focus on a)dynamical differences, namely thermal acceleration of the outflow in GRB

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sources vs. cosmological deceleration; b) nuclear composition differences assynthesis of light elements in the early Universe and possible destruction ofheavy elements in GRB plasma; c) different physical conditions during lastscattering of photons by electrons in both cases leading to nearly perfect blackbody spectrum of the microwave background radiation vs. non thermal spec-trum of the photospheric emission in GRBs.

3.2. Photospheric emission from ultrarelativisticoutflows

We study the photospheric emission from ultrarelativistic outflows in Ap-pendix E.

Two popular models of optically thick relativistic outflows exist: the windand the shell. In this work the optical depth during the acceleration andcoasting phases is computed analytically within both models and its variousasymptotic limits are derived. In particular we show that quite counterin-tuitively a geometrically thin shell may appear as a thick wind for photonspropagating inside it. For this reason we introduce notions of photon thickand photon thin outflows, which appear more general and better physicallymotivated with respect to winds and shells.

We study the geometry of dynamic photospheres emerging from relativis-tic outflows. The photosphere of photon thin outflow has always a convexshape. In the photon thick one it is initially convex since there is always aphoton thin layer in any outflow, and then it becomes concave. Assumingthat photons are emitted with comoving thermal spectrum at the momentwhen the optical depth decreases to unity we compute the observed flux andinstantaneous spectra for both photon thick and photon thin outflows. It isour main finding that the photospheric emission from the photon thin out-flow produces non thermal time integrated spectra, which may be describedby the Band function well known in the GRB literature.

We find that energetic GRBs should produce photon thin outflows and onlytime integrated spectra may be observed from such GRBs. In other words, ob-served Band spectrum is obtained quite naturally from the comoving thermalone by the integration over the photosphere.

3.3. Correlation dynamics in cosmology

Two fundamental processes are known to occur in a self-gravitating systemof collisionless massive particles: gravitational instability and violent relax-ation. A new analytic approach is proposed in Appendix F aimed in describ-ing these two apparently distinct phenomena as different manifistations ofessentially the same physical process: gravitational structure formation. This

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3.4. Neutrinos in cosmology

approach is based on application of two averaging schemes: spatial averag-ing and coarse-graining. A master equation for spatially averaged coarse-grained distribution function of dark matter is constructed and its limitingcases are analyzed. Discussion of the related works, such as the recent workof J. Einasto et al., (2011) discussing phase synchronization in the large scalestructure is presented.


Many observational facts make it clear that luminous matter alone cannotaccount for the whole matter content of the Universe. Among them thereis the cosmic background radiation anisotropy spectrum, that is well fittedby a cosmological model in which just a small fraction of the total density issupported by baryons.

In particular, the best fit to the observed spectrum is given by a flat ΛCDMmodel, namely a model in which the main contribution to the energy densityof the Universe comes from vacuum energy and cold dark matter. This resultis confirmed by other observational data, like the power spectrum of largescale structures.

Another strong evidence for the presence of dark matter is given by therotation curves of galaxies. In fact, if we assume a spherical or ellipsoidalmass distribution inside the galaxy, the orbital velocity at a radius r is givenby Newton’s equation of motion. The peculiar velocity of stars beyond thevisible edge of the galaxy should then decrease as 1/r. What is instead ob-served is that the velocity stays nearly constant with r. This requires a haloof invisible, dark, matter to be present outside the edge. Galactic size shouldthen be extended beyond the visible edge. From observations is follows thatthe halo radius is at least 10 times larger than the radius of visible part of thegalaxy. Then it follows that a halo is at least 10 times more massive than allstars in a galaxy.

Neutrinos were considered as the best candidate for dark matter abouttwenty years ago. Indeed, it was shown that if these particles have a smallmass mν ∼ 30 eV, they provide a large energy density contribution up to crit-ical density. Tremaine and Gunn (1979) have claimed, however, that massiveneutrinos cannot be considered as dark matter. Their paper was very influen-tial and turned most of cosmologists away from neutrinos as cosmologicallyimportant particles.

Tremaine and Gunn paper was based on estimation of lower and upperbounds for neutrino mass; when contradiction with these bounds was found,the conclusion was made that neutrinos cannot supply dark matter. The up-per bound was given by cosmological considerations, but compared with theenergy density of clustered matter. It is possible, however, that a fraction ofneutrinos lays outside galaxies.

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Moreover, their lower bound was found on the basis of considerations ofgalactic halos and derived on the ground of the classical Maxwell-Boltzmannstatistics. Gao and Ruffini (1980) established a lower limit on the neutrinomass by the assumption that galactic halos are composed by degenerateneutrinos. Subsequent development of their approach Arbolino and Ruffini(1988) has shown that contradiction with two limits can be avoided.

At the same time, in 1977 the paper by Lee and Weinberg (1977) appeared,in which authors turned their attention to massive neutrinos with mν >>2 GeV. Such particles could also provide a large contribution into the energydensity of the Universe, in spite of much smaller value of number density.

Recent experimental results from laboratory (see Dolgov (2002) for a re-view) rule out massive neutrinos with mν > 2 GeV. However, the paper byLee and Weinberg was among the first where very massive particles wereconsidered as candidates for dark matter. This can be considered as the firstof cold dark matter models.

Today the interest toward neutrinos as a candidate for dark matter camedown, since from one side, the laboratory limit on its mass do not allow forsignificant contribution to the density of the Universe, and from other side,conventional neutrino dominated models have problems with formation ofstructure on small scales. However, in these scenarios the role of the chem-ical potential of neutrinos was overlooked, while it could help solving bothproblems.

3.4.1. Massive neutrino and structure formation

Lattanzi et al. (2003) have studied the possible role of massive neutrinos inthe large scale structure formation. Although now it is clear, that massivelight neutrinos cannot be the dominant part of the dark matter, their influ-ence on the large scale structure formation should not be underestimated. Inparticular, large lepton asymmetry, still allowed by observations, can affectcosmological constraints on neutrino mass.

3.4.2. Cellular structure of the Universe

One of the interesting possibilities, from a conceptual point of view, is thechange from the description of the physical properties by a continuous func-tion, to a new picture by introducing a self-similar fractal structure. Thisapproach has been relevant, since the concept of homogeneity and isotropyformerly apply to any geometrical point in space and leads to the concept ofa Universe observer-homogeneous (Ruffini (1989)). Calzetti et al. (1987), Gi-avalisco (1992), Calzetti et al. (1988) have defined the correlation length of afractal

r0 =(

1− γ

3

)1/γRS, (3.4.1)

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Figure 3.1.: Cellular structure of the Universe.

where RS is the sample size, γ = 3− D, and D is the Hausdorff dimensionof the fractal. Most challenging was the merging of the concepts of fractal,Jeans mass of dark matter and the cellular structure in the Universe, ad-vanced by Ruffini et al. (1988). The cellular structure emerging from thisstudy is represented in Figure 3.1. There the upper cutoff in the fractalstructure Rcutoff ≈ 100 Mpc, was associated to the Jeans mass of the ”ino”

Mcell =(

mplmino

)2mpl.

3.4.3. Lepton asymmetry of the Universe

Lattanzi et al. (2005), Lattanzi et al. (2006) studied how the cosmological con-straints on neutrino mass are affected by the presence of a lepton asymmetry.The main conclusion is that while constraints on neutrino mass do not changeby the inclusion into the cosmological model the dimensional chemical po-tential of neutrino, as an additional parameter, the value of lepton asymmetryallowed by the present cosmological data is surprisingly large, being

L = ∑ν

nν − nν

nγ. 0.9. (3.4.2)

Therefore, large lepton asymmetry is not ruled out by the current cosmologi-cal data.

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3.5. Semidegenerate self-gravitating system offermions as a model for dark matter halos anduniversality laws

The problem of dark matter distribution in galactic halos has traditionallybeen treated in the realm of newtonian physics in view of the low velocities ofthe stars in the galaxies, like the simulations from Navarro, Frenk and White(1997). In the meantime, phenomenological profiles of dark matter have beenadvanced by Einasto (1965); Burkert (1995), and universal properties of thedark matter distribution have been inferred from dwarf galaxies and proba-bly globular clusters all the way to very massive galaxies (Gentile et al., 2009;Donato et al., 2009; Walker et al., 2010; Boyarsky et al., 2009). However, aproblem arises: while simulations like those from NFW point to a cuspedhalo, observations from various types of galaxies seem to show cored halos(Salucci et al., 2011). This discrepancy between theory and observations is notyet fully understood, but could show a problem with the simulations doneso far.

Different approach to the study of properties of dark matter is coming fromcosmology and specially from theories of galaxy formation and evolution.While studies of CMB allow to constrain number and masses of light neu-trinos representing hot dark matter (Giusarma et al., 2011), properties andmasses of warm and cold dark matter are constrained by the total mass den-sity of the Universe (Lee and Weinberg, 1977) and galactic halos structure(Tremaine and Gunn, 1979) and substructure (Polisensky and Ricotti, 2011).The Lee-Weinberg bound (Lee and Weinberg, 1977) limits the mass of darkmatter particles with given coupling constant from above implying that par-ticle was in thermal equilibrium in early Universe. For typical weak inter-action coupling GF mass is constrained to be more than ∼ 2 GeV/c2 andsuch particles is referred to as Weakly Interacting Massive Particles, WIMPs.Bound by Tremaine and Gunn (1979) and its improvement by many authors(Hogan and Dalcanton, 2000; de Vega and Sanchez, 2010) are based on as-sumption of fermionic nature of dark matter and nondegeneracy of galactichaloes of dwarf galaxies, that leads to lower limit on mass ∼ 0.41 keV/c2.

In a completely unrelated field (as of yet), the physics of Active GalacticNuclei (AGN) and quasars has been recognized for more than 50 years asdominated by relativistic gravitational effects of a black hole. The formationof these black holes is not yet fully understood, although different black holesmass estimates for AGNs and quasars show masses up to 1010 M all the wayto z ≈ 6.4 (Peterson, 2010; Vestergaard, 2002; Vestergaard and Peterson, 2006;Targett et al., 2011). Due to the lack of understanding on the energetics ofAGNs and quasars and on the formation of the black holes, the possibility ofan extended object in the core of galaxies has been advanced by Viollier et al.(1993).

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3.6. Constraining cosmological models with CMB observations

The aim of this work is to present a unified approach to the dark matterdistribution in the galactic halos and also in the galactic center. In order to dothat, some assumptions have been made:

1. The treatment must be a general relativistic one from the beginning, inorder to explain both the galactic nuclei and galactic haloes.

2. The matter particles are assumed to be semi-degenerated fermions andso obey the Fermi-Dirac statistics.

3. Configurations are in relativistic thermal equilibrium√

g00T = const.

We show how the distribution of Dark Matter (DM) in galaxies can be ex-plained within a model based on a semidegenerate self-gravitating systemof fermions in General Relativity. We reproduce the observed properties ofgalaxies as the core, the halo, as well as the flattening of the rotation curves.In order to account for the evaporation phenomena (the escape velocity) weintroduced a cut-off in the fermion momentum space. The model providesphysical interpretation of phenomenological pseudo-isothermal sphere andBurkert DM profiles. It is consistent with a mass of the DM particle of the or-der of 14 KeV, compatible with a possible sterile neutrino interpretation. Wehave also extended the application of the model to general spiral, ellipticaland group of galaxies, by explaining the phenomenological features of DMhalos, i.e. the Universality laws found by Gentile et al. (2009); Donato et al.(2009) and Walker et al. (2010) in two different scopes, describing univer-sality of galactic surface densities within a Dark Matter scale length, and byBoyarsky et al. (2009), providing an analogous Universality law but extendedin the DM halo mass range. For details see Appendix G.

3.6. Constraining cosmological models with CMBobservations

Precision measurement of the cosmological observables have led to believethat we leave in a flat Friedmann Universe, seeded by nearly scale-invariantadiabatic primordial fluctuations (Komatsu et al., 2009, 2011). The major-ity (∼ 70%) of the energy density of the Universe is in the form of a fluidwith a cosmological constant-like equation of state (w ∼ −1), dubbed darkenergy, that is responsible for the observed acceleration of the Universe (Frie-man et al., 2008). This so-called “concordance model” is adequately describedby just six parameters, namely the baryon density, the cold dark matter den-sity, the Hubble constant, the reionization optical depth, the amplitude andthe spectral index of the primordial spectrum of density fluctuations. Theseparameters are measured to a very high precision (Komatsu et al., 2009, 2011).

21


However, even if the concordance model gives a very satisfactory fit ofall available data, it is worth to consider extended models and to constrainttheir parameters. In some cases these extended models simply arise whenconsidering properties that, to a first approximation, can be neglected wheninterpreting cosmological data. This is the case for parameters like the neu-trino mass and the curvature of the Universe. Both are very small and canbe put to zero as a first approximation; however, allowing them to vary al-lows to put useful constraints on their value. For recent constraints on theneutrino mass from cosmology, see e.g. Melchiorri et al. (2010) and Archidia-cono et al. (2010). The existence of sterile neutrinos can also be probed usingcosmological data (Giusarma et al., 2011).

Another example is given by the reionization history: in the concordancemodel, this is assumed to happen istantaneously. A more realistic descriptionis definetely in order. These more realistic, and more general, reionizationscenarios can be constrained by the observations (Pandolfi et al., 2011). It isalso important to check how considering more general models impacts thedetermination of the other parameters, like for example the neutrino mass(Archidiacono et al., 2011), or the inflationary parameters (Pandolfi et al.,2010b,c).

It is also important to study the characteristics of the spectrum of primor-dial density perturbations. Although the observations are consistent with anearly scale-invariant spectrum with a spectral index ns ± 0.967± 0.014 (Ko-matsu et al., 2011), nevertheless more general shapes of the power spectrum,that arise in particular inflationary models. Models with a non-standardspectrum of primordial perturbations (specifically, with a broken power-lawform) have been considered by Pandolfi et al. (2010a), also in relation to previ-ous claims that in this class of models the CMB observations can be fitted withΩΛ = 0. Benetti et al. (2011) have instead the considered the possibility thatslow-roll is briefly violated during inflation; this naturally happens in theo-ries with many interacting scalar fields, as it is the case, for example, in a classof multifield, supergravity-inspired models, where supersymmetry-breakingphase transitions occur during inflation. These phase transitions correspondto sudden changes in the inflaton effective mass and can be modeled as stepsin the inflationary potential, leading in turn to the appearance of characterisc-tic oscillatory features in the primordial perturbation spectrum. Thus CMBobservations, like those of the WMAP and ACT experiments, and in futureof the Planck satellite, can be used to constrain such models (Benetti et al.,2011).

Models with a dynamical dark energy have been considered by Serra et al.(2009). Models in which the fundamental constants are allowed to vary withtime were instead studied, also in the presence of dark energy, by Menegoniet al. (2009); Martins et al. (2010); Menegoni et al. (2010); Menegoni (2010);Calabrese et al. (2011). For details see Appendix H.

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4. Publications

4.1. Publications before 2005

1. R. Ruffini, D. J. Song, and L. Stella, “On the statistical distribution ofmassive fermions and bosons in a Friedmann universe” Astronomy andAstrophysics, Vol. 125, (1983) pp. 265-270.

The distribution function of massive Fermi and Bose particles in an expandinguniverse is considered as well as some associated thermodynamic quantities,pressure and energy density. These considerations are then applied to cosmo-logical neutrinos. A new limit is derived for the degeneracy of a cosmologicalgas of massive neutrinos.

2. R. Ruffini and D. J. Song, “On the Jeans mass of weakly interacting neu-tral massive leptons”, in Gamow cosmology, eds. F. Melchiorri and R.Ruffini, (1986) pp. 370–385.

The cosmological limits on the abundances and masses of weakly interactingneutral particles are strongly affected by the nonzero chemical potentials ofthese leptons. For heavy leptons (mx > GeV), the value of the chemical po-tential must be much smaller than unity in order not to give very high valuesof the cosmological density parameter and the mass of heavy leptons, or theywill be unstable. The Jeans’ mass of weakly interacting neutral particles couldgive the scale of cosmological structure and the masses of astrophysical ob-jects. For a mass of the order 10 eV, the Jeans’ mass could give the scenarioof galaxy formation, the supercluster forming first and then the smaller scales,such as clusters and galaxies, could form inside the large supercluster.

3. D. Calzetti, M. Giavalisco, R. Ruffini, J. Einasto, and E. Saar, “The corre-lation function of galaxies in the direction of the Coma cluster”, Astro-physics and Space Science, Vol. 137 (1987) pp. 101-106.

Data obtained by Einasto et al. (1986) on the amplitude of the correlation func-tion of galaxies in the direction of the Coma cluster are compared with theo-retical predictions of a model derived for a self-similar observer-homogeneousstructure. The observational samples can be approximated by cones of angu-lar width alpha of about 77 deg. Eliminating sources of large observationalerror, and by making a specified correction, the observational data are foundto agree very well with the theoretical predictions of Calzetti et al. (1987).

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4. Publications

4. R. Ruffini, D. J. Song, and S. Taraglio, “The ’ino’ mass and the cellu-lar large-scale structure of the universe”, Astronomy and Astrophysics,Vol. 190, (1988) pp. 1-9.

Within the theoretical framework of a Gamow cosmology with massive ”inos”,the authors show how the observed correlation functions between galaxiesand between clusters of galaxies naturally lead to a ”cellular” structure for theUniverse. From the size of the ”elementary cells” they derive constraints onthe value of the masses and chemical potentials of the cosmological ”inos”.They outline a procedure to estimate the ”effective” average mass density ofthe Universe. They also predict the angular size of the inhomogeneities to beexpected in the cosmological black body radiation as remnants of this cellularstructure. A possible relationship between the model and a fractal structure isindicated.

5. D. Calzetti, M. Giavalisco, and R. Ruffini, “The normalization of thecorrelation functions for extragalactic structures”, Astronomy and As-trophysics, Vol. 198 (1988), pp. 1-15.

It is shown that the spatial two-point correlation functions for galaxies, clus-ters and superclusters depend explicitly on the spatial volume of the statisticalsample considered. Rules for the normalization of the correlation functions aregiven and the traditional classification of galaxies into field galaxies, clustersand superclusters is replaced by the introduction of a single fractal structure,with a lower cut-off at galactic scales. The roles played by random and stochas-tic fractal components in the galaxy distribution are discussed in detail.

6. M. V. Arbolino and R. Ruffini, “The ratio between the mass of the haloand visible matter in spiral galaxies and limits on the neutrino mass”,Astronomy and Astrophysics, Vol. 192, (1988) pp. 107-116.

Observed rotation curves for galaxies with values of the visible mass rangingover three orders of magnitude together with considerations involving equi-librium configurations of massive neutrinos, impose constraints on the ratiobetween the masses of visible and dark halo comporents in spiral galaxies.Upper and lower limits are derived for the mass of the particles making up thedark matter.

7. A. Bianconi, H. W. Lee, and R. Ruffini, “Limits from cosmological nu-cleosynthesis on the leptonic numbers of the universe”, Astronomy andAstrophysics, Vol. 241 (1991) pp. 343-357.

Constraints on chemical potentials and masses of ’inos’ are calculated usingcosmological standard nucleosynthesis processes. It is shown that the elec-tron neutrino chemical potential (ENCP) should not be greater than a value ofthe order of 1, and that the possible effective chemical potential of the otherneutrino species should be about 10 times the ENCP in order not to conflict

24

4.1. Publications before 2005

with observational data. The allowed region (consistent with the He-4 abun-dance observations) is insensitive to the baryon to proton ratio η, while thoseimposed by other light elements strongly depend on η.

8. R. Ruffini, J. D. Salmonson, J. R. Wilson, and S.-S. Xue, “On the pairelectromagnetic pulse of a black hole with electromagnetic structure”,Astronomy and Astrophysics, Vol. 350 (1999) pp. 334-343.

We study the relativistically expanding electron-positron pair plasma formedby the process of vacuum polarization around an electromagnetic black hole(EMBH). Such processes can occur for EMBH’s with mass all the way up to6 · 105M. Beginning with a idealized model of a Reissner-Nordstrom EMBHwith charge to mass ratio ξ = 0.1, numerical hydrodynamic calculations aremade to model the expansion of the pair-electromagnetic pulse (PEM pulse)to the point that the system is transparent to photons. Three idealized specialrelativistic models have been compared and contrasted with the results of thenumerically integrated general relativistic hydrodynamic equations. One ofthe three models has been validated: a PEM pulse of constant thickness in thelaboratory frame is shown to be in excellent agreement with results of the gen-eral relativistic hydrodynamic code. It is remarkable that this precise model,starting from the fundamental parameters of the EMBH, leads uniquely to theexplicit evaluation of the parameters of the PEM pulse, including the energyspectrum and the astrophysically unprecedented large Lorentz factors (up to6 · 103 for a 103 M EMBH). The observed photon energy at the peak of thephoton spectrum at the moment of photon decoupling is shown to range from0.1 MeV to 4 MeV as a function of the EMBH mass. Correspondingly the totalenergy in photons is in the range of 1052 to 1054 ergs, consistent with observedgamma-ray bursts. In these computations we neglect the presence of baryonicmatter which will be the subject of forthcoming publications.

9. R. Ruffini, J. D. Salmonson, J. R. Wilson, and S.-S. Xue, “On the pair-electromagnetic pulse from an electromagnetic black hole surroundedby a baryonic remnant”, Astronomy and Astrophysics, Vol. 359 (2000)pp. 855-864.

The interaction of an expanding Pair-Electromagnetic pulse (PEM pulse) witha shell of baryonic matter surrounding a Black Hole with electromagneticstructure (EMBH) is analyzed for selected values of the baryonic mass at se-lected distances well outside the dyadosphere of an EMBH. The dyadosphere,the region in which a super critical field exists for the creation of e+e− pairs,is here considered in the special case of a Reissner-Nordstrom geometry. Theinteraction of the PEM pulse with the baryonic matter is described using a sim-plified model of a slab of constant thickness in the laboratory frame (constant-thickness approximation) as well as performing the integration of the generalrelativistic hydrodynamical equations. Te validation of the constant-thicknessapproximation, already presented in a previous paper Ruffini et al. (1999) for a

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4. Publications

PEM pulse in vacuum, is here generalized to the presence of baryonic matter.It is found that for a baryonic shell of mass-energy less than 1% of the totalenergy of the dyadosphere, the constant-thickness approximation is in excel-lent agreement with full general relativistic computations. The approximationbreaks down for larger values of the baryonic shell mass, however such casesare of less interest for observed Gamma Ray Bursts (GRBs). On the basis ofnumerical computations of the slab model for PEM pulses, we describe (i) theproperties of relativistic evolution of a PEM pulse colliding with a baryonicshell; (ii) the details of the expected emission energy and observed tempera-ture of the associated GRBs for a given value of the EMBH mass; 103M, andfor baryonic mass-energies in the range 10−8 to 10−2 the total energy of thedyadosphere.

10. M. Lattanzi, R. Ruffini, and G. Vereshchagin, “On the possible role ofmassive neutrinos in cosmological structure formation”, in Cosmologyand Gravitation, eds. M. Novello and S. E. Perez Bergliaffa, Vol. 668 ofAIP Conference Series, (2003) pp. 263–287.

In addition to the problem of galaxy formation, one of the greatest open ques-tions of cosmology is represented by the existence of an asymmetry betweenmatter and antimatter in the baryonic component of the Universe. We believethat a net lepton number for the three neutrino species can be used to under-stand this asymmetry. This also implies an asymmetry in the matter-antimattercomponent of the leptons. The existence of a nonnull lepton number for theneutrinos can easily explain a cosmological abundance of neutrinos consistentwith the one needed to explain both the rotation curves of galaxies and theflatness of the Universe. Some propedeutic results are presented in order toattack this problem.

4.2. Publications (2005 – 2010)

1. A.G. Aksenov, R. Ruffini and G.V. Vereshchagin, “Pair plasma relax-ation time scales”, Physical Review E, Vol. 81 (2010) 046401.

By numerically solving the relativistic Boltzmann equations, we compute thetime scale for relaxation to thermal equilibrium for an optically thick electron-positron plasma with baryon loading. We focus on the time scales of elec-tromagnetic interactions. The collisional integrals are obtained directly fromthe corresponding QED matrix elements. Thermalization time scales are com-puted for a wide range of values of both the total energy density (over 10 or-ders of magnitude) and of the baryonic loading parameter (over 6 orders ofmagnitude). This also allows us to study such interesting limiting cases as thealmost purely electron-positron plasma or electron-proton plasma as well asintermediate cases. These results appear to be important both for laboratoryexperiments aimed at generating optically thick pair plasmas as well as for

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4.2. Publications (2005 – 2010)

astrophysical models in which electron-positron pair plasmas play a relevantrole.

2. R. Ruffini, G.V. Vereshchagin and S.-S. Xue, “Electron-positron pairs inphysics and astrophysics: from heavy nuclei to black holes” PhysicsReports, Vol. 487 (2010) No 1-4, pp. 1-140.

From the interaction of physics and astrophysics we are witnessing in theseyears a splendid synthesis of theoretical, experimental and observational re-sults originating from three fundametal physical processes. They were origi-nally proposed by Dirac, by Breit and Wheeler and by Sauter, Heisenberg, Eu-ler and Schwinger. For almost seventy years they have all three been followedby a continued effort of experimental verification on Earth-based experiments.The Dirac process, e+e− → 2γ, has been by far the most successful. It has ob-tained extremely accurate experimental verification and has led as well to anenormous number of new physics in possibly one of the most fruitful experi-mental avenue by introduction of storage rings in Frascati and followed by thelargest accelerators worldwide: DESY, SLAC etc. The Breit-Wheeler process,2γ → e+e−, although conceptually simple, being the inverse process of theDirac one, has been by far one of the most difficult to be verified experimen-tally. Only recently, through the technology based on free electron X-ray laserand its numerous applications in Earth-based experiments, some first indica-tions of its possible verification have been reached. The vacuum polarizationprocess in strong electromagnetic field, pioneered by Sauter, Heisenberg, Eulerand Schwinger, introduced the concept of critical electric field Ec = m2

e c3/eh.It has been searched without success for more than forty years by heavy-ioncollisions in many of the leading particle accelerators worldwide. The novelsituation today is that these same processes can be studied on a much moregrandiose scale during the gravitational collapse leading to the formation of ablack hole being observed in Gamma Ray Bursts (GRBs). This report is ded-icated to the scientific race in act. The theoretical and experimental work de-veloped in Earth-based laboratories is confronted with the theoretical interpre-tation of space-based observations of phenomena originating on cosmologicalscales. What has become clear in the last ten years is that all the three abovementioned processes, duly extended in the general relativistic framework, arenecessary for the understanding of the physics of the gravitational collapse to ablack hole. Vice versa, the natural arena where these processes can be observedin mutual interaction and on an unprecedented scale, is indeed the realm of rel-ativistic astrophysics. We systematically analyze the conceptual developmentswhich have followed the basic work of Dirac and Breit-Wheeler. We also recallhow the seminal work of Born and Infeld inspired the work by Sauter, Heisen-berg and Euler on effective Lagrangian leading to the estimate of the rate forthe process of electron-positron production in a constant electric field. In ad-dition of reviewing the intuitive semi-classical treatment of quantum mechan-ical tunneling for describing the process of electron-positron production, we

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4. Publications

recall the calculations in Quantum Electro-Dynamics of the Schwinger rate andeffective Lagrangian for constant electromagnetic fields. We also review theelectron-positron production in both time-alternating electromagnetic fields,studied by Brezin, Itzykson, Popov, Nikishov and Narozhny, and the corre-sponding processes relevant for pair production at the focus of coherent laserbeams as well as electron beam-laser collision. We finally report some cur-rent developments based on the general JWKB approach which allows to com-pute the Schwinger rate in spatially varying and time varying electromagneticfields. We also recall the pioneering work of Landau and Lifshitz, and Racahon the collision of charged particles as well as experimental success of AdAand ADONE in the production of electron-positron pairs. We then turn to thepossible experimental verification of these phenomena. We review: A) the ex-perimental verification of the e+e− → 2γ process studied by Dirac. We alsobriefly recall the very successful experiments of e+e− annihilation to hadronicchannels, in addition to the Dirac electromagnetic channel; B) ongoing Earthbased experiments to detect electron-positron production in strong fields byfocusing coherent laser beams and by electron beam-laser collisions; and C) themultiyear attempts to detect electron-positron production in Coulomb fieldsfor a large atomic number Z > 137 in heavy ion collisions. These attemptsfollow the classical theoretical work of Popov and Zeldovich, and Greiner andtheir schools. We then turn to astrophysics. We first review the basic workon the energetics and electrodynamical properties of an electromagnetic blackhole and the application of the Schwinger formula around Kerr-Newman blackholes as pioneered by Damour and Ruffini. We only focus on black hole masseslarger than the critical mass of neutron stars, for convenience assumed to coin-cide with the Rhoades and Ruffini upper limit of 3.2M. In this case the elec-tron Compton wavelength is much smaller than the spacetime curvature andall previous results invariantly expressed can be applied following well estab-lished rules of the equivalence principle. We derive the corresponding rate ofelectron-positron pair production and the introduction of the concept of Dya-dosphere. We review recent progress in describing the evolution of opticallythick electron-positron plasma in presence of supercritical electric field, whichis relevant both in astrophysics as well as ongoing laser beam experiments. Inparticular we review recent progress based on the Vlasov-Boltzmann-Maxwellequations to study the feedback of the created electron-positron pairs on theoriginal constant electric field. We evidence the existence of plasma oscillationsand its interaction with photons leading to energy and number equipartitionof photons, electrons and positrons. We finally review the recent progress ob-tained by using the Boltzmann equations to study the evolution of an electron-positron-photon plasma towards thermal equilibrium and determination ofits characteristic timescales. The crucial difference introduced by the correctevaluation of the role of two and three body collisions, direct and inverse, isespecially evidenced. We then present some general conclusions. The resultsreviewed in this report are going to be submitted to decisive tests in the forth-

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4.2. Publications (2005 – 2010)

coming years both in physics and astrophysics. To mention only a few of thefundamental steps in testing in physics we recall the starting of experimentalfacilities at the National Ignition Facility at the Lawrence Livermore NationalLaboratory as well as corresponding French Laser the Mega Joule project. Inastrophysics these results will be tested in galactic and extragalactic black holesobserved in binary X-ray sources, active galactic nuclei, microquasars and inthe process of gravitational collapse to a neutron star and also of two neutronstars to a black hole giving origin to GRBs. The astrophysical description ofthe stellar precursors and the initial physical conditions leading to a gravita-tional collapse process will be the subject of a forthcoming report. As of todayno theoretical description has yet been found to explain either the emission ofthe remnant for supernova or the formation of a charged black hole for GRBs.Important current progress toward the understanding of such phenomena aswell as of the electrodynamical structure of neutron stars, the supernova ex-plosion and the theories of GRBs will be discussed in the above mentionedforthcoming report. What is important to recall at this stage is only that boththe supernovae and GRBs processes are among the most energetic and tran-sient phenomena ever observed in the Universe: a supernova can reach energyof ˜1054 ergs on a time scale of a few months and GRBs can have emission ofup to ˜1054 ergs in a time scale as short as of a few seconds. The central roleof neutron stars in the description of supernovae, as well as of black holes andthe electron-positron plasma, in the description of GRBs, pioneered by one ofus (RR) in 1975, are widely recognized. Only the theoretical basis to addressthese topics are discussed in the present report.

3. A. G. Aksenov, R. Ruffini, and G. V. Vereshchagin, “Kinetics of theMildly Relativistic Plasma and GRBs” in the Proceedings of “The Sun,the stars, the Universe and General Relativity” meeting in honor of 95thAnniversary of Ya. B. Zeldovich in Minsk, AIP Conference Proceedings1205 (2010) 11-16.

We consider optically thick photon-pair-proton plasma in the framework ofBoltzmann equations. For the sake of simplicity we consider the uniform andisotropic plasma. It has been shown that arbitrary initial distribution functionsevolve to the thermal equilibrium state through so called kinetic equilibriumstate with common temperature of all particles and nonzero chemical poten-tials. For the plasma temperature 0.1− 10 MeV relevant for GRB (Gamma-RayBurst) sources we evaluate the thermalization time scale as function of totalenergy density and baryonic loading parameter.

4. D. Cumberbatch, M. Lattanzi, J. Silk, “Signatures of clumpy dark matterin the global 21 cm background signal ”, in Phys. Rev. D82, 103508(2010).

We examine the extent to which the self-annihilation of supersymmetric neu-tralino dark matter, as well as light dark matter, influences the rate of heating,

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4. Publications

ionisation and Lyman-α pumping of interstellar hydrogen and helium and theextent to which this is manifested in the 21 cm global background signal. Wefully consider the enhancements to the annihilation rate from DM halos andsubstructures within them. We find that the influence of such structures can re-sult in significant changes in the differential brightness temperature, δTb. Thechanges at redshifts z < 25 are likely to be undetectable due to the presence ofthe astrophysical signal; however, in the most favourable cases, deviations inδTb, relative to its value in the absence of self-annihilating DM, of up to ' 20mK at z = 30 can occur. Thus we conclude that, in order to exclude thesemodels, experiments measuring the global 21 cm signal, such as EDGES andCORE, will need to reduce the systematics at 50 MHz to below 20 mK.

5. M. Lattanzi, S. Mercuri, “A solution of the strong CP problem via thePeccei-Quinn mechanism through the Nieh-Yan modified gravity andcosmological implications” in Phys. Rev. D81, 125015 (2010). By identi-fying the recently introduced Barbero–Immirzi field with the QCD axion, thestrong CP problem can be solved through the Peccei–Quinn mechanism. Aspecific energy scale for the Peccei–Quinn symmetry breaking is naturally pre-dicted by this model. This provides a complete dynamical setting to evalu-ate the contribution of such an axion to the cold dark matter content of theUniverse. Furthermore, a tight upper bound on the tensor-to-scalar ratio pro-duction of primordial gravitational waves can be fixed, representing a strongexperimental test for this model.

6. S. Pandolfi, E. Giusarma, M. Lattanzi, A. Melchiorri, “Inflation with pri-mordial broken power law spectrum as an alternative to the concor-dance cosmological model” in Phys. Rev. D81, 103007 (2010).

We consider cosmological models with a non scale-invariant spectrum of pri-mordial perturbations and assess whether they represent a viable alternativeto the concordance ΛCDM model. We find that in the framework of a modelselection analysis, the WMAP and 2dF data do not provide any conclusive evi-dence in favour of one or the other kind of model. However, when a marginal-ization over the entire space of nuisance parameters is performed, models witha modified primordial spectrum and ΩΛ = 0 are strongly disfavoured.

7. M. Lattanzi, “The majoron: a new dark matter candidate ”in J. Kor.Phys. Soc 56, 1677 (2010).

We review our recent proposal of the majoron as a suitable dark matter candi-date. The majoron is the Goldstone boson associated to the spontaneous break-ing of ungauged lepton number, one of the mechanisms proposed to give riseto neutrino masses. The majoron can acquire a mass through quantum gravityeffects, and can possibly account for the observed dark matter component ofthe Universe. The majoron dark matter scenario is consistent with the currentobservations of the cosmic microwave background anisotropy provided that

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4.2. Publications (2005 – 2010)

its lifetime τ & 250 Gyr. In the case of thermal production, the majoron shouldlie in the range 0.13 keV < mJ < 0.17 keV, although these limits are modifiedin the non-thermal case. Applying this results to a given seesaw model forthe generation of neutrino masses, it is found that the energy scale for the lep-ton number breaking phase transition is constrained to be EL & 106 GeV. Wethus find that the majoron decaying dark matter (DDM) scenario fits nicely inmodels where neutrino masses arise a la seesaw, and may lead to other possiblecosmological implications.

8. M. Archidiacono, A. Cooray, A. Melchiorri, S. Pandolfi, “CMB neutrinomass bounds and reionization”, Phys. Rev. D 82, 087302 (2010).

Abstract: Current cosmic microwave background (CMB) bounds on the sumof the neutrino masses assume a sudden reionization scenario described by asingle parameter that determines the onset of reionization. We investigate thebounds on the neutrino mass in a more general reionization scenario based ona principal component approach. We found the constraint on the sum of theneutrino masses from CMB data can be relaxed by a ∼ 40% in a generalizedreionization scenario. Moreover, the amplitude of the r.m.s. mass fluctuationsσ8is also considerably lower providing a better consistency with a low ampli-tude of the Sunyaev-Zel’dovich signal.

9. S. Pandolfi, A.Cooray, E.Giusarma, E.W.Kolb, A.Melchiorri, O.Menaand P.Serra, “Harrison-Zel’dovich primordial spectrum is consistentwith observations”, Phys. Rev. D 81, 123509 (2010).

Abstract: Inflation predicts primordial scalar perturbations with a nearly scale-invariant spectrum and a spectral index approximately unity (the Harrison–Zel’dovich (HZ) spectrum). The first important step for inflationary cosmol-ogy is to check the consistency of the HZ primordial spectrum with currentobservations. Recent analyses have claimed that a HZ primordial spectrum isexcluded at more than 99% c.l.. Here we show that the HZ spectrum is onlymarginally disfavored if one considers a more general reionization scenario.Data from the Planck mission will settle the issue.

10. P. Serra, A. Cooray, D. E. Holz, A. Melchiorri, S. Pandolfi, and D. Sarkar,“No evidence for dark energy dynamics from a global analysis of cos-mological data”, Phys. Rev. D 80, 121302 (2009).

Abstract: We use a variant of principal component analysis to investigate thepossible temporal evolution of the dark energy equation of state, w(z). Weconstrain w(z) in multiple redshift bins, utilizing the most recent data fromType Ia supernovae, the cosmic microwave background, baryon acoustic oscil-lations, the integrated Sachs-Wolfe effect, galaxy clustering, and weak lensingdata. Unlike other recent analyses, we find no significant evidence for evolvingdark energy; the data remains completely consistent with a cosmological con-stant. We also study the extent to which the time-evolution of the equation of

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state would be constrained by a combination of current- and future-generationsurveys, such as Planck and the Joint Dark Energy Mission.

11. E. Menegoni, S. Pandolfi, S. Galli, M. Lattanzi, A. Melchiorri “Con-straints on the dark energy equation of state in presence of a varyingfine structure constant” in Int. J. Mod. Phys D19, 507 (2010).

We discuss the cosmological constraints on the dark energy equation of statein the presence of primordial variations in the fine structure constant. Wefind that the constraints from CMB data alone on w and the Hubble constantare much weaker when variations in the fine structure constant are permitted.Vice versa, constraints on the fine structure constant are relaxed by morethan 50% when dark energy models different from a cosmological constant areconsidered.

12. C.J.A.P. Martins, E. Menegoni, S. Galli and A. Melchiorri, “Varying cou-plings in the early universe: correlated variations of α and G, PhysicalReview D 82 023532 (2010)

The cosmic microwave background anisotropies provide a unique opportu-nity to constrain simultaneous variations of the fine-structure constant α andNewton’s gravitational constant G. Those correlated variations are possiblein a wide class of theoretical models. In this brief paper we show that thecurrent data, assuming that particle masses are constant, give no clear indi-cation for such variations, but already prefer that any relative variations in α

should be of the same sign of those of G for variations of 1%. We also showthat a cosmic complementarity is present with big bang nucleosynthesis andthat a combination of current CMB and big bang nucleosynthesis data stronglyconstraints simultaneous variations in α and G. We finally discuss the futurebounds achievable by the Planck satellite mission.

13. E. Menegoni, “New Constraints on Variations of Fine Structure Con-stant from Cosmic Microwave Background Anisotropies”, GRAVITA-TIONAL PHYSICS: TESTING GRAVITY FROM SUBMILLIMETER TOCOSMIC: Proceedings of the VIII Mexican School on Gravitation andMathematical Physics. AIP Conference Proceedings, Volume 1256, pp.288-292 (2010).

The recent measurements of Cosmic Microwave Background temperature andpolarization anisotropy made by the ACBAR, QUAD and BICEP experimentssubstantially improve the cosmological constraints on possible variations ofthe fine structure constant in the early universe. In this work I analyze thisrecent data obtaining the constraint α/α0 = 0.987+/-0.012 at 68% c.l.. The in-clusion of the new HST constraints on the Hubble constant further increasesthe bound to α/α0 = 1.001+/-0.007 at 68% c.l., bringing possible deviationsfrom the current value below the 1% level.

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4.2. Publications (2005 – 2010)

14. A. Melchiorri, F. De Bernardis, E. Menegoni, “Limits on the neutrinomass from cosmology”. GRAVITATIONAL PHYSICS: TESTING GRAV-ITY FROM SUBMILLIMETER TO COSMIC: Proceedings of the VIIIMexican School on Gravitation and Mathematical Physics. AIP Con-ference Proceedings, Volume 1256, pp. 96-106 (2010).

We use measurements of luminosity-dependent galaxy bias at several differentredshifts, SDSS at z = 0.05, DEEP2 at z = 1 and LBGs at z = 3.8, combined withWMAP five-year cosmic microwave background anisotropy data and SDSSRed Luminous Galaxy survey three-dimensional clustering power spectrumto put constraints on cosmological parameters.

15. A.G. Aksenov, R. Ruffini and G.V. Vereshchagin, “Thermalization of themildly relativistic plasma”, Physical Review D, Vol. 79 (2009) 043008.

In the recent Letter Aksenov et al. (2007) we considered the approach ofnonequilibrium pair plasma towards thermal equilibrium state adopting a ki-netic treatment and solving numerically the relativistic Boltzmann equations.It was shown that plasma in the energy range 0.1-10 MeV first reaches kineticequilibrium, on a timescale tk . 10−14 sec, with detailed balance betweenbinary interactions such as Compton, Bhabha and Møller scattering, and pairproduction and annihilation. Later the electron-positron-photon plasma ap-proaches thermal equilibrium on a timescale tth . 10−12 sec, with detailedbalance for all direct and inverse reactions. In the present paper we system-atically present details of the computational scheme used in Aksenov et al.(2007), as well as generalize our treatment, considering proton loading ofthe pair plasma. When proton loading is large, protons thermalize first byproton-proton scattering, and then with the electron-positron-photon plasmaby proton-electron scattering. In the opposite case of small proton loadingproton-electron scattering dominates over proton-proton one. Thus in all casesthe plasma, even with proton admixture, reaches thermal equilibrium config-uration on a timescale tth . 10−11 sec. We show that it is crucial to accountfor not only binary but also triple direct and inverse interactions between elec-trons, positrons, photons and protons. Several explicit examples are given andthe corresponding timescales for reaching kinetic and thermal equilibria aredetermined.

16. A. G. Aksenov, R. Ruffini, and G. V. Vereshchagin, “Thermalization ofpair plasma with proton loading” in the Proceedings of “PROBINGSTELLAR POPULATIONS OUT TO THE DISTANT UNIVERSE” meet-ing, AIP Conference Proceedings 1111 (2009) 344-350.

We study kinetic evolution of nonequilibrium optically thick electron-positronplasma towards thermal equilibrium solving numerically relativistic Boltz-mann equations with energy per particle ranging from 0.1 to 10 MeV. We gen-eralize our results presented in Aksenov et al. (2007), considering proton load-ing of the pair plasma. Proton loading introduces new characteristic timescales

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essentially due to proton-proton and proton-electron Coulomb collisions. Tak-ing into account not only binary but also triple direct and inverse interactionsbetween electrons, positrons, photons and protons we show that thermal equi-librium is reached on a timescale tth ' 10−11 sec.

17. M. Lattanzi, J. Silk “Can the WIMP annihilation boost factor be boostedby the Sommerfeld enhancement? ”, in Phys. Rev. D79, 083523 (2009).

We demonstrate that the Sommerfeld correction to cold dark matter (CDM)annihilations can be appreciable if even a small component of the dark matteris extremely cold. Subhalo substructure provides such a possibility given thatthe smallest clumps are relatively cold and contain even colder substructuredue to incomplete phase space mixing. Leptonic channels can be enhancedfor plausible models and the solar neighbourhood boost required to accountfor PAMELA/ATIC data is plausibly obtained, especially in the case of a fewTeV mass neutralino for which the Sommerfeld-corrected boost is found tobe ∼ 104 − 105. Saturation of the Sommerfeld effect is shown to occur belowβ ∼ 10−4, thereby making this result largely independent on the presence ofsubstructures below ∼ 105M. We find that the associated diffuse gamma raysignal from annihilations would exceed EGRET constraints unless the chan-nels annihilating to heavy quarks or to gauge bosons are suppressed. Thelepton channel gamma rays are potentially detectable by the FERMI satellite,not from the inner galaxy where substructures are tidally disrupted, but ratheras a quasi-isotropic background from the outer halo, unless the outer substruc-tures are much less concentrated than the inner substructures and/or the CDMdensity profile out to the virial radius steepens significantly.

18. L. Pieri, M. Lattanzi, J. Silk “Constraining the Sommerfeld enhancementwith Cherenkov telescope observations of dwarf galaxies”, in Mon.Not. Roy. Astron. Soc., 399, 2033 (2009).

The presence of dark matter in the halo of our galaxy could be revealedthrough indirect detection of its annihilation products. Dark matter annihila-tion is one possible interpretation of the recently measured excesses in positronand electron fluxes, provided that boost factors of the order of 103 or moreare taken into account. Such boost factors are actually achievable throughthe velocity-dependent Sommerfeld enhancement of the annihilation cross-section. Here we study the expected γ-ray flux from two local dwarf galax-ies for which Cherenkov Telescope measurements are available, namely Dracoand Sagittarius. We use recent stellar kinematical measurements to model thedark matter halos of the dwarfs, and the results of numerical simulations tomodel the presence of an associated population of subhalos. We incorporatethe Sommerfeld enhancement of the annihilation cross-section. We compareour predictions with the observations of Draco and Sagittarius performed byMAGIC and HESS, respectively, and derive exclusion limits on the effectiveannihilation cross-section. We also study the sensitivities of Fermi and of the

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future Cherenkov Telescope Array to cross-section enhancements. We find thatthe boost factor due to the Sommerfeld enhancement is already constrained bythe MAGIC and HESS data, with enhancements greater than ∼ 104 being ex-cluded.

19. M. Lattanzi, “Mass Varying Neutrinos: A model-independent ap-proach”, in Nucl. Phys. Proc. Suppl. 188, 40, (2009).

In Mass Varying Neutrinos (MaVaNs) models, the neutrinos are coupled withthe quintessence field supposed to be responsible for the acceleration of theUniverse. Here we propose a new parameterization for the neutrino massvariation that is independent on the details of the scalar field potential and stillcaptures the essential of most MaVaNs models. We also find an upper limit onthe mass variation in the case of decreasing mass models, independent of theparticular parameterization.

20. U. Franca, M. Lattanzi, J. Lesgourgues, S. Pastor “Model independentconstraints on mass-varying neutrino scenarios”, in Phys. Rev. D80,083506 (2009).

Models of dark energy in which neutrinos interact with the scalar field sup-posed to be responsible for the acceleration of the universe usually imply avariation of the neutrino masses on cosmological time scales. In this work wepropose a parameterization for the neutrino mass variation that captures theessentials of those scenarios and allows to constrain them in a model indepen-dent way, that is, without resorting to any particular scalar field model. Us-ing WMAP 5yr data combined with the matter power spectrum of SDSS and2dFGRS, the limit on the present value of the neutrino mass is m0 ≡ mν(z =

0) < 0.43 (0.28) eV at 95% C.L. for the case in which the neutrino mass waslighter (heavier) in the past, a result competitive with the ones imposed forstandard (i.e., constant mass) neutrinos. Moreover, for the ratio of the massvariation of the neutrino mass ∆mν over the current mass m0 we found thatlog[|∆mν|/m0] < −1.3 (−2.7) at 95% C.L. for ∆mν < 0 (∆mν > 0), totallyconsistent with no mass variation.

21. A.G. Aksenov, R. Ruffini and G.V. Vereshchagin, “Thermalization ofnonequilibrium electron-positron-photon plasmas”, Physical ReviewLetters, Vol. 99 (2007) No 12, 125003.

Starting from a nonequilibrium configuration we analyze the role of the di-rect and the inverse binary and triple interactions in reaching thermal equi-librium in a homogeneous isotropic pair plasma. We focus on energies in therange 0.1− 10 MeV. We numerically integrate the relativistic Boltzmann equa-tion with the exact QED collisional integrals taking into account all binary andtriple interactions. We show that first, when a detailed balance is reached forall binary interactions on a time scale tk < 10−14 sec, photons and electron-positron pairs establish kinetic equilibrium. Subsequently, when triple inter-

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actions satisfy the detailed balance on a time scale teq < 10−12 sec, the plasmareaches thermal equilibrium. It is shown that neglecting the inverse triple in-teractions prevents reaching thermal equilibrium. Our results obtained in thetheoretical physics domain also find application in astrophysics and cosmol-ogy.

22. C.L. Bianco, R. Ruffini, G.V. Vereshchagin and S.-S. Xue, “Equations ofMotion and Initial and Boundary Conditions for Gamma-ray Burst”,Journal of the Korean Physical Society, Vol. 49 (2006) No. 2, pp. 722-731.

We compare and contrast the different approaches to the optically thick adia-batic phase of GRB all the way to the transparency. Special attention is givento the role of the rate equation to be self consistently solved with the rela-tivistic hydrodynamic equations. The works of Shemi and Piran (1990), Piran,Shemi and Narayan (1993), Meszaros, Laguna and Rees (1993) and Ruffini,Salmonson, Wilson and Xue (1999,2000) are compared and contrasted. The roleof the baryonic loading in these three treatments is pointed out. Constraintson initial conditions for the fireball produced by electro-magnetic black holeare obtained.

23. P. Singh, K. Vandersloot and G.V. Vereshchagin, “Nonsingular bouncinguniverses in loop quantum cosmology”, Physical Review D, Vol. 74(2006) 043510.

Nonperturbative quantum geometric effects in loop quantum cosmology(LQC) predict a ρ2 modification to the Friedmann equation at high energies.The quadratic term is negative definite and can lead to generic bounces whenthe matter energy density becomes equal to a critical value of the order ofthe Planck density. The nonsingular bounce is achieved for arbitrary matterwithout violation of positive energy conditions. By performing a qualitativeanalysis we explore the nature of the bounce for inflationary and cyclic modelpotentials. For the former we show that inflationary trajectories are attractorsof the dynamics after the bounce implying that inflation can be harmoniouslyembedded in LQC. For the latter difficulties associated with singularities incyclic models can be overcome. We show that nonsingular cyclic models canbe constructed with a small variation in the original cyclic model potential bymaking it slightly positive in the regime where scalar field is negative.

24. M. Lattanzi, R. Ruffini and G.V. Vereshchagin, “Joint constraints on thelepton asymmetry of the Universe and neutrino mass from the Wilkin-son Microwave Anisotropy Probe”, Physical Review D, Vol. 72 (2005)063003.

We use the Wilkinson Microwave Anisotropy Probe (WMAP) data on the spec-trum of cosmic microwave background anisotropies to put constraints on thepresent amount of lepton asymmetry L, parametrized by the dimensionless

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4.2. Publications (2005 – 2010)

chemical potential (also called degeneracy parameter) xi and on the effectivenumber of relativistic particle species. We assume a flat cosmological modelwith three thermally distributed neutrino species having all the same massand chemical potential, plus an additional amount of effectively massless ex-otic particle species. The extra energy density associated to these species isparametrized through an effective number of additional species ∆Nothers

e f f .We find that 0 < |ξ| < 1.1 and correspondingly 0 < |L| < 0.9 at 2σ, so thatWMAP data alone cannot firmly rule out scenarios with a large lepton number;moreover, a small preference for this kind of scenarios is actually found. Wealso discuss the effect of the asymmetry on the estimation of other parametersand, in particular, of the neutrino mass. In the case of perfect lepton symmetry,we obtain the standard results. When the amount of asymmetry is left free, wefind at 2sigma. Finally we study how the determination of |L| is affected by

the assumptions on ∆Ne f fothers. We find that lower values of the extra energy

density allow for larger values of the lepton asymmetry, effectively ruling out,

at 2sigma level, lepton symmetric models with ∆Ne f fothers ' 0.

25. G.V. Vereshchagin, “Gauge Theories of Gravity with the Scalar Field inCosmology”, in “Frontiers in Field Theory”, edited by O. Kovras, NovaScience Publishers, New York, (2005), pp. 213-255 (ISBN: 1-59454-127-2).

Brief introduction into gauge theories of gravity is presented. The most generalgravitational lagrangian including quadratic on curvature, torsion and non-metricity invariants for metric-affine gravity is given. Cosmological implica-tions of gauge gravity are considered. The problem of cosmological singularityis discussed within the framework of general relativity as well as gauge theo-ries of gravity. We consider the role of scalar field in connection to this prob-lem. Initial conditions for nonsingular homogeneous isotropic Universe filledby single scalar field are discussed within the framework of gauge theories ofgravity. Homogeneous isotropic cosmological models including ultrarelativis-tic matter and scalar field with gravitational coupling are investigated. Weconsider different symmetry states of effective potential of the scalar field, inparticular restored symmetry at high temperatures and broken symmetry. Ob-tained bouncing solutions can be divided in two groups, namely nonsingularinflationary andoscillating solutions. It is shown that inflationary solutions exist for quite gen-eral initial conditions like in the case of general relativity. However, the phasespace of the dynamical system, corresponding to the cosmological equationsis bounded. Violation of the uniqueness of solutions on the boundaries of thephase space takes place. As a result, it is impossible to define either the pastor the future for a given solution. However, definitely there are singular solu-tions and therefore the problem of cosmological singularity cannot be solvedin models with the scalar field within gauge theories of gravity.

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26. R. Ruffini, M. G. Bernardini, C. L. Bianco, L. Caito, P. Chardonnet, M.G. Dainotti, F. Fraschetti, R. Guida, M. Rotondo, G. Vereshchagin, L.Vitagliano, S.-S. Xue,”The Blackholic energy and the canonical Gamma-Ray Burst” in Cos-mology and Gravitation: XIIth Brazilian School of Cosmology andGravitation, edited by M. Novello and S.E. Perez Bergliaffa, AIP Con-ference Proceedings, Vol. 910, Melville, New York, 2007, pp. 55-217.

Gamma-Ray Bursts (GRBs) represent very likely “the” most extensive com-putational, theoretical and observational effort ever carried out successfullyin physics and astrophysics. The extensive campaign of observation fromspace based X-ray and γ-ray observatory, such as the Vela, CGRO, Bep-poSAX, HETE-II, INTEGRAL, Swift, R-XTE, Chandra, XMM satellites, havebeen matched by complementary observations in the radio wavelength (e.g.by the VLA) and in the optical band (e.g. by VLT, Keck, ROSAT). The netresult is unprecedented accuracy in the received data allowing the determina-tion of the energetics, the time variability and the spectral properties of theseGRB sources. The very fortunate situation occurs that these data can be con-fronted with a mature theoretical development. Theoretical interpretation ofthe above data allows progress in three different frontiers of knowledge: a) theultrarelativistic regimes of a macroscopic source moving at Lorentz gammafactors up to ∼ 400; b) the occurrence of vacuum polarization process verify-ing some of the yet untested regimes of ultrarelativistic quantum field theo-ries; and c) the first evidence for extracting, during the process of gravitationalcollapse leading to the formation of a black hole, amounts of energies up to1055 ergs of blackholic energy — a new form of energy in physics and as-trophysics. We outline how this progress leads to the confirmation of threeinterpretation paradigms for GRBs proposed in July 2001. Thanks mainly tothe observations by Swift and the optical observations by VLT, the outcome ofthis analysis points to the existence of a “canonical” GRB, originating from avariety of different initial astrophysical scenarios. The communality of theseGRBs appears to be that they all are emitted in the process of formation of ablack hole with a negligible value of its angular momentum. The followingsequence of events appears to be canonical: the vacuum polarization processin the dyadosphere with the creation of the optically thick self acceleratingelectron-positron plasma; the engulfment of baryonic mass during the plasmaexpansion; adiabatic expansion of the optically thick “fireshell” of electron-positron-baryon plasma up to the transparency; the interaction of the accel-erated baryonic matter with the interstellar medium (ISM). This leads to thecanonical GRB composed of a proper GRB (P-GRB), emitted at the momentof transparency, followed by an extended afterglow. The sole parameters inthis scenario are the total energy of the dyadosphere Edya, the fireshell baryonloading MB defined by the dimensionless parameter B = MBc2/Edya, and theISM filamentary distribution around the source. In the limit B −→ 0 the total

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energy is radiated in the P-GRB with a vanishing contribution in the afterglow.In this limit, the canonical GRBs explain as well the short GRBs. In these lec-ture notes we systematically outline the main results of our model comparingand contrasting them with the ones in the current literature. In both cases, wehave limited ourselves to review already published results in refereed pub-lications. We emphasize as well the role of GRBs in testing yet unexploredgrounds in the foundations of general relativity and relativistic field theories.

27. M. Lattanzi, R. Ruffini and G.V. Vereshchagin, ”Do WMAP data con-straint the lepton asymmetry of the Universe to be zero?” in Albert Ein-stein Century International Conference, edited by J.-M. Alimi, and A.Fuzfa, AIP Conference Proceedings, Vol. 861, Melville, New York, 2006,pp.912-919.

It is shown that extended flat ΛCDM models with massive neutrinos, a size-able lepton asymmetry and an additional contribution to the radiation contentof the Universe, are not excluded by the Wilkinson Microwave AnisotropyProbe (WMAP) first year data. We assume a flat cosmological model withthree thermally distributed neutrino species having all the same mass andchemical potential, plus an additional amount of effectively massless exoticparticle species X. After maximizing over seven other cosmological parame-ters, we derive from WMAP first year data the following constraints for thelepton asymmetry L of the Universe (95% CL): 0 < |L| < 0.9, so that WMAPdata alone cannot firmly rule out scenarios with a large lepton number; more-over, a small preference for this kind of scenarios is actually found. We alsofind for the neutrino mass mν < 1.2eV and for the effective number of rela-tivistic particle species −0.45 < ∆Ne f f < 2.10, both at 95% CL. The limit on∆Ne f f is more restrictive man others found in the literature, but we argue thatthis is due to our choice of priors.

28. R. Ruffini, C.L. Bianco, G.V. Vereshchagin, S.-S. Xue “Baryonic loadingand e+e− rate equation in GRB sources” to appear in the proceedingsof ”Relativistic Astrophysics and Cosmology - Einstein’s Legacy” Meet-ing, November 7-11, 2005, Munich, Germany.

The expansion of the electron-positron plasma in the GRB phenomenon iscompared and contrasted in the treatments of Meszaros, Laguna and Rees, ofShemi, Piran and Narayan, and of Ruffini et al. The role of the correct numeri-cal integration of the hydrodynamical equations, as well as of the rate equationfor the electron-positron plasma loaded with a baryonic mass, are outlined andconfronted for crucial differences.

29. G.V. Vereshchagin, M. Lattanzi, H.W. Lee, R. Ruffini, ”Cosmologicalmassive neutrinos with nonzero chemical potential: I. Perturbations incosmological models with neutrino in ideal fluid approximation”, in

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proceedings of the Xth Marcel Grossmann Meeting on Recent Develop-ments in Theoretical and Experimental General Relativity, World Scien-tific: Singapore, 2005, vol. 2, pp. 1246-1248.

Recent constraints on neutrino mass and chemical potential are discussed withapplication to large scale structure formation. Power spectra in cosmologi-cal model with hot and cold dark matter, baryons and cosmological term arecalculated in newtonian approximation using linear perturbation theory. Allcomponents are considered to be ideal fluids. Dissipative processes are takeninto account by initial spectrum of perturbations so the problem is reduced toa simple system of equations. Our results are in good agreement with thoseobtained before using more complicated treatments.

30. M. Lattanzi, H.W. Lee, R. Ruffini, G.V. Vereshchagin, ”Cosmologicalmassive neutrinos with nonzero chemical potential: II. Effect on the es-timation of cosmological parameters”, in proceedings of the Xth MarcelGrossmann Meeting on Recent Developments in Theoretical and Exper-imental General Relativity, World Scientific: Singapore, 2005, vol. 2, pp.1255-1257.

The recent analysis of the cosmic microwave background data carried out bythe WMAP team seems to show that the sum of the neutrino mass is ¡ 0.7 eV.However, this result is not model-independent, depending on precise assump-tions on the cosmological model. We study how this result is modified whenthe assumption of perfect lepton symmetry is dropped out.

31. R. Ruffini, M. Lattanzi and G. Vereshchagin, ”On the possible role ofmassive neutrinos in cosmological structure formation” in Cosmologyand Gravitation: Xth Brazilian School of Cosmology and Gravitation,edited by M. Novello and S.E. Perez Bergliaffa, AIP Conference Pro-ceedings, Vol. 668, Melville, New York, 2003, pp.263-287.

In addition to the problem of galaxy formation, one of the greatest open ques-tions of cosmology is represented by the existence of an asymmetry betweenmatter and antimatter in the baryonic component of the Universe. We believethat a net lepton number for the three neutrino species can be used to under-stand this asymmetry. This also implies an asymmetry in the matter-antimattercomponent of the leptons. The existence of a nonnull lepton number for theneutrinos can easily explain a cosmological abundance of neutrinos consistentwith the one needed to explain both the rotation curves of galaxies and theflatness of the Universe. Some propedeutic results are presented in order toattack this problem.

32. A.G. Aksenov, C.L. Bianco, R. Ruffini and G.V. Vereshchagin, “GRBsand the thermalization process of electron-positron plasmas” in the Pro-ceedings of the ”Gamma Ray Bursts 2007” meeting, AIP Conf.Proc.1000 (2008) 309-312.

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We discuss temporal evolution of the pair plasma, created in Gamma-RayBursts sources. A particular attention is paid to the relaxation of plasma intothermal equilibrium. We also discuss the connection between the dynamics ofexpansion and spatial geometry of plasma. The role of the baryonic loadingparameter is emphasized.

33. A. G. Aksenov, R. Ruffini, and G. V. Vereshchagin, “Thermalization ofElectron-Positron-Photon Plasmas with an Application to GRB” in REL-ATIVISTIC ASTROPHYSICS: 4th Italian-Sino Workshop, AIP Confer-ence Proceedings, Vol. 966, Melville, New York, 2008, pp. 191-196.

The pair plasma with photon energies in the range 0.1− 10MeV is believedto play crucial role in cosmic Gamma-Ray Bursts. Starting from a nonequilib-rium configuration we analyze the role of the direct and the inverse binary andtriple interactions in reaching thermal equilibrium in a homogeneous isotropicpair plasma.We numerically integrate the relativistic Boltzmann equation withthe exact QED collisional integrals taking into account all binary and triple in-teractions. We show that first, when a detailed balance is reached for all bi-nary interactions on a time scale tk= 10−14sec , photons and electronpositronpairs establish kinetic equilibrium. Subsequently, when triple interactions sat-isfy the detailed balance on a time scale teq= 10−12sec , the plasma reachesthermal equilibrium. It is shown that neglecting the inverse triple interactionsprevents reaching thermal equilibrium. Our results obtained in the theoreticalphysics domain also find application in astrophysics and cosmology.

34. R. Ruffini, G. V. Vereshchagin and S.-S. Xue, “Vacuum Polarizationand Electron-Positron Plasma Oscillations” in RELATIVISTIC ASTRO-PHYSICS: 4th Italian-Sino Workshop, AIP Conference Proceedings, Vol.966, Melville, New York, 2008, pp. 207-212.

We study plasma oscillations of electrons-positron pairs created by the vacuumpolarization in an uniform electric field. Our treatment, encompassing thecase of E > Ec, shows also in the case E < Ecthe existence of a maximumLorentz factor acquired by electrons and positrons and allows determinationof the a maximal length of oscillation. We quantitatively estimate how plasmaoscillations reduce the rate of pair creation and increase the time scale of thepair production.


1. A. Benedetti, W.-B. Han, R. Ruffini and G.V. Vereshchagin, “On the fre-quency of oscillations in the pair plasma generated by a strong electricfield”, Physics Letters B, Vol. 698 (2011) 75-79.

We study the frequency of the plasma oscillations of electron-positron pairscreated by the vacuum polarization in a uniform electric field with strength E

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in the range 0.2Ec < E < 10Ec. Following the approach adopted in Ruffini etal. (2007) we work out one second order ordinary differential equation for avariable related to the velocity from which we can recover the classical plasmaoscillation equation when E→0. Thereby, we focus our attention on its evo-lution in time studying how this oscillation frequency approaches the plasmafrequency. The time-scale needed to approach to the plasma frequency andthe power spectrum of these oscillations are computed. The characteristic fre-quency of the power spectrum is determined uniquely from the initial valueof the electric field strength. The effects of plasma degeneracy and pair anni-hilation are discussed.

2. B. Patricelli, M.G. Bernardini, C.L. Bianco, L. Caito, L. Izzo, R. Ruffiniand G.V. Vereshchagin, “A New Spectral Energy Distribution of Pho-tons in the Fireshell Model of GRBs”, International Journal of ModernPhysics D, Vol. 20 (2011) 1983-1987.

The analysis of various Gamma-Ray Bursts (GRBs) having a low energeticswithin the fireshell model has shown how the N(E) spectrum of their promptemission can be reproduced in a satisfactory way by a convolution of ther-mal spectra. Nevertheless, from the study of very energetic bursts such as,for example, GRB 080319B, some discrepancies between the numerical simu-lations and the observational data have been observed. We investigate a dif-ferent spectrum of photons in the comoving frame of the fireshell in order tobetter reproduce the spectral properties of GRB prompt emission within thefireshell model. We introduce a phenomenologically modified thermal spec-trum: a thermal spectrum characterized by a different asymptotic power-lawindex in the low energy region. Such an index depends on a free parameter α,so that the pure thermal spectrum corresponds to the case α = 0. We test thisspectrum by comparing the numerical simulations with the observed promptemission spectra of various GRBs. From this analysis it has emerged that theobservational data can be correctly reproduced by assuming a modified ther-mal spectrum with α = −1.8.

3. R. Ruffni, I. A. Siutsou and G. V. Vereshchagin, “Theory of photosphericemission from relativistic outflows”, submitted to the AstrophysicalJournal (2011).

Two popular models of optically thick relativistic outflows exist: the wind andthe shell. In this paper the optical depth during the acceleration and coastingphases is computed analytically within both models and its various asymp-totic limits are derived. In particular we show that quite counterintuitively ageometrically thin shell may appear as a thick wind for photons propagatinginside it. For this reason we introduce notions of photon thick and photon thinoutflows, which appear more general and better physically motivated withrespect to winds and shells.

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We study the geometry of dynamic photospheres emerging from relativisticoutflows. The photosphere of photon thin outflow has always a convex shape.In the photon thick one it is initially convex since there is always a photon thinlayer in any outflow, and then it becomes concave. Assuming that photonsare emitted with comoving thermal spectrum at the moment when the opti-cal depth decreases to unity we compute the observed flux and instantaneousspectra for both photon thick and photon thin outflows. It is our main findingthat the photospheric emission from the photon thin outflow produces nonthermal time integrated spectra, which may be described by the Band functionwell known in the GRB literature.

We find that energetic GRBs should produce photon thin outflows and onlytime integrated spectra may be observed from such GRBs. In other words, ob-served Band spectrum is obtained quite naturally from the comoving thermalone by the integration over the photosphere.

4. Stefania Pandolfi, Andrea Ferrara, T. Roy Choudhury, Alessandro Mel-chiorri, Sourav Mitra, ”Data-constrained reionization and its effect oncosmological parameters”, submitted to PRD, arXiv:1111.3570v1 [astro-ph.CO]

We perform an analysis of the recent WMAP7 data considering physically mo-tivated and viable reionization scenarios with the aim of assessing their ef-fects on cosmological parameter determinations. The main novelties are: (i)the combination of CMB data with astrophysical results from quasar absorp-tion line experiments; (ii) the joint variation of both the cosmological and as-trophysical [governing the evolution of the free electron fraction x e(z)] pa-rameters. Including a realistic, data-constrained reionization history in theanalysis induces appreciable changes in the cosmological parameter valuesdeduced through a standard WMAP7 analysis. Particularly noteworthy arethe variations in Ωbh2 = 0.02258 + 0.00057− 0.00056 (WMAP7) vs. Ωbh2 =

0.02183± 0.00054 (WMAP7 + ASTRO), and the new constraints for the scalarspectral index, for which WMAP7 + ASTRO excludes the Harrison-Zel’dovichvalue ns = 1 at more than 3σ. Finally, the e.s. optical depth value is consider-ably decreased with respect to the standard WMAP7, i.e. τe = 0.080± 0.012.We conclude that the inclusion of astrophysical datasets, allowing to robustlyconstrain the reionization history, in the extraction procedure of cosmologicalparameters leads to relatively important differences in the final determinationof their values.

5. Elena Giusarma, Martina Corsi, Maria Archidiacono, Roland de Putter,Alessandro Melchiorri, Olga Mena, Stefania Pandolfi. ”Constraints onmassive sterile neutrino species from current and future cosmologicaldata”, Phys.Rev. D83, 115023 (2011)

Sterile massive neutrinos are a natural extension of the standard model of ele-mentary particles. The energy density of the extra sterile massive states affects

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cosmological measurements in an analogous way to that of active neutrinospecies. We perform here an analysis of current cosmological data and derivebounds on the masses of the active and the sterile neutrino states, as well ason the number of sterile states. The so-called (3+2) models, with three sub-eV active massive neutrinos plus two sub-eV massive sterile species, is wellwithin the 95% CL allowed regions when considering cosmological data only.If the two extra sterile states have thermal abundances at decoupling, big bangnucleosynthesis bounds compromise the viability of (3+2) models. Forecastsfrom future cosmological data on the active and sterile neutrino parametersare also presented. Independent measurements of the neutrino mass from tri-tium beta-decay experiments and of the Hubble constant could shed light onsub-eV massive sterile neutrino scenarios.

6. M. Archidiacono, A. Melchiorri, S. Pandolfi, ”The impact of Reioniza-tion modelling on CMB Neutrino Mass Bounds”, Nuclear Physics B,Proceedings Supplements, Volume 217, Issue 1, p. 65-67. (2011)

We investigate the bounds on the neutrino mass in a general reionization sce-nario based on a principal component approach. We found the constraint onthe sum of the neutrino masses from CMB data can be relaxed by a ∼ 40 % ina generalized reionization scenario.

7. Erminia Calabrese, Eloisa Menegoni, C. J. A. P. Martins, AlessandroMelchiorri, and Graca Rocha, ”Constraining variations in the fine struc-ture constant in the presence of early dark energy”, Phys.Rev. D84(2011) 023518.

We discuss present and future cosmological constraints on variations of thefine structure constant α induced by an early dark energy component hav-ing the simplest allowed (linear) coupling to electromagnetism. We find thatcurrent cosmological data show no variation of the fine structure constant atrecombination respect to the present-day value, with α/α0 = 0.975± 0.020 at95% c.l., constraining the energy density in early dark energy to Ωe < 0.060 at95% c.l. Moreover, we consider constraints on the parameter quantifying thestrength of the coupling by the scalar field. We find that current cosmologicalconstraints on the coupling are about 20 times weaker than those obtainablelocally (which come from Equivalence Principle tests). However forthcomingor future missions, such as Planck Surveyor and CMBPol, can match and pos-sibly even surpass the sensitivity of current local tests.

8. Micol Benetti, Massimiliano Lattanzi, Erminia Calabrese, AlessandroMelchiorri, ”Features in the primordial spectrum: new constraints fromWMAP7+ACT data and prospects for Planck”, Phys. Rev. D 84, 063509(2011)

We update the constraints on possible features in the primordial inflationarydensity perturbation spectrum by using the latest data from the WMAP7 and

44

4.4. Invited talks at international conferences

ACT Cosmic Microwave Background experiments. The inclusion of new datasignificantly improves the constraints with respect to older work, especially tosmaller angular scales. While we found no clear statistical evidence in the datafor extensions to the simplest, featureless, inflationary model, models witha step provide a significantly better fit than standard featureless power-lawspectra. We show that the possibility of a step in the inflationary potentiallike the one preferred by current data will soon be tested by the forthcomingtemperature and polarization data from the Planck satellite mission.

9. Stefania Pandolfi, Elena Giusarma, Edward W. Kolb, MassimilianoLattanzi, Alessandro Melchiorri, Olga Mena, Manuel Pena, AsanthaCooray, Paolo Serra, ”Impact of general reionization scenarios on ex-traction of inflationary parameters”, Phys.Rev. D82, 123527, (2010).

Determination of whether the Harrison–Zel’dovich spectrum for primordialscalar perturbations is consistent with observations is sensitive to assumptionsabout the reionization scenario. In light of this result, we revisit constraintson inflationary models using more general reionization scenarios. While thebounds on the tensor-to-scalar ratio are largely unmodified, when differentreionization schemes are addressed, hybrid models are back into the inflation-ary game. In the general reionization picture, we reconstruct both the shapeand amplitude of the inflaton potential. We find a broader spectrum of poten-tial shapes when relaxing the simple reionization restriction. An upper limit of1016 GeV to the amplitude of the potential is found, regardless of the assump-tions on the reionization history.


1. ”Photospheric emission from thermally accelerated relativistic out-flows”

GRBs, their progenitors and the role of thermal emission, Les Houches,France, 2-7 October, 2011

2. “Thermalization of the pair plasma”

(G.V. Vereshchagin, A.G. Aksenov and R. Ruffini)

From Nuclei to White Dwarfs and Neutron Stars, Les Houches, France,3-8 April, 2011

3. ”Photospheric emission from relativistic outflows: 1DHD”

(G.V. Vereshchagin, R. Ruffini and I.A. Siutsou)

Recent News from the MeV, GeV and TeV Gamma-Ray Domains,Pescara, Italy, 21-26 March, 2011

45

4. Publications

4. ”Thermalization of degenerate electron-positron plasma”

I.A. Siutsou, A.G. Aksenov, R. Ruffini and G.V. Vereshchagin

IRAP Ph.D. Erasmus Mundus School—May 27, 2011, Nice, France

5. ”Semidegenerate self-gravitating systems of fermions as central objectsand dark matter halos in galaxies”

(I. A. Siutsou, A. Geralico and R. Ruffini)

Recent News from the MeV, GeV and TeV Gamma-Ray Domains, March24, 2011, Pescara, Italy

6. ”Thermalization of degenerate electron-positron plasma”

(I.A. Siutsou, A.G. Aksenov, G.V. Vereshchagin and R. Ruffini)

3rd Galileo-Xu Guangqi Meeting—October 12, 2011, Beijing, China

7. ”Photospheric emission of relativistically expanding outflows”

(I.A. Siutsou, G.V. Vereshchagin and R. Ruffini)

12th Italian-Korean Symposium on Relativistic Astrophysics—July 5,2011, Pescara, Italy

8. On the frequency of oscillations in the pair plasma generated by astrong electric field.

(Alberto Benedetti, W.-B. Han, R. Ruffini, G.V. Vereshchagin)IRAP Ph.D. Erasmus Mundus Workshop, April 5, 2011, Pescara (Italy)

9. Oscillations in the pair plasma generated by a strong electric field

(Alberto Benedetti, W.-B. Han, R. Ruffini, G.V. Vereshchagin)Italian-Korean Meeting, July 4-9, 2011, Pescara (Italy)

10. Electron-Positron plasma oscillations: hydro-electrodynamic and ki-netic approaches

(Alberto Benedetti, R. Ruffini, G.V. Vereshchagin)IRAP Ph.D. Erasmus Mundus School, September 7, 2011, Nice (France)

11. Boltzmann equation: from an interacting plasma toward the photo-spheric emission of a GRB

(Alberto Benedetti, A. Aksenov, R. Ruffini, I. Siotsou, G.V. Vereshcha-gin)IRAP Ph.D. Erasmus Mundus Workshop, October 6, 2011, Les Houches(France)

46


12. Electron-Positron plasma oscillations: hydro-electrodynamic and ki-netic approaches.

(Alberto Benedetti, A. Aksenov, R. Ruffini, I. Siutsou, G.V. Vereshcha-gin)Galileo-Xu Guanqui Meeting, October 12, 2011, Beijing (China)

13. “Inflation in a general reionization scenario ”

(S. Pandolfi)

Essential Cosmology for the Next Generation, Puerto Vallarta , Mexico,January 10-14, 2011

14. “Constraints on Inflation in extended cosmological scenarios ”

(S. Pandolfi)

28 January 2011, Dark Cosmlogy Center, Copenhagen, Denmark.

15. “Theoretical Development toward the Planck mission ”

(S. Pandolfi)

IRAP PhD and Erasmus MundusWorkshop: Workshop on RecentNews from the GeV and TeV Gamma-RayDomains: Results andInterpretations,21-26 March 2011, ICRANet (Pescara), Italy.

16. “Joint Astrophysical and Cosmological constrains on reionization ”

(S. Pandolfi)

DAVID WORKSHOP VI, Scuola Normale Superiore, Pisa, October 18-20 2011

17. “Features in the spectrum of primordial perturbations: new constraintsfrom WMAP7+ACT data and prospects for Planck ”

(M. Lattanzi)

12th Italian-Korean Symposium on Relativistic Astrophysics, Pescara(Italy), July 4-8, 2011.

18. “New constraints on features in the primordial spectrum ”

(M. Benetti)

3rd Galileo- Xu Guangqi meeting, Beijing (China), October 11-15, 2011.


(G.V. Vereshchagin with A.G. Aksenov and R. Ruffini)

Korean Physical Society 2010 Fall Meeting, Pyeong-chang, Korea, 20-22October, 2010.

47

4. Publications

20. “The spatial structure of expanding optically thick relativistic plasmaand the onset of GRBs”

(G.V. Vereshchagin with A.G. Aksenov, G. de Barros and R. Ruffini)

GRB 2010 / Dall’eV al TeV tutti i colori dei GRB, Secondo CongressoItaliano sui Gamma-ray Burst, Cefalu’ 15-18 June 2010.

21. “From thermalization mechanisms to emission processes in GRBs”

(G.V. Vereshchagin)

XII Marcel Grossmann Meeting, Paris, 12-18 July 2009.

22. “Kinetics of the mildly relativistic plasma and GRBs”

(A.G. Aksenov R. Ruffini, and G.V. Vereshchagin)

“The Sun, the Stars, the Universe, and General Relativity” - Interna-tional conference in honor of Ya. B. Zeldovich 95th Anniversary, Minsk,Belarus, April 19-23, 2009.

23. “Pair plasma around compact astrophysical sources: kinetics, electro-dynamics and hydrodynamics”

(G.V. Vereshchagin and R. Ruffini)

Invited seminar at RMKI, Budapest, February 24, 2009.

24. “Thermalization of the pair plasma with proton loading”

(G.V. Vereshchagin, R. Ruffini, and A.G. Aksenov)

Probing Stellar Populations out to the Distant Universe, Cefalu’, Italy,September 7-19, 2008.

25. “Thermalization of the pair plasma with proton loading”


3rd Stueckelberg Workshop, Pescara, Italy, 8-18 July, 2008.



27. “Non-singular solutions in Loop Quantum Cosmology”

(G.V. Vereshchagin)

2nd Stueckelberg Workshop, Pescara, Italy, 3-7 September, 2007.

28. “(From) massive neutrinos and inos and the upper cutoff to the fractalstructure of the Universe (to recent progress in theoretical cosmology)”

(G.V. Vereshchagin, M. Lattanzi and R. Ruffini)

A Century of Cosmology, San Servolo, Venice, Italy, 27-31 August, 2007.

48


29. “Pair creation and plasma oscillations”

(G.V. Vereshchagin, R. Ruffini, and S.-S. Xue)4th Italian-Sino Workshop on Relativistic Astrophysics, Pescara, Italy,20-29 July, 2007.

30. “Thermalization of electron-positron plasma in GRB sources”

(G.V. Vereshchagin, R. Ruffini, and A.G. Aksenov)Xth Italian-Korean Symposium on Relativistic Astrophysics, Pescara,Italy, 25-30 June, 2007.

31. “Kinetics and hydrodynamics of the pair plasma”

(G.V. Vereshchagin, R. Ruffini, C.L. Bianco, A.G. Aksenov)

32. “Pair creation and plasma oscillations”

(G.V. Vereshchagin, R. Ruffini and S.-S. Xue)Cesare Lattes Meeting on GRBs, Black Holes and Supernovae,Mangaratiba-Portobello, Brazil, 26 February - 3 March 2007.

33. “Cavallo-Rees classification revisited”

(G.V. Vereshchagin, R.Ruffini and S.-S. Xue)

On recent developments in theoretical and experimental general rela-tivity, gravitation and relativistic field theories: XIth Marcel GrossmannMeeting, Berlin, Germany, 23-29 July, 2006.

34. “Kinetic and thermal equilibria in the pair plasma”

(G.V. Vereshchagin)

The 1st Bego scientific rencontre, Nice, 5-16 February 2006.

35. “From semi-classical LQC to Friedmann Universe”

(G.V. Vereshchagin)

Loops ’05, Potsdam, Golm, Max-Plank Institut fur Gravitationsphysik(Albert-Einstein-Institut), 10-14 October 2005.

36. “Equations of motion, initial and boundary conditions for GRBs”

(G.V. Vereshchagin, R. Ruffini and S.-S. Xue)

IXth Italian-Korean Symposium on Relativistic Astrophysics, Seoul, Mt.Kumgang, Korea, 19-24 July 2005.

37. “On the Cavallo-Rees classification and GRBs”

(G.V. Vereshchagin, R. Ruffini and S.-S. Xue)

II Italian-Sino Workshop on Relativistic Astrophysics, Pescara, Italy, 10-20 June, 2005.

49

4. Publications

38. “Primordial gravitional waves as a probe of the cosmic expansion his-tory”

(M. Lattanzi)

16th International Symposium on Particles, Strings and Cosmology, Va-lencia, Spain, 19-23 July 2010.

39. “Detecting Signatures of the Cosmic Thermal History through PulsarObservations”

(M. Lattanzi)

14th Gravitational Waves Data Analysis Workshop, Rome, Italy, 26-29January 2010.

40. “On the Propagation of Gravitational Waves across the Universe: Inter-action with the Neutrino Component”

(M. Lattanzi)

2nd Italian-Pakistani Workshop on Relativistic Astrophysics, Pescara,Italy, 8-10 July 2009.

41. “Enhancement of the Darl Matter Annihilation Cross-Section in ColdSubstructures”

(M. Lattanzi)

12th Marcel Grossmann Meeting on General Relativity, Paris, France,12-18 July 2009.

42. “On the Propagation of Gravitational Waves across the Universe: Inter-action with the Neutrino Component”

(M. Lattanzi)

12th Marcel Grossmann Meeting on General Relativity, Paris, France,12-18 July 2009.

43. “Constraining Dark Matter Models Through 21cm Observations”

(M. Lattanzi)

2nd Universenet School and Meeting, Oxford, UK, 22-26 September2008.

44. “Constraints on Mass-Varying Neutrino Scenarios”

(M. Lattanzi)

Neutrino Oscillation Workshop 2008, Otranto (Lecce), Italy, 6-13September 2008.

50


45. “Constraining Dark Matter Models Through 21cm Observations”

(M. Lattanzi)

3rd Stueckelberg Workshop on Quantum Field Theories, Pescara, Italy,8-18 July 2008.

46. “Cosmological Constraints on Neutrino Physics”

(M. Lattanzi)

Theta13 Half Day Meeting, Oxford, UK, 24 September 2007.

47. “Decaying warm dark matter, neutrino masses and the cosmic mi-crowave background”

(M. Lattanzi)

2nd Meeting of the “Red Nacional Tem‡tica de Astroparticulas” (RE-NATA), Valencia, Spain, 17-19 September 2007.

48. “Decaying majoron dark matter and neutrino masses”

(M. Lattanzi)

Workshop “The Path to Neutrino Mass”, Aarhus, Denmark, 3-6September 2007.


(M. Lattanzi)

4rd Italian-Sino Workshop on Relativistic Astrophysics, Pescara, Italy,20-30 July 2007.


(M. Lattanzi)

10th Italian-Korean Symposium on Relativistic Astrophysics, Pescara,Italy, 25-30 June 2007.

51. “Constraints on the neutrino asymmetry of the Universe from cosmo-logical data”

(M. Lattanzi)

11th Marcel Grossmann Meeting, Berlin, Germany, 23-29 July 2006.

52. “Effect of cosmological neutrinos on the propagation of primordialgravitational waves”

(M. Lattanzi)

11th Marcel Grossmann Meeting, Berlin, Germany, 23-29 July 2006.

51

4. Publications

53. “Does WMAP data constrain the lepton asymmetry of the Universe tobe zero?”

(M. Lattanzi)

“Albert Einstein Century” International Conference, Paris, France, 18-22 July 2005.

54. “On the interaction bewteen relic neutrinos and primordial gravita-tional waves”

(M. Lattanzi)

II Sino-Italian Workshop on Cosmology and Relativistic Astrophysics,Pescara, Italy, 10-20 June 2005.

55. Impact of general reionization scenarios on inflation

(S. Pandolfi)

Horiba International Conference, COSMO/CosPa 2010, 30th September2010, at The University of Tokyo, Tokyo, Japan.

56. Impact of general reionization scenarios on inflation

(S. Pandolfi)

Cosmolo Meeting, 8th September 2010, at IFIC, Instituto de Fisica Cor-puscular, Valencia, Spain.

57. Inflation in general reionization scenarios

(S. Pandolfi)

Summer School in Cosmology, 19-31 July 2010, at ICTP–the AbdusSalam International Centre for Theoretical Physics, Trieste, Italy.

58. Harrison Zel’dovich spectrum is consistent with observation

(S. Pandolfi)

2nd Galileo-XuGuangqi meeting, 12-17 July 2010, Giardini BotaniciHanbury, Ventimiglia, Italy

59. Inflation in a general reionization scenario

(S. Pandolfi)

Xth School of Cosmology, 5-10 July 2010 at IESC, Cargese, Corse, France

60. Harrison-Zel’dovich primordial spectrum is consistent with observa-tions

(S. Pandolfi)

10th Great Lakes Cosmology Workshop, 14-16 June 2010, KICP at theUniversity of Chicago (IL), USA.

52

4.5. Posters

61. Inflation and Reionization

(S. Pandolfi)

62. University of Michigan, 23rd June, Ann Arbor (MI), USA

63. Inflation with the CMB

(S. Pandolfi)

64. Brookhaven National Laboratory, 8th June 2010

65. Inflation in a General Reionization Scenario

(S. Pandolfi)

66. IberiCos2010 (5th Iberian Cosmology Meeting), Porto, Portugal, 29-31March 2010

4.5. Posters

1. “Constraints on the cosmological lepton asymmetry”

(M. Lattanzi)

XIXmes Rencontres de Blois: “Matter and Energy in the Universe: fromnucleosynthesis to cosmology”, Blois, France, 20-25

May, 2007.

2. “The interaction between relic neutrinos and cosmological gravitationalwaves: implication for interferometric detectors”

(M. Lattanzi)

“Albert Einstein Century” International Conference, Paris, France, 18-22 July 2005.

4.6. Lecture courses

1. ”Pair plasma in GRBs and cosmology”

2 lectures, IRAP Ph.D. Erasmus Mundus September school, 12 – 23September, 2011, University of Nice Sophia Antipolis, Nice, France.

2. “Relativistic kinetic theory and its applications in astrophysics and cos-mology”

(G.V. Vereshchagin)

Lecture course for International Relativistic Astrophysics PhD, ErasmusMundus Joint Doctorate Program from the

53

4. Publications

European Commission, September 6-24, 2010, University of NiceSophia Antipolis, Nice, France.

3. “Relativistic kinetic theory and its applications”, IRAP Ph.D. lectures

(G.V. Vereshchagin)

February 1-19, 2010, Observatoire de la Cote d’Azur, Nice, France.

4. Inflationary Constraints and reionization

(S. Pandolfi)

IRAP Ph.D. Lectures in Nice, Observatoire de la Cote d’Azur, 12-16February 2010

54

5. APPENDICES

55

5. APPENDICES

56

A. Bose enhancement and Pauliblocking in the pair plasma

A.1. Introduction

The description of processes involving electron-positron pairs is required inmany phenomena in physics and astrophysics (Ruffini et al., 2010). The stan-dard cosmological model includes lepton era with electron-positron plasmaat high temperature and initially in thermal equilibrium (see e.g. Weinberg(2008)). Strong electromagnetic fields are generated in laser experiments aim-ing at production of electron-positron pairs (Gerstner, 2010; Chen et al., 2009).When electromagnetic field invariants E2 − H2 and E ·H approach criticalvalue vacuum polarization leads to copious pair production ultimately lead-ing to formation of electron-positron plasma (Mustafa and Kampfer, 2009).Strong electromagnetic fields are thought to occur in astrophysical condi-tions, near such compact objects as black holes (Damour and Ruffini, 1975),hypothetical strange stars (Alcock et al., 1986; Usov, 1998) and possibly neu-tron stars (Rotondo et al., 2011).

Relaxation of electron-positron plasma to thermal equilibrium has beenconsidered in Aksenov et al. (2007, 2009). There relativistic Boltzmann equa-tions with exact QED collisional integrals taking into account all relevanttwo-particle (scatterings and pair creation/annihilation, etc.) and three-particle (e.g. relativistic bremsstrahlung, double Compton scattering etc.) in-teractions were solved numerically. It was confirmed that a metastable statecalled ”kinetic equilibrium” (Pilla and Shaham, 1997) exists in such plasma,which is characterized by the same temperature of all particles, but nonnullchemical potentials. Such state occurs when the detailed balance of all two-particle reactions is established. It was pointed out in Aksenov et al. (2007,2009) that direct and inverse 3p interactions are essential in bringing electron-positron plasma to thermal equilibrium. In Aksenov et al. (2010) relaxationtimescales for optically thick electron-positron plasma in a wide range of tem-peratures and proton loadings were computed numerically using the kineticcode developed in Aksenov et al. (2007, 2009). These timescales were pre-viously estimated in the literature by order of magnitude arguments usingthe reaction rates of the dominant processes (Gould, 1981; Stepney, 1983). Itwas shown that these numerically obtained timescales differ from previousestimations by several orders of magnitude.

57

A. Bose enhancement and Pauli blocking in the pair plasma

Notice that temperature range considered in Aksenov et al. (2007, 2009,2010)

0.1 <kT

mec2 < 10 (A.1.1)

was selected in order to avoid production of other particles such as neutrino(Ruffini et al., 2010). In the description of plasma Boltzmann statistics of par-ticles was used in these works.

If the temperature of fermions decreases for a given concentration of par-ticles, they may become degenerate (Landau and Lifshitz, 1980). The samephenomenon occurs when concentration grows, but the temperature is fixed.

The temperature, above which the gas can be described by Boltzmannstatistics, and below which it becomes degenerate, is defined by

θF ≡EF

mec2 ,

where EF is Fermi energy, corresponding to Fermi momentum

pF = h(

3n8π

) 13

.

For relativistic gas of electrons and positrons E2 = p2c2 + m2e c4 and

θF =

[(h

mec

)2( 3n8π

) 23

+ 1

]1/2

− 1.

In this way the degeneracy parameter

D =θ

θF< 1 (A.1.2)

determines the temperature where relativistic degeneracy occurs. Thermalelectron-positron plasma becomes degenerate at kT & 3mec2. It should benoted that the average occupation numbers for kinetic equilibrium state arenot high even in the ultrarelativistic limit with chemical potential µ = 0: 8.7 %for electrons and positrons and 36.8 % for photons.

In this work we extend the previous results on thermalization timescaleof pair plasma, including exact QED treatment of three-particle interactionand quantum corrections to collision integrals by the corresponding Pauliblocking and Bose enhancement factors. In section A.2 relativistic Boltzmannequation with quantum corrections is introduced for both two- and three-particle interactions. In section 3 the details od adopted numerical schemeare given. In section 4 we report our results. Discussion and conclusionsfollow.

58

A.2. Boltzmann equation with QED collisional integrals

A.2. Boltzmann equation with QED collisionalintegrals

In uniform isotropic pair plasma relativistic Boltzmann equation for distribu-tion function fα of the particle specie α, where α stands for electron, positron,and photon, normalized as

nα(t) =∫

fα(~p, t)d3~p, (A.2.1)

where nα is the number density of particles α, has the following form (Ak-senov et al., 2007):

1c

ddt

fα(p, t) = ∑q

(η

qα − χ

qα fα(p, t)

), (A.2.2)

where the sum is taken over all two- and three-particle reactions q, ηqα and χ

qα

are, respectively, the emission and absorption coefficients, c is the speed oflight.

Collisional integrals describe various interactions occurring in relativisticelectron-positron-photon plasma, as listed in Table A.1. Such interactions aregenerally divided in two groups: two-particle interactions and three-particleinteractions.

A.2.1. Two-particle interactions

Consider interaction of two incoming particles of species α and β havingmomenta in the phase space volumes d3pa around pa and d3pb around pb,respectively, resulting in two outgoing particles of species γ and δ havingmomenta in the phase space volumes d3pi around pi and d3pj around pj,respectively. The indexes of particle species will be omitted where possible.Energy and momentum conservations read

εa + εb = εi + εj, ~pa + ~pb = ~pi + ~pj. (A.2.3)

The differential emission and absorption coefficients for this reaction are

χα fα(pa, t)d3pa = χβ fβ(pb, t)d3pb = ηγ(pi, t)d3pi = ηδ(pj, t)d3pj =

d3pid3pjd3pad3pb ×Wpa,pb;pi,pj

× fα(pa, t) fβ(pb, t)(

1± fγ(pi, t)/h3) (

1± fδ(pj, t)/h3)

, (A.2.4)

where W is the transition function and(1± f (p, t)/h3) are respectively Bose

enhancement (sign ”+”) and Pauli blocking (sign ”−”) factors (Uehling and

59


Uhlenbeck, 1933; Uehling, 1934).

In a specific case when the particles α and β undergo scattering from initialstates a and b into new states i and j, respectively, the emission coefficient ofparticle α is

ηα(pi, t) =∫

d3pjd3pad3pb ×Wpa,pb;pi,pj

× fα(pa, t) fβ(pb, t)(

1± fγ(pi, t)/h3) (

1± fδ(pj, t)/h3)

. (A.2.5)

Considering the inverse process, corresponding to exchange of indices a↔ i,b↔ j, we obtain the absorption coefficient

χα fα(pi, t) =∫

d3pjd3pad3pb ×Wpi,pj;pa,pb

× fα(pi, t) fβ(pj, t)(

1± fα(pa, t)/h3) (

1± fβ(pb, t)/h3)

. (A.2.6)

Then Boltzmann equation for particle α reads

1c

ddt

fα(pi, t) =∫

d3pjd3pad3pb

×[

Wpa,pb;pi,pj × fα(pa, t) fβ(pb, t)(

1± fα(pi, t)/h3) (

1± fβ(pj, t)/h3)

−Wpi,pj;pa,pb × fα(pi, t) fβ(pj, t)(

1± fα(pa, t)/h3) (

1± fβ(pb, t)/h3)]

.

(A.2.7)

Taking into account the detailed balance condition Wpa,pb;pi,pj = Wpi,pj;pa,pb ,classical form of Boltzmann equation is recovered

1c

ddt

fα(pi, t) =∫

d3pjd3pad3pb ×Wpa,pb;pi,pj

×[

fα(pa, t) fβ(pb, t)(

1± fα(pi, t)/h3) (

1± fβ(pj, t)/h3)

− fα(pi, t) fβ(pj, t)(

1± fα(pa, t)/h3) (

1± fβ(pb, t)/h3)]

. (A.2.8)

The Boltzmann equation for particle β is obtained in complete analogy.

When the process is not a scattering, but, for instance, pair annihilation intotwo photons, the Boltzmann equation is identical to (A.2.8). In the case ofMøller scattering of only one specie (electron or positron) there is additional1/2 multiplier in front of the integral due to the presence of two indistin-guishable particles before and after the interaction.

60

A.2. Boltzmann equation with QED collisional integrals

A.2.2. Three-particle interactions

Consider now three-particle interactions. Let two incoming particles ofspecies α and β have momenta in the phase space volumes d3pa around paand d3pb around pb, respectively. Three outgoing particles of species γ, δ,and ε have momenta in the phase space volumes d3pi around pi, d3pj aroundpj, and d3pk around pk, respectively.

The energy and momentum conservation read

εa + εb = εi + εj + εk, ~pa + ~pb = ~pi + ~pj + ~pk. (A.2.9)

The differential emission and absorption coefficients for this reaction are

ηγ(pi, t)d3pi = ηδ(pj, t)d3pj = ηε(pk, t)d3pk =

χα fα(pa, t)d3pa = χβ fβ(pb, t)d3pb =

d3pad3pbd3pid3pjd3pk ×Wpa,pb;pi,pj,pk × fα(pa, t) fβ(pb, t)

×(

1± fγ(pi, t)/h3) (

1± fδ(pj, t)/h3) (

1± fε(pk, t)/h3)

, (A.2.10)

and for the inverse one

χγ fγ(pi, t)d3pi = χδ fδ(pj, t)d3pj = χε fε(pk, t)d3pk =

ηα(pa, t)d3pa = ηβ(pb, t)d3pb =

d3pad3pbd3pid3pjd3pk ×Wpi,pj,pk;pa,pb × fγ(pi, t) fδ(pj, t) fε(pk, t)

×(

1± fα(pa, t)/h3) (

1± fβ(pb, t)/h3)

. (A.2.11)

Consequently Boltzmann equation for particle specie α reads

1c

ddt

fα(pa, t) =∫

d3pid3pjd3pkd3pb

×[−W

pαa ,pβ

b ;pγi ,pδ

j ,pεk× fα(pa, t) fβ(pb, t)

×(

1± fγ(pi, t)/h3) (

1± fδ(pj, t)/h3) (

1± fε(pk, t)/h3)

+ Wpγ

i ,pδj ,pε

k ;pαa ,pβ

b× fγ(pi, t) fδ(pj, t) fε(pk, t)

×(

1± fα(pa, t)/h3) (

1± fβ(pb, t)/h3)]

. (A.2.12)

It may happen that some particle specie is present among both incomingand outgoing particles, for instance α coincides with γ. There are the fol-lowing emission and absorption terms for this particle specie: absorption of

61


incoming particle α and emission of outgoing particle γ, corresponding tothe direct reaction; and in addition emission of outgoing particle α and ab-sorption of incoming particle γ, corresponding to the inverse reaction. Sinceparticles α and γ are of the same specie, the collisional integral for that parti-cle consists of four terms instead of two. Formally two additional terms maybe obtained by interchanging particle states a ↔ i. Boltzmann equation inthis case reads

1c

ddt

fα(pa, t) =∫

d3pid3pjd3pkd3pb

×[−Wpa,pb;pi,pj,pk × fα(pa, t) fβ(pb, t)

×(

1± fα(pi, t)/h3) (

1± fδ(pj, t)/h3) (

1± fε(pk, t)/h3)

+ Wpi,pj,pk;pa,pb × fα(pi, t) fδ(pj, t) fε(pk, t)

×(

1± fα(pa, t)/h3) (

1± fβ(pb, t)/h3)

+ Wpα

i ,pβb ;pα

a ,pδj ,pε

k× fα(pi, t) fβ(pb, t)

×(

1± fα(pa, t)/h3) (

1± fδ(pj, t)/h3) (

1± fε(pk, t)/h3)

−Wpα

a ,pδj ,pε

k ;pαi ,pβ

b× fγ(pa, t) fδ(pj, t) fε(pk, t)

×(

1± fα(pi, t)/h3) (

1± fβ(pb, t)/h3)]

. (A.2.13)

Generally speaking, such four terms should be present in collisional inte-gral for a particle specie which is present both among incoming and outgoingparticles, unless the process is a scattering. This state is valid for arbitrarynumber of incoming and outgoing particles. It is not limited to the case ofquantum electrodynamics but applies to kinetic theory in general.

As for QED processes considered in this work the corresponding collisionalintegrals in many three-particle interactions contain four terms instead oftwo, as can be seen from table A.1. It should be noted that the equilibriumconditions may be obtained by only two terms in collisional integrals with-out interchanging the states a and i. However, due to different structure of allfour emission and absorption coefficients their presence is essential in Boltz-mann equation (A.2.13).

A.3. The numerical scheme

The main difficulty arising from computing collisional integrals is that therate of particle emission/absorbtion now depends not only on the distribu-tion functions of incoming particles, but also on the ones of outgoing par-

62

A.3. The numerical scheme

Two-particle interactions Three-particle interactionsCompton scattering Double Compton

e±γ−→e±′γ′ e±γ←→e±′γ′γ′′

Coulomb, Møller and Bhabha scattering Bremmstrahlunge±1 e±2 −→ e±′1 e±′2 e±1 e±2 ←→e±′1 e±′2 γe+e− −→ e+′e−′ e+e−←→e+′e−′γ

Creation/annihilation Three-photon annihilatione+e− ←→ γ1γ2 e+e−←→γ1γ2γ3

Pair creation/annihilationγ1γ2←→e+e−γ′

e±γ←→e±′e+e−

Table A.1.: Particle interactions in the pair plasma.

ticles. Therefore we adopt a new approach which we refer to as ”reaction-oriented” instead of previous ”particle-oriented” one.

Recall that the finite difference conservative scheme used in Aksenov et al.(2007, 2009, 2010) instead of distribution functions operates with spectral en-ergy densities

Ei(εi) =dρ

dε=

4πε3i βi fi

c3 , (A.3.1)

where βi =√

1− (mic2/εi)2 (mi, is the mass of i-th particle specie), in theenergy phase space εi. The number density of particle i is given by

ni =∫

fidpi =∫ Ei

εidεi, dni = fidpi, (A.3.2)

while the corresponding energy density is

ρi =∫

εi fidpi =∫

Eidεi.

In these variables the Boltzmann equations (A.2.2) read

1c

dEi

dt= ∑

q(η

qi − χ

qi Ei), (A.3.3)

where ηqi = (4πε3

i βi/c3)ηqi .

To obtain emission and absorbtion coefficients the computational grids areintroduced in the phase space εi, µi, φi, where µi = cos ϑi, ϑi and φi areangles in spherical coordinates of particle momentum space pi. The zoneboundaries are εi,ω∓1/2, µk∓1/2, φl∓1/2 for 1 ≤ ω ≤ ωmax, 1 ≤ k ≤ kmax,1 ≤ l ≤ lmax. The length of the i-th interval is ∆εi,ω ≡ εi,ω+1/2 − εi,ω−1/2. On

63


the finite grid the functions (A.3.1) become

Ea = Ei,ω ≡1

∆εi,ω

∫∆εi,ω

dε Ei(ε), (A.3.4)

where for simplification of formulae we use collective indices a = i, ω.The collisional integrals in (A.3.3) are replaced by the corresponding sums.

When particles are treated classically we have for time derivative of each vari-able the following expression

Ea = ∑b,c

A(b,c|a,d)EbEc −∑b,c

B(a,b|c,d)EaEb, (A.3.5)

where first sum on the right side is for emission in reaction b + c→ a + d andsecond is for absorbtion in reaction a + b → c + d. There is no third summa-tion (by index d) because of delta-function in the initial integrals originatingfrom the energy conservation. This can be effectively rewritten as just onesum

Ea = ∑b,c

Aab,cEbEc, (A.3.6)

Aab,c = A(b,c|a,d) − δc

a ∑e

B(a,b|e,d), (A.3.7)

and this sum can be found by direct computation.When the quantum statistics effects are included we have instead

Ea = ∑b,c

(1± Ea/Ea)(1± Ed/Ed)A(b,c|a,d)EbEc

−∑b,c

(1± Ed/Ed)(1± Ec/Ec)B(a,b|c,d)EaEb, (A.3.8)

which Ea is the critical spectral energy density for which quantum statisticscorrection becomes relevant (corresponding to occupation numbers equal tounity)

Ea =∫

ag ε

d3~ph3 , (A.3.9)

where g is the number of helicity states for the particle. It turns out that whilethe sums on the right-hand side of (A.3.8) can be reduced to one sum only,but due to different structure of Bose enhancement and Pauli blocking factorsthe numerical scheme based on the resulting expression will not be optimal.

Instead noticing that the phase space blocking/enhancement factors (1±Ed/Ed)(1± Ee/Ee) are the same for all four particles involved in the processb + c → d + e (a is one of the b, c, d, e), the corresponding parts of collisionalintegrals arising in the above-mentioned sums can be computed only once

64

A.4. Results

instead of four times. As a result it is convenient not to fix a and sum over allpossible b, c, as in (A.3.8), which we refer to as ”particle-oriented” approach,but instead sum over all possible reactions, which we refer to as ”reaction-oriented” approach. It means that at each step of calculations we fix b, c andfor all possible reaction results the emission rates of outcomes d and e and theabsorbtion rates of incomes b, c are added to array of derivatives Ea. This ap-proach considerably reduces the computational time and memory consump-tion.

In our method exact energy and number of particles conservation laws aresatisfied, as we adopt interpolation of grid functions Ea inside the energyintervals. The number of energy intervals is 15, while internal grid of angleshas 32 points in µi and 64 in φi. Most time-consuming part of the numericalsolution is initial integration over particle angles.

A.4. Results

We solved numerically Boltzmann equation (A.2.2) in two cases. Initiallyonly photons are present with constant spectral energy density and total en-ergy density ρ = 1023 erg/cm3 and ρ = 1029 erg/cm3. Such energy densitiescorresponds to the temperature θ in thermal equilibrium of 0.3 and 8, respec-tively.

In Figs. A.1 and A.2 we present number density, energy density, tempera-ture and chemical potential of photons and pairs in both cases. We also showthe difference between quantum and Boltzmann statistics by including andomitting the Pauli blocking and Bose enhancement factors in evolution equa-tions (A.3.8). Time is expressed in units of Compton time

τC =1

σTn±c, (A.4.1)

where n± is number density of pairs in thermal equilibrium, σT is Thomsoncross section.

Timescales of relaxation to thermal equilibrium for quantum (classical)statistics nearly coincide: 15τC (18τC) for ρ = 1023 erg/cm3, and 27τC (23τC)for ρ = 1029 erg/cm3. Inspection of Figs. A.1 and A.2 indicates that bothtemperatures and chemical potentials of leptonic and photon components be-come nearly equal when the total number density of particles shown by bluecurves is almost constant. This fact indicates that three-particle interactionsbecome relevant when almost detailed balance (kinetic equilibrium) is es-tablished by two-particle interactions Aksenov et al. (2007). Notice, however,that due to energy dependence of reaction rates the characteristic timescale onwhich kinetic equilibrium is established is larger than Compton time (A.4.1).For the same reason the characteristic timescale on which thermal equilib-

65


-3 -2 -1 0 1 2 30

2

4

6

8

10

12

log10 tΤC

n,10

28cm

-3

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

log10 tΤC

Ρ,10

23er

g×cm

-3

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

log10 tΤC

Θ

-3 -2 -1 0 1 2 3

-14

-12

-10

-8

-6

-4

-2

0

log10 tΤC

ΝΘ

Figure A.1.: Comparison of evolution of number n and energy ρ densities,dimensionless temperature θ, chemical potential ν/θ, for quantum (solidcurves) and classical (dotted curves) statistics with total energy density ρ =1023 erg/cm3. Black and red curves correspond to photons and pairs, respec-tively, blue curve gives the sum of densities.

rium is established is smaller than the simple estimate α−1τC. Thus the ratioof the timescales of kinetic and thermal equilibrium is no longer α but larger.This indicates the necessity of exact treatment of three-particle interactions,especially for high energy densities.

In Figs. A.3 and A.4 we show spectral evolution for both our initial con-ditions. The final spectra shown for t = 103τC are in good agreement withPlanck/Fermi-Dirac distribution functions, correspondingly, obtained for thegiven energy density, typically within 5 % accuracy. Notice that at the Comp-ton time both electron/positron and photon spectra are far from equilibriumshape, with the only exception of leptonic spectrum for ρ = 1023 erg/cm3.This quick relaxation of leptonic component is due to large Coulomb loga-rithm for non-relativistic temperatures.

A.5. Conclusions

In this work we consider relaxation of nonequilibrium optically thick pairplasma to complete thermal equilibrium by integrating numerically relativis-

66

A.5. Conclusions

-3 -2 -1 0 1 2 30

1

2

3

4

5

6

log10 tΤC

n,10

33cm

-3

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

log10 tΤC

Ρ,10

29er

g×cm

-3

-3 -2 -1 0 1 2 30

5

10

15

20

log10 tΤC

Θ

-3 -2 -1 0 1 2 3

-6

-4

-2

0

log10 tΤC

ΝΘ

Figure A.2.: The same as in Fig. A.1, but for total energy density ρ =1029 erg/cm3.

0.1 0.2 0.5 1.0 2.01026

1027

1028

1026

1027

1028

1029

1026

1027

1028

1026

1027

1028

1029

Ε

dΡ

±

dΕ

,cm

-3

dΡ

Γ

dΕ

,cm

-3

0.1 0.2 0.5 1.0 2.01026

1027

1028

1026

1027

1028

1029

1026

1027

1028

1026

1027

1028

1029

Ε

dΡ

±

dΕ

,cm

-3

dΡ

Γ

dΕ

,cm

-3

Figure A.3.: Numerical spectral energy densities of photons (black line) andpairs (red line) at t = τC (left) and at t = 103τC (right) for ρ = 1023 erg/cm3.Thick curves show the corresponding Bose-Einstein and Fermi-Dirac distri-butions with the same number and energy densities, respectively. Dashedthin line shows initial photon spectrum.

67


2 5 10 20 50 1001030

1031

1032

1030

1031

1032

1033

1030

1031

1032

1030

1031

1032

1033

Ε

dΡ

±

dΕ

,cm

-3

dΡ

Γ

dΕ

,cm

-3

2 5 10 20 50 1001030

1031

1032

1030

1031

1032

1033

1030

1031

1032

1030

1031

1032

1033

Ε

dΡ

±

dΕ

,cm

-3

dΡ

Γ

dΕ

,cm

-3

Figure A.4.: The same as in Fig. A.3 for ρ = 1029 erg/cm3.

tic Boltzmann equations with exact QED two-particle and three-particle colli-sional integrals. Quantum nature of particle statistics is accounted for in col-lisional integrals by the corresponding Bose enhancement and Pauli blockingfactors.

We point out that unlike classical Boltzmann equation for binary interac-tions such as scattering, more general interactions are typically described byfour collisional integrals for each particle that appears both among incomingand outgoing particles.

Our numerical results indicate that the rates of three-particle interactionsbecome comparable to those of two-particle ones for temperatures exceedingthe electron rest-mass energy. Thus three particle interactions such as rela-tivistic bremsstrahlung, double Compton scattering and radiative pair cre-ation become essential not only for establishment of thermal equilibrium, butalso for correct estimation of interaction rates, energy losses etc.

68

B. Dynamics and emission frommildly relativistic plasma

B.1. Introduction

Electron-positron plasma are expected to be present in compact astrophysi-cal objects (Damour and Ruffini, 1975), (Alcock et al., 1986), (Usov, 1998) andplays crucial role in the Gamma-Ray Burst (GRB) phenomenon (Ruffini et al.,2009b), (Ruffini et al., 2010). Electron-positron pairs are detected in the centerof our Galaxy (Churazov et al., 2005), leaving their characteristic imprint inthe observed radiation spectra. The observed annihilation line 511 keV nearthe Earth indicates that pair plasma can be created in the upper atmosphereat the thunderstorm (Briggs et al., 2011). Also interest in electron-positronplasmas is due to the exciting possibility of generating such plasmas in lab-oratory facilities using ultraintense focused short laser pulses, (Ruffini et al.,2010; Blaschke et al., 2006; Myatt et al., 2009; Thoma, 2009; Chen et al., 2009;Wilks et al., 1992).

Electrons-positron plasmas in the laboratory experiment or in thunder-storms have low density and are consequenly optically thin (Mustafa andKampfer, 2009; Katz, 2000). In contrast, in GRB source pair plasma is denseand optically thick (Ruffini et al., 2009b; Goodman, 1986; Piran, 1999). Suchoptically thick plasma is expected to expand, cool down and eventually be-come transparent for radiation, producing the characteristic flash of quasi-thermal radiation, the so called photospheric emission. Many recent worksare devoted to study of this phenomenon, see e.g. (Beloborodov, 2011), (Pe’erand Ryde, 2011), (Toma et al., 2011).

For the simulation of the photospheric emission of expanding plasma onecan use suitable Monte Carlo techniques based on well known reaction rates(Bisnovatyi-Kogan et al., 1971; Lightman, 1982; Svensson, 1982; Guilbert andStepney, 1985). While in the optically thick case usually hydrodynamic ap-proach is postulated (Piran, 1999). Such approximation is justified for largeoptical depth, or in the beginning of expansion. Due to complexity of the cal-culations only very few work adopt the kinetic approach for the descriptionof the plasma and try to calculate the spectra when the optical depth is notvery far from one (Beloborodov, 2010).

In this paper we report on our study of mildly relativistic plasma, whichis initially optically thick. The description of plasma is based on relativistic

69

B. Dynamics and emission from mildly relativistic plasma

Boltzmann equations. By means of this instrument we verify the applica-tion of hydrodynamic description of the plasma. We focus on the differencebetween the hydrodynamic description and more detailed kinetic one. Inparticular, we consider the optical depth, the number density and averageparticle energy evolution with time. We also report photon spectra from thisexpanding electron-positron plasma, which allows us to describe the photo-spheric emission in great details.

B.2. Formulation of the problem

Table B.1.: Physical Processes Included in Simulations.Binary interactions Radiative and pair producing variants

Møller and Bhabha scattering Bremsstrahlunge±1 e±2 −→ e±′1 e±′2 e±1 e±2 ←→ e±′1 e±′2 γe±e∓ −→ e±′e∓′ e±e∓ ←→ e±′e∓′γ

Single Compton scattering Double Compton scatteringe±γ −→ e±γ′ e±γ←→ e±′γ′γ′′

Pair production Radiative pair productionand annihilation and three photon annihilation

γγ′ ←→ e±e∓ γγ′ ←→ e±e∓γ′′

e±e∓ ←→ γγ′γ′′

e±γ←→ e±′e∓e±′′

We use the relativistic Boltzmann equations for the e± pairs and photons.The distribution function for the particles of type i, fi(|p|, µ, r, t) (i = e±, γ),satisfies (Mihalas and Mihalas, 1984)

1c

∂ fi

∂t+ βi

(µ

∂ fi

∂r+

1− µ2

r∂ fi

∂µ

)= ∑

q(η

qi − χ

qi fi), (B.2.1)

where µ = (rp)/(|r||p|), βi = vi/c =√

1− (mic2/εi)2, εi = c√

p2 + (mic2).η

qi is the emission coefficient for the production of a particle of type i via

the physical process by q, and χqi is the corresponding absorption coefficient.

The summation runs over all considered physical processes that involve aparticle of type i. We take into account all two-body interactions, see e.g.Berestetskii et al. (1982), Greiner and Reinhardt (2003), and correspondingtriple interactions(Svensson, 1984) written in Table B.1.

The particle concentrations and energy densities are

ni(r, t) =∫

fi(p, r, t)dp, ρi(r, t) =∫

fi(p, r, t)εdp. (B.2.2)

70

B.3. Computational method

For the GRB source plasma, for example, the corresponding temperaturesin thermal equilibrium are (Aksenov et al., 2007)

0.1 ≤ Tth ≤ 10 MeV. (B.2.3)

At such conditions the number of particles in Debye sphere is large. We canuse one-particle distributions functions. Another important property is thatsuch pair plasma can not generate additional particles like neutrinos.

B.3. Computational method

To make the scheme conservative we use, instead of fi, the spectral energydensities

Ei(εi, µ, r, t) =2πε3

i βi fi

c3 , (B.3.1)

because the energy density in our phase space (ε, µ, r, t) is

εi fidrdp =2πε3βi fi

c3 rdεidµ = Eirdεidµ. (B.3.2)

For Ei the Boltzmann equation has the conservative form

1c

∂Ei

∂t+

µ

r2∂

∂r(r2βiEi) +

1r

∂

∂µ[(1− µ2)βiEi)] = ∑

q(η

qi − χ

qi fi). (B.3.3)

We use the lines method to solve above task (Aksenov et al., 2004). We in-troduced grid in phase space εω+1/2, µk+1/2, rj+1/2. After the replacing of allderivatives except the derivative over time in Eq. (B.3.3) by finite differences(Aksenov et al., 2004) and collisional integrals by sums (Aksenov et al., 2009)we have the set of ordinary differential equations for grid values

Ei,ω,k,j ≡

∫∆εω∆µk∆(r3

j /3) dεdµd(r3/3)Ei(ε, µ, r)

∆εω∆µk∆(r3j /3)

. (B.3.4)

To carry out numerical evolution in the optically thick regions we use im-plicit Gear’s method to solve the stiff system of ordinary differential equa-tions (Hall and Watt, 1976).

We evaluate numerically exact expressions for collisional integrals withmatrix elements from QED (Aksenov et al., 2009). Our investigations showsthe timescale of triple interaction is α−1 ' 137 times larger, compared tothe timescale of binary interactions. Consequently plasma reaches so calledkinetic equilibrium with the common temperature but nonnull chemical po-tentials before triple interactions become important. This fact allows us to

71


simplify calculations of triple interactions in the nondegenerate cases as de-scribed in Aksenov et al. (2009).

For very large optical depths reactions rates cannot be resolved with arough angle grid. For this reason we adopt the following method for comput-ing reaction rates in a such region τ 1. First we compute hydrodynamicvelocity β from the known distribution functions, and transform our quanti-ties in the comoving frame by using the following Lorentz transformations,see e.g. Mihalas and Mihalas (1984), p. 414

(ε, µ, φ, r, t) =

Γ(

ε′ +Vc

cp′µ′)

,p′µ′ + V

cε′c√(

ε′c + V

c p′µ′)2− m2c2

Γ2

, φ′, r′, t′

.

Then we calculate average values n′i in the comoving frame, which do notdepend on angles. Then in comoving frame we evaluate average absorptioncoefficients χ′ = const. The emission coefficients η′ in the comoving frameare taken to be proportional to equilibrium intensities. Finally we transformthe emission and absorption coefficients back into the laboratory frame by

I(ε, µ) =ε2p

ε′2p′I′(ε′, µ′), E(ε, µ, r, t) ≡ 2πε3β f

c3 ∝ I,

η(ε, µ) =εp

ε′p′η′(ε′, µ′), χ(ε, µ) =

ε′

εχ′(ε′, µ′),

also preserving exact energy and momentum conservation on the finite grid.

B.4. Results

We consider kinetic evolution of nonequilibrium optically thick plasma con-sisting at the moment t = 0 of electron-positron pairs with number densityn = 1030 cm−3 in the small region with radius R0 = 200 cm. Althoughsuch parameters are far from both laboratory conditions and the real GRBsources, we consider this choice of parameters important since it providessome new insights with respect to the traditional hydrodynamic description(Goodman, 1986), (Piran et al., 1993), (Meszaros et al., 1993), (Mustafa andKampfer, 2009), (Yaresko et al., 2010). Figs. B.1–B.5 show the plasma evolu-tion with time, and Fig. B.6 shows spectra of photons near the maximum ofemission at t = 7 · 10−7 sec, crossing the sphere with radius 2.2 · 104 cm.

Initially nonequilibrium plasma relaxes to the thermal state on thetimescale 5 · 10−11 sec, and it starts to expand, on the dynamical timecaleR0/c ' 6 · 10−9 sec, see Fig. B.3. The concentrations (Fig. B.2) and the opticaldepths of both electrons/positrons and photons (Fig. B.1) decrease with time.

72

B.4. Results

102 103 10410-1

100

101

102

103

104

105

106

107

1.7 10-7

10-7

3 10-8

7 10-9

2 10-15

, e

r, cmFigure B.1.: Mean optical depth for photons (solid) and pairs (dashed) as afunction of the radius at different time moments, as marked on the curves.

102 103 1041020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

, e

n g, ne

r, cmFigure B.2.: Photon number density (solid) and pair number density (dashed)as a function of the radius at different time moments, as in Fig. B.1.

73


102 103 104

1

2

3

4

, e

r, cmFigure B.3.: Photon (solid) and pair (dashed) radial velocity as a function ofthe radius at different time moments, as in Fig. B.1.

102 103 1041034

1035

1036

1037

1038

1039

1040

, e

L, e

rg/s

r, cmFigure B.4.: Rates of energy flow in photons (solid), in pairs (dashed) throughthe surface at radius r as a function of the radius at different time moments,as in Fig. B.1.

74

B.4. Results

102 103 104101

102

103

, e

<>,

keV

r, cmFigure B.5.: Average energy of photons (solid) and of pairs (dashed) as a func-tion of the radius at different time moments, as in Fig. B.1.

100 10001033

1034

1035

1036

1.3 10-6

8.7 10-7

7 10-7

4 10-7

dL/d

, e

rg/s

*keV

, keV

Figure B.6.: Energy spectrum of emerging photons in the laboratory referenceframe, at the radius 2.2 · 104 cm at selected time moments near the maximumluminosity.

75


As the temperature becomes smaller than mec2 the energy density in pairsstarts to decrease exponentially, since they remain in thermal equilibrium.When the optical depth of photons is large, we find hydrodynamic descrip-tion accurate enough to calculate the photons spectra. At t = 1.5 · 10−7 sec theoptical depth for photons decreases below unity, see Fig. B.1. After this mo-ment the radial distribution of photons in the expanding shell bocomes fixed,see Fig. B.4. The average energy shown in Fig. B.5 indicates that the peakof the photon spectrum is in the range 10− 1000 keV. Photon spectra aroundthe maximal luminosity are nonthermal, see Fig. B.6. They possess an expo-nential cut-off at high energy, and the spectral energy density is almost flat inits low energy part. This result may be an explanation of additional power inlow energy spectra observed in GRBs (Patricelli et al., 2011).

B.5. Conclusions and future perspectives

In this work we obtained for the first time the detailed kinetic information in-cluding photon spectra from initially optically thick outflow, composed ofelectron-positron pairs and photons. Initial conditions are selected in theintermediate region between conditions which are expected to be reachedin laboratory experiments, and those thought to occur in astrophysical set-tings. This allows us to study the radial self-acceleration of electron-positronplasma up to mildly relativistic velocities. We found that photon spectra nearthe peak of the luminosity are non-thermal, they possess an exponential cut-off at high energy, and are almost flat in the low energy part. This resultmay shed some light on the issue why GRBs have nonthermal spectra, andin particular why the low energy part of the spectrum contains more powerwith respect to Planck spectrum. We hope to extend our analysis for ultra-relativistic case which is necessary to be explored for realistic description ofGamma-Ray Bursts phenomenon.

76

C. Evolution of the pair plasmagenerated by a strong electricfield

Introduction

Electron-positron pairs can be produced by vacuum polarization in strongelectric field E if it satisfies the following relation

E ≥ m2e c3

eh= Ec , (C.0.1)

where me is the electron mass, c is the speed of light, h is the Planck constantand Ec is the so called ”critical” electric field (Ruffini et al., 2010).

In (Benedetti et al., 2011) we addressed the problem of pair creationin a strong electric by simple treatment based on continuity and energy-momentum conservation equations. A second order ordinary differentialequation has been worked out and all the other physical quantities of inter-est can be obtained from its solution. According to previous works (Klugeret al., 1991),(Ruffini et al., 2003b),(Benedetti et al., 2011), the pairs move backand electric field oscillates as well with the same frequency but with a dif-ferent phase. This approach allowed us to study the behavior of the systembeyond the asymptotic time τa, when the pairs oscillation frequency is closeto the plasma frequency; at this stage almost all the energy density of the ini-tial electric field is converted into rest energy density of the pairs. The opticaldepth has been computed in order to estimate the time scale τγ after whichthe interactions can not be neglected. We found that τγ τa for all the con-sidered initial conditions, concluding that the role of the interactions shouldbe taken into account in a long run. Besides, we assumed that pairs are orig-inally produced at rest even though the most general rate of pair productionalready gives a specific distribution in the momentum space. In particularthey are in a momentum state such that there is motion only in orthogonaldirection to the initial electric field.

We now extend and generalize the results of the previous work (Benedettiet al., 2011) with the intention to investigate the effects of pairs interactionsand the dynamical role of a most general rate of pair production. Collisionscan be naturally described within the kinetic approach. There is invariance

77

C. Evolution of the pair plasma generated by a strong electric field

under rotations around the direction of the electric field. In this perspectivewe solve numerically the relativistic Boltzmann-Vlasov equation in a uni-form and homogeneous physical space but with an axially symmetric mo-mentum space. The pairs interaction is accounted for by collision integralscomputed from the exact QED cross sections for the two particle interactions,namely electron-positron annihilation into two photons and its inverse pro-cess, Bhabha, Moller and Compton scatterings. As a consequence, also pho-tons are described by the relativistic Boltzmann equation. The three particleinteractions are not taken into account because their cross section is roughlyα times smaller than the two particle ones.

In the adopted kinetic scheme the Distribution Function (DF) is the basicobject we are dealing with because all the fundamental physical quantitiescan be extracted from it; among them the number density, the rest and ki-netic energy densities, the bulk momentum. Since it is well known that apair plasma thermalizes in a very short time scale (Aksenov et al., 2009), weare also interested in this characteristic time for different initial conditions.The temperature in kinetic equilibrium may be found from the total energyand number densities of the plasma (Aksenov et al., 2007). Moreover, wealready know that the electric field accelerates electrons and positrons up tohigh Lorentz factors of the order of hundreds. Along the direction orthogonalto that, pairs as they are produced have Lorentz factors at most of the order ofthe initial electric field. It means that distribution of electrons and positronsin the momentum space is strongly anisotropic being elongated in the direc-tion of initial electric field. So we can think about system that is characterizedby different temperatures T‖ and T⊥, in respectively parallel and orthogonaldirections to the electric field. For this reason we introduce a Relativistic TwoTemperatures Distribution Function (R2TDF) such that electrons, positronsand photons can be described. A complete equilibrium in the system is ex-pected when a unique temperature T = T‖ = T⊥ is established.

C.1. Cylindrical coordinates in the momentumspace

Based on the symmetry of the problem we consider axially symmetric mo-mentum space. Hence, the momentum of the particle is described by twocomponents, one parallel (p‖) and one orthogonal (p⊥) to the direction of theinitial electric field, and the angle (φ) between this preferred direction and theactual momentum direction. These momentum space coordinates are definedin the intervals

φ ∈ [0, 2π] , p⊥ ∈ [0,+∞) , p‖ ∈ (−∞,+∞) . (C.1.1)

78

C.2. The Distribution Function

Within the chosen phase space configuration, the prescription for the integralover the entire momentum space is∫

d3p =∫ 2π

0dφ∫ +∞

−∞dp‖

∫ +∞

0dp⊥ p⊥ , (C.1.2)

and the relativistic energy is given by the following equation

ε =√

p2‖ + p2

⊥ + m2 , (C.1.3)

where m is the mass of the considered particle.

C.2. The Distribution Function

Instead of using the usual DF f such that the number density is given by theintegral over the momentum space

n =∫

d3p f = 2π∫ +∞

−∞dp‖

∫ +∞

0dp⊥ p⊥ f , (C.2.1)

we introduce a new DF F such that the energy density is given by the follow-ing integral role

ρ =∫ +∞

−∞dp‖

∫ +∞

0dp⊥ F . (C.2.2)

In isotropic momentum space this DF is reduced to the spectral energy den-sity dρ/dε. Hence we can recover f from F using the following prescription

f =F

2π ε p⊥. (C.2.3)

Besides, because of the axial symmetry, F does not depend on φ, it dependsonly on the two components of the momentum, that is

F = F(p‖, p⊥) . (C.2.4)

79


C.3. Two temperature DF

Neglecting the relativistic degeneracy an ensamble of particles in thermalequilibrium can be described by the Relativistic Maxwellian DF

f (p‖, p⊥, T) = (C.3.1)

=mec2/kBT

4πK2(mec2/kBT)exp

−me c2

kB T

√1 +

( p‖me c

)2

+

(p⊥

me c

)2 ,

where T is the temperature and K2 is the modified Bessel function of the sec-ond kind. Therefore we expect that after a characteristic time scale, depen-dent on the initial conditions, the distribution function of each sort of particleshould achieve the limit given by Eq. (C.3.1).

On the other hand, we are dealing with two independent components ofthe momentum and the rate Eq. (C.4.2) indicates that initially the DF of elec-trons and positrons is strongly anisotropic. For that reason we describe thedifferent particles populations with a Two Temperature DF before the ther-mal equilibrium is reached. In this lines we adopt the following fiducial DF

f (p‖, p⊥, T‖, T⊥, p0‖ , k) = (C.3.2)

= k exp

−me c2

kB T

√√√√1 +

(TT‖

)2( p‖ − p0‖

me c

)2

+

(T

T⊥

)2( p⊥me c

)2

,

where T = (T‖ + T⊥)/2, k is a numerical factor which eventually dependson T and p0‖ is the centering value for the parallel momentum. In the lit-erature there is no such a definition, however it recovers two well knownDFs in the relativistic and non-relativistic domains. First of all, if we equalthe parallel and orthogonal temperatures, the argument of the exponential inEq. (C.3.2) equals the one corresponding to the relativistic Maxwellian of Eq.(C.3.1). Furthermore, if we compute again the argument of the exponentialin the non-relativistic limit we get the classical two temperatures Maxwelliandistribution function given by

f (p‖, p⊥, τ‖, τ⊥, p0‖ , k′) = (C.3.3)

= k′ exp

[−me c2

kB

(1τ‖

( p‖ − p0‖

me c

)2

+1

τ⊥

(p⊥

me c

)2)]

,

80

C.4. Boltzmann equations

where the new temperatures are defined as follows

τ‖ =2 T2‖

T‖ + T⊥, τ⊥ =

2 T2⊥

T‖ + T⊥. (C.3.4)

C.4. Boltzmann equations

The relativistic Boltzmann equation for electrons and positrons can be writtenin our framework as follows

12π ε p⊥

(∂F±(p‖, p⊥)

∂t± E

∂F±(p‖, p⊥)∂p‖

)=

= η±(p‖, p⊥)− χ±(p‖, p⊥) F±(p‖, p⊥) + S(p‖, p⊥, E) , (C.4.1)

where E = E/Ec is the dimensionless electric field, η±, χ± are the emissionand absorption coefficients due to the interactions, and S is the rate of pairproduction. In particular the electron-positron DF in Eq. (C.4.1), varies dueto the acceleration by the electric Field, the pairs creation due to vacuumpolarization and the creation or annihilation of particles generated by theinteractions.

The rate of pair production already distributes particles in the momentumspace accordingly to

S(p‖, p⊥, E) = −(me

h

)4|E| log

[1− exp

(−π(1 + (p⊥/(mec))2)

|E|

)]δ(p‖) .

(C.4.2)From the previous equation it is clear that if E < 1 this rate is exponentiallysuppressed. Going further, we see that for a given overcritical field E therange of orthogonal momentum on which pairs will be distributed is approx-imatively

[0, E me c

]. Then the Boltzmann equation for photons is

12π ε p⊥

∂Fγ(p‖, p⊥)∂t

= ηγ(p‖, p⊥)− χγ(p‖, p⊥) Fγ(p‖, p⊥) , (C.4.3)

because their DF changes accordingly to the collisional term only. In moredetail, photons must be produced first from the pairs annihilation, then theyaffect the electron-positron DF through Compton and finally they can anni-hilate producing one electron-positron pair.

81


C.5. Computational scheme

C.5.1. Acceleration and electric field evolution

Once electrons and positrons are produced, they are accelerated by the elec-tric field toward opposite directions. The time derivative of the electron orpositron parallel momentum dp±‖ in the presence of an electric field E is givenby the equation of motion

dp±‖dt

= ± e E , (C.5.1)

where the sign + (−) refer to the positron (electron) and −e is the electroncharge. Numerically, we move particles from one cell to another one suchthat the number of particles is conserved and Eq. (C.5.1) is satisfied.

Also the electric field evolves according to the Maxwell equations. Oncethe currents of the moving pairs are computed, the time derivative of theelectric field is known. Consequently a new ordinary differential equationmust be added to the system of Eqs. (C.4.1) and (C.4.3). However, due tothe uniformity and homogeneity of the physical space, we can describe theelectric field simply using the energy conservation law.

C.5.2. Emission and absorption coefficients

While the numerical description of the acceleration is quite easy, it is not sofor the ”collisional term”: in fact the emission and absorption coefficientscontain several integrals over the momentum space of the distribution func-tions. Since we describe a given interaction denoting the involved particlesas follows

1 + 2→ 3 + 4 , (C.5.2)

it means that the particle 1 and 2 are absorbed while particle 3 and 4 areproduced, then we have χ1, χ2 and η3, η4. In order to obtain these emissionand absorption coefficients for each particle we make use of the followingquantity

C =∫

d3p1

∫d3p2

∫d3p3

∫d3p4 w1,2;3,4

F1(p1‖ , p1⊥) F2(p2‖ , p2⊥)

(2π)2 ε1 p1⊥ ε2 p2⊥, (C.5.3)

where

w1,2;3,4 =h2c6

(2π)2 δ(ε) δ(3)(p)

∣∣M f i∣∣2

16ε1ε2ε3ε4(C.5.4)

contains the matrix element for the considered interaction.The interaction cross section is invariant by rotations around an arbitrary

axis, therefore the angle φ1 can be set to be constant. Then we have 4 con-

82


servation laws, energy and momentum, given by the Dirac deltas in w. Ourchoice is to eliminate the three integrals over the particle 4 by δ(3)(p) andeliminate the integration over φ2 using the δ(ε).

Rewriting eq. (C.5.3) in our computational scheme we get

C =∫

dp1‖

∫dp1⊥

∫dp2‖

∫dp2⊥

∫dp3‖

∫dp3⊥× (C.5.5)

× R(p1‖ , p1⊥ , p2‖ , p2⊥ , p3‖ , p3⊥) F1(p1‖ , p1⊥) F2(p2‖ , p2⊥) ,

where

R(p1‖ , p1⊥ , p2‖ , p2⊥ , p3‖ , p3⊥) =h2c6

2(4π)3

∫dφ3

p3⊥ε2

1 ε22 ε3 ε4

J∣∣M f i

∣∣2 , (C.5.6)

and the Jacobian is given by

J =ε4/p2⊥

sin φ2 (p3⊥ cos φ3 − p1⊥ cos φ1) + cos φ2 (p1⊥ sin φ1 − p3⊥ sin φ3).

(C.5.7)

All the information about each reaction are stored in R and the integral overφ3 within its expression can be computed. For each particle we have differentcoefficients bU1, bU2, aU3, aU4 and each of them depends on p1‖ , p1⊥ , p2‖ , p2⊥ ,p3‖ , p3⊥ , defined in such a way that

χ1(p1‖ , p1⊥) =∫

dp2‖

∫dp2⊥

∫dp3‖

∫dp3⊥× (C.5.8)

× bU1(p1‖ , p1⊥ , p2‖ , p2⊥ , p3‖ , p3⊥) F2(p2‖ , p2⊥)

χ2(p2‖ , p2⊥) =∫

dp1‖

∫dp1⊥

∫dp3‖

∫dp3⊥× (C.5.9)

× bU2(p2‖ , p2⊥ , p1‖ , p1⊥ , p3‖ , p3⊥) F1(p1‖ , p1⊥)

η3(p3‖ , p3⊥) =∫

dp1‖

∫dp1⊥

∫dp2‖

∫dp2⊥× (C.5.10)

× aU3(p3‖ , p3⊥ , p1‖ , p1⊥ , p2‖ , p2⊥) F1(p1‖ , p1⊥) F2(p2‖ , p2⊥)

η4(pm‖ , pn⊥) =∫

dp1‖

∫dp1⊥

∫dp2‖

∫dp2⊥× (C.5.11)

× aU4(pm‖ , pn⊥ , p1‖ , p1⊥ , p2‖ , p2⊥) F1(p1‖ , p1⊥) F2(p2‖ , p2⊥) .

By definition, the particles 1, 2 and 3 exactly on the momentum points ofthe grid, as a consequence it is not guaranteed that the particle 4 will be on thegrid point as well. Nevertheless, we need to conserve the number of particles,the energy and the orthogonal and parallel components of the momentum.

83


Therefore we have to distribute it also on in the nearby cells as follows

∂F4(p4‖ , p4⊥)

∂t= ∑

m∑n

xmn∂F4(pm‖ , pn⊥)

∂t, (C.5.12)

where xmn are given by the conservation laws.

C.5.3. Two particle kinematics

The interaction between two particles, 1 and 2, that gives the particle 3 and 4as a product can be represented schematically as follows

1 + 2→ 3 + 4 . (C.5.13)

The i-th particle has 3 degrees of freedom (pi‖ , pi⊥ , φi) in the momentum spaceand summing all of them we get 12. Since the interaction process conservesmomentum and energy, they reduce to 8 independent degrees of freedom.That means that 4 quantities can be determined uniquely once the others arespecified. For us, the 8 independent variables are p1‖ , p1⊥ , φ1, p2‖ , p2⊥ , p3‖ ,p3⊥ , φ3; then p4‖ , p4⊥ , φ4, φ2 are functions of the previous ones.

The conservation of the parallel momentum gives use easily the corre-sponding component of the 4-th particle

p4‖ = p1‖ + p2‖ − p3‖ . (C.5.14)

The orthogonal momentum for the same particle can be worked out usingthe energy conservation law

ε1 + ε2 = ε3 + ε4 (C.5.15)

and using the definition of the energy given by Eq. (C.1.3) we get

p4⊥ =

√(ε1 + ε2 − ε3)

2 − p24‖−m2

4 , (C.5.16)

where p4‖ has been obtained in Eq. (C.5.15).From the conservation of the orthogonal momentum we have the following

relations

p1⊥ cos(φ1) + p2⊥ cos(φ2) = p3⊥ cos(φ3) + p4⊥ cos(φ4) (C.5.17)p1⊥ sin(φ1) + p2⊥ sin(φ2) = p3⊥ sin(φ3) + p4⊥ sin(φ4) (C.5.18)

from which we can write down analytical expressions for φ4 and φ2. Unfor-tunately for the previous system of equations we have two valid solutions.

84


For φ2 we have the following equation

a cos(φ2) + b sin(φ2) + c = 0 , (C.5.19)

where the coefficients are given by

a = p1⊥ cos(φ1)− p3⊥ cos(φ3) ; (C.5.20)b = p1⊥ sin(φ1)− p3⊥ sin(φ3) ; (C.5.21)

c =p2

1⊥+ p2

2⊥+ p2

3⊥− p1⊥ p3⊥ cos(φ1 − φ3)

2p2⊥. (C.5.22)

The solution of eq. (C.5.19) are given by the following conditions

• if b 6= 0, a2 + b2 6= 0, c = a ⇒ φ2 = −2 arctan(

ab

)+ 2nπ ;

• if b = 0, c = a ⇒ φ2 = π + 2nπ ;

• if a 6= c, a2 + b2 − ac− b√

a2 + b2 − c2 6= 0

⇒ φ2 = 2 arctan(

b−√

a2 + b2 − c2a− c

);

• if a 6= c, a2 + b2 − ac + b√

a2 + b2 − c2 6= 0

⇒ φ2 = 2 arctan(

b +√

a2 + b2 − c2a− c

).

Once φ2 has been chosen, φ4 can be easily obtained from the eqs. (C.5.17)and (C.5.18).

The Jacobian of Eq. (C.5.7) has been computed using using the followingidentity for the Dirac Delta

δ( f (x)) = ∑i

δ(x− xi)

|(d f /dx)xi |(C.5.23)

where f is a function such that f (xi) = 0. In our framework the functioninside the Dirac delta is given by the energy conservation

f (φ2) = ε1 + ε2 − ε3 − ε4(φ2) (C.5.24)

where φ2 is now the independent variable. From the previous equation wecompute its derivative with respect to φ2 and the value φ∗2 such that f (φ∗2) =0. Rewriting explicitly Eq. (C.5.23) we have that

δ( f (φ2)) =δ(φ2 − φ∗2)

|(d f /dφ2)φ∗2| = J δ(φ2 − φ∗2) . (C.5.25)

85


C.6. Results

C.6.1. Non interacting systems

1 10 100

Time (tc)

0.001

0.01

0.1

1

n / n

_max

Boltzmann equationPrevious result

0.028

Figure C.1.. The evolutions of the number density with timefound in (Benedetti et al., 2011) (dashed) and in our treatmentwithout interactions (solid) are compared for Ei = 10. A dif-ferent behavior occurs after the first oscillation: in the formerit continues to grow almost linearly but in the latter it satu-rates after a few oscillations.

In this section we present the results obtained from the relativistic Boltz-mann equation (C.4.1) for electrons and positrons with χ± = η± = 0. In thissense we neglect the interactions of electron positrons pairs. We expect thisresult to be closely related to those reported in (Benedetti et al., 2011).

We solved numerically the Boltzmann equation for different initial electricfields (Ei = 1, 3, 10, 30, 100) and with no pairs at the beginning; consequentlythe electrons and positrons are produced exclusively by the vacuum polar-ization process.

86

C.6. Results

0 50 100 150 200

Time (tc)

-40

-20

0

20

40

60

80

Electric Field (Ec)Electron bulk parallel momentum (m_e*c)

Figure C.2.. Electric field and bulk parallel momentum ob-tained for the non-interacting system with initial conditionEi = 10. It is evident how the electric field magnitude dropswithin several oscillations in contrast with the result obtainedin (Benedetti et al., 2011). The bulk parallel momentum is de-creasing rapidly with time as well, but we stress that its mag-nitude in the first two peaks is very similar to those attainedin (Benedetti et al., 2011)

For all the explored initial conditions, there are important analogies be-tween the approach adopted in (Benedetti et al., 2011) and the one presentedin this work. We remind that the maximum achievable pairs number densityis

nmax =E2

16πmec2 , (C.6.1)

which corresponds to the case of maximum conversion of the initial energydensity into electron-positron rest energy density. In Fig. C.1 we see the ratioof the pairs number density of pairs nmax obtained in these two treatments.The two curves are very close up to the first zero of the electric field identifiedby the first plateau since particles can not be produced.

From the electrons and positrons DFs we can extrapolate the bulk parallel

87


-200 -100 0 100 200

parallel momentum (m_e*c)

0

1x1046

2x1046

3x1046

4x1046

num

ber

of p

artic

les

(arb

itrar

y un

its)

1st e-2nd e-3rd e-4th e-5th e-6th e-7th e-8th e-9th e-9th e+

Figure C.3.. In this plot the integral along the orthogonal mo-mentum of the distribution function is shown. Each curvecorresponds to a different zero of the electric field; becauseof that they are snapshots of the DF when a maxima thebulk parallel momentum is achieved. This figure put in evi-dence how the distribution function evolves toward a station-ary configuration characterized by a bulk parallel momentumvery close to zero.

momentum of electrons and positrons as follows

p‖± =

∫dp‖

∫dp⊥ F±(p‖, p⊥) p‖/ε∫

dp‖∫

dp⊥ F±(p‖, p⊥)/ε, (C.6.2)

From the symmetry of our problem it must be that p‖− = − p‖+ identically.This quantity is plotted in Fig. C.2 and it is very close to that obtained in(Benedetti et al., 2011) during this initial time interval.

Nevertheless, after this short period, the results of the two treatments di-verge. As it was found in (Benedetti et al., 2011), the subsequent oscillationsoccur with a quickly increasing frequency. In our case, the frequency of theoscillations increases as well but slowly.

Besides, in (Benedetti et al., 2011) we found that the absolute peak value

88

C.6. Results

of the electric field decreases slowly with time. Now we find that this peakelectric field decreases much faster and it does not exceed a few percent ofthe starting value. Subsequently, as it follows from Eq. (C.4.2) the pair pro-duction saturates as illustrated in Fig. C.1.

There is a third consequence of the strong damping of the electric field.Let us consider two different populations of electrons (or positrons): the firstwith very small orthogonal momentum and the second one with very largeorthogonal momentum. If we compute their bulk parallel momentum psmall

‖

and plarge‖ separately at the end of the run, we find that the former is approxi-

matively zero while the latter is not. This fact can be grasped from Eq. (C.4.2).In fact electrons and positrons are originally generated with null parallel mo-mentum, and the smaller the electric field the shorter the range of orthogonalmomentum in which pairs are produced. The new born pairs gives moreweight to the particles with p‖ = 0 and its average p‖ is consequently low-ered. Again, this effect manifests for particles with orthogonal momentum inthe range given by Eq. (C.4.2).

In Fig. C.2 we see how p‖ decreases rapidly and it attains a non relativisticvalue at the end of the run. Each maximum of | p‖| has a corresponding curvein Fig. C.3 where we show the following ”marginal”DF

dN(p‖)dp‖

=∫ +∞

0dp⊥

F(p‖, p⊥)ε

. (C.6.3)

In Fig C.4 we show various normalized energy densities. The total energydensity of the pairs ρ which is related to the actual and initial electric fields Eand Ei by the energy conservation law

ρ =E2 − E2

i8π

. (C.6.4)

The rest energy of the pairs, namely the sum of the electrons and positronsrest energy densities, is

ρrest = ρ− + ρ+ = (n− + n+)me c2 , (C.6.5)

where n− and n+ are the electrons and positrons number densities. By defi-nition the plot of this quantity reproduces exactly the behavior of the numberdensity of pairs in Fig. C.1. Then we define the kinetic energy density

ρkin = (n− + n+)me c2

√( p‖me c

)2

+ 1− 1

, (C.6.6)

which is the energy given by particles moving with the same magnitude for

89


the parallel momentum p‖ and with p⊥ = 0. We stress that the rest energydensity has been already subtracted in its definition. We see how its valuedrops to very small values when the condition p‖ = 0 is fulfilled. The dif-ference between the total energy density and all the others defined above istaken to be the internal energy density

ρin = ρ− ρrest − ρkin . (C.6.7)

We call this ”internal” since it is related to the dispersion of the DF around agiven point with coordinates (p‖ , p⊥) in the momentum space.

The fundamental aspect which comes out from Fig. C.4 is the relative im-portance of ρkin and ρin with time. From their definition we understand whythere is correspondence between their maximum and minimum values. Mostimportant is that at the end the kinetic energy is not larger than a few percentof the internal one. This is also an indication toward a possible thermal de-scription of such a system.

Even if we have no interactions such that the thermal equilibrium can oc-cur, we tried to fit the final DF with our fiducial 2TDF given by Eq. (C.3.2).We found that the main part can be well fitted putting p0‖ ' 0, but there isalways a huge difference between parallel and orthogonal temperatures dueto large anisotropy in the momentum space. We summarized these values forthe temperature in Tab. C.1. From the table we see that the higher the electricfield the higher is the orthogonal temperature. In fact, according to Eq. (C.4.2)the higher is the electric field the larger is the orthogonal momentum rangein which pairs are distributed. However, the same trend is not followed bythe parallel temperature. Indeed, even if we have large uncertainties in itsestimate, it apparently has a minimum between 1 and 10 Ec. For this reasonwe inserted in the table the peak of the parallel momentum p‖1 at the firstzero of the electric field obtained in (Benedetti et al., 2011). This quantity alsoshows a minimum in the same range of initial electric field.

In this table we compare two different number densities n1 and ns. Thefirst is the number density of pairs at the first zero of the electric field foundin (Benedetti et al., 2011). The second is the saturation number density ofpairs in our computation. Both have a maximum in the range between 1 and10 Ec.

C.6.2. Interacting systems

Up to now, we can present only preliminary results for the interacting casesbecause they are computationally time consuming and an optimization mustbe introduced in order to execute long time runs. Yet from Eq. (C.4.2) it isclear that the larger the initial electric field the higher is the achieved num-ber density and consequently the larger is the interaction probability betweenparticles. Following this logic, we analyzed first the the case under consid-

90

C.6. Results

Table C.1.: Dimensionless parallel and orthogonal temperatures T‖ and T⊥,normalized parallel momentum p‖1 and number density n1 of pairs at thefirst zero of the electric field obtained in (Benedetti et al., 2011), normalizedsaturation number density ns for the non-interacting system and for differentinitial electric field.

Ei kBT⊥/mec2 kBT‖/mec2 p‖1/mec n1/nmax ns/nmax

1 0.2 70 146 0.0068 0.01 - 0.02

3 0.3 120 73 0.0137 0.01 - 0.02

10 0.5 45 83 0.0120 0.03

30 0.8 65 125 0.0080 0.015

100 1.5 110 217 0.0046 0.01

eration with highest electric field where the interactions are expected to playsoon an important role.

Then, we solved numerically Eq. (C.4.1) and (C.4.3) for Ei = 100 with allthe two particle QED interactions. The first striking outcome is that after∆t∗ = 1.27 · 10−2 tC from the beginning, the energy density of the photons isroughly 10% of the electrons or positrons ones. This points out that at thisstage, also the presence of the photons can not be neglected anymore.

Again, looking at Fig. C.5 we realize that the photons DF is perfectly sym-metric with respect to the plane identified by the condition p‖ = 0. The situa-tion changes slightly for electrons and positrons because the electric field thataccelerate them toward opposite directions, namely the regions characterizedby p‖ > 0 and p‖ < 0 respectively. However the differences with a perfectlysymmetric DF are very small and this fact can be understood easily having inmind the time laps during which the electric field acted on the pairs.

Another consequence due to the relative scatterings between particles isthat the bulk parallel momentum of the pairs is smaller that the one achievedafter the same evolution time in the non-interacting system. The latter canbe readily estimated since the electric field is nearly constant during the timeinterval ∆t∗, then the corresponding variation of the momentum should beabout ∆p∗ni

‖ ' 1.27 me c while for the interacting one the same quantity is

∆p∗i‖ ' 0.64 me c. Comparing these two numerical values we understandthat electrons and positrons moving toward opposite directions exchange animportant fraction of their parallel momentum previously gained from theelectric field acceleration; in that sense, scatterings act like friction betweenelectrons and positrons.

91


The example presented above is a clear evidence that interactions can playa very important role. As it was expected, the higher the field the more dra-matic are their effects.

C.7. Conclusions

There are many analogies between the results we obtain using the two meth-ods, namely the time dependence during the first half oscillation of all thequantities involved as pairs number density, bulk parallel momentum andelectric field. However, after this short period, the two methods give sub-stantially different results.

In the non-interacting cases and for all the considered parameter sets wefind that after several oscillations the magnitude of the electric field becomesmuch less than the starting value; as a consequence, acceleration and pairproduction are strongly suppressed. The bulk momentum parallel to the ex-ternal field becomes very small, while the number density of the pairs satu-rates to a small fraction of the maximum achievable one, well below 10 per-cent for all cases considered. This result comes out only when kinetic treat-ment is adopted, and consequently distribution of particles in momentumspace is accounted for. We find that production of pairs and their accelera-tion in the same electric field simultaneously produce a peculiar distributionfunction of particles which quickly settles down in a sort of equilibrium.

This equilibrium is well described using a relativistic two-temperature dis-tribution function, with the temperature along the electric field being muchlarger than the one in orthogonal direction. Therefore we find that substan-tial part of total energy, initially stored in the electric field is converted afterfew oscillations into internal energy. This effect could not be obtained withinthe simple treatment we used before, since all particles were assumed to havesingle momentum (delta-function distribution in momentum space).

For the interacting system with Ei = 100 we found that the interactionsbecome important soon, as it was expected. Interactions act like a frictionbetween pairs since the achieved parallel momentum is always less than thatattained in the collisionless approximation. Besides, the pairs annihilationprocess becomes so important that the photons energy density becomes com-parable to the electrons and positrons one within a fraction of Compton time.At least in this specific case, it implies that interactions can not be neglectedeven for a very short time scale. We expect that interactions assume an im-portant role also for a smaller initial electric field, but at later times.

The photons DF appears to be perfectly symmetric with respect to theplane with null parallel momentum while the electrons and positrons DFsare nearly symmetric. In fact the interactions scatter particles all over the mo-mentum space even if the external electric field is accelerating electrons andpositrons toward opposite directions. It was not so for the non-interacting

92

C.7. Conclusions

system where electrons and positrons were initially distributed over disjointregions in the momentum space as soon as they were produced.

93


0.1 1 10

time (tc)

0.0001

0.001

0.01

0.1

1

ener

gy d

ensi

ty

totrestkineticthermal

Figure C.4.. The rest (red), kinetic (purple), internal (lightblue) energy densities of the pairs and their sum (black) forthe non-interacting system with Ei = 100. Since the rest en-ergy density is simply the number density n multiplied byme c2, its behavior reproduces exactly the solid curve depictedin Fig. C.1. The kinetic one is the related to the bulk motionas defined in Eq. (C.6.2). The internal energy is related to thedispersion of the DF in the momentum space. At the end, theelectric field is about 1 Ec.

94

C.7. Conclusions

30 20 10 0 10 20 30

5

10

15

20

25

30

p mec

p m

ec

Log10 fΓ

6.8

22.5

Figure C.5.. Density plot of the photons DF fγ in the mo-mentum space in the case with Ei = 100 after 1.27 · 10−2 tC.At this instant the energy densities of photons and electrons(or positrons) are ργ = 1.13 · 1024 erg/cm3, ρ− = ρ+ =1.12 · 1025 erg/cm3 and the electric field is E = 99.99848.

95


96

D. Electron-positron plasma inGRBs and in cosmology

D.1. Introduction

Electron-positron plasmas are discussed in connection with astrophysi-cal phenomena such as Galactic Center, microquasars, Gamma-Ray Bursts(GRBs), as well as laboratory experiments with high power lasers, for detailssee Ruffini et al. (2010). According to the standard cosmological model, suchplasma existed also in the early Universe. It is naturally characterized by theenergy scale given by the electron rest mass energy, 511 keV. It is interest-ing that at the epoch when Universe had this temperature, several importantphenomena took place almost contemporarily: electron-positron pair annihi-lation, the Big Bang Nucleosynthesis (BBN) and neutrino decoupling.

Electron-positron plasma also is thought to play an essential role in GRBsources, where simple estimates for the initial temperature give values inMeV region. Such plasma is energy dominated and optically thick due toboth Compton scattering and electron-positron pair creation, and relaxes tothermal equilibrium on a time scale less than 10−11 sec, see Aksenov et al.(2007). The latter condition results in self-accelerated expansion of the plasmauntil it becomes either transparent or matter dominated.

In the literature there have been several qualitative arguments mentioningpossible similarities between electron-positron plasmas in the early Universeand in GRB sources. However, until now there is no dedicated study whichdraws analogies and differences between these two cases. This paper aims inconfronting dynamics and physical conditions in both cases.

D.2. General equations

The framework which describes electron-positron plasma both in cosmologyand in GRB sources is General Relativity. Both dynamics of expansion ofthe Universe, and the process of energy release in the source of GRB shouldbe considered within that framework. Hydrodynamic expansion of GRBsources may, however, be studied within much simplier formalism of Spe-cial Relativity.

97

D. Electron-positron plasma in GRBs and in cosmology

We start with Einstein equations

Rµν −12

gµνR =8πG

c4 Tµν, (D.2.1)

where Rµν, gµν and Tµν are respectively Ricci, metric and energy-momentumtensors, G is Newton’s constant, c is the speed of light, and the energy-momentum conservation, following from (D.2.1)

(Tµ

ν)

;ν =∂(√−g Tµ

ν)

∂xν+√−g Γµ

νλTνλ = 0, (D.2.2)

where Γµνλ are Cristoffel symbols and g is determinant of the metric tensor.

We assume for the energy-momentum tensor

Tµν = p gµν + ωUµUν, (D.2.3)

where Uµ, is four-velocity, ω = ρ + p is proper enthalpy, p is proper pressureand ρ is proper energy density.

When plasma is optically thick, radiation is trapped in it and entropy con-servation applies. It may be obtained multiplying (D.2.2) by four-velocity

−Uµ(Tµ

ν)

;ν = Uµρ;µ + ωUµ;µ = 0. (D.2.4)

Using the second law of thermodynamics

d(ω

n

)= Td

(σ

n

)+

1n

dp, (D.2.5)

where σ = ω/T is proper entropy density, T is temperature, one may rewrite(D.2.4) as

(σUµ);µ = Uµσ;µ + σUµ;µ = 0. (D.2.6)

Baryon number conservation equation has exactly the same form

(nUµ);µ = Uµn;µ + nUµ;µ = 0. (D.2.7)

Now recalling that Uµ ∂∂xµ = d

dτ and Uµ;µ = d ln V/dτ, where V is comov-

ing volume, τ is the proper time, from (D.2.4) and (D.2.7) we get

dρ + ωd ln V = 0, d ln n + d ln V = 0, (D.2.8)

Finally, introducing the thermal index γ = 1 + pρ restricted by the inequality

1 ≤ γ ≤ 4/3 we obtain from (D.2.4) the following scaling laws

ρVγ = const, nV = const. (D.2.9)

98


Both these conservations laws are valid for the early Universe and GRB plas-mas.

One can obtain the corresponding scaling laws for comoving temperatureby splitting the total energy density into nonrelativistic (with γ = 1) and ul-trarelativistic (with γ = 4/3) parts ρ → nmc2 + ε, where m is the mass ofparticles1, ε is proper internal energy density. The entropy of the ultrarela-tivistic component is then σ = 4

3εT , and (D.2.6) gives

εVT

= const. (D.2.10)

For ε nmc2, which is the energy dominance condition, internal energyplays dynamical role by influencing the laws of expansion. For ε nmc2,which is the matter dominance condition, internal energy does not play anydynamical role, but determines the scaling law of the temperature. In orderto understand the dynamics of thermodynamic quantities in both early Uni-verse and in GRBs, one should write down the corresponding equations ofmotion.

D.2.1. Early Universe

For the description of the early Universe we take the Robertson-Walker met-ric with the interval

ds2 = −c2dt2 + a2 (t)[

dr2

1− kr2 + r2dϑ2 + r2 sin2 ϑdϕ2]

, (D.2.11)

where a (t) is the scale factor and k = 0,±1 stands for the spatial curvature.In homogeneous and isotropic space described by (D.2.11), Einstein equa-tions (D.2.1) are reduced to Firedmann equations together with the continuityequation (

dadt

)2

+ c2k =8πG3c2 ρa2, (D.2.12)

2ad2adt2 +

(dadt

)2

+ c2k = −8πGc2 pa2, (D.2.13)

dρ

dt+

3a

dadt

(ρ + p) = 0, (D.2.14)

where a is the scale factor. Notice, that only two equations in the systemabove are independent. The continuity equation (D.2.14) follows from the

1Nonrelativistic component is represented by baryons. For simplicity we assume only onesort of baryons, say protons, having mass m. Ultrarelativistic component is representedby photons and electron-positron pairs.

99


Einstein equations (D.2.12) and (D.2.13) as the energy conservation. In fact,(D.2.14) may be also obtained from the entropy conservation (D.2.4). Thecomoving volume in Friedmann’s Universe is V = a3, so (D.2.14) and thefirst equality in (D.2.9) are equivalent.

On the radiation dominated stage of Universe expansion one has

ρ ∝ V−4/3 ∝ a−4, n ∝ V−1 ∝ a−3, (D.2.15)

while on the matter dominated stage

ρ ∝ n ∝ V−1 ∝ a−3. (D.2.16)

Entropy conservation (D.2.10) leads to the unique temperature dependenceon the scale factor

T ∝ V−1/3 ∝ a−1. (D.2.17)

The corresponding time dependence of thermodynamic quantities may beobtained from solutions of Friedmann equation (D.2.12) and continuity equa-tion (D.2.14).

D.2.2. GRBs

Different situation takes place for the sources of GRBs. Assuming sphericalsymmetry for the case of GRB the interval2 is

ds2 = −c2dt2 + dr2 + r2dϑ2 + r2 sin2 ϑdϕ2. (D.2.18)

Optically thick to Compton scattering and pair production electron-positronplasma in GRB sources is radiation dominated. Its equations of motion followfrom the energy-momentum conservation law (D.2.2) and baryon numberconservation law (D.2.7). Initially plasma expands with acceleration drivenby the radiative pressure.

In spherically symmetric case the number conservation equation (D.2.7) is

∂ (nΓ)∂t

+1r2

∂

∂r

(r2n√

Γ2 − 1)= 0, (D.2.19)

Integrating this equation over the volume from certain ri(t) to re(t) which weassume to be comoving with the fluid dri(t)

dt = β(ri, t), dre(t)dt = β(re, t), and

ignoring a factor 4π we have

2General Relativity effects may be included by taking Schwarzschild or Kerr-Newman met-ric. However, we are interested in optically thick plasma which expands with accelerationand propagates far from its source, where the spatial curvature effects may be neglected.For this reason we simplify the treatment and adopt a spatially flat metric.

100


re∫ri

∂ (nΓ)∂t

r2dr +re∫

ri

∂

∂r

(r2n√

Γ2 − 1)

dr = (D.2.20)

∂

∂t

re∫ri

(nΓ) r2dr− dre

dtn(re, t)Γ(re, t)r2

e +dri

dtn(ri, t)Γ(ri, t)r2

i +

+r2e n(re, t)

√Γ2(re, t)− 1− r2

i n(ri, t)√

Γ2(ri, t)− 1 =

=ddt

re∫ri

(nΓ) r2dr = 0,

Since we deal with arbitrary comoving boundaries, this means that the totalnumber of particles integrated over all differential shells is conserved

N = 4π

R(t)∫0

nΓr2dr = const, (D.2.21)

where R(t) is the external radius of the shell.Following Piran et al. (1993) one can transform (D.2.19) from the variables

(t, r) to the new variables (s = t− r, r) and then show that

1r2

∂

∂r

(r2n√

Γ2 − 1)= − ∂

∂s

(n

Γ +√

Γ2 − 1

). (D.2.22)

For ultrarelativistic expansion velocity Γ 1, the RHS in (D.2.22) tends tozero, and then the number of particles in each differential shell between theboundaries ri(t) and re(t) is also conserved with a good approximation, i.e.

dN = 4πnΓr2dr ≈ const. (D.2.23)

Relations (D.2.21) and (D.2.23) then imply

4π

re∫ri

(nΓr2

)dr = 4π

[n(r, t)Γ(r, t)r2

] re∫ri

dr (D.2.24)

= 4π(

nΓr2)

∆ ≈ const,

where the first argument of functions n(r, t) and Γ(r, t) is restricted to the in-terval ri < r < re, and consequently ∆ ≡ re − ri ≈ const. Taking into accountthat ri(t) and re(t) are arbitrary, this means that ultrarelativistically expand-

101


ing shell preserves its width measured in the laboratory reference frame. Thisfact has been used by Ruffini et al. (2000) and referred there as the constantthickness approximation.

The volume element measured in the laboratory reference frame is dV =4πr2dr, while the volume element measured in the reference frame comov-ing with the shell is dV = 4πΓr2dr. Comoving volume of the expandingultrarelativistic shell with Γ ' const will be

V = 4πΓ∫ r

r−∆r2dr ' 4πΓr2∆. (D.2.25)

Then we rewrite the conservation equations (D.2.9) as

ρ1γ Γr2 = const, nΓr2 = const, (D.2.26)

Unlike the early Universe, where both energy and entropy conservations re-duce to (D.2.14), in the case of GRBs the energy conservation is a separateequation coming from the zeroth component of (D.2.2) as

(T0ν) ;ν = ωU0Uν

;ν + Uν (ωU0);ν = 0. (D.2.27)

which givesρΓ2r2 = const. (D.2.28)

From (D.2.26) and (D.2.28) we then find

Γ ∝ r2(γ−1)

2−γ , n ∝ r−2

2−γ , ρ ∝ r−2γ

2−γ . (D.2.29)

For the ultrarelativistic equation of state with γ = 4/3 we immediately obtain

Γ ∝ r, n ∝ r−3, ρ ∝ r−4. (D.2.30)

Taking into account that the relation between the comoving and the physi-cal coordinates in cosmology is given by the scale factor a, it follows from(D.2.30) that both energy density and baryonic number density behave asin the radiation dominated Universe, see (D.2.15). This analogy betweenthe GRB source and the Friedmann Universe is noticed by Shemi and Piran(1990), Piran et al. (1993).

In the presence of baryons as the pressure decreases, plasma becomes mat-ter dominated and expansion velocity saturates. Hence for the nonrelativisticequation of state with γ = 1 different scaling laws come out

Γ = const, n ∝ r−2, ρ ∝ r−2. (D.2.31)

Transition between the two regimes (D.2.30) and (D.2.31) occurs at the radiusRc = B−1R0, where R0 is initial size of plasma.

102

D.3. Heavy elements

Therefore, one may reach the conclusion that for comoving observerthe radiation-dominated plasma looks indistinguishable from a portion ofradiation-dominated Universe. However, this is true only in the absence ofpressure gradients. Strong gradients are likely present in GRB sources, andthey should produce local acceleration in the radiation-dominated electron-positron plasma, making it distinct from the early Universe, where matterinhomogeneities are known to be weak.

It is easy to get from (D.2.26) and (D.2.28) for internal energy density andtemperature

ε ∝ r−4, T ∝ r−1, R0 < r < Rc, (D.2.32)

andε ∝ r−8/3, T ∝ r−2/3, Rc < r < Rtr, (D.2.33)

where Rtr is the radius at which the outflow becomes transparent. The out-flow may become transparent for photons also at the acceleration phase, pro-vided that Rtr < Rc. For instance, a pure electron-positron plasma gets trans-parent at the acceleration phase.

D.3. Heavy elements

Cosmological nucleosynthesis is a well established branch of cosmology.Classical computations made in the middle of the XXth century revealed thatheavy elements cannot be built in the early Universe. Hydrogen and heliumcontribute approximately 3/4 and 1/4, leaving some room, much less than 1per cent for deuterium, tritium and lithium. All the heavier elements musthave been produced in stars.

Some of these stars, as indicated by observations, end their life as progeni-tors of GRBs. For this reason it is likely that initially in the source of GRBs el-ements heavier than hydrogen are present. In this section we consider chem-ical evolution of plasma in the sources of GRBs.

Assume that in the source of a GRB the amount of energy E0 is releasedin the volume with linear size R0 during the time ∆t, making this regionoptically thick to Compton scattering and pair production. The amount ofbaryons which may be present as well is parametrized by

B '

Mc2

E0, ∆t R0/c,

Mc2

L, ∆t R0/c,

(D.3.1)

where L = dE/dt is the luminosity, M = dM/dt is the mass ejection rateand M is total baryonic mass. Ultrarelativistic outflow is generated through

103


thermal acceleration of baryons by the radiative pressure if plasma is initiallyenergy dominated, i.e.

B 1. (D.3.2)

In the case of instant energy release with ∆t R0/c initial temperature inthe source of GRB may be estimated neglecting the baryonic contribution,provided (D.3.2) is satisfied as

T0 '(

3E0

4πaR30

)1/4

' 6.5E1/454 R−3/4

8 MeV, (D.3.3)

where a = 4σSB/c, σSB is the Stefan-Boltzmann constant and the last valueis obtained by substituting numerical values for E0 = 1054E54 erg and R0 =108R8 cm.

As it has been shown in Kafexhiu (2010) for temperatures above 1 MeVeven low density plasma with density n = 1018 cm−3 quite quickly destroysall heavier nuclei, and the final state contains just protons and neutrons andsome small traces of Deuterium and 4He. The timescale of this process (∼10−2 sec for T0 = 1 MeV) strongly depends on temperature, but the ratesof almost all reactions increase with temperature, and correspondingly theabundances of nuclei evolve much faster. Therefore, nuclei disintegration isfast enough to occur before plasma starts to expand and cool on the timescaleR0/c.

During early stages of plasma expansion its temperature decreases in thesame way as it happened in the early Universe. Therefore similar synthesisof light elements to BBN occurs also in sources of GRBs. Most important is,however, another similarity with the early Universe: it is well known thatpractically all free neutrons have been captured into elements heavier thanhydrogen. So we do not expect dynamically important free neutrons presentin GRB plasma after it started to expand and cool down unless they are en-gulfed by the expanding plasma later. The role of such free neutrons havebeen considered in the literature, see e.g. Derishev et al. (2000).

D.4. Recombination

On the radiation dominated phase both in the early Universe and in GRBplasma entropy conservation (D.2.4) results in decrease of temperature(D.2.17). In the early Universe, after the BBN epoch and electron-positronannihilation, cosmological plasma consists of fully ionized hydrogen, heliumand small admixture of other light elements. The temperature continues todecrease until it gets sufficiently low to allow formation of neutral atoms:that is the moment in the cosmic history where the formation of the CosmicMicrowave Background Radiation (CMB) happens.

104

D.4. Recombination

The theory of cosmological recombination of hydrogen, based on threelevel approximation, has been developed by Zeldovich, Kurt and SunyaevZeldovich et al. (1968) and Peebles Peebles (1968) in the late 60s. The onlymodification that such theory undergone in the later years is the account fordark matter and addition of more levels to the model, currently about 300.There is a basic difference with respect to the equilibrium recombination es-sentially by the process e + p↔ H + γ, described by the Saha equation

nenp

nH=

gegp

gH

(2πmekT)3/2

h3 exp(− Ei

kT

), (D.4.1)

where gi are statistical weights, h is Planck’s constant, Ei = 13.6 eV is hydro-gen ionization energy. This difference is due to the presence of the 2p quan-tum level, which produces Ly-α photons. The absorption of such photons isvery strong. However, ionization from the 2p level requires only Ei/4. There-fore the formation of neutral hydrogen proceed through the 2s− 1s transitionin the presence of abundant Ly-α photons.

In fact, the early Universe would become transparent for radiation even ifformation of hydrogen would have been forbidden, see e.g. Naselsky et al.(2011). The optical depth to Thomson scattering is

τ =∫ t0

tσTnbcdt '

4× 10−2 ΩbΩm

h[

ΩΛ + Ωm (1 + z)3]1/2− 1

, (D.4.2)

where σT is the Thomson cross section, Ωi = ρi/ρc, ρc = 3H2c2/8πG,H = 100h km s/Mpc and b, m, Λ stand for, respectively baryons, dark mat-ter and cosmological constant contributions to the total energy density of theUniverse. For large z we have

τ (z∗) = 1, z∗ ' 8.4Ω−2/3b Ω1/3

m h−2/3.

For typical values Ωbh2 ' 0.02, Ωm ' 0.3, and h ' 0.7 we have z∗ ' 60.At such redshift the Universe would be expected to become transparent toThomson scattering. That is exactly what happens in plasma in GRB sources.Below we show that, unlike radiation-dominated cosmological expansionwhere comoving quantities also fulfill relations (D.2.32), the comoving tem-perature in GRB outflows remains always high enough to prevent recombi-nation of hydrogen.

From (D.2.32) and (D.2.33) one can see that during both acceleration andcoasting phases the comoving temperature decreases. However, when theoutflow reaches the radius Rs = B−2R0, the comoving temperature saturates.It can be shown, see Ruffini et al. (2011) for details, using the expression for

105


the photospheric radius

Rtr =

(σE0B

4πmpc2

)1/2

, (D.4.3)

where mp and σ are proton mass and Thompson cross section, respectively.Indeed, using (D.3.3), (D.4.3), (D.2.32) and (D.2.33) we have

Tmin = BT0

(Rc

Rtr

)2/3

= (D.4.4)

=

(3

4πa

)1/4( σ

4πmpc2

)−1/3

(E0R0)−1/12 . (D.4.5)

Notice how extremely insensitive this value is with respect to both E0 and R0!Expressed in units of typical energy and size

T(s)min ' 42 (E54R8)

−1/12 eV. (D.4.6)

In the case of gradual energy release with with ∆t R0/c and constantluminosity and mass ejection rate the initial temperature is

T0 '(

3L16πσSBR2

0

)1/4

, (D.4.7)

and similar expression to (D.4.4) may be derived

Tmin =

(3

16πσSB

)1/4( σ

4πmpc2

)−1/3

L−1/12R1/60 ∆t−1/3, (D.4.8)

which may be rewritten, introducing L = 1050L50 erg/s and ∆t = 1∆t1 s, as

T(w)min ' 22L−1/12

50 R1/68 ∆t−1/3

1 eV. (D.4.9)

Even if (D.4.9) appears to be less stringent that (D.4.6), they are both quiteinsensitive to initial parameters. As a result, even if the comoving tempera-ture decreases very much compared to its initial value, typically of the orderor MeV, it saturates well above the ionization energy of hydrogen Ei = 13.6eV, thus preventing formation of neutral hydrogen. In fact, such hydrogenformed would not allow formation of the afterglow of GRBs. Indeed, evenhaving ultrarelativistic velocities of expansion neutral hydrogen would notinteract with the interstellar medium!

A simplified way to look at this lower bound on the comoving temperatureis to say that if a fraction ε of solar mass is released in the volume having

106

D.5. Conclusions

radius δ solar Schwarzschild radii, then its minimum comoving temperaturebefore transparency is

T(s)min ' 66 (εδ)−1/12 eV, (D.4.10)

in the case of instant energy release and

T(w)min ' 3.7ε−1/12δ1/6∆t−1/4

1 eV, (D.4.11)

in the case of gradual energy release during time ∆t1. Clearly in both casesδ > 1, and likely ε < 1. Notice, that while in the case of instant energy re-lease the lower bound on temperature decreases with increasing δ, it insteadincreases in the case of gradual energy release.

Therefore, we have reached the conclusion that hydrogen recombinationwhich is responsible for transparency of cosmological plasma does not hap-pen in GRB plasma. This difference in physical conditions may result indeviations from black body spectrum, as observed in GRBs. Recently wepresented a theory of photospheric emission from relativistic outflows, seeRuffini et al. (2011). Assuming that the spectrum of radiation in the comov-ing reference frame is the perfect black body one, we have shown that thespectrum seen by a distant observer may be essentially nonthermal due toboth geometric and dynamical special relativistic effects. The possibility thatthe spectrum of photospheric emission is nonthermal also in the comovingframe should be investigated.

D.5. Conclusions

Regarding the dynamical aspects, there is an apparent similarity between theelectron-positron plasma in the early Universe and the one in GRB sources.For an observer comoving with the radiation-dominated plasma in GRBsource it may look indistinguishable from a portion of radiation-dominatedUniverse. However, this is true only in the absence of pressure gradients.Strong gradients are likely present in GRB sources, and they should pro-duce local acceleration in the radiation-dominated electron-positron plasma,making it distinct from the early Universe, where matter inhomogeneities areknown to be weak.

There is also an apparent similarity with respect to the nucleosynthesisphenomenon. Given that the temperature reached in GRB sources, see Eq.(D.3.3), may be as high as several MeV, nuclear reactions are expected to op-erate on timescales of 10−2 sec or shorter. That is on the order of magnitudeof dynamical timescale of the GRB sources. It means that reprocessing of nu-clear abundances may likely take place in GRB sources. Since observationsimply that GRBs may originate from compact stellar objects elements heavier

107


than helium are likely to be present in GRB sources. Such heavy elements arethen destroyed, resulting mainly in protons with small admixture of helium.Thus, similarly to the early Universe, we do not expect dynamically impor-tant free neutrons present in GRB plasma after it started to expand and cooldown unless they are engulfed by the expanding plasma later.

Finally, there is an important difference between the electron-positronplasma in the early Universe and the one in GRB sources. We show in thispaper that unlike the primordial plasma which recombines to form neutralhydrogen, and emits the Cosmic Microwave Background Radiation, GRBplasma does not cool down enough to recombine. Therefore GRB plasma be-comes transparent due to Compton scattering. The kinetics of this processis different from the cosmic recombination. In particular, cosmic recombina-tion is a photon non-conserving process, which results in an almost perfectblack body spectrum of CMB. At the same time, Compton scattering, whichis a dominant process at GRB transparency is a photon conserving process.This difference may result in deviations from the perfect thermal spectrum ofemission to be observed when GRB plasma becomes transparent.

108

E. Photospheric emission fromrelativistic outflows

E.1. Introduction

High Lorentz factors of the bulk motion of various outflows are common inrelativistic astrophysics. The best known examples are Active Galactic Nuclei(Maraschi, 2003), microquasars and Gamma-Ray Bursts (GRBs) (Piran, 2004).In the latter case outflows indeed reach ultrarelativistic velocities. For thisreason GRB emission a) originates far from the source of energy release andb) is observed as a transient event with a typical duration on the order ofseconds.

Various models are suggested to explain the acceleration of outflows to ul-trarelativistic velocities. The electromagnetic model (Lyutikov, 2006) assumesthat the energy in the source of GRB is converted into electromagnetic energywhich is transported in the form of a Poynting flux. We adopt here anotherpopular idea that the energy release leads to creation of an optically thicksource which expands due to thermal acceleration. This idea is the basis ofboth the fireball (Piran, 1999) and the fireshell (Ruffini et al., 2009a) models.

In pioneer works by Goodman (1986) who considered an instant explosion,and by Paczynski (1986) who discussed a gradual energy release, a conclu-sion was reached that the electron-positron plasma is created in the sourceof GRB. Assuming further that the plasma reaches thermal equilibrium theyfocused on hydrodynamic expansion in such models and gave photometricand spectroscopic predictions for GRBs. Later, baryonic loading of fireballswas considered for explosions by Shemi and Piran (1990) and for winds byPaczynski (1990). Abramowicz et al. (1991) considered the appearance of thephotosphere of the relativistic wind to a distant observer showing that it isconcave.

The interest to photospheric emission from relativistic winds has been re-vived recently in papers by Daigne and Mochkovitch (2002), Pe’er et al.(2007), Beloborodov (2011), Ryde et al. (2011), Pe’er and Ryde (2011), andothers.

In the fireshell model, which assumes an explosive energy release, the firstpotentially visible component of any GRB, the Proper GRB (Bianco et al.,2001), comes from the thermal flash of photons emitted when the outflowbecomes transparent for photons.

109

E. Photospheric emission from relativistic outflows

In this paper we readdress the issue of the photospheres of relativistic out-flows using a simple model valid for both instant and gradual energy release.We rederive analytic expression for the optical depth, give asymptotic solu-tions for the photospheric radius providing clear physical interpretation ofour results. Then we present the analytic treatment and fitting formulae forobserved flux of photospheric emission assuming isotropic thermal distribu-tion of photons in the comoving frame. Finally, we compute both instanta-neous and time-integrated spectra of photospheric emission. These resultsare then applied within both the shell and the wind models of GRBs.

The structure of the paper is as follows. In Section 2 we discuss, compareand contrast an impulsive explosion and gradual energy release, giving riserespectively to an ultrarelativistic shell and wind. In Section 3 we computethe optical depth and the radius of photosphere of a portion of relativisticwind. Section 4 discusses the geometry and dynamics of the photospherein the relativistic outflow. In Section 5 we treat the observed flux emergingfrom the photosphere. Observed spectra are computed in Section 6. Then weapply these results to GRBs in Section 7. Conclusions follow.

E.2. Optically thick relativistic outflows: wind vs.explosion

Relativistic outflows are generally classified as winds or shells depending onhow fast the energy in their sources is released. Consider energy release ina source of linear dimension R0. If the time scale of energy release is short∆t R0/c, an explosion occurs, which may be characterized by the sizeR0, total energy released E0 and total baryonic mass M. When the energy isreleased gradually, on a time scale ∆t R0/c, but the source luminosity Lexceeds the Eddington limit, a wind is formed, which is characterized by itsactivity time ∆t, luminosity L and mass ejection rate M.

It is possible that the temperature in the source of the relativistic outflowis large enough for electron-positron pair creation. In that case e+e− pairsmake an additional contribution to the optical depth. Whether the outflowbecomes relativistic or not depends on the entropy in the region where theenergy is released. Both the wind and explosion cases can be parametrized(Ruffini et al., 2000) by a dimensionless baryonic loading parameter

B '

Mc2

E0, explosion,

Mc2

L, wind.

(E.2.1)

When the baryonic loading is sufficiently small the baryons will be acceler-

110

E.2. Optically thick relativistic outflows: wind vs. explosion

ated to a relativistic velocity v of bulk motion, attaining large Lorentz factors

Γ =[1− (v/c)2

]−1/2, while in the opposite case of large baryonic loading

the outflow remains nonrelativistic:

Γ = Γm ' B−1, B 1 (E.2.2)

v ' c√

B/2, B 1. (E.2.3)

In what follows we consider only ultrarelativistic spherically symmetricoutflows with Γ 1. In the simplest cases of wind or explosion in vacuum,the dynamics of the outflow is divided into an acceleration phase and a coast-ing phase (Piran, 1999) with respectively

Γ ' RR0

, n ' n0

(RR0

)−3

, R0 < R <R0

B, (E.2.4)

Γ ' B−1 = const, n ' n0

(RR0

)−2

,R0

B< R < Rtr, (E.2.5)

where Rtr is the radius at which the outflow becomes transparent to Thomsonscattering and n is the comoving number density of baryons in the outflow.The outflow may also become transparent for photons at the accelerationphase, provided that Rtr < B−1R0. Notice that in the case of an impulsiveexplosion for r R0 the matter and energy appear to a distant observer tobe concentrated in a geometrically thin shell having width l ∼ R0 due to therelativistic contraction.

It is important to stress that both an infinitely long wind with a time-independent mass ejection rate and luminosity on the one hand, and an in-finitely thin shell originating from an instantaneous explosion in infinitely thinregion represent two limiting cases for the energy release.

During both acceleration and coasting phases the continuity equation forthe laboratory number density reduces to nΓ ∝ r−2. We parametrize genericdensity profile as

nΓ =

n0

(R0

r

)2

f (ξ), Ro(t)− l < r < Ro(t),

0, otherwise,

(E.2.6)

where Ro(t) is the equation of motion of the outer boundary of the outflow,f (ξ) is an arbitrary function of the depth ξ of the outflow measured from theouter boundary. Analogously one may introduce the corresponding func-tions for temperature and Lorentz factor dependences within expanding out-flows.

In this paper we focus on a simplest case with constant number density,

111


Figure E.1.: The laboratory number density is shown for a shell with initiallyconstant density and a wind with finite duration at two different times t = t1and t = t2. While the wind preserves the r−2 shape, the shape of the shellchanges making its density profile steeper with time.

temperature and Lorentz factor measured by an observer with fixed labora-tory radius. This profile corresponds to a portion of relativistic wind withf (ξ) = 1. Such an outflow may be produced by a gradual energy releasewith constant luminosity and mass ejection rate on a finite time ∆t and wewill refer to it as the wind.

Below we show that both the shell and the wind defined above may appearfor photons emitted inside it as long wind or as thin shell, depending on theinitial conditions that specify respectively their width l and activity duration∆t. It is crucial to keep in mind that photons emitted inside the expandingoutflow propagate in a medium whose laboratory number density dependsboth on the radial coordinate and on time Γ (r, t) n (r, t). For photons prop-agating in the wind the spatial dependence of the number density plays thekey role, while for photons propagating in the shell its time dependence iscrucial.

In the next section we compute the optical depth of the portion of wind.

E.3. Optical depth along the line of sight

In this section we compute the optical depth of relativistic outflows and de-termine the corresponding photospheric radii along the line of sight. Theoptical depth along the photon world line L is

τ =∫L

σjµdxµ, (E.3.1)

112


where σ is the cross section, jµ is the 4-current of particles, on which thephoton scatters, and dxµ is the element of the photon world line.

Consider a spherically symmetric expanding outflow with an ultrarela-tivistic velocity v = βc ' 1 − 1/2Γ2. Assume that the photon is emittedat the interior boundary r = R of the outflow. The optical depth computedalong the photon path from (E.3.1) is (see e.g. Abramowicz et al. (1991))

τ =∫ R+∆R

RσnΓ (1− β cos θ)

drcos θ

, (E.3.2)

where R+∆R is the radial coordinate at which the photon leaves the outflow,and θ is the angle between the velocity vector of the outflow and the directionof propagation of the photon, n is the comoving number density of electronsand positrons, which may be present due to pair production. Along the lineof sight (E.3.2) reduces to

τ =∫ R+∆R

Rσn(R)Γ (1− β) dr ≈

∫ R+∆R

Rσ

n2Γ

dr. (E.3.3)

In the remaining part of this section we will consider the optical depth underdifferent conditions characterizing the outflow and derive expressions for thephotospheric radius along the line of sight.

E.3.1. Pure electron-positron plasma

A pure electron-positron plasma reaches thermal equilibrium before expan-sion and remains thermal and accelerating until it becomes transparent toradiation (Aksenov et al., 2007, 2009, 2010). Comoving number density ofelectrons and positrons is a function of their comoving temperature, whichis decreasing during accelerating adiabatic expansion (e.g. Shemi and Piran(1990)) as

T(R) = T0R0

R, (E.3.4)

where T0 is the temperature of plasma in the source. Since the transparencytemperature is nonrelativistic, Ttr mec2/k, we can treat electron-positronpairs as nondegenerate and nonrelativistic. Their number density is then

n(T) =1√2

(kTme

πh2

)3/2

exp(−mec2

kT

). (E.3.5)

113


From (E.3.3), (E.3.4), and (E.3.5) we obtain the optical depth along the line ofsight

τ =∫ R+∆R

R

σ√2

(kT0me

πh2

)3/2(R0

r

)5/2

exp(−mec2r

kT0R0

)dr. (E.3.6)

Initial temperature T0 can be expressed in terms of initial size and energy.Then by equating (E.3.6) to unity we obtain the radius at which the outflowbecomes completely transparent along the line of sight, the transparency radiusRtr.

Due to exponential dependence on the radial coordinate in (E.3.6), trans-parency is reached at

kT± ' 0.040mec2 (E.3.7)

independent of the initial conditions. Note that the optical depth for an ex-panding electron-positron-photon shell computed by Shemi and Piran (1990)is incorrect since it uses photon thin asymptotics, see (E.3.23) below, whichnever applies to the pure e+e− outflows. The formula (E.3.7) is in agreementwith works of Grimsrud and Wasserman (1998) and Li and Sari (2008).

In the case of the shell with l ' R0

Rtr =1

T±

(3E0l4πa

)1/4

, (E.3.8)

where a = 4σSB/c and σSB is the Stefan-Boltzmann constant. Analogously inthe case of the wind

Rtr =1

T±

(E0R2

04πla

)1/4

=1

T±

(LR2

016πσSB

)1/4

, (E.3.9)

where we used a relation

E0 = L∆t ' Ll/c. (E.3.10)

E.3.2. Acceleration phase

Now we focus on the acceleration phase described by (E.2.4), assuming thatthe optical depth (E.3.2) along the line of sight corresponds to electrons withdensity equal to the baryon density n. Here and below we take for the labo-ratory density a profile (E.2.6) with f (ξ) = 1 which gives

nΓ =

n0

(R0

r

)2

, Ro(t)− l < r < Ro(t),

0, otherwise.(E.3.11)

114


Actually positrons give a contribution to the optical depth even as comov-ing temperature decreases below (E.3.7) due to their freeze out. Their contri-bution evaluates to

n±n

=4πlr2

±n±Γ±N

, (E.3.12)

where quantities with subscript ± refer to the temperature (E.3.7), and totalnumber of baryons is obtained from (E.3.11) as

N = 4πn0R20l. (E.3.13)

In what follows we neglect the electron-positron pair contribution, assumingn± n.

Assuming that the cross section is constant, and using (E.2.4), (E.3.3) and(E.3.11) one has

τ =∫ R+∆R

Rσn0

(R0

r

)2(

R20

2r2

)dr =

16

σn0R40

(1

R3 −1

(R + ∆R)3

). (E.3.14)

In the ultrarelativistic approximation from equations of motion of the photonand of the outflow one can see that such photon cannot leave the outflowduring acceleration phase. Therefore (E.3.14) reduces to

τ =σn0R4

06R3 =

σNR20

24πR3l. (E.3.15)

We then obtain the following expression for the transparency radius of theaccelerating outflow:

Rtr =

(σNR2

024πl

)1/3

. (E.3.16)

Finally we can express the number of baryons in the outflow for either theshell or wind, assuming baryons are represented by protons with mass mp.For the wind N = M∆t/mp ' Ml/(mpc), where the outflow width is givenby l ' c∆t, while for the shell with l ' R0 the number of baryons is N =M/mp. Then using (E.2.1) we have

N =

E0Bmpc2 , shell,

LBlmpc3 , wind.

(E.3.17)

115


As result we obtain for the shell

Rtr =

(σE0BR2

024πmpc2l

)1/3

, (E.3.18)

and for the wind

Rtr =

(σLBR2

024πmpc3

)1/3

. (E.3.19)

E.3.3. Coasting phase

At the coasting phase described by (E.2.5), the equation of motion for theexternal boundary of the outflow is

Ro(t) = R + l + cβt, (E.3.20)

so the photon “sees” the expanding outflow 2Γ2 times thicker than its widthmeasured in the laboratory frame

∆R =l

1− β' 2Γ2l. (E.3.21)

The optical depth along the line of sight computed along the path of the pho-ton is again given by (E.3.3)

τ =∫ R+∆R

R

σn0

2Γ2

(R0

r

)2

dr =σn0R2

02Γ2

[1R− 1

R + 2Γ2l

]. (E.3.22)

This expression has two asymptotics, which using (E.3.13) can be written asfollows

τ =

σn0R2

02Γ2R

=σN

8πΓ2lR, 2Γ2l R,

σn0R20l

R2 =σN

4πR2 , 2Γ2l R.

(E.3.23)

Finally the transparency radius for the shell is

Rtr =

σE0B3

8πmpc2l, 2Γ2l Rtr,

(σE0B

4πmpc2

)1/2

, 2Γ2l Rtr,

(E.3.24)

116


while for the wind it is

Rtr =

σLB3

8πmpc3 , 2Γ2c∆t Rtr,

(σLB∆t4πmpc2

)1/2

, 2Γ2c∆t Rtr.

(E.3.25)

E.3.4. Discussion

Let us interpret the results obtained above. On the one hand, Eq. (E.3.15) andthe first line of (E.3.23) with τ ∝ R−3 and τ ∝ R−1 correspond to the casewhen a photon emitted inside the outflow stays there for a significant timeso that it feels its density decreasing with time (or radius). In this respect thephoton “sees” the outflow as a long wind, even if the laboratory thickness ofthe outflow is small, l R. We refer to this case as a photon thick outflow.On the other hand, the second line in (E.3.23) with τ ∝ R−2 corresponds tothe case when the photon spends too little time inside the outflow to feelits density decreasing with radius (for the wind) or time (for the shell). Thephoton “sees” the outflow as a thin shell even if the duration of explosioncould be long and a wind was launched. We refer to this case as a photon thinoutflow. In other words, a geometrically thin ultrarelativistically expandingshell may be both thin or thick as viewed by the photon propagating insideit.

Similar consideration may be applied to a photon emitted form any dis-tance ξ from the outer boundary of the outflow. It is clear then, that even ina photon thick outflow there is always a photon thin layer located near theouter boundary. During acceleration phase such a photon thin part accountsfor a fraction not larger than B of the entire width of the outflow.

In the derivation of (E.3.23) we used the approximation (E.3.11) represent-ing a portion of a relativistic wind. The results for both density profiles areequal, provided that R l.

The expressions for the radius of the photosphere of a relativistic windwere obtained in e.g. Meszaros and Rees (2000). These formulas coincide upto a numerical factor (which comes from the integration over the radial coor-dinate) with our (E.3.19) and (E.3.25). It should be noted, however, that onlythe photon thick asymptotic limit is applied in Meszaros and Rees (2000).

The photon thin asymptotic limit may also be valid for relativistic windsin the coasting phase, provided that l R/(2Γ2). This is an independentcondition from ∆t R0/c and it is therefore possible to give the followingconstraints for ∆t under which the outflow takes the form of a wind, but it isphoton thin:

R0

c ∆t σLB5

16πmpc4 . (E.3.26)

117


Transparency radius for photon thick and photon thin asymptotics for a por-tion of the coasting relativistic wind (E.3.25) was obtained by Daigne andMochkovitch (2002).

Similar considerations apply to an ultrarelativistic shell which is consid-ered e.g. in Shemi and Piran (1990) and Meszaros et al. (1993) and by Ruffiniet al. (2000) in the photon thin approximation. The corresponding conditionthat the shell appears to be photon thin is

c∆t R0 (

σE0B5

16πmpc2

) 12

.

It is possible, however, that initial conditions satisfy the opposite constraint

R0 c∆t, R0 (

σE0B5

16πmpc3

) 12

,

which results in a photon thick shell. This photon thick asymptotic limit hasbeen used by Nakar et al. (2005) following Grimsrud and Wasserman (1998)without any discussion of its applicability. All cases discussed above havebeen considered by Meszaros et al. (2002), except the transparency of pureelectron-positron outflow. Finally, Toma et al. (2011) consider all asymptoticsolutions, applying them to a relativistic wind.

Numerical hydrodynamic simulations produce complex density, tempera-ture and Lorentz factor profiles of the outflow. In particular, Piran et al. (1993)and Meszaros et al. (1993) considered an explosion in a compact region withradius R0 and studied numerically the hydrodynamic evolution of an opti-cally thick plasma with various initial conditions. They have shown that theplasma forms a relativistically expanding shell with some density and ve-locity profiles. The characteristic width of the density profile appears to beconstant up to large radii, but later it increases linearly with radius due to thefact that the Lorentz factor appears to be monotonically increasing within theshell

l ' R0, R < B−2R0, (E.3.27)

l ∝ R, R > B−2R0. (E.3.28)

Such a spreading in density profile may result in a substantial increase in thewidth l R0 of the shell when it becomes transparent to radiation. Note thatcondition (E.3.28) coincides with the condition in the second line of (E.3.23),which corresponds to the case of a photon thin outflow. Numerical integra-tion of (E.3.3) shows nevertheless that the photon thin asymptotics (E.3.23) isvalid even for the shell undergoing such spreading.

118

E.4. Geometry and dynamics of the photosphere


Unlike traditional static sources usually dealt with in astrophysics, relativisticoutflows have strongly time-varying photospheres. For a given laboratorytime t the photosphere geometry r = r(µ) is obtained by equating (E.3.2) tounity.

For the portion of wind (E.3.11) the optical depth can be calculated analyt-ically both at acceleration and coasting phases for photon thin and photonthick outflows:

τ(r, θ, t) = σn0R20

1

r sin θ

[θ − tan−1

(r sin θ

cT + r cos θ

)]

−(

1− 12Γ2

m

)(1r− 1√

(cT + r cos θ)2 + (r sin θ)2

)

+R2

06

(1r3 −

1

[(cT + r cos θ)2 + (r sin θ)2]3/2

), (E.4.1)

where T is the time interval during which photon remains inside the outflow,determined by the equations of motion of the photon and of the outflow, andΓm is defined by (E.2.2).

For the infinitely long relativistic wind with β = const (Abramowicz et al.,1991) ∆t, T → ∞, the photosphere is a static surface

r = σn0R20

(θ

sin θ− β

), (E.4.2)

which has a concave shape for β > 2/3, see Fig. E.2. It appears for a distantobserver as a static spot with radius

Robs = πσn0R20 (E.4.3)

and brightness decreasing from the center to the edge. For the acceleratinginfinite wind the photosphere is also a static surface and its curvature is largerthan that of the coasting wind (see Fig. E.2). These static photospheres repre-sent asymptotic limits of the dynamic photospheres of coasting and acceler-ating photon thick outflows, respectively.

Inside the relativistic beaming cone θ = arccos β the photosphere of aninfinitely thin shell at a fixed laboratory time is an infinitely thin ring. Thecollection of such rings for all laboratory times represents a surface

r =(

σΣ1− β cos θ

| cos θ − β|

)1/2

, (E.4.4)

119


where Σ = n0R20l is the shell surface density. The curvature of this surface is

even larger than that of accelerating wind, as can be seen from Fig. E.2 andthe small angle θ 1/Γ expansion of (E.4.1)

τ =

σNl24πR3

(1 +

R2θ2

l2

), R ΓR0,

σN8πlRΓ2

(1 +

13

Γ2θ2)

, ΓR0 R 2Γ2l,

σN4πR2

(1 + 2Γ2θ2) , R 2Γ2l.

(E.4.5)

The surface (E.4.4) represents asymptotic limit of photospheres of photon thinoutflows.

In ultrarelativistic case the surfaces (E.4.2) and (E.4.4) give the position ofthe corresponding photon thick and photon thin outflow photospheres withvery good accuracy.

The part of dynamic photosphere seen by a distant observer at a given in-stant of arrival time ta represents equitemporal surface (EQTS) of the photo-spheric emission. EQTS has been initially introduced for the GRBs by Biancoet al. (2001). We will refer to that surfaces as Photospheric EQTS, (PhE).

E.4.1. Coasting photon thin case

Geometrical shape of dynamical PhE of the photon thin outflow is similar toEQTS of infinitesimally thin constantly emitting relativistic shell consideredfirstly by Couderc (1939) and then by Rees (1966, 1967). The EQTS of thisshell appears to a distant observer as an ellipsoid with axes ratio equal toΓ. However the PhE of photon thin outflow is not the entire ellipsoid: it isonly a part of that surface, see Fig. E.6. The external boundary of the surfacefor a given ta is defined by the condition τ(ξ = 0, µ, ta) = 1, which meansthat the optical depth of photons emitted from the outermost layer of theoutflow equals unity. In the photon thin asymptotics this surface coincideswith the relativistic beaming cone µ = β. As soon as the outflow reachesthe transparency radius given by the second lines of (E.3.24) for the shell or(E.3.25) for the wind an internal boundary appears as well. The surface ofinternal boundaries is described by (E.4.4), see also Fig. E.6.

E.4.2. Accelerating and coasting photon thick cases

As we already mentioned in Sec. E.3.4, in a photon thick outflow there isalways an external photon thin layer. Since this layer gets transparent first,initial evolution of PhE in the photon thick case is similar to the photon thin

120


5 6 7 8 90.0

0.2

0.4

0.6

0.8

x

y Μ=Β

Rtr

Figure E.2.: The shape of photospheres of infinitely long coasting (dashedcurve) and accelerating (dotted curve) winds as well as time-integrated pho-tosphere of infinitely thin shell (solid curve) in logarithmic coordinates x =cos θ × log10 rtr, y = sin θ × log10 rtr for β = 1− 10−3.

121


one. When transition to the photon thick asymptotics occurs, initially convexPhE of the photon thin layer of the outflow transforms to a concave PhE of itsphoton thick part, see Fig. E.4 and Fig. E.7 for accelerating and coasting cases,respectively. This concave PhE in both cases approaches the photosphere ofinfinitely long wind. In the coasting case the approach to the surface (E.4.2)is only asymptotic, while in the accelerating case the photosphere actuallyreaches it at finite arrival time. Similarly to the photon thin case, there ex-ists an external boundary of the PhE determined by the same condition, andshown in Fig. E.7. Notice that this boundary is wider than the relativisticbeaming surface (these are tube and cone for accelerating and coasting out-flows, respectively). As soon as the innermost part of the outflow reaches thetransparency radius, i.e. observer sees the switching off of the wind, the in-ner boundary of the PhE appears with increasing θ(ta). The surface of theseboundaries is given by (E.4.2) in the case of coasting photon thick outflow.

E.5. Observed flux

In this section we consider the appearance of the outflow photosphere to adistant observer and describe time evolution of luminosity in adiabatic ap-proximation.

E.5.1. Adiabatic approximation for evaluation of observedflux

Take a volume element dV located at the laboratory radial distance ξ fromthe outer boundary of the outflow at angular separation θ from the line ofsight, see Fig. E.3. It is convenient to introduce the notation µ = cos θ. Thebasic assumption of our method is that photons are locked inside this volumeelement for τ(ξ, µ, t) > 1 and stream freely when the optical depth decreasesbelow unity. In this approximation we neglect diffusion of photons from theoptically thick outflow. Besides, we assume that all photons at given (ξ, µ)are emitted simultaneously. It means that the last scattering surface definedby the condition τ = 1 is infinitely thin. The probability of photon emissionis δ function in (ξ, µ). Laboratory volume of this element is then

dV = 2πr(t)2dµdξ. (E.5.1)

Photons are emitted to the observer from the element at the laboratory timete(ξ, µ) defined by

τ(ξ, µ, te) = 1, (E.5.2)

when the radial coordinate of the element, determined from the equation ofmotion at given te, is r = Re(ξ, µ). Emitted energy is obtained from the energy

122

E.5. Observed flux

Ξ

dΞ

Ro

Θ rRobs

Figure E.3.: Geometry and illustration of variables used for calculation of theflux received from relativistically expanding outflow. Observer is located tothe right at infinity.

density per unit solid angle u(Ω) as

dEobs = u(Ω)dVdΩ =Ic

dVdΩ, (E.5.3)

where dΩ is the solid angle of the observer’s detector as seen from the photo-sphere in the laboratory frame and I is bolometric intensity of radiation (seee.g. Eq. (1.6) of Rybicki and Lightman (1979)). Assuming isotropic photondistribution in the comoving frame Ic = const we get in the laboratory frame(see e.g. Eq. (4.97b) of Rybicki and Lightman (1979))

I(µ) =Ic

Γ4(1− βµ)4 = Λ−4 Ic, (E.5.4)

where Λ = Γ(1− βµ), and finally changing from Ic to the laboratory energydensity ε we arrive to (see e.g. Eq. (382) of Pauli (1958))

dEobs =dΩ4π

ε

Γ2(1 + β2/3)Λ4 dV ' 3ε

16πΓ2Λ4 dVdΩ. (E.5.5)

Define the arrival time of that radiation as

ta = te − µRe/c, (E.5.6)

i.e. as time delay with respect to a ”trigger” photon emitted at te = 0, Re = 0(see e.g. Pe’er and Ryde (2011)). The observed flux at a given ta is obtainedby integration of dF = dEobs

dtaover the PhE.

123


E.5.2. Acceleration phase

When transparency is achieved at acceleration phase (E.2.4) the equation ofmotion of the element dV is

r(ξ, t) =√

R20 + (ct− ξ)2, t > ξ/c. (E.5.7)

Entropy conservation gives for laboratory energy density

ε = ε0(R0/r)2. (E.5.8)

Using µ as integration variable over the outflow PhE, the element of observedbolometric flux becomes

dF =dEobs

dξ

∂ξ

∂ta=

38

(R0

Re

)6 ε0R20dΩdµ

[1− β(Re)µ]4

∂ξ

∂ta, (E.5.9)

where β(Re) =√

1− R20/R2

e .

The observed flux as function of arrival time is represented in Fig. E.4 bythick red curve. The characteristic raising and decaying time is

δt = R20/(Rtrc). (E.5.10)

There is no simple analytic expression describing the light curve. For largeenough ∆t δt the light curve has almost rectangular shape due to the factthat its increase and decay times are much shorter than ∆t.

The outflow appears to a distant observer s a spot with size Robs =√R2

0 − (ta/c)2, for −R0/c ≤ ta ≤ 0. As soon as the PhE reaches the cor-responding accelerating infinitely long wind photosphere at ta = 0 the spotsize starts to increase almost linearly Robs ' R0 + cta, see Fig. E.4. Finally,as the innermost part of the outflow reaches the transparency radius the spottransforms to a ring with rapidly decreasing width and brightness.

E.5.3. Coasting phase

At the coasting phase the element of the outflow has the following equationof motion

r(t) = βct− ξ, µ(t) = µ = const. (E.5.11)

Arrival time of the photospheric emission (E.5.6) from that element

ta =Re

βc(1− βµ) +

ξ

βc(E.5.12)

124

E.5. Observed flux

0 50 100 150 2000

1

2

3-4.0 -2.0 0.0 2.0 4.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

xR0

0.0 0.5 1.0 1.5 2.0

TT0

yR

0

ta∆t

FF

max

Figure E.4.: Evolution of PhE and the light curve of photospheric emis-sion (red thick curve) from the photon thick accelerating outflow. Thickblack curve represents the photosphere of infinitely long accelerating wind.PhEs shown by thin curves correspond from right to left to arrival timesta = (−2−10δt,−2−9δt, . . . ), see (E.5.10). The surface µ = β is given byRobs = R0 and it is shown by thick black line. Dashed curves show the max-imal visible Robs for ta = (2−10δt, 2−9δt, . . . ) from bottom to top. The PhE atthat arrival times is a part of the wind photosphere limited by the correspond-ing curves. Observed temperature of photospheric emission is illustrated bycolor, see legend. Here Rtr = 100R0, and ∆t = 2δt.

125


depends on time of emission and µ as

te =Re(ξ, µ) + ξ

βc. (E.5.13)

Entropy conservation at the coasting phase gives (see e.g. Piran et al.(1993))

ε = ε0(R0/r)8/3. (E.5.14)

Change of variables can be made from µ to ta and the observed bolometricflux from the element dV is then

dF =dEobs

dµ

∂µ

∂ta=

=38

R8/30 dΩc4Γ6

ε0R10/3e dξ(

ta − ξβc

)4 ∣∣∣Rec −

d ln Redµ

(ta − ξ

βc

)∣∣∣ . (E.5.15)

The first parenthesis in denominator is the consequence of relativistic beam-ing, while the second one describes geometry of the photosphere. The ob-served flux from the outflow is obtained by integration over ξ from 0 to ξmaxwhich is obtained from the position of the PhE at the line of sight, see Fig.E.5.

Now we compute the photospheric emission from the coasting photon thinand photon thick outflows. In the case of the portion of wind (E.3.11) ξmax canbe found analytically:

ξmax =β

2(1− β)

[ta(2− β) + D]

−√[ta(2− β) + D]2 − 4t2

a(1− β)

, (E.5.16)

whereD = σn0r2

0/(2Γ2)2. (E.5.17)

The function Re(ξ, µ) defined by (E.5.2) cannot be found in closed analyticform. Nevertheless the flux can be evaluated numerically and it can be wellfitted by the following expression

dF '

0, ta < t0,

F0

[1 +

(Dξ

)5/6] (

t0− ξβc

ta− ξβc

)3

dξ, ta > t0,(E.5.18)

126

E.5. Observed flux

R R+l0.0

0.5

1.0

1.5

r

Τ nG

Ξmax

Figure E.5.: Optical depth along the line of sight (black solid line) and labora-tory density profile (red dashed line) at the coasting phase for density profile(E.3.11) with Γ = 20 at t = const.

127


where

t0(ξ) =ξ

βc

[1 +

β

2

(√1 +

4Dβξ− 1

)](E.5.19)

is the minimal arrival time of photons emitted from the outflow at given ξalong the line of sight. This form of pulses is generally referenced in the lit-erature as FRED-like, i.e. fast rise, exponential decay pulses (Fenimore et al.,1996), although decay is not exponential but has a power-law shape. Integra-tion of (E.5.18) over ξ gives the form of the light curve of the photosphericemission.

The photon thin outflow initially appears to a distant observer as a brightspot with increasing radius

Robs = Γcta (E.5.20)

and bolometric fluxF ∝ t1/3

a , (E.5.21)

see the raising part of the light curve in Fig. E.6. The peak flux is reached at

tp 'Rtr

2Γ2c, (E.5.22)

when the observer sees photons emitted from the innermost part of the out-flow. Then the spot transforms to a ring with steeply decreasing thicknessand brightness resulting in the observed flux

F ∝ t−3a , (E.5.23)

see the decaying part of the light curve in Fig. E.6.The observed photospheric emission of the photon thick outflow starts sim-

ilarly to the previous case, namely a spot with flux increasing as (E.5.21), dueto the presence of the outer photon thin layer. Then transition to the photonthick asymptotics results in the corresponding change of flux to

F = Fmax

[1− (tp/ta)

2]

, (E.5.24)

i.e. increase up to the saturation value Fmax ∝ L, see the raising part of thelight curve in Fig. E.7. As soon as the arrival time exceeds tp + ∆t the fluxstarts to decrease rapidly

F ∝ t2p

[1

(ta − ∆t)2 −1t2a

], (E.5.25)

and for ta ∆t the flux again behaves like (E.5.23), see the decreasing partof the light curve in Fig. E.7.

The photon thick outflow is observed initially as a spot with radius increas-

128

E.5. Observed flux

0 500 1000 1500 2000 25000

20406080

100120

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

xR0

1.0 2.5 4.0 5.5 7.0

TT0

yR

0

tatp

FF

max

Figure E.6.: Evolution of PhE and the light curve (thick red curve) of photo-spheric emission from the photon thin coasting outflow. PhEs shown by thincurves correspond from left to right to arrival times ta = (tp/5, 2tp/5, . . . ),see (E.5.22). Thick black curves bounding PhEs correspond to the position oftransparency of the innermost part of the outflow ξ = l (lower curve) and thesurface µ = β (upper curve). Observed temperature of photospheric emis-sion is illustrated by color, see legend. Here β = 1− 10−3.

129


ing linearly in ta as (E.5.20), and then reaching its maximal size (E.4.3). Againas the innermost part of the outflow approaches the wind photosphere (E.4.2)along the line of sight, the spot transforms to a ring, see Fig. E.7.

Similarly to the accelerating photon thick outflow the light curve for ∆ttp has almost rectangular shape due to the fact that its increase and decaytimes are much shorter than ∆t.

E.5.4. Discussion

It should be noted that expression (E.5.23) is known in the literature for thebolometric flux of instantaneously flashing shell (e.g. Fenimore et al. (1996);Granot et al. (1999a,b); Woods and Loeb (1999)). It may be obtained from theexpression for detected spectral flux (see Granot et al. (1999b,a))

Fν(ta) ∝∫ ∞

02πr2dr

∫ 1

−1dµ Λ−2 j′ν′(~n

′, r, ta + rµ/c), (E.5.26)

where j′ is the emissivity of the matter in its comoving frame (the energyper unit time per unit volume per unit frequency per unit solid angle), ν′ =Λν is the corresponding frequency of emission, and ~n′ is the direction to thedetector in that frame. Assuming isotropic emissivity in the comoving frameand integrating over the spectrum one arrives to the bolometric flux

F(ta) ∝∫ ∞

02πr2dr

∫ 1

−1dµ Λ−3

∫ ∞

0j′(ν′, r, ta + rµ/c)dν′, (E.5.27)

and for instantaneously flashing shell with∫j′(ν′, r, ta + rµ/c)dν′ ∝ δ(r− R)δ

(ta +

rµ

c− R

βc

)(E.5.28)

expression (E.5.23) is recovered.Actually (E.5.26) is incorrect. All the above mentioned works did not ac-

count for the transformation in the computation of observed flux from thelaboratory time t to the arrival time ta for dynamic emitter (see, e.g. page141 of Rybicki and Lightman (1979)). Accounting for that effect we have anadditional multiplier (1− βµ)−1 appearing in the integral (E.5.26) and corre-spondingly in Eq. (3) of Granot et al. (1999a), that gives for the bolometricflux after the integration over ν

F(ta) = F0t−4a ,

Rβc

(1− β) < ta <Rβc

(1 + β) . (E.5.29)

The same result can be actually obtained from (E.5.18). Analogous mistakewas done in the work of Woods and Loeb (1999), and all the subsequent arti-

130

E.5. Observed flux

0.0 0.5 1.0 1.5 2.00.00

0.04

0.08

0.12

0.160.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

xRtr

0.0 0.5 1.0 1.5 2.0

TT0

yR

tr

taHDt+tpL

FF

max

Figure E.7.: Evolution of PhE and the light curve (thick red curve) ofphotospheric emission of a photon thick coasting outflow. PhEs shownby thin curves correspond from left to right to arrival times ta =(10−2tp, 10−1.75tp, 10−1.5tp, . . . ), see (E.5.22). Thick black curves boundingPhEs correspond to the position of infinite wind photosphere (lower curve)and the maximal visible angles at given ta (upper curve). Notice that theseangles exceed the relativistic beaming angle µ = β, shown by dashed redline. Observed temperature of photospheric emission is illustrated by color,see legend. Here β = 1− 10−3 and ∆t = 5tp.

131


cles are suffering from the same problem, see e.g. Genet and Granot (2009).Correct expressions for spectral and bolometric fluxes were obtained by

Chiang and Dermer (1999) and by Ruffini et al. (2002). Using transformationlaws for intensity of radiation from comoving to laboratory frame (see Equa-tion before (381b) of Pauli (1958)) the expression for spectral flux instead of(E.5.26) becomes

Fν(ta) ∝∫ ∞

02πr2dr

∫ 1

−1dµ Λ−3 j′ν′(~n

′, r, ta + rµ/c). (E.5.30)

Finally, we discuss the applicability of our adiabatic approximation. Be-loborodov (2011) considered the photospheric emission from infinitely longwind both at acceleration and coasting phases and solved the correspond-ing radiative transfer equation. His main conclusion is that in addition tousual relativistic beaming leading to anisotropy of radiation in laboratoryframe (E.5.4), in the coasting wind there develops another anisotropy in thecomoving frame of the outflow. This comoving anisotropy results from thefraction of photons which already underwent their last scattering in the bulkphoton field of the outflow. The anisotropy of such photons grows with in-creasing radius for geometrical reasons. Since the amount of such photonsincreases with radius the intensity of the entire photon field becomes increas-ingly anisotropic. Such phenomenon is referred to as ”fuzzy photosphere”.

We expect this additional anisotropy in the case of photon thick coastingoutflows. In other words, in photon thick outflows the probability distribu-tion of last scattering positions is wide in radius and angles, see also Pe’er(2008). For photon thin outflows instead such effect is not relevant. It is pos-sible to show that practically all photons leave the outflow before the abovementioned anisotropy develops. It means that the corresponding probabilitydistribution is sharp in radius.

In order to describe the late-time photospheric emission of switching offrelativistic wind Pe’er (2008) considered emission from a single differentiallayer. He proposed a model for calculation of observed flux and spectrumbased on the probability density function describing the last scattering ofphotons. Actually he computes not the traditional energy flux understoodas energy crossing unit area in unit time, but photon flux as number of pho-tons crossing unit area in unit time. For this reason his decay law for photonflux at late times is

Fob(ta) ∝ t−2a . (E.5.31)

Lorentz transformation of the photon energy from the comoving frame to thelaboratory one results in additional multiplier (1− βµ)−1 in the energy fluxthat leads to the observed flux F ∝ t−3

a , which agrees with our (E.5.18).We conclude that adiabatic approximation used in our computation is

clearly applicable for photon thin outflows due to the presence of well de-fined, sharp last scattering surface. As for photon thick outflows our sim-

132

E.6. Instantaneous and time-integrated spectra from the photosphere

ple adiabatic approximation gives the same results for the bolometric fluxas more accurate treatments of radiation transfer (Beloborodov, 2011) and ofprobability density function (Pe’er, 2008).

E.6. Instantaneous and time-integrated spectrafrom the photosphere

In the adiabatic approximation the observed temperature Tobs of the PhE de-pends on angle θ and arrival time ta:

Tobs(ta, Re(µ)) =Tc(Re(µ))

Γ(1− βµ), (E.6.1)

where Tc(Re) is the comoving temperature on the PhE. When temperaturedependence on radius is given, the distribution of observed temperaturesacross the PhE may be obtained from (E.6.1).

Luminosity of photospheric emission is proportional to the volume of theoutflow that becomes transparent to radiation at angle θ between ta and ta +dta, i.e. to

vb =dξ

dta

∣∣∣∣µ=const

, (E.6.2)

which we will refer to as the blooming velocity. Finally, assuming thermalspectrum in the comoving frame the differential spectral flux is

dFν =dEobsdνdta

=2hc3

ν3dΩ

exp(

hν

kTobs

)− 1

2πR2e vbdµ. (E.6.3)

Blooming velocity vb can be obtained from equations of motion of PhE andthat of the outflow. In the case of photon thick outflow it is ultrarelativisticand is approximately given by the velocity of the outflow itself since its PhEquickly becomes almost static. From (E.5.12) and (E.5.14) for transparency ataccelerating phase we have

Tobs(ta, µ) =

(R0

Re

)2 T0

1− µ

√1−

(R0Re

)2, (E.6.4)

while at coasting phase

Tobs(ta, µ) =

(R0

Re

)2/3 T0

Γ(1− βµ)=

T0R2/30 R1/3

e

Γ(taβc− ξ). (E.6.5)

133


E.6.1. Photon thin outflow

Instantaneous spectrum of a photon thin outflow integrated over PhE is veryclose to the thermal one for both accelerating and coasting phases. At ac-celerating phase observed temperature does not vary with arrival time. Atcoasting phase observed temperature decreases as

T ∝ t−2/3a , (E.6.6)

while emitting area increases as

A ∝ t3a, (E.6.7)

leading to the increase of flux (E.5.21). For ta > tp the emitting area starts todecrease

A ∝ t−1/3a , (E.6.8)

leading to diminishing flux as described by (E.5.23).

It can be seen from Fig. E.6, that temperature across a given PhE is practi-cally constant. Actually, the radial coordinates of the PhE change only twicebetween θ = 0 and θ = 1/Γ. That is converted by (E.6.5) to even smaller tem-perature interval: from the temperature at the line of sight TLOS to TLOS/ 3

√2

at θ = 1/Γ.

Integrating these instantaneous spectra over ta a non thermal spectrum isobtained. It may be represented as Band function (Band et al., 1993) with low-energy power law index α ' 0.5 and high-energy power law index β = −4,see Fig. E.8. There is an additional cutoff in that spectrum at a very highfrequency corresponding to initial temperature of the outflow T0.

Evolution of instantaneous spectrum with arrival time depends on as-sumed outflow density profile (E.3.11). Time evolution of observed tempera-ture is determined by entropy conservation so it does not depend on assumeddensity profile. But emitting area dependence on arrival time actually doesdepend on it due to change in the blooming velocity. Low-energy power lawindex is quite insensitive to the form of both density and temperature pro-files. However high-energy part of the time-integrated spectrum of photonthin outflows strongly depends on it. In particular, for power law densityprofiles (E.2.6) with slope δ the high-energy part of the spectrum is

Fν ∝ ν−3

1+δ . (E.6.9)

The resulting β parameter of the Band spectrum may vary from −∞ (expo-nential cutoff) for δ = −1 to −1 for δ = ∞.

134

E.6. Instantaneous and time-integrated spectra from the photosphere

0.01 1 100 104 106 ΝΝ010-15

10-12

10-9

10-6

0.001

1

ΝFΝ

Figure E.8.: Time-integrated spectrum of photospheric emission of photonthin outflow (thick curve, Γ = 100), superimposed with two instantaneousspectra of that emission, corresponding to arrival time of photons emittedat the moment of transition from acceleration to coasting (thin red curve onthe right) and to arrival time of complete LOS transparency of the outflow atta = tp (thin green curve on the left).

135


E.6.2. Photon thick outflow

The photospheric spectrum of both accelerating and coasting photon thickoutflows evolves through three phases. Initially photon thin layer of the out-flow gets transparent, which is described in the previous subsection. Thenthe PhE becomes concave and it approaches the corresponding static surface,see Figs. E.4 and E.7 for accelerating and coasting phases respectively. Bothaccelerating and coasting photon thick outflows exhibit spectra close to ther-mal ones, see Figs. E.9 and E.10. The spectra do not evolve until observerdetects emission from the innermost part of the outflow, when third phaseof spectral evolution begins with very fast decrease of both temperature andemitting area.

Considering time-integrated spectrum we find that as characteristic timesof the first and third phases are much less than that of the second one, theydo not affect its main part, and the spectrum is close to the thermal one.

10-4 10-3 10-2 10-1 100 101 ΝΝ0

10-12

10-9

10-6

10-3

100

ΝFΝ

Figure E.9.: Instantaneous spectrum of photospheric emission of accelerat-ing photon thick outflow (thick curve, Lorentz factor at transparency radiusΓ = 100). Dashed red curve represents the thermal spectrum with the tem-perature TLOS.

E.6.3. Discussion

Goodman (1986) was the first to obtain the observed spectrum of outflow re-sulting from instant explosion without any baryonic loading. He found that

136

E.7. Application to GRBs

10-4 10-3 10-2 10-1 100 101 ΝΝ0

10-12

10-9

10-6

10-3

100

ΝFΝ

Figure E.10.: Instantaneous spectrum of photospheric emission of coastingphoton thick outflow (thick curve, Lorentz factor Γ = 100). Dashed red curverepresents the thermal spectrum with the temperature TLOS.

it is slightly broader than the pure thermal spectrum. Similar conclusion waslater reached by Grimsrud and Wasserman (1998) who considered a staticwind created by electron-positron pairs. The observed spectrum of photo-spheric emission from relativistic wind with baryonic loading and variableLorentz factor was obtained by Daigne and Mochkovitch (2002). Again theyfound it slightly broader than the thermal one. The results obtained with ouradiabatic approximation are in agreement with the above mentioned works.

Using the probability density function method Pe’er and Ryde (2011) havecomputed the observed spectrum form switching off relativistic wind. Theyfound that at late times it becomes flat over a wide spectral range. This ef-fect comes from wide distribution of probability of photon last-scattering, thesame effect as ”fuzzy photosphere” of Beloborodov (2011). Therefore we ex-pect this effect to be present in photon-thick outflows at ta > tp + ∆t.


Now we apply the theory of photospheric emission from ultrarelativistic out-flows developed in previous sections to GRBs. For simplicity we will ne-glect cosmological corrections which reduce the observed energy by a factor(1 + z) and increase the arrival time by the same factor.

137


E.7.1. Shell model

In this subsection we focus on instantaneous explosion producing an ultrarel-ativistic shell, and apply our results considering typical parameters of GRBsexpressing their total energy as E0 = 1054E54 erg, initial size as R0 = 108R8cm and baryonic loading parameter as B = 10−2B−2.

Summarizing the results obtained above, and taking for σ the Thomsoncross section we have for the radius of transparency the following asymptoticsolutions together with domains of their applicability

Rtr =

1Ttr

(3E0R0

4πa

)1/4

= 4.4× 1010 (E54R8)1/4 cm,

E54 4.8× 10−20B−4−2R−1

8 ,

(σE0BR0

24πmpc2

)1/3

= 1.8× 1012 (E54B−2R8)1/3 cm,

4.8× 10−20B−4−2R−1

8 E54 3.2× 10−8B−4−2R2

8,

σE0B3

8πmpc2R0= 1.8× 1017E54B3

−2R−18 cm,

3.2× 10−8B−4−2R2

8 E54 1.1× 10−5B−5−2R2

8,

(σE0B

4πmpc2

)1/2

= 5.9× 1014 (E54B−2)1/2 cm,

E54 1.1× 10−5B−5−2R2

8.

(E.7.1)

For very small baryonic loading, or in other words for a pure electron-positron plasma, the transparency radius does not depend on B parameter,then it increases as B1/3 (accelerating photon thick solution), then it steeplyincreases as B3 (coasting photon thick solution), and finally it increases asB1/2 (coasting photon thin solution), see Fig. E.12. In Fig. E.12 we alsoshow as function of the baryonic loading parameter the following quanti-ties computed at the transparency radius: the Lorentz factor, the observedand comoving LOS temperatures, fraction of energy emitted from the pho-tosphere to the total energy, for different values of the total energy E0. It isclear that the highest Lorentz factors at transparency radius are attained in

138


photon thick asymptotics. The largest transparency radii are reached insteadin photon thin asymptotics.

Fig. E.11 shows the energy-baryonic loading diagram, where the regions ofvalidity of the asymptotics discussed above are indicated explicitly for typicalparameters of GRBs. For all the relevant range of GRBs parameters 1048 erg <E0 < 1055 erg and 106 cm < R0 < 1012 cm all four asymptotics are present inthe interval 10−10 < B < 10−1.

Using (E.3.12) we obtain for the electron-positron contribution to opacity

n±/n = 2.5 · 10−5R3/48 B−1

2 E−1/454 , (E.7.2)

after their decoupling, so it is clear that this contribution may be safely ne-glected for, say, B > 10−4.

In photon thin case, expressing the peak arrival time (E.5.22) in units oftotal energy and baryonic loading, we find

tp ' 1E1/254 B5/2

−2 s, (E.7.3)

This expression gives an estimate for duration of photospheric emission inarrival time, and we conclude that typically this duration should be smallerthan one second, given that typically measured bulk Lorentz factors exceed102. Notice that tp does not depend on R0 and, consequently, on the lightcrossing time of the outflow. Currently time resolved spectra of GRBs withgood statistics cannot resolve such small time scales, and therefore we expectthat only time integrated spectra may be observed from GRBs which havetheir parameters corresponding to the photon thin case. Thus, starting fromcomoving thermal spectrum for the photospheric emission we obtain for thefirst time an observed spectrum which may be well described by the Bandfunction with high energy power law index β being determined by the den-sity profile of the outflow. We find this result quite remarkable. Notice thatthe idea of convolution over time has been introduced for GRBs by Blinnikovet al. (1999). Double convolution over EQTS and arrival time is also one ofthe key ideas in the fireshell model (Ruffini et al., 2003a).

It is even more remarkable that GRBs appear to be the only objects in natureable to reach the photon thin asymptotics in their ultrarelativistic expansion.For thermally accelerated relativistic plasmas which are discussed in connec-tion with their possible synthesis in ground based laboratories (e.g. Ruffiniet al. (2010)) it is unreachable. In order to quantify what should be initial op-tical depth τ0 = σTn0l in order to reach the photon thin asymptotics one mayuse (E.3.23) to get

τ0 4Γ4 = 4× 108B−4−2. (E.7.4)

139


GRBs clearly satisfy this constraint with the contribution of baryons only

τ0 =3σTBE0

4πmpc2l2 ' 1014E54B−2R−28 . (E.7.5)

which is increased even further by the presence of electron-positron pairs.In contrast, in photon thick case the corresponding observed spectrum, as it

is well known in the literature, is a black body with small deviations presentat both low and high energy parts of the spectrum. It remains close to ther-mal also when variation of the Lorentz factor through the outflow is present(Daigne and Mochkovitch, 2002).

When the outflow becomes transparent in the transition from photon thickto photon thin conditions, the observed spectrum will contain the Band com-ponent produced by the photon thin layer, with an almost black body comingfrom the photon thick part superimposed. As we already mentioned, in anyoutflow there is always a photon thin layer. So when the photon thick partbecomes dominant, the spectrum is expected to be dominated by the blackbody component. This may be the reason why in most cases analysed byRyde and Pe’er (2009) there are both power law and black body components:transparency occurs at the boundary between photon thin and photon thickconditions.

In Sec. E.5.3 we give analytic expressions for the photon flux for the sim-ple model of the portion of wind (E.3.11). With more complex density pro-file composed of presumably many shells the light curve is expected to bevariable and arbitrarily complex. It is necessary to emphasize however, thatthe decaying part of the light curve is determined solely by the geometryof the limiting surfaces in photon thick (E.4.2) and photon thin (E.4.4) cases,respectively. So when the photon flux is observed with faster than t−3 de-cay it should be concluded that the emission does not come from the pho-tosphere. The photospheric emission may be additionally identified by thespectral analysis. What we have shown here, though, is that the observedspectrum may not necessarily be close to thermal, but in the photon thin caseit may look very different.

E.7.2. Wind model

In the case of gradual energy release resulting in relativistic wind an addi-tional parameter is present, that is the duration of energy release, which weparametrize as ∆t = ∆t1 s. Instead of the total energy E0 the luminosityL = 1050L50 erg/s will be used.

From formulae (E.3.9), (E.3.19), and (E.3.25) we obtain for the transparency

140


10-4 10-3 10-21048

1050

1052

1054

10-4 10-3 10-2

1048

1050

1052

1054

B

E0,

erg

Accelerating

Photon Thick

Coasting

Photon ThinCoastingPhoton Thick

Figure E.11.: The energy-baryonic loading diagram showing the validity ofthe various asymptotic solutions for the transparency radius for typical pa-rameters of GRBs with l = R0 = 108 cm.

141


radius

Rtr =

1Ttr

(LR2

04πca

)1/4

= 8.1× 108L1/450 R1/2

8 cm,

L50 5.3× 10−15B−4−2R−2

8 ,

(σLBR2

024πmpc3

)1/3

= 1.3× 1010 (L50B−2R28)1/3 cm,

5.3× 10−15B−4−2R−2

8 L50 9.8× 10−2B−4−2R8,

σLB3

8πmpc3 = 5.9× 1010L50B3−2 cm,

9.8× 10−2B−4−2R8 L50 105B−5

−2∆t1,

(σLB∆t4πmpc2

)1/2

= 5.9× 1010 (L50∆t1B−2)1/2 cm,

L50 105B−5−2∆t1.

(E.7.6)

In Fig. E.13 we show as function of the baryonic loading parameter thefollowing quantities computed at the transparency radius: the Lorentz factor,the observed and comoving LOS temperatures, fraction of energy emittedfrom the photosphere to the total energy, for different duration of the windwith the total energy E0 = 1051 erg, and inner boundary radius R0 = 108 cm.Wind duration ranges from 10 ms to 10 s. The corresponding wind lumi-nosity varies from 1053 erg/s to 1050 erg/s. Fig. E.14 shows the luminosity-baryonic loading diagram where the regions of validity of the asymptoticsdiscussed above are indicated. Photon thin asymptotics is not common forthe typical parameters of wind considered in the literature, but still can berelevant for energetic winds of short duration.

All the discussion on the complexity of light curves for shell models is validfor wind models as well. Due to smaller range of applicability of photonthin asymptotics we expect that relativistic winds produce spectrum whichis close to thermal one.

Generally speaking, for both shell and wind models with more complexnumber density, energy density and Lorentz factor distributions within the

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s,eV

log 1

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trlo

g 10

Rtr,c

m

1049 erg

1055 erg

1055 erg1049 erg

1049 erg

1055 erg

1049 erg

1055 erg

1049 erg

1055 erg

Figure E.12.: Transparency radius Rtr, Lorentz factor Γtr, observed Tobs andcomoving Ttr temperatures, and ratio of the energy emitted to the total energyEPh/E0 at transparency radius as functions of baryonic loading B for shellswith different total energy E0 but the same width l = R0 = 108 cm. Allfour regimes with different asymptotics are clearly visible and dashed linescorresponding to their domain of validity from Eq. (E.7.1) are shown. Curvesare drawn for E0 equal to 1049 erg (green), 1051 erg (blue), 1053 erg (violet),and 1055 erg (red).

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10 s

0.01 s

10 s0.01 s

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0.01 s

Figure E.13.: The same as in Fig. E.12 for winds with different duration, butthe same total energy E0 = 1051 erg and radius of origin R0 = 108 cm. Allfour regimes with different asymptotics are clearly visible. Curves are drawnfor ∆t from to 10−2 s (green) to 10 s (red) in steps of one order of magnitude.

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10-4 10-3 10-21048

1049

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1048

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1054

B

L,e

rgs

Accelerating

Photon Thick

CoastingPhoton Thick

Figure E.14.: The luminosity-baryonic loading diagram showing the valid-ity of the various asymptotic solutions for transparency radius of wind withduration ∆t = 0.1 s. Notation is the same as on Fig. E.11.

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outflow various light curves and observed spectra of photospheric emissionmay be obtained, with the geometrical constraints discussed in previous sec-tions. The theory developed in this paper may be used for the calculationof these observed quantities using the corresponding profiles obtained frome.g. hydrodynamic simulations. We plan to report on this study elsewhere(Ruffini et al., 2011).

E.8. Conclusions

In summary, in this paper we readdressed the issue of photospheric emissionfrom spherically symmetric relativistic outflows. Two types of outflows arediscussed: winds resulting from gradual energy release, and shells producedby instant energy release. We evaluated the optical depth and the correspond-ing photospheric radius for each type of the outflow. We demonstrated thatthere are two asymptotic solutions for a coasting shell, similarly to a coastingwind. In particular we showed that due to ultrarelativistic motion a geomet-rically thin shell may appear as thick for photons propagating inside it. Forthis reason we suggest to refer to any relativistic outflow as photon thick,when the photon spends enough time inside it to feel its density decreasingwith time or radius. In the opposite case we refer to the outflow as photonthin.

We also studied geometry and dynamics of photospheres of relativistic out-flows. As we are interested in appearance of the photosphere to a distant ob-server, we introduced the notion of photospheric equitemporal surface, thePhE. While in general the PhE is dynamically evolving, we showed that itis well approximated by a part of an ellipsoid in the case of the photon thinoutflow. The photon thick PhE initially looks the same, due to the presence ofphoton thin layer in any outflow. Then it becomes concave and approachesasymptotically the photosphere of infinitely long wind.

Then, assuming that photons are trapped in the outflow with thermal andisotropic distribution for optical depth greater than one, and are release tofreely stream when opacity decreases below one, we computed both photonflux and observed spectra of photon thick and photon thin outflows. Ouradiabatic approximation corresponds to an assumption that the last scatter-ing surface of photons is well defined. This assumption is shown to work inthe photon thin case. As for photon thick case, our results for observed fluxare shown to agree with other results in the literature, obtained with moreaccurate methods (Beloborodov, 2011), see also (Pe’er and Ryde, 2011).

We find that instantaneous spectra of both photon thick and photon thinoutflows are close to the thermal one. This result is in agreement with thecorresponding results in the literature for relativistic winds, since such windswith long enough duration must indeed be photon thick. However, the timeintegrated spectrum from photon thin outflows appears to be non thermal,

146

E.8. Conclusions

due to strong temperature and luminosity evolution in arrival time. Thisspectrum may be described by the Band function (Band et al., 1993) withsome specific low energy and high energy power law indices, and an ex-ponential cutoff corresponding to the temperature in the source of energyrelease.

Our results are then applied to GRBs. We show in particular, that instanta-neous spectra cannot be observed in the case of photon thin outflows, and wepredict therefore that typical observed spectrum originating from the photo-spheric emission should be the Band spectrum. Our analysis for simple den-sity profile gives values for the low energy power law index α ' 0.5 and thehigh-energy power law index β = −4, which are different from the typicalvalues observed in GRBs. The low energy index is insensitive to both densityand temperature profiles adopted. The high energy index strongly dependson the density profile, and it may vary from −∞ to −1.

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148

F. Correlation dynamics incosmology

IntroductionAccording to the modern cosmology gravitational instability of dark matter

is the main mechanism for generation of large-scale structure of the Universe,see e.g. Dodelson (2003). This structure is thought to be formed dynamicallyfrom initial Gaussian seeds seen in cosmic microwave background radiatione.g. by WMAP satellite Komatsu et al. (2011) all the way to the present dayobserved galaxies, clusters and superclusters of galaxies along with voids asseen by galaxy surveys such as 2dF Cole et al. (2005) and SDSS Percival et al.(2007).

While the structure around us up to superclusters scales is strongly in-homogeneous, the distribution of matter becomes homogeneous on largerscales Hogg et al. (2005),Yadav et al. (2005). This point supports the cosmo-logical principle which is needed to match theoretical cosmology with wellestablished background dynamics to observations. It is in fact believed thateven highly nonlinear structures seen on scales from galaxies to clusters donot influence global dynamics expressed by the (simple) Friedmann back-gound solution of (complex) Einstein equations. However, phenomenolog-ical ΛCDM model, being in agreement with observations, lacks predictivepower and cannot neither explain values of numerous cosmological param-eters nor answer more fundamental questions such as what is the nature ofrecently discovered cosmic acceleration. There is growing concern in the lit-erature Wiltshire (2007),Mattsson (2010),Ellis (2008) that the picture describedabove is an adequate one. We believe that in the epoch when the Universe isdominated by highly nonlinear structures the concept of background space-time dynamics may have limited physical meaning. Dynamical theory ofstructure formation is required, but not yet developed.

Standard cosmological structure formation model distinguishes two dif-ferent physical regimes. Initially fluctuations of matter density are small atall scales and they are considered as linear perturbations to a backgroundhomogeneous isotropic solution. This linear regime of structure formationis well understood. On the contrary, the second, nonlinear, regime of thestructure formation lacks a self-consistent theoretical treatment. The onlyexisting approaches are strictly limited to N-body simulations Davis et al.(1985),Springel et al. (2005),Klypin et al. (2011). Recent study Einasto et al.(2011) of the large-scale structure using wavelet technique provides impor-

149

F. Correlation dynamics in cosmology

tant insights into clustering phenomenon. However, such approach is notsuitable for dynamical description of structure formation.

The key concept of the structure formation theory is that of the averagematter density which is assumed to be a source for the underlying back-ground spacetime cosmological solution. Therefore, an averaged descriptionof the cosmological evolution is a critical issue in cosmology, see also SylosLabini et al. (2009). A consistent covariant spacetime averaging of the gravi-tational field within this approach naturally leads to appearence of the gravi-tational correlation terms which modify the structure of the averaged mattersource Zalaletdinov (1992),Zalaletdinov (1993),Zalaletdinov (1997).

In this connection it is important to stress that observed nonlinear struc-tures are usually described not by dark matter density contrast, but by itsspatial correlations as observed e.g. in the two-point correlation function ξ(r)Peebles (1993). This two-point correlation function is interpreted as manifes-tation of the fractal structure Ruffini et al. (1988). There is no satisfactorydynamical theory of cosmological correlations describing in particular evo-lution of the correlation function ξ(r, t).

The aim of this Letter is to propose a possible direction for the developmentof the theoretical model which describes the evolution of the Universe fromthe beginning of the gravitational instability up to the currently observednonlinear structures which is spatially homogeneous and isotropic on largescales as dynamics of gravitational correlations.

Basic equations and initial conditionsDark matter is assumed to consist of weakly interacting massive particles

which are non-relativistic when decouple from the thermodynamic equilib-rium with cosmic plasma. Therefore, it should be described at kinetic level,as collisionless self-gravitating system of particles. On scales of interest, fromgalaxies to superclusters, curvature effects are negligible and Newtonianapproximation is adopted. The fundamental equations consist of coupledBoltzmann-Vlasov and Poisson equations Binney and Tremaine (1987),Lon-gair (1998)

∂ f∂t

+ v·∂ f∂r− ∂ϕ

∂r·∂ f∂v

= 0, (F.0.1)

∇2ϕ = 4πG∫

f d3v, (F.0.2)

where there are two unknown functions: f (r, v, t) is the distribution functionof dark matter particles and ϕ(r, t) is the gravitational potential; r and v arephase space coordinates, G is Newton’s constant.

Initially the distribution function is assumed to be Maxwellian with smallfluctuations f (r, v, t) = f0(|v|) [1 + δ(r, v, t)], δ 1. These perturbationsare assumed to be Gaussian, having Harrison-Zeldovich power spectrum|δ(k, t0)|2 = Ak, where A is a normalization constant, k is a wavenumber,

150

δ(r, v, t) = V(2π)3

∫δ(k, v, t) exp (−ik · r) d3k. In comoving coordinates the

distribution function depends implicitly on time due to cosmic expansion.Distribution function, spatial averaging and coarse grainingThe whole dynamics can, in principle, be extracted from the solution of

(F.0.1),(F.0.2) or equivalent N-body problem Ma and Bertschinger (2004). Inpractice, however, analytic solution of the system (F.0.1),(F.0.2) is hardly pos-sible, while N-body approach provides limited information.

On the linear phase of the structure formation, when δ < 1, fluctuations ofthe distribution function grow essentially in the configuration space since

∂ϕ

∂r·∂δ(r, v, t)

∂v v·∂δ(r, v, t)

∂r, (F.0.3)

This phase is well studied and is described in a hydrodynamic-like ap-proximation Gilbert (1966),Stewart (1972),Bisnovatyi-Kogan and Zel’Dovich(1971), where the term 1

f0

∂ f0∂v

∂ϕ∂r plays the role of the source term in linearized

equation (F.0.1). In the short wavelengths limit fluctuations experience Lan-dau damping, while in the opposite limit perfect fluid oscillatory behav-ior is recovered. The critical wavelength separating these two regimes, isk J(m) = −4πGm

∫ +∞−∞

1v

∂ f0∂v dv. This characteristic scale depends on parti-

cle’s mass. However, candidates for CDM particles have so large massesm >GeV that their Jeans mass is irrelevant for cosmological structure forma-tion. In contrast, during nonlinear phase, when δ > 1, the condition (F.0.3)does not hold, distortions of the distribution function in velocity space be-come significant, so that the distribution function is driven out of equilib-rium. This out of equilibrium function is then subject to the mean field re-laxation Lynden-Bell (1967),Severne and Luwel (1980),White (1996),Arad andLynden-Bell (2005),Chavanis and Bouchet (2005) in strongly time-varyinggravitational potential. From the one hand, during nonlinear phase equa-tion (F.0.1) is still satisfied, but it becomes untractable since f (r, v, t) remainsalways out of equilibrium and time dependent. From the other hand, thecoarse-grained distribution function relaxes locally to a stationary functionfLB(r, v) characterizing Lynden-Bell equilibrium. Relaxed distribution func-tion fLB(r, v) is thought to describe virialized objects which are in deep non-linear regime, such as elliptical galaxies, however it certainly does not applyto unvirialized object such as superclusters of galaxies, which are still in lin-ear regime today.

Existence of several types of objects with the corresponding different scales,which are in different stages of their evolution calls for development of a newdynamical scheme in the theory of structure formation. This scheme shouldbe designed in order to disentangle two manifistations of the gravitationalinteraction in collisionless system of particles: gravitational instability andmean field relaxation.

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Important observation, which helps to construct such a scheme, is thatthe distribution function f is not directly accessible to measurements Bin-ney and Tremaine (1987). Potentially measurable are, instead, mass densityρ ≡ m

∫f d3v, m is the mass of dark matter particle, as well as the grav-

itational potential ϕ averaged on some spatial scale d. Consequently, keydynamical variables are spatially averaged distribution function and gravita-tional potential

〈 f (r, v, t)〉 ≡ 1VS

∫S

f (r + r′, v, t)d3r′, (F.0.4)

〈ϕ(r, t)〉 ≡ 1VS

∫S

ϕ(r + r′, t)d3r′, (F.0.5)

where VS is the volume space region S (see Zalaletdinov and Coley (2002),Zalaletdinov and Coley (2002) for defintion and properties of space volumeaverage tensor fields). A compact space regions S of the space volume aver-aging is determined by its characteristic size d, so that the phase space cellvolume is VS ' d3.

Substituting (F.0.4) and (F.0.5) into (F.0.1),(F.0.2) one finds

∂ 〈 f 〉∂t

+ v·∂ 〈 f 〉∂r− ∂ 〈ϕ〉

∂r·∂ 〈 f 〉

∂v= −Φ, (F.0.6)

∇2 〈ϕ〉 = 4πG∫〈 f 〉 d3v, (F.0.7)

where the right hand side of the averaged Boltzmann equations contains nowa correlation operator, defined as

Φ ≡⟨

∂ϕ

∂r·∂ f∂v

⟩− ∂ 〈ϕ〉

∂r·∂ 〈 f 〉

∂v. (F.0.8)

The correlation operator depends on the averaging scale d as a free parameter.

While equations (F.0.6),(F.0.7) are consistent with the linear phase of struc-ture formation, where the correlation operator Φ is negligible, the spatialaveraging (F.0.4),(F.0.5) loses the statistical meaning in the nonlinear phasesince the distribution function f (r, v, t) is driven out of equilibrium. In or-der to restore the statistical meaning of the spatial averaging, additional stepis required. It is proposed that the dynamical variables should be further

152

coarse-grained

⟨f (r, v, t)

⟩≡ 1

V∆Γ

∫∆Γ

⟨f (r + r′, v + v′, t)

⟩d3r′d3v′, (F.0.9)

〈ϕ(r, t)〉 ≡ 1V∆Γ

∫∆Γ

⟨ϕ(r + r′, t)

⟩d3r′. (F.0.10)

The properties of a typical cell of the coarse grained phase space are deter-mined by its characteristic size in coordinate space scale ∆γx and velocityspace scale ∆γv, so that the phase space cell volume is V∆Γ ' (∆γx)3(∆γv)3.

The fundamental relation between the three characteristic space scales,∆γx, d and L which represents a coarse-grained self-gravitating collisionlesssystem of particles under a space volume averaging is given by the relation

∆γx d L. (F.0.11)

This is conceptual relaization of description for a classical physical systemwith the microscopic scale ∆γx and the macroscopic scale L.

For a scalar quantity p(r, v, t) defined on the 6-dimensional phase-space Γ,the operations of the space volume averaging over a compact space region Sand the coarse graining over a phase-space cell ∆Γ commute, 〈 p〉 = 〈p〉. Thisfact is expected to play a fundamental role in the theory of structure forma-tion. It is worth noting that the integration over velocity space also commuteswith the above mentioned operations. This property readily permits buildingup moments of the distribution function

⟨f⟩.

As the result of application of the proposed averaging scheme, dark mat-ter is assumed to be described by the coupled space-volume averaged andcoarse-grained system of the Boltzmann-Vlasov and Poisson equations

∂⟨

f⟩

∂t+ v·

∂⟨

f⟩

∂r− ∂ 〈ϕ〉

∂r·∂⟨

f⟩

∂v= −Φ, (F.0.12)

∇2 〈ϕ〉 = 4πG∫ ⟨

f⟩

d3v. (F.0.13)

The structure of the correlation operator, being a functional of the coarsegrained gravitational potential and distribution fuction Φ

[ϕ, f

], and depend-

ing on two free parameters γx and d, completely determines dynamics ofstructure formation on both linear and nonlinear phases.

Structure formation as correlation dynamicsThe main task in the scheme presented above is the construction of the cor-

relation operator Φ. An attempt to study correlation dynamics via BBGKYhierarchy Prigogine and Severne (1966),Fall and Saslaw (1976) fails since un-like a plasma, weak coupling approximation cannot be adopted for correla-tions in self-gravitating system. Other assumptions such as simple relations

153


between three-point and two-point correlation functions Davis and Peebles(1977) do not work. There are two possible approaches in counstructing thecorrelation operator. Firstly, it can be found from first principles. Secondly, aphenomenological approach can be followed. The structure of the correlationoperator has been studied by applying the first and second laws of thermody-namics and attempting a representation by means of Fokker-Planck diffusioncoefficients Chavanis and Bouchet (2005). In such an approach interpretationof the diffusion coefficients should be given.

On the linear phase of structure formation Φ is negligible, since fluctu-ations are assumed to be Gaussian, and equations (F.0.6),(F.0.7) reduce to(F.0.1),(F.0.2). Therefore, linear theory of structure formation is recovered inthe proposed scheme as the limit in which the correlation operator (F.0.8) van-ishes and

⟨f⟩−→ f . As expected, both spatial averaging and coarse-graining

have no role since gradients of density and gravitational potential are small,along with time variations of the gravitational potential.

Spatial correlations, represented by the correlation operator Φ, becomeimportant once fluctuations grow to become nonlinear on the dynamicaltimescale τ ' (Gρ)−1/2. Since mean field relaxation operates essentially onthe same timescale, description of nonlinear structures should be given interms of

⟨f⟩, satisfying equations (F.0.12),(F.0.13). It is then natural to identify

the scale where fluctuations reach nonlinearity RNL(t) with the microscopicscale

∆γx ' RNL(t). (F.0.14)

We stress that the dynamics of fine-grained distribution function f remainsessentailly of non-equilibrium nature. Therefore, only

⟨f⟩

can reach station-ary form fLB(r, v) which results from smearing out microscopic evolution ofthe distribution function f on scales l < ∆γx, v < ∆γv. Severne and LuwelSeverne and Luwel (1980) in an attempt to construct dynamical theory of col-lisionless relaxation, derived an equation for f by linearizing (F.0.12),(F.0.13).Instead of our spatial averaging they used an ensemble averaging. Theirequation, containing the source term proportional to f 3, gives as a solutionthe Linden-Bell distribution function Kadomtsev and Pogutse (1970).

There are two issues here of fundamenal significance. Firstly, upon a meanfield relaxation the distribution function remain inhomogeneous in space,that is, the particles of a self-gravitating collisionless system should be ex-pected to form a stationary spatially distributed structure. Secondly, sucha stationary spatial structure must be of universal nature since there is noseparation by masses of particles by virtue of their absence in the Boltzmann-Vlasov-Poisson equations. The universal character of simulated dark matterhalo profiles Navarro et al. (1997) velocity distributions Merrall and Henrik-sen (2003), observed two-point correlation function ξ Jones et al. (2004) andnumerical experiments Levin et al. (2008) seems to give a clear evidence ofthat fundamental phenomenon. Both these facts should be reflected in the

154

form and structure of the correlation operator.The phenomemon of mean field relaxation is usually attributed to the for-

mation process of elliptical galaxies. Unlike a single galaxy, the large scalestructure is thought to originate as the interrelated process of hierarchicalhalo formation and subsequent merging Press and Schechter (1974), the lat-ter operate simultaneously with the mean field relaxation. In other words,at scales where fluctuations of dark matter density are nonlinear the gravita-tional instability ends and dark matter experiences mean field relaxation. Itis important to stress therefore, that both gravitational instability and meanfield relaxation are generic, universal features of collisionless self-gravitatingmatter. They are just different manifestations of the same physical law. De-spite many models of nonlinear clustering have been proposed, for reviewsee Jones et al. (2004),Bernardeau et al. (2002),Malik and Wands (2009) up tonow there is no model bridging the gap between these two phenomena. Thescheme proposed in this paper naturally closes this gap, suggesting unifieddescription of the structure formation, from initial small perturbations all theway to presently observed structures.

We were focusing on the Newtonian cosmology in this paper, giving usefulinsight into mechanisms underlying structure formation. However, in suchan approach there remain several limitations. In particular, 1) Jeans swin-dle should be adopted; 2) average density cannot be defined and should betreated as free parameter; 3) effects of General Relativity, and topology areneglected.

ConclusionsDynamical theory of structure formation is still missing. Various ap-

proaches to nonlinear structure formation can capture limited number of phe-nomena and do not provide a unified picture of the structure formation fromseed perturbations all the way to the presently observed large scale struc-ture. It is generally accepted that on such large scales the only importantinteraction is the gravitational one. In this work we propose to generalize thewidely accepted treatment of large scale clustering based on Vlasov-Poissonequations for collisionless dark matter particles, by combining two averagingprocedures: spatial averaging and coarse graining. We show that in this wayboth phenomena of gravitational instability and mean field relaxation findnatural and unified description.

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156

G. Semidegenerate self-gravitatingsystem of fermions as a modelfor dark matter halos anduniversality laws

G.1. Introduction

Dark matter properties are constrained both from cosmology and astro-physics. Cosmological bounds are based on the assumptions on cross sec-tions of interaction between the dark matter particles and usual matter plusinteraction between dark matter particles themselves. Usually it appears thatdark matter decouple from normal matter at thermal equilibrium and at thesame time or earlier interaction between dark matter particles themselvescome to a halt, so that they form collisionless dark matter. In this case massdensity of the particles can be found and compared to the known dark mat-ter cosmological density. This provides different limits on the mass of darkmatter particles, two of them being Gershtein-Zeldovich limit for the sum ofneutrino masses (Gershtein and Zel’Dovich, 1966)

m < (G.1.1)

Different approach to the study of properties of dark matter is coming fromcosmology and specially from theories of galaxy formation and evolution.While studies of CMB allow to constrain number and masses of light neu-trinos representing hot dark matter (Giusarma et al., 2011), properties andmasses of warm and cold dark matter are constrained by the total mass den-sity of the Universe (Lee and Weinberg, 1977) and galactic halos structure(Tremaine and Gunn, 1979) and substructure (Polisensky and Ricotti, 2011).The Lee-Weinberg bound (Lee and Weinberg, 1977) limits the mass of darkmatter particles with given coupling constant from above implying that par-ticle was in thermal equilibrium in early Universe. For typical weak inter-action coupling GF mass is constrained to be more than ∼ 2 GeV/c2 andsuch particles is referred to as Weakly Interacting Massive Particles, WIMPs.Bound by Tremaine and Gunn (1979) and its improvement by many authors(Hogan and Dalcanton, 2000; de Vega and Sanchez, 2010) are based on as-

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G. Semidegenerate self-gravitating system of fermions as a model for darkmatter halos and universality laws

sumption of fermionic nature of dark matter and nondegeneracy of galactichaloes of dwarf galaxies, that leads to lower limit on mass ∼ 0.41 keV/c2.

The problem of dark matter distribution in galactic halos has traditionallybeen treated in the realm of newtonian physics in view of the low velocities ofthe stars in the galaxies, like the simulations from Navarro, Frenk and White(1997). In the meantime, phenomenological profiles of dark matter have beenadvanced by Einasto (1965); Burkert (1995), and universal properties of thedark matter distribution have been inferred from dwarf galaxies and proba-bly globular clusters all the way to very massive galaxies (Gentile et al., 2009;Donato et al., 2009; Walker et al., 2010; Boyarsky et al., 2009). However, aproblem arises: while simulations like those from NFW point to a cuspedhalo, observations from various types of galaxies seem to show cored halos(Salucci et al., 2011). This discrepancy between theory and observations is notyet fully understood, but could show a problem with the simulations doneso far.

In a completely unrelated field (as of yet), the physics of Active GalacticNuclei (AGN) and quasars has been recognized for more than 50 years asdominated by relativistic gravitational effects of a black hole. The formationof these black holes is not yet fully understood, although different black holesmass estimates for AGNs and quasars show masses up to 1010 M all the wayto z ≈ 6.4 (Peterson, 2010; Vestergaard, 2002; Vestergaard and Peterson, 2006;Targett et al., 2011). Due to the lack of understanding on the energetics ofAGNs and quasars and on the formation of the black holes, the possibility ofan extended object in the core of galaxies has been advanced by Viollier et al.(1993).

The aim of this work is to present a unified approach to the dark matterdistribution in the galactic halos and also in the galactic center. In order to dothat, some assumptions have been made:

1. The treatment must be a general relativistic one from the beginning, inorder to explain both the galactic nuclei and galactic haloes.

2. The matter particles are assumed to be semi-degenerated fermions andso obey the Fermi-Dirac statistics.

3. Configurations are in relativistic thermal equilibrium√

g00T = constor sufficiently close to it.

The aim of this paper is to present a unified approach to the dark matterdistribution in the galactic halos and also in the galactic center. In order to dothat, some assumptions have been made:

1. The treatment must be a fully relativistic one from the beginning, inorder to explain both the galactic nuclei and galactic haloes.

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G.2. Model

2. The matter particles are semi-degenerated fermions and so obey theFermi-Dirac statistics, together with the relativistic thermodynamicalequilibrium conditions.

G.2. Model

The equilibrium configurations of a self-gravitating semi-degenerate systemof fermions were studied by Ruffini and Stella (1983) in Newtonian gravityand by Gao et al. (1990) in general relativity. It is shown that in any such sys-tem the density at large radii scales as r−2 quite independently of the valuesof the central density, providing flat rotation curve. Then solution was ex-tended to an energy cutoff in the distribution function (Ingrosso et al., 1992).

Following Gao et al. (1990) we are considering spherical symmetry, the lineelement is written in standard Schwarzschild coordinates as

ds2 = −eν(r)dt2 + eλ(r)dr2 + r2(dθ2 + sin2 θdφ2). (G.2.1)

The equilibrium equations are given by

dPdr

= −Gc2

(P + ρc2)(M(r) + 4πρr3)

r(rc2 − 2GM(r))(G.2.2)

dMdr

= 4πρr2, (G.2.3)

with M the mass within a radius r, ρ and P the energy density and the pres-sure respectively, given by

ρ = mgh3

∫ εc

0

(1 +

ε

mc2

) 1− e(ε−εc)/kT

e(ε−µ)/kT + 1d3p (G.2.4)

P =2g3h3

∫ εc

0

(1 +

ε

2mc2

) (1 +

ε

mc2

)−1 (1− e(ε−εc)/kT)ε

e(ε−µ)/KT + 1d3p, (G.2.5)

with εc being the cutoff energy, g = 2s + 1 being the multiplicity factor, mbeing the mass of the particle, and T being the temperature and µ the chem-ical potential. The volume element in momentum space can be expressed interms of the kinetic energy ε of the particles as

d3p = 4πp2dp = 4√

2m3c3√

1 + ε/2mc2(1 + ε/mc2)√

ε/mc2 d(ε/mc2).(G.2.6)

The particle energy is a constant of motion, so

(ε + mc2)eν/2 = const, (G.2.7)

while thermodynamical equilibrium (Tolman condition and Klein integral

159


(Rueda et al., 2011)) implies

(µ + mc2)eν/2 = (µR + mc2)eνR/2 (G.2.8)

Teν/2 = TReνR/2, (G.2.9)

where the quantities with subscript “R” refer to the boundary of the configu-ration. For the cutoff energy we have

(εc + mc2)eν/2 = mc2eνR/2, (G.2.10)

since εc(R) = 0.

Introducing the function W = εc/kT and the temperature parameter at theboundary βR = kTR/mc2, and using eqs. (G.2.9) and (G.2.10) we can find that

mc2

kT=

1− βRWβR

. (G.2.11)

Note that the condition 0 ≤ βRW < 1 has to be fulfilled. Using eq. (G.2.9)to substitute the temperature in eq. (G.2.8) we get the relation between themetric function ν and W:

eν = eνR(1− βRW)2 (G.2.12)

so now the spacetime metric is completely determined:

eν = eνR [1− βRW]2, eλ =

(1− 2GM

rc2

)−1

(G.2.13)

with νR + λR = 0.

Differentiating eq. (G.2.12) and using the conservation of the energy mo-mentum tensor

dPdr

= −12(P + ρc2)

dν

dr(G.2.14)

givesdPdr

=βR(P + ρc2)

1− βRWdWdr

(G.2.15)

and we can write eq. (G.2.2) as

dWdr

= −Gc2

[1− βRW

βR

]Mc2 + 4πPr3

r(rc2 − 2GM)(G.2.16)

In order to numerically integrate the final set of equations (G.2.3) and(G.2.16) with initial conditions W(0) = W0 and M(0) = 0, it is useful to

160

G.2. Model

transform all of our physical variables into dimensionless ones:

ρ =c2

Gχ2 ρ (G.2.17)

P =c4

Gχ2 P (G.2.18)

M =c2χ

GM (G.2.19)

r = χr, (G.2.20)

where

χ =h

mc

(mp

m

)(8π3

g

)1/2

(G.2.21)

has dimension of length and mp = (hc/G)1/2 is the Planck mass.It is instructive to write down characteristic length χ, that is inversely pro-

portional to square of the mass of the particle, in conventional units

χ = 0.870m−2 pc, (G.2.22)

where m is measured in keV/c2, and unit of mass is

c2χ

G= 1.820× 1013m−2M, (G.2.23)

where M = 1.989 · 1033 g is mass of the Sun.We then obtain the dimensionless equations

dWdr

= −[

1− βRWβR

]M(r) + 4πPr3

r(r− 2M(r))dM(r)

dr= 4πρr2, (G.2.24)

where

ρ = 4√

2π

[βR

1− βRW

]3/2 ∫ W

0

[1 +

βRx/21− βRW

]1/2 [1 +

βRx1− βRW

]2 1− ex−W

ex−θ + 1x1/2dx

P =8√

23

π

[βR

1− βRW

]5/2 ∫ W

0

[1 +

βRx/21− βRW

]3/2 1− ew−W

ex−θ + 1x3/2dx, (G.2.25)

where θ = µ/kT is the degeneracy parameter and we introduced the variablex = ε/kT. We have for this variable

ε

mc2 =βRx

1− βRW. (G.2.26)

161


The relation between the degeneracy parameter θ and W is

W = θ − θR, (G.2.27)

so that W(R) = 0, where θR is the value of the degeneracy parameter at theboundary. We can relate the parameters in the boundary with those in thecenter

θR = θ0 −W0 βR =β0

1 + β0W0(G.2.28)

so that βR ≈ β0 for β0 << 1. Besides that we have

1− βRWβr

=1− β0(W −W0)

β0. (G.2.29)

Now the system can be completely solved (numerically) by solving theequations eq. (G.2.24) together with

θ = θ0 + W −W0 (G.2.30)

and using eq. (G.2.25) with three independent parameters: W0, θ0 and β0.The only remaining free parameter is the mass of the particle, which occursonly in the definition of β and the characteristic length χ.

G.2.1. Properties of the equilibrium configurations

We have solved numerically the system of integral-differential equationsgiven by (G.2.16), the two equations corresponding to β and θ and (G.2.25),with a set of initial conditions M0, W0, β0 and θ0. Galactic dark matter haloshave asymptotic rotation velocities of the order of ten to thousands km/s, i.e.,they are not relativistic. As that velocities are of the same order as thermalvelocities of fermionic particles forming the halo, this means that βR 1and consequently β0 1. For semidegenerate configurations θ0 & 20, and inthis case we have three regions of halo (fig. G.1): a degenerate core of almostconstant density, an inner halo also with almost constant density and a tailwhere density scales as r−2 until the cutoff.

On the velocity curve, we can see 4 characteristic regions (fig. G.2):

• Part I: The core with constant density, where v ∝ r;

• Part II: The first part of the inner halo, where the mass of the core pre-vails over the mass of the halo and v ∝ r−1/2;

• Part III: Second part of the inner halo, where now the mass of the haloprevails and again v ∝ r;

162

G.3. Comparison with other DM profiles

Figure G.1.: Density profile of the model for β0 = 10−8, θ0 = 32 and W0 = 92.

Figure G.2.: Velocity curve for the same parameter values as before

• Part IV: The outer halo, where the velocity tends to a constant value v0after some oscillations of diminishing magnitude.


To compare results obtained with known Dark Matter properties we needto find out the correspondence between fits of circular velocity, much like itwas suggested in Boyarsky et al. (2009). There is some controversy in cur-rent literature about the undisturbed profile of dark matter in Galaxies and

163


clusters. Cold dark matter simulations suggest the so-called Navarro-Frenk-White profile (Navarro et al., 1997)

ρ =ρNFW

r/rNFW(1 + r/rNFW)2 (G.3.1)

and Einasto profile (Navarro et al., 2010) introduced by J. Einasto for model-ing of matter distribution in Milky Way (Einasto, 1965)

ρ = ρE exp(−2

α[(r/rE)

α − 1])

(G.3.2)

while phenomenological pseudoisothermal sphere

ρ =ρiso

1 + (r/riso)2 (G.3.3)

and Burkert profile (Burkert, 1995)

ρ =ρB

(1 + r/rB)(1 + (r/rB)2). (G.3.4)

are commonly used for fitting. Comparing profiles of circular velocity for allthese profiles with the one of the semidegenerate solution, we came to con-clusion that NFW and Burkert profiles, having wrong asymptotics as r → ∞,better reproduce the characteristic ”bump” in the circular velocity near theedge of inner halo (fig. G.3). The best reproduction is obtained for Burkertprofile. As most of papers dealing with rotational curve fitting find out thatBurkert or other cored profiles are the best fits for dark matter distribution,and that characteristic scale rB of the fitted profile is comparable to the fulllength of fitting range (see, e.g., Gentile et al. (2004)), that means that semide-generate fermion halo can provide the same quality of fits for that galaxies.

As we move outside from the border of inner/outer halo, the fits by pseu-doisothermal sphere became better than that of other profiles (fig. G.4). Thefits by other profiles suffer due to their different outer slope, so constant circu-lar velocity can be only approximated in a finite range of radii by a decreasingfunction. As a result we have systematic deviations from the real flat curvein the beginning and the end of fitting range.

The result obtained means that all fits of rotational curves by Burkert, NFWand pseudoisothermal sphere profiles could be fitted as well by semidegen-erate fermion halo.

164


10 000 20 000 30 000 40 000r, pc

1.0

1.2

1.4

1.6

105vcc

Figure G.3.: Dependence of vc/c on radius r (black thick line) for β =10−10, θ0 = 20, m = 9.3 keV/c2 near the edge of the inner halo and its fits byNFW (red dashed line), Burkert (blue thin line) and pseudoisothermal sphere(green dot-dashed line) profiles. The radius range used for fitting is shownby grey shading (r = 103 to r = 104 pc).

100 000 200 000 300 000 400 000r, pc

0.8

1.0

1.2

1.4

1.6105vcc

Figure G.4.: Dependence of vc/c on radius r (black thick line) for β =10−10, θ0 = 20, m = 9.3 keV/c2 in the outer halo and its fits by NFW (reddashed line), Burkert (blue thin line) and pseudoisothermal sphere (greendot-dashed line) profiles. The radius range used for fitting is shown by greyshading (r = 104 to r = 105 pc).

165


G.4. Scaling Laws

The solutions obtained show remarkable self-similarity properties. The char-acteristics of solutions obey five scaling laws against the free parameters ofthe model β0, θ0, W0 and m f . These are laws for the asymptotic velocity ofthe rotation curves, for the core mass, for the core radius, for the halo massand for the halo radius.

Asymptotic rotation velocity scaling law

v0 = 4.47× 105√β0 km/s (G.4.1)

show dependence on the temperature of the configuration only. This is es-sentially the same scaling law that appears in the case of isothermal sphere(Binney and Tremaine, 1987).

Core is defined as region from the center of the system till the first maxi-mum of the rotation velocity curve (region I in fig. G.2). Near that point thedensity of fermions decreases fast. Scalings of mass and radius of the core are

Mc = 1.96× 1012(β0θ0)0.75m−2

f M, (G.4.2)

rc = 0.180(β0θ0)−0.25m−2

f pc, (G.4.3)

where m f is in units of keV/c2. Characteristics of core is dependent not onβ0 and θ0 separately, but only on their product β0θ0, i.e. on the chemicalpotential µ at the semidegenerate center of configuration, in accordance withresults of Narain et al. (2006). From the laws an important relation could beobtained, involving only Mc, rc and m

Mc

r3c

= 1.15× 1010m4f

Mpc3 . (G.4.4)

Halo is defined as region up r0, where density drops to 1/4 of that in min-imum rotational velocity point ρ = ρ(rh) =

ρ04 (regions II and III in fig. G.2).

Scaling laws of halo properties are

Mh = 1.49× 1013β00.77100.16 θ0m−2

f M, (G.4.5)

rh = 0.35β0−0.25100.16 θ0m−2

f pc. (G.4.6)

Notice that instead of power-law dependance on θ0 in core properties, haloproperties depend on central degeneracy of configuration exponentially.

The equations above are exact in m f and hold in the following physicalrange of the other parameters: log10 β0 ∈ [−10,−5], θ0 ∈ [20, 200], W0 ∈

166

G.4. Scaling Laws

[110, 200]. Note that W0 does not appear in the scaling laws, so that its valuedoes not change inner structure of configuration.

G.4.1. Solving the scaling laws system of equations

In order to find out the order of semidegenerate halo parameters correspond-ing to observed ones, we adopted the four most reliable observed character-istics of the Milky Way, i.e. its asymptotic rotational velocity v0 ≈ 220 km/s,mass of the central object in the Galactic center Mc ≈ 4× 106 M, radius ofGalactic halo rh ≡ r0 ≈ 14× 103 pc, (r0: one dark halo scale length) and itsmass Mh ≈ 2× 1011 M.

We take three equations corresponding to the number of three unknownparameters, namely (G.4.1), (G.4.5), and (G.4.2). Then solving the systemusing the data presented above we arrive to

β0 = 10−6.6, (G.4.7)θ0 = 32.4 (W0 = 130), (G.4.8)

m f = 12.6 KeV. (G.4.9)

Finally we can take the law (G.4.3) to obtain rc = 2.2× 10−2 pc. Althoughthis value is quite far from the size of very compact region known as SgrA*,it is still in the very inner region of the bulge, in accordance with the ob-servations of Ghez et al. (2008) and Gillessen et al. (2009) made for orbits ofS2(blue) stars, where can be seen that at a radius of around 1× 10−2 pc theenclosed mass for the orbit must be around 4× 106 M.

G.4.2. Application to spiral, elliptical and group of galaxies

The application of the model to galaxies has been inspired by the recent ob-servations showing that the mean dark matter surface density within onedark halo scale/length r0 (within this radius the volume density profile ofdark matter remains approximately flat) is constant across a wide range ofgalaxies (Gentile et al., 2009; Donato et al., 2009), or equivalently, an almostconstant DM acceleration at r0: aDM ≈ 1 km2

s2·pc is observed (Walker et al., 2010).This Universality laws have also been extended by Boyarsky et al. (2009) formatter distributions at all observed scales.

Spiral galaxies

We based our analysis within the phenomenological work made by Walkeret al. (2010) from the MO7 data, consisting in HI rotation curves, and thenisolating the DM contribution to an observed rotation curve substracting

167


contributions from stellar and HI regions. This data is based on 686 inde-pendent resolved rotation velocities measurements for 60 galaxies, spanningradii between 1 kpc ≤ rh ≤ 75 kpc. The important feature is that this sam-ple covers virtually the entire range of spiral properties; ie. circular velocities50 km/s ≤ v0 ≤ 300 km/s within others. With this data they could finda constant acceleration due to DM aDM = 0.9+0.9

−0.5km2

s2·pc = 3+3−2 × 10−9 cm

s2 , incomplete accordance with the Universality law found by Gentile et al. (2009).

In the context of our model, and in order to reproduce the aDM with thecorresponding scatter in a plot aDM vs rh, we proceeded as follows:

i) From (G.4.1) we obtained the corresponding b0 for the lowest observedvalue of the circular velocity v0 = 50 km

s ; log10 β0 = −7.9;ii) From (G.4.6) and using the lowest limit for β0, log10 β0 = −7.9, m f =

13.5 KeV and the two observed border values for rh (1 kpc till 75 kpc) weobtained the entire range of validity for θ0, namely 23.6 ≤ θ0 ≤ 35.3;

iii) The same as in i), but now for the highest value of the circular velocityv0 = 300 km/s, and then finding the corresponding log10 β0 = −6.34;

iv) The same as in ii), but now with the highest limit of β0, obtaining nowa new range of validity for θ0, 26 ≤ θ0 ≤ 38;

v) Considering that aDM = GMhrh

2 and making use of the equations (G.4.5)and (G.4.6), we obtained the corresponding scaling law for the accelerationdue to DM as

log10aDM

km2/(s2 · pc)= 11.786 + 1.27 log10 β0 − 0.16 θ0 + 2 log10

mkeV/c2

(G.4.10)vi) Finally, with (G.4.6) and (G.4.10) and the physical ranges for β0 and θ0

obtained above, we made a plot aDM vs rh, that shows that all the data pointsof the Universality law for aDM made by Walker et al. (2010) are included bythe physical range of parameters of our model. The two sloped lines indicatesthe β0 range (ie. the velocity range), and the length of these lines indicatesthe θ0 range; while the two horizontal lines are the limits related to the scatterfrom the observations.

The comparison between the two graphics is shown in Fig. G.5.

Spiral, elliptical and group of galaxies

The application of the model in this section was inspired in the phenomeno-logical work made by Boyarsky et al. (2009) based on analogous observationsas in the former case mentioned in the work. The objects covered in this caseare: 124 spiral galaxies, 10 elliptical ones and 25 groups of galaxies (withinspecific selection criteria explained in that paper).

168

G.4. Scaling Laws

0.01

0.1

1

10

10 100 1000 10000 100000

aDM

(km

2 /(s2 pc

1 ))

r (pc)

model boundslower observational limitupper observational limit

Figure G.5.: The acceleration due to dark matter vs the halo scale-length rh.The left bound corresponds to v0 = 50 km/s (log β0 = −7.9), while the rightcorresponds to v0 = 300 km/s (log β0 = −6.34).

Authors present the Universality law in terms of the DM column density

S =2r2

0

∫ r0

0rdr

∫dzρDM

√r2 + z2 ≈

Mcyl

πr20

, (G.4.11)

so that the column density can be approximated as the mass within a cylin-der of radius r0 (meaning this as always for our purpose, the one halo scale-length). This quantity is also proportional to ρ0r0, so it should be constant fora wide span of magnitudes.

Interpreting Mcyl as the mass of the halo, and using the formula mentioned

in (Boyarsky et al., 2009), S ≈ Mcyl

π.r20

and noting that S G π = aDM we find from

(G.4.5) and (G.4.6) the corresponding scaling law in this case as

log10S

M/pc2 = 16.115 + 1.27 log10 β0 − 0.16 θ0 + 2 log10m f

keV/c2 . (G.4.12)

Then, in order to reproduce the log10 S with the corresponding scatter ina plot log10 S vs Mh, we proceeded in a complete analogous way as in thepoints i)-iv) for the former case.

The comparison between the two graphics is shown in Fig. G.6.Two important remarks concerning the richness of the model are: i) we

don’t take into account the data of Dwarf Galaxies due to the very high errorbars in the observations, and ii) as it can be seen from making a direct com-parison between the two graphics for the second case, although the groups ofgalaxies are covered by the range of parameters mentioned above, the modelexclude the clusters of galaxies as a part of a Universality law.

So we have showed within the model that always exists a physical rangeof parameters β0, θ0 and W0 that reproduces the graphics showing the Uni-

169


0

1

2

3

4

5

6

107 108 109 1010 1011 1012 1013 1014 1015 1016

DM column density, lg (S/M

sun pc

-2)

DM halo mass [Msun]

Clusters of galaxiesGroups of galaxiesSpiral galaxiesElliptical galaxiesdSphsIsolated halos from ΛCDM N-body simulationsSubhalos from Aquarius simulation

0

1

2

3

4

5

6

1e+07 1e+08 1e+09 1e+10 1e+11 1e+12 1e+13 1e+14 1e+15 1e+16

log

S (

Msu

n/pc

2 )

Mh (Msun)

model boundsupper observational limitlower observational limit

Figure G.6.: The dark matter column density S vs the halo mass Mh. The leftbound from the model corresponds to v0 = 50 km/s (log β0 = −7.9) and theright one to v0 = 1400 km/s (log β0 = −5). The dot is the milky way.

versality laws in the different scopes.

170

H. Features in the primordialspectrum: new constraints fromWMAP7+ACT data andprospects for Planck

H.1. Introduction

Current cosmological observations can be explained in terms of the so-calledconcordance ΛCDM model in which the primordial fluctuations are createdduring an early period of inflationary expansion of the Universe. In par-ticular, the spectrum of anisotropies of the cosmic microwave background(CMB) is in excellent agreement with the inflationary prediction of adiabaticprimordial perturbations with a nearly scale-invariant power spectrum Ko-matsu et al. (2011); Larson et al. (2011); Das et al. (2011); Dunkley et al. (2011);Hlozek et al. (2011). In its simplest implementation, inflation is driven bythe potential energy of a single scalar field, the inflaton, slowly rolling downtowards a minimum of its potential; curvature perturbations, that constitutethe primordial seeds for structure formation, are originated during the slowroll from quantum fluctuations in the inflaton itself. The scale invariance ofthe spectrum is directly related to the flatness and smoothness of the infla-ton potential, that are necessary to ensure that the slow-roll phase lasts longenough to solve the paradoxes of the Big Bang model.

However, in more general inflationary models, there is the possibility thatslow roll is briefly violated. This naturally happens in theories with manyinteracting scalar fields, as it is the case, for example, in a class of multifield,supergravity-inspired models Adams et al. (1997b,a), where supersymmetry-breaking phase transitions occur during inflation. These phase transitionscorrespond to sudden changes in the inflaton effective mass and can be mod-eled as steps in the inflationary potential. If the transition is very strong, itcan stop the inflationary phase as it happens in the usual hybrid inflationscenario; on the contrary, inflation can continue but the inflationary pertur-bations and thus the shape of the primordial power spectrum are affected.Departures from the standard power-law behaviour can also be caused bychanges in the initial conditions due to trans-planckian physics Branden-berger and Martin (2001); Easther et al. (2002); Martin and Brandenberger

171

H. Features in the primordial spectrum: new constraints fromWMAP7+ACT data and prospects for Planck

(2003) or to unusual initial field dynamics Burgess et al. (2003); Contaldi et al.(2003)

A violation of slow-roll will possibly lead to detectable effects on the cos-mological observables, or at least to the opportunity to constraint these mod-els by the absence of such effects. In particular, step-like features in the pri-mordial power spectrum have been shown Adams et al. (2001); Hunt andSarkar (2004) to lead to characteristic localized oscillations in the power spec-trum of the primordial curvature perturbation. Such oscillations have beenconsidered as a possible explanation to the “glitches” observed by the Wilkin-son Microwave Anisotropy Probe (WMAP) in the temperature anisotropyspectrum of the CMB, although the WMAP team notes that these could becaused simply by having neglected beam asymmetry, the gravitational lens-ing of the CMB, non-gaussianity in the CMB maps and other “small” (. 1%)contributions to the covariance matrix. In the following we will assume thatthese features have indeed a cosmological origin as in the class of extendedmodels described above and we will use CMB data to constrain the phe-nomenological parameters describing the step in the inflaton potential.

Constraints on oscillation in the primordial perturbation spectrum, as wellas best-fit values for the step parameters, have been previously derived inRefs. Peiris et al. (2003); Covi et al. (2006); Hamann et al. (2007); Mortonsonet al. (2009); Hazra et al. (2010). Here we improve on the previous analyses inseveral aspects. First, we use more recent CMB data, in particular the WMAP7-year and the Atacama Cosmology Telescope (ACT) data. This allows usto derive tighter constraints on the parameters; in particular we get an up-per limit on the step height (related to the amplitude of oscillations) that isindependent on the position of the step itself in the prior range considered.We also find a clear correlation between the position and the height of thestep. Secondly, we generate mock data corresponding to the model provid-ing the best-fit to the WMAP data, and use these data to assess the abilityof the Planck satellite to detect the presence of oscillations in the primordialspectrum.

The paper is organized as follows: in Section II we describe the evolution ofperturbations in interrupted slow roll and the phenomenological model usedto describe a step in the inflationary potential. In Section III we discuss theanalysis method adopted. In Section IV we present the results and in SectionV we derive our conclusions.

H.2. Inflationary perturbations in models withinterrupted slow roll

Steps in the potential can naturally appear in “multiple inflation” models,where the inflaton field φ is gravitationally coupled to a “flat direction” field

172

H.2. Inflationary perturbations in models with interrupted slow roll

ρ (belonging to the visible sector of the theory), i.e. a direction in field spacealong which the potential vanishes. The ρ-field can undergo a symmetry-breaking phase transition and acquire a vacuum expectation value 〈ρ〉. Thegravitational coupling between the ρ and the inflaton field will cause the ef-fective mass-squared of the latter to change; for example, in the case in whichthe coupling between the two fields is described by a term λφ2ρ2/2 in theLagrangian, the inflaton mass-squared after the phase transition will becomem2

eff(φ) = m20 + λ〈ρ2〉. It is worth noticing that the presence of flat field direc-

tions also opens the possibility to have inflation with a curved trajectory infield space; however, in the following, we will disregard this scenario.

The exact behaviour of the inflaton mass will depend by the dynamics ofthe phase transition; however, this is so fast that the ρ-field reaches the min-imum of its potential very rapidly. It is then very reasonable to model theinflaton mass in a phenomenological way as

m2eff(φ) = m2

[1 + c tanh

(φ− b

d

)]. (H.2.1)

Here, the parameter b is of the order of the critical value of the inflaton fieldfor which the phase transition occurs, c is the height of the step (related tothe change in the inflaton mass) and d is its width (related to the duration ofthe phase transition). In the following we shall work in reduced Planck units(c = h = 8πG = 1), so that all dimensional quantities like m, b and d shouldbe multiplied by the reduced Planck mass Mp = 2.435× 1018 GeV in order toget their values in physical units.

Let us now briefly recall how to compute the spectrum of primordial per-turbations, as discussed in details by Adams et al. (2001). For the moment,we do not specify the exact form of the inflaton potential V(φ); we will returnon this in the next Section. In the case of scalar perturbations, it is useful todefine the gauge-invariant quantity Stewart and Lyth (1993) u ≡ −zR, wherez = aφ/H, a is the scale factor, H is the Hubble parameter, R is the curvatureperturbation, and dots denote derivatives with respect to the cosmologicaltime t. The Fourier components of u evolve according to:

u′′k +

(k2 − z′′

z

)uk = 0 , (H.2.2)

where k is the wavenumber of the mode, and primes denote derivatives withrespect to conformal time η. When k2 z′′/z, the solution to the aboveequation tends to the free-field solution uk = e−ikη/

√2k.

In the slow-roll approximation, z′′2H2. However, in the models consideredhere this expectation can be grossly violated near the phase transition, andthe time evolution of z has to be derived by solving the equations for the

173


inflaton field and for the Hubble parameter:

φ + 3Hφ +dVdφ

= 0 , (H.2.3)

3H2 =φ2

2+ V(φ) . (H.2.4)

Once the form of the potential is given, these can be integrated to get H andφ, and thus z, as a function of time. At this point, it is possible to integrate Eq.(H.2.2) to get uk(η) for free-field initial conditions when k2 z′′/z. Finally,knowing the solution for the mode k, the power spectrum of the curvatureperturbation PR can be computed by means of

PR =k3

2π

∣∣∣ukz

∣∣∣2 (H.2.5)

evaluated when the mode crosses the horizon. The resulting spectrum formodels with a step in the potential is essentially a power-law with superim-posed oscillations; thus, asymptotically, the spectrum will recover the famil-iar kns−1 form typical of slow-roll inflationary models.

In practice, however, one has to relate the horizon size at the step witha physical wavenumber. For a general wavenumber k? one can write k? ≡a?H? = aende−N? H?, where a? and H? are the scale factor and the Hubbleparameter at the time the mode crossed the horizon, aend is the scale factorat the end of inflation, and N? is the number of e-fold taking place after themode left the horizon. We choose N? = 50 for the pivot wavenumber k? =k0 = 0.0025 Mpc−1. A different choice would correspond to a translation inthe position of the step in φ and would thus be highly degenerate with b. Forthis reason we do not treat N? as a free parameter, consistent with what hasbeen done in previous studies Covi et al. (2006); Hamann et al. (2007).

H.3. Analysis Method

We compare the theoretical predictions of a class of inflationary models witha step in the inflaton potential with observational data. We use a modifiedversion of the CAMB code that solves Eqs. (H.2.2)–(H.2.4) numerically us-ing a Bulirsch-Stoer algorithm in order to compute the initial perturbationspectrum (H.2.5) and, from that, the CMB anisotropy spectrum for given val-ues of the relevant parameters describing the model. CAMB is then interfacedwith a modified version of the Markov Chain Monte Carlo (MCMC) packageCosmoMC Lewis and Bridle (2002), that we use to find the best-fit value of theparameters, to reconstruct their posterior probability density function, and toinfer constraints on the parameter themselves.

Models. We consider a chaotic inflation potential of the form V(φ) =

174


19.6

19.8

20

20.2

20.4

14.2 14.25 14.3

V(φ)/(10

16 GeV)4

φ/MPl

Figure H.1.: Inflationary potential (H.3.1) for m = 7.5× 10−6. The solid thinblack line corresponds to a smooth (c = 0) chaotic potential m2φ2/2. Thelong-dashed red curve has b = 14.23, c = 0.001 and d = 0.025 and roughlycorresponds to the spectrum giving the best fit to the WMAP7 data (see Sec.H.4 below). The other curves correspond to b = 14.23, c = 0.005, d = 0.025(blue short-dashed), b = 14.23, c = 0.005, d = 0.05 (green dotted) and b =14.25, c = 0.005, d = 0.025 (magenta dot-dashed).

m2eff(φ)φ

2/2. Using Eq. (H.2.1), this corresponds to a potential

V(φ) =12

m2φ2[

1 + c tanh(

φ− bd

)]. (H.3.1)

In Fig. H.1 we show the shape of this potential for m = 7.5× 10−6 and differ-ent values of the step parameters (close to the best-fit values), compared to asmooth m2φ2/2 potential (c = 0).

The potential (H.3.1) uniquely defines the spectrum of perturbations PR.The parameters that define the primordial spectrum and the initial conditionsfor the evolution of cosmological perturbations are then the inflaton mass mand the step parameters b, c and d. The inflaton mass sets the overall scale forthe potential and consequently for the amplitude of the perturbations; it canthen be traded, in the Monte Carlo analysis, for the more familiar parameterAs, i.e., the amplitude of the primordial spectrum at the pivot wavenumberk0 = 0.0025 Mpc−1. On the other hand, as already noted above, a step in the

175


0

1e-09

2e-09

3e-09

4e-09

5e-09

6e-09

0.0001 0.001 0.01 0.1

Δ2R(k)

k [Mpc-1]

Figure H.2.: Primordial power spectrum for an inflationary potential of theform (H.3.1). The values of the step parameters are the same as in Fig. (H.1),namely: b = 14.23, c = 0.001, d = 0.025 (red long-dashed), b = 14.23, c =0.005, d = 0.025 (blue short-dashed), b = 14.23, c = 0.005, d = 0.05 (greendotted) and b = 14.25, c = 0.005, d = 0.025 (magenta dot-dashed).

potential produces a perturbation spectrum with oscillations superimposedover a smooth power law. In the case of the potential (H.3.1), the underlyingpower-law has a fixed spectral index ns = 0.96. In Fig. H.2 we show theprimordal spectrum for different values of the step parameters.

The results obtained in the case of a specific potential will be, by definition,model-dependent. However, as argued in Ref. Hamann et al. (2007), the issueof model dependence can be alleviated in a phenomenological way by restor-ing the spectral index as a free parameter, i.e., by defining the “generalized”spectrum P

genR as

PgenR (k) = Pch

R (k)×(

kk0

)ns−0.96

, (H.3.2)

where PchR (k) is the spectrum induced by the chaotic potential (H.3.1). Since

the latter has a overall tilt of 0.96, ns will describe the overall tilt of the gener-alized spectrum.

Summarising, we consider two classes of models. Models belonging tothe first class (referred to as class A) corresponding to the potential (H.3.1),

176


are described by eight parameters: the physical baryon and cold dark matterdensities ωb = Ωbh2 and ωc = Ωch2, the ratio θ between the sound hori-zon and the angular diameter distance at decoupling, the optical depth toreionization τ, the parameters b, c and d of the step-inflation model, and theoverall normalization of the primordial power spectrum As (equivalent tospecifying m2 as discussed above). Models in the second class, referred to asclass B, correspond to the generalized step model (H.3.2) and are describedby the effective tilt ns in addition to the eight parameters of the first class. Inboth cases, we consider purely adiabatic initial conditions, impose flatnessand neglect neutrino masses. We limit our analysis to scalar perturbations.

Priors. Apart from the hard-coded priors of CosmoMC on H0 (40 km s−1

Mpc−1 < H0 <100 km s−1 Mpc−1) and the age of the Universe (10 Gyr < t0 <20 Gyr), we impose flat priors on ωb, ωc, θ, τ and, when considered, ns anda logarithmic prior on As. As we shall see, for these parameters the widthof the posterior is much smaller than the prior range, so that the latter isnot really relevant. For the step parameters, the situation is complicated bythe fact that the likelihood (and the posterior) does not go to zero in certaindirections of the subspace. This happens in particular for very small values ofc, for which the spectrum becomes indistinguishable from a power law, andfor values of b either too large or too small so that the features in the spectrumare moved outside the range of observable scales. Then we choose for b a flatprior 13 ≤ b ≤ 15, that roughly encompasses said range. In the case of c andd, since we do not have any a priori information on these parameters, not evenon their order of magnitude, we find convenient to consider a logarithmicprior on both of them. Hence, we take (in the following, log x denotes thebase 10 logarithm) −6 ≤ log c ≤ −1 and −2.5 ≤ log d ≤ −0.5. Additionally,since the combination c/d2 is better constrained by the data than d alone, wealso impose a prior −5 ≤ log(c/d2) ≤ 3. Finally, we recall that, since theposteriors for b, log c and log d do not necessarily vanish at the edge of theprior range, all integrals of the probability density function depend on theextremes of integration and are thus somewhat ill-defined. Care should thenbe taken when quoting confidence limits in the b, log c and log d subspaces.

Datasets. We perform the statistical analysis for each of the models by com-paring the theoretical predictions to two different datasets. The first includesthe WMAP 7-year temperature and polarization anisotropy data (WMAP7).The likelihood is computed using the the WMAP likelihood code publiclyavailable at the LAMBDA website1. We marginalize over the amplitude of theSunyaev-Zel’dovich signal. The second dataset includes the WMAP7 datawith the addition of the small-scale CMB temperature anisotropy data fromthe ACT experiment. For the ACT dataset we also consider two extra param-eters accounting for the Poisson and clustering point sources foregroundscomponents. The ACT dataset is considered up to `max = 2500.

1http://lambda.gsfc.nasa.gov/

177


Other than deriving the limits on the models from existing data, we alsoassess the ability of future experiments, in particular of the Planck satellite,to improve these constraints. In order to do this, we simulate “mock” datacorresponding to the step model that yields the best-fit to the WMAP 7 andthen perform a statistical analysis on these data as if they were real. Theforecast method we use is identical to the one presented in Galli et al. (2010)and we refer to this paper for further details and references. The syntheticdataset is generated by considering for each C` a noise spectrum given by:

N` = w−1 exp(`(`+ 1)8 ln 2/θ2b), (H.3.3)

where θb is the full width at half maximum (FWHM) of the beam assuminga Gaussian profile and where w−1 is the experimental power noise related tothe detectors sensitivity σ by w−1 = (θbσ)2. The experimental parameters arereported in Table H.1.

Experiment Channel FWHM ∆T/T ∆P/TPlanck 70 14’ 4.7 6.7

fsky = 0.85 100 10’ 2.5 4.0143 7.1’ 2.2 4.2

Table H.1.: Planck Collaboration (2006) experimental specifications of Plancksatellite. Channel frequency is given in GHz, FWHM in arcminutes and noiseper pixel for the Stokes I (∆T/T), Q and U parameters (∆P/T) is in [µK/K],where T = TCMB = 2.725 K. In the analysis, we assume that beam uncertain-ties and foreground uncertainties are smaller than the statistical errors.

Together with the primary anisotropy signal we also take into account in-formation from CMB weak lensing, considering the power spectrum of thedeflection field Cdd

` and its cross correlation with temperature maps CTd` .

Analysis. We derive our constraints from parallel chains generated usingthe Metropolis-Hastings algorithm. We use the Gelman and Rubin R param-eter to evaluate the convergence of the chains, demanding that R− 1 < 0.03.The one- and two-dimensional posteriors are derived by marginalizing overthe other parameters.

H.4. Results and Discussion

We first consider the WMAP7 and WMAP7+ACT datasets. We find that theΛCDM fit to both datasets can be improved by the inclusion of a step in theinflationary potential, in both cases when the scalar spectral index is beingfixed to ns = 0.96 (Model A), and when it is being treated as a free parameter

178

H.4. Results and Discussion

0

0.2

0.4

0.6

0.8

1

13.6 13.8 14 14.2 14.4 14.6 14.8 15

L/L m

ax

b

Figure H.3.: Model likelihood as a function of b for model A (thin curves) andB (thick curves) using WMAP7 data (dashed curves) and the WMAP7+ACTdataset (solid curves)

Table H.2.: Best-fit values for the parameters of the primordial spectrum.Model A Model A Model B Model B

Parameter WMAP7 WMAP7+ACT WMAP7 W7+ACTb 14.23 14.25 14.24 14.25

log c -3.11 -2.71 -2.97 -2.67log d -1.58 -1.60 -1.65 -1.45

ns – – 0.953 0.959ln[1010As] 3.08 3.06 3.07 3.08

χ2 7469.4 7489.6 7467.9 7491.4

179


13 13.5 14 14.5 15

Post

erio

r Pr

obab

ilit

y

b

Model A, WMAP7Model B, WMAP7

Model A, WMAP7+ACTModel B, WMAP7+ACT

-4 -3.5 -3 -2.5 -2 -1.5 -1

Post

erio

r Pr

obab

ilit

y

Log[c]



-2.5 -2 -1.5 -1 -0.5

Post

erio

r Pr

obab

ilit

y

Log[d]



Figure H.4.: One-dimensional posterior probability density for b (left panel),log c (middle panel) and log d (right panel) for model A (thin curves) and B(thick curves) using WMAP7 data (dashed curves) and the WMAP7+ACTdataset (solid curves)

(Model B). The best-fit values for the step parameters are reported in TableH.2. We also show the full likelihood for b in Fig. H.3. It can be seen that in allcases the maximum in the likelihood occurs for b ' 14.2; as we show below,this is due to oscillations placed in correspondence to the WMAP glitches at` ∼ 20 and ` ∼ 40 and thus able to improve, for suitable values of the otherparameters, the goodness-of-fit with respect to the vanilla ΛCDM model. Wefound that in case of the WMAP7 analysis the best-fit vanilla Λ-CDM modelis at about ∆χ2

eff ∼ 6 from the global best fit with features.As long as bayesian statistics is concerned, the actual probability density

distribution for a parameter is not given by the likelihood (the probability ofthe data given the parameters) but instead by the posterior (the probabilityof the parameters given the data). In Fig. H.4 we show the one-dimensionalposterior distributions for the step parameters b, log c and log d. It can benoted that the posterior for b has a peculiar shape, presenting a peak forb ' 14.2 and a fairly wide dip for b . 14. The peak traces the peak in thelikelihood discussed above. The decrease for b < 14 is instead due to the factthat, lowering b, the oscillations are moved to larger multipoles where theytend to spoil the ΛCDM fit unless c is set to a very small value.

This is clearly illustrated in Fig. H.5, where we compare the WMAP7data with three realizations of the CMB spectrum: the ΛCDM best fit tothe WMAP data, the generalized step model best fit to the same data (cor-responding to the third column of Table H.2), and a generalized step modelwith the same parameters as the best fit, with the exception of b that is set tob = 13.9. It is clear, especially from the second panel, that for b = 14.2 the os-cillations improve the fit in the region 20 . ` . 50. On the other hand, whenb = 13.9 the height of the first peak is diminished so that the predicted spec-trum is completely at variance with the data. The posterior does not drop tozero because it still exist a fair amount of parameter space, i.e., models withlow c, than can fit the data even with the oscillations placed in the “wrong”place. The posterior going to a constant value at the edges of the prior rangeis instead related to the oscillations being moved out of the observable scales.The inclusion of the ACT data in addition to WMAP7 helps in constraining

180

H.5. Concluding Remarks

small values of b, i.e., oscillations at small scales (large `’s).The shape of the log c posterior is typical of a quantity parametrizing the

amplitude of a non-standard effect: it is constant for “small” values of theparameter (when the step model becomes indistinguishable from standardΛCDM), and then rapidly vanishes above a critical value. It can be seenthat the probability density becomes half of its asymptotic value at c = 0for c ≥ 10−2. Finally, the posterior for log c clearly shows that this parameteris largely unconstrained by data.

We do not quote one-dimensional confidence limits on the parameters be-cause, as noted in Sec. H.3, the posteriors do not vanish at the edge of theprior range and in this case the confidence limits depend on the integrationrange chosen. However, for illustrative purposes, in Fig. H.6 we show the 2-dimensional 95% confidence regions, computed assuming that the posteriorvanishes outside the prior range, in the (b− log c) plane. It is clear from theplots that there is a region below b = 14 where the data are more sensitive tothe value of c; this is related as noted above to the oscillations being placed inthe region where the data are more accurate and favour a smooth spectrumover one with oscillations.

The results presented here are fully compatible with the analysis madeby Mortonson et al. (2009) where the WMAP5 dataset was considered. Theapparently different value for the best-fit b parameter found in that paperis due to the different choice of the pivot scale (k0 = 0.05 Mpc−1 insteadof k0 = 0.0025 Mpc−1 as assumed in our analysis). We have checked thatperforming the analysis on the WMAP7 dataset with the assumption ofk0 = 0.05 Mpc−1 results in a best-fit value of b ∼ 14.7 in agreement withthe results of Mortonson et al. (2009).

Finally, we show our results on the sensitivity of Planck to the step param-eters. We have assumed as a fiducial model a generalized step model withb = 14.2, log c = −2.97, log d = 1.65, ns = 0.953, As = 2.16× 10−9 (basi-cally corresponding to the Model B best-fit to the WMAP7 data, i.e, the thirdcolumn of Tab. H.2). The one-dimensional posteriors for b, log c and log dare shown in Fig. H.7, while in Tab. H.3 we report the mean values for theprimordial spectrum parameters together with their 2σ error. As we can seethe prior range dependence goes away with Planck data and we can quotemarginalised credible intervals. We also show the two-dimensional posteri-ors for the step parameters in Fig. H.8. It is evident that the Planck data willgreatly increase the precision to which the step parameters can be measured;in particular, a detection of oscillations will be possible.


We have considered inflation models with a small-amplitude step-like fea-ture in the inflaton potential. Features of these kind can be due for example

181


0

1000

2000

3000

4000

5000

6000

1 10 100 1000

l(l+

1)C l/2

π

l

ΛCDM best fitStep model best fit

b=13.9WMAP data

600

800

1000

1200

1400

1600

10 20 30 40 50 60

l(l+

1)C l/2

π

l

ΛCDM best fitStep model best fit

b=13.9WMAP data

Figure H.5.: Upper panel: CMB anisotropy spectrum for the ΛCDM (red solidline) and generalized step model (blue long dashed line) best fits, and fora step model with b = 13.9 (black short dashed line), compared with theWMAP7 data. Lower panel: Zoom of the region ` ≤ 60, showing the im-proved fit of the step model.

182


b

log

c

12 13 14 15 16−6

−5

−4

−3

−2

b

log

c

12 13 14 15 16−6

−5

−4

−3

−2

b

log

c

12 13 14 15 16−6

−5

−4

−3

−2

−1

b

log

c

12 13 14 15 16−6

−5

−4

−3

−2

−1

Figure H.6.: 95% two-dimensional confidence region in the (b–log c) plane.The four panels correspond to, from left to right and from top to bottom: classA, WMAP7+ACT; class B, WMAP7+ACT; class A, WMAP7; class B, WMAP7.

0

0.2

0.4

0.6

0.8 1 14.18 14.185

14.19 14.195

14.2 14.205

14.21 14.215

14.22 14.225

Posterior probability

b

0

0.2

0.4

0.6

0.8 1-3.8-3.6

-3.4-3.2

-3-2.8

-2.6-2.4


logc

0

0.2

0.4

0.6

0.8 1-2.1-2

-1.9-1.8

-1.7-1.6

-1.5-1.4

-1.3-1.2


logd

Figure H.7.: One-dimensional posterior probability density for b (left panel),log c (middle panel) and log d (right panel) derived from the mock Planckdata, for models of class A (dashed curves) and B (solid curves).

Table H.3.: Parameter constraints from Planck.Parameter Model A Model B

b 14.200± 0.010 14.200± 0.011log c −3.00± 0.32 −3.00± 0.34log d −1.66± 0.22 −1.64± 0.23

ns 0.96 (fixed) 0.957± 0.007ln[1010As] 3.073± 0.016 3.074± 0.016

183


b

log

d

14.2 14.22

−2

−1.8

−1.6

−1.4

log d

log

c

−2 −1.5

−3.6

−3.4

−3.2

−3

−2.8

−2.6

b

log

c

14.2 14.22

−3.6

−3.4

−3.2

−3

−2.8

−2.6

b

log

c

14.1914.214.21−3.6

−3.4

−3.2

−3

−2.8

−2.6

log d

log

c

−2 −1.5−3.6

−3.4

−3.2

−3

−2.8

−2.6

blo

g d

14.1914.214.21

−2

−1.8

−1.6

−1.4

Figure H.8.: Two-dimensional posteriors in the (b–log c) (left), (log d–log c)(middle) and (b–log d) (right) planes, for models of class A (top) and B (bot-tom), derived from the mock Planck data. The shaded areas correspond tothe 68% (light blue) and 95% (dark blue) confidence regions.

184


to phase transitions occurring during the slow roll in multi-field inflationarymodels. In these models the primordial perturbation spectrum has the formof a power-law (as in the standard featureless case) with superimposed oscil-lations, localized in a finite range of scales that basically depends on the posi-tion of the step in the potential. We have compared the theoretical predictionsof a specific model, i.e., chaotic inflation, and of a more general phenomeno-logical model to the WMAP7 and ACT data, in order to find constraints onthe parameter describing the model. We have also studied the possibilityof detecting the oscillations with the upcoming Planck data in the case theyreally exist.

We have found that models with features can improve the fit to theWMAP7 data when the step in the potential is placed in such a way as toproduce oscillations in the region 20 . ` . 60, where the WMAP7 datashows some glitches. We found no further evidence for small scales glitchesfrom the recent ACT data, this is fully consistent with the recent analysis ofHlozek et al. (2011). We have also found that models with too high a step areexcluded by the data. Finally, assuming as a fiducial model the generalizedstep model that provides the best fit to the WMAP7 data, we have found thatthe Planck data will allow to measure the parameters of the model with re-markable precision, possibly confirming the presence of glitches in the region20 . ` . 60.

185


186

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greatest open

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