The clustering of galaxies in the completed SDSS-III ... · Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 13 July 2016 (MN LATEX style ﬁle v2.2) The clustering of galaxies

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 13 July 2016 (MN LATEX style file v2.2)

The clustering of galaxies in the completed SDSS-III BaryonOscillation Spectroscopic Survey: double-probe measurements fromBOSS galaxy clustering & Planck data – towards an analysiswithout informative priors

Marcos Pellejero-Ibanez1,2,3,4?, Chia-Hsun Chuang3,4†, J. A. Rubino-Martın1,2, Anto-nio J. Cuesta5, Yuting Wang6,7, Gong-bo Zhao6,7, Ashley J. Ross8,7, Sergio Rodrıguez-Torres3,9,10, Francisco Prada3,9,11, Anze Slosar12, Jose A. Vazquez12, Shadab Alam13,14,Florian Beutler15,7, Daniel J. Eisenstein16, Hector Gil-Marın17,18, Jan Niklas Grieb19,20,Shirley Ho13,14,15,21, Francisco-Shu Kitaura4,15,21, Will J. Percival7, Graziano Rossi22,Salvador Salazar-Albornoz19,20, Lado Samushia23,24,7, Ariel G. Sanchez20, SiddharthSatpathy13,14, Hee-Jong Seo25, Jeremy L. Tinker26, Rita Tojeiro27, Mariana Vargas-Magana28, Joel R. Brownstein29, Robert C Nichol7, Matthew D Olmstead30

1 Instituto de Astrofısica de Canarias (IAC), C/Vıa Lactea, s/n, E-38200, La Laguna, Tenerife, Spain2 Departamento Astrofısica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain3 Instituto de Fısica Teorica, (UAM/CSIC), Universidad Autonoma de Madrid, Cantoblanco, E-28049 Madrid, Spain4 Leibniz-Institut fur Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany5 Institut de Ciencies del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martı i Franques 1, E08028 Barcelona, Spain6 National Astronomy Observatories, Chinese Academy of Science, Beijing, 100012, P.R.China7 Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth PO1 3FX, UK8 Center for Cosmology and Astroparticle Physics, Department of Physics, The Ohio State University, OH 43210, USA9 Campus of International Excellence UAM+CSIC, Cantoblanco, E-28049 Madrid, Spain10 Departamento de Fısica Teorica M8, Universidad Autonoma de Madrid (UAM), Cantoblanco, E-28049, Madrid, Spain11 Instituto de Astrofısica de Andalucıa (CSIC), Glorieta de la Astronomıa, E-18080 Granada, Spain12 Brookhaven National Laboratory, Upton, NY 1197313 Department of Physics, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, USA14 The McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 1521315 Lawrence Berkeley National Lab, 1 Cyclotron Rd, Berkeley CA 94720, USA16 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA17 Sorbonne Universits, Institut Lagrange de Paris (ILP), 98 bis Boulevard Arago, 75014 Paris, France18 Laboratoire de Physique Nuclaire et de Hautes Energies, Universite Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France19 Universitats-Sternwarte Munchen, Scheinerstrasse 1, 81679, Munich, Germany20 Max-Planck-Institut fur extraterrestrische Physik, Postfach 1312, Giessenbachstr., 85741 Garching, Germany21 Departments of Physics and Astronomy, University of California, Berkeley, CA 94720, USA22Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea23 Kansas State University, Manhattan KS 66506, USA24 National Abastumani Astrophysical Observatory, Ilia State University, 2A Kazbegi Ave., GE-1060 Tbilisi, Georgia25Department of Physics and Astronomy, Ohio University, 251B Clippinger Labs, Athens, OH 45701, USA26 Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA27 School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK28 Instituto de Fısica, Universidad Nacional Autonoma de Mexico, Apdo. Postal 20-364, Mexico29 Department of Physics and Astronomy, University of Utah, 115 S 1400 E, Salt Lake City, UT 84112, USA30 Department of Chemistry and Physics, King’s College, 133 North River St, Wilkes Barre, PA 18711, USA

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? E-mail: [email protected]† E-mail: [email protected]

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ABSTRACTWe develop a new methodology called double-probe analysis with the aim of minimizinginformative priors in the estimation of cosmological parameters. Using our new methodology,we extract the dark-energy-model-independent cosmological constraints from the joint datasets of Baryon Oscillation Spectroscopic Survey (BOSS) galaxy sample and Planck cosmicmicrowave background (CMB) measurement. We measure the mean values and covariancematrix of R, la, Ωbh

2, ns, log(As), Ωk, H(z), DA(z), f(z)σ8(z), which give an efficientsummary of Planck data and 2-point statistics from BOSS galaxy sample. The CMB shiftparameters are R =

√ΩmH2

0 r(z∗), and la = πr(z∗)/rs(z∗), where z∗ is the redshift at thelast scattering surface, and r(z∗) and rs(z∗) denote our comoving distance to z∗ and soundhorizon at z∗ respectively; Ωb is the baryon fraction at z = 0. The advantage of this methodis that we do not need to put informative priors on the cosmological parameters that galaxyclustering is not able to constrain well, i.e. Ωbh

2 and ns.Using our double-probe results, we obtain Ωm = 0.304 ± 0.009, H0 = 68.2 ± 0.7,

and σ8 = 0.806 ± 0.014 assuming ΛCDM; Ωk = 0.002 ± 0.003 assuming oCDM; w =−1.04 ± 0.06 assuming wCDM; Ωk = 0.002 ± 0.003 and w = −1.00 ± 0.07 assumingowCDM; and w0 = −0.84 ± 0.22 and wa = −0.66 ± 0.68 assuming w0waCDM. Theresults show no tension with the flat ΛCDM cosmological paradigm. By comparing with thefull-likelihood analyses with fixed dark energy models, we demonstrate that the double-probemethod provides robust cosmological parameter constraints which can be conveniently usedto study dark energy models.

We extend our study to measure the sum of neutrino mass using different methodolo-gies including double probe analysis (introduced in this study), the full-likelihood analysis,and single probe analysis. From the double probe analysis, we obtain Σmν < 0.10/0.22(68%/95%) assuming ΛCDM and Σmν < 0.26/0.52 (68%/95%) assuming wCDM. Thispaper is part of a set that analyses the final galaxy clustering dataset from BOSS.

Key words: cosmology: observations - distance scale - large-scale structure of Universe -cosmological parameters

1 INTRODUCTION

We have entered the era of precision cosmology along with thedramatically increasing amount of sky surveys, including the cos-mic microwave background (CMB; e.g., Bennett et al. 2013; Adeet al. 2014a), supernovae (SNe; Riess et al. 1998; Perlmutter et al.1999), weak lensing (e.g., see Van Waerbeke & Mellier 2003 fora review), and large-scale structure from galaxy redshift surveys,e.g. 2dF Galaxy Redshift Survey (2dFGRS; Colless et al. 2001,2003, Sloan Digital Sky Survey (SDSS, York et al. 2000; Abaza-jian et al. 2009, WiggleZ (Drinkwater et al. 2010; Parkinson et al.2012), and the Baryon Oscillation Spectroscopic Survey (BOSS;Dawson et al. 2013; Alam et al. 2015) of the SDSS-III (Eisensteinet al. 2011). The future galaxy redshift surveys, e.g. Euclid1 (Lau-reijs et al. 2011), Dark Energy Spectroscopic Instrument 2 (DESI;Schlegel et al. 2011), and WFIRST3 (Green et al. 2012), will collectdata at least an order of magnitude more. It is critical to develop themethodologies which could reliably extract the cosmological infor-mation from such large amount of data.

The galaxy redshifts samples have been analysed studied in acosmological context (see, e.g., Tegmark et al. 2004; Hutsi 2005;Padmanabhan et al. 2007; Blake et al. 2007; Percival et al. 2007,2010; Reid et al. 2010; Montesano et al. 2012; Eisenstein et al.2005; Okumura et al. 2008; Cabre & Gaztanaga 2009; Martinezet al. 2009; Sanchez et al. 2009; Kazin et al. 2010; Chuang et al.2012; Samushia et al. 2012; Padmanabhan et al. 2012; Xu et al.

1 http://sci.esa.int/euclid2 http://desi.lbl.gov/3 http://wfirst.gsfc.nasa.gov/

2013; Anderson et al. 2013; Manera et al. 2012; Nuza et al. 2013;Reid et al. 2012; Samushia et al. 2013; Tojeiro et al. 2012; Ander-son et al. 2014b; Chuang et al. 2013a; Sanchez et al. 2013; Kazinet al. 2013; Wang 2014; Anderson et al. 2014a; Beutler et al. 2014b;Samushia et al. 2014; Chuang et al. 2013b; Sanchez et al. 2014;Ross et al. 2014; Tojeiro et al. 2014; Reid et al. 2014; Alam et al.2015; Gil-Marın et al. 2015a,b; Cuesta et al. 2015).

Eisenstein et al. (2005) demonstrated the feasibility of mea-suring Ωmh

2 and an effective distance, DV (z) from the SDSSDR3 (Abazajian et al. 2005) LRGs, where DV (z) corresponds to acombination of Hubble expansion rate H(z) and angular-diameterdistance DA(z). Chuang & Wang (2012) demonstrated the fea-sibility of measuring H(z) and DA(z) simultaneously using thegalaxy clustering data from the two dimensional two-point correla-tion function of SDSS DR7 (Abazajian et al. 2009) LRGs and it hasbeen improved later on in Chuang & Wang (2013b,a) upgrading themethodology and modelling to measureH(z),DA(z), the normal-ized growth rate f(z)σ8(z), and the physical matter density Ωmh

2

from the same data. Analyses have been perform to measureH(z),DA(z), and f(z)σ8(z) from earlier data release of SDSS BOSSgalaxy sample Reid et al. (2012); Chuang et al. (2013a); Wang(2014); Anderson et al. (2014a); Beutler et al. (2014b); Chuanget al. (2013b); Samushia et al. (2014).

There are some cosmological parameters, e.g. Ωbh2 (the phys-

ical baryon fraction) and ns (the scalar index of the power lawprimordial fluctuation), not well constrained by galaxy clusteringanalysis. We usually use priors adopted from CMB measurementsor fix those to the best fit values obtained from CMB while do-ing Markov Chain Monte Carlo (MCMC) analysis. There would be

c© 0000 RAS, MNRAS 000, 000–000

Double-Probe Measurements from BOSS & Planck 3

some concern of missing weak degeneracies between these param-eters and those measured. These could lead to incorrect constraintsif models with very different predictions are tested, or double-counting when combining with CMB measurements. One mightadd some systematics error budget to be safe from the potentialbias (e.g., see Anderson et al. (2014a)). An alternative approachis to use a very wide priors, e.g. 5 or 10 σ flat priors from CMB,to minimize the potential systematics bias from priors (e.g., seeChuang et al. (2012); Chuang & Wang (2012)). However, the ap-proach would obtain weaker constraints due to the wide priors. Inthis study, we test the ways in which LSS constraints are com-bined with CMB data, focussing on the information content, andthe priors used when analysing LSS data. Since CMB data canbe summarized with few parameters (e.g., see Wang & Mukherjee(2007)), we use the joint data set from Planck and BOSS to extractthe cosmological constraints without fixing dark energy models.By combining the CMB data and the BOSS data in the upstreamof the data analysis to constrain the cosmological constraints, wecall our method ”double-probe analysis”. Our companion paper,Chuang et al. (2016), constrains geometric and growth informationfrom the BOSS data alone independent of the CMB data, therebydubbed ”single-probe”, and combines with the CMB data in thedownstream of the analysis. Note that we assume there is no earlytime dark energy or dark energy clustering in this study. Ωbh

2 andns will be well constrained by CMB so that we will obtain thecosmological constraints without concerning the problem of pri-ors. The only input parameter which is not well constrained by ouranalysis is the galaxy bias on which is applied a wide flat prior.In principle, our methodology extract the cosmological constraintsfrom the joint data set with the optimal way since we do not needto include the uncertainty introduced by the priors.

In addition to constraining dark energy model parameters,we extend our study to constrain neutrino masses. High energyphysics experiments provides with the squared of mass differ-ences between neutrino species from oscillation neutrino exper-iments. Latest results are ∆m2

21 = 7.53 ± 0.18 × 10−5eV 2

and ∆m232 = 2.44 ± 0.06 × 10−3eV 2 for the normal hierarchy

(m3 m2 ' m1) and ∆m232 = 2.52± 0.07× 10−3eV 2 for the

inverted mass hierarchy (m3 m2 ' m1) (Olive & Group 2014).Cosmology shows as a unique tool for the measurement of the sumof neutrino masses Σmν , since this quantity affects the expansionrate and the way structures form and evolve. Σmν estimations fromgalaxy clustering has been widely studied theoretically (see Huet al. 1998; Lesgourgues & Pastor 2006 for a review) and with dif-ferent samples such as WiggleZ (see Riemer-Sørensen et al. 2014;Cuesta et al. 2015) or SDSS data (see Aubourg et al. 2015; Beut-ler et al. 2014a; Reid et al. 2010; Thomas et al. 2010; Zhao et al.2013). At late times, massive neutrinos can damp the formation ofcosmic structure on small scales due to the free-streaming effect(Dolgov 2002). Existing in the form of radiation in the early Uni-verse, neutrinos shift the epoch of the matter-radiation equality thuschanging the shape of the cosmic microwave background (CMB)angular power spectrum. They affect CMB via the so called EarlyIntegrated Sachs Wolfe Effect and they influence gravitational lens-ing measurements (e.g., see Lesgourgues et al. 2006). Recent pub-lications have attempted to constrain Σmν , imposing upper limits(Seljak et al. 2006; Hinshaw et al. 2009; Dunkley et al. 2009; Reidet al. 2010; Komatsu et al. 2011; Saito et al. 2011; Tereno et al.2009; Gong et al. 2008; Ichiki et al. 2009; Li et al. 2009; de Putteret al. 2012; Xia et al. 2012; Sanchez et al. 2012; Giusarma et al.2013) and some hints of lower limits using cluster abundance re-sults (Ade et al. 2014b; Battye & Moss 2014; Wyman et al. 2014;

Burenin 2013; Rozo et al. 2013). We measure the sum of neutrinomass using different methodologies including double probe analy-sis (introduced in this study), the full-likelihood analysis, and singleprobe analysis (Chuang et al. 2016; companion paper).

This paper is organized as follows. In Section 2, we intro-duce the Planck data, the SDSS-III/BOSS DR12 galaxy sample andmock catalogues used in our study. In Section 3, we describe the de-tails of the methodology that constrains cosmological parametersfrom our joint CMB and galaxy clustering analysis. In Section 4,we present our double-probe cosmological measurements. In Sec-tion 5, we demonstrate how to derive cosmological constraints fromour measurements with some given dark energy model. In Sec-tion 6, opposite to the manner of dark energy model independentmethod, we present the results from the full-likelihood analysiswith fixing dark energy models. In Section 7, we measure the sumof neutrino mass with different methodologies. We summarize andconclude in Section 8.

2 DATA SETS & MOCKS

2.1 The SDSS-III/BOSS Galaxy Catalogues

The Sloan Digital Sky Survey (SDSS; Fukugita et al. 1996; Gunnet al. 1998; York et al. 2000; Smee et al. 2013) mapped over onequarter of the sky using the dedicated 2.5 m Sloan Telescope (Gunnet al. 2006). The Baryon Oscillation Sky Survey (BOSS, Eisen-stein et al. 2011; Bolton et al. 2012; Dawson et al. 2013) is partof the SDSS-III survey. It is collecting the spectra and redshiftsfor 1.5 million galaxies, 160,000 quasars and 100,000 ancillary tar-gets. The Data Release 12 (Alam et al. 2015) has been made pub-licly available4. We use galaxies from the SDSS-III BOSS DR12CMASS catalogue in the redshift range 0.43 < z < 0.75 andLOWZ catalogue in the range 0.15 < z < 0.43. CMASS samplesare selected with an approximately constant stellar mass threshold(Eisenstein et al. 2011); LOWZ sample consists of red galaxies atz < 0.4 from the SDSS DR8 (Aihara et al. 2011) image data. Weare using 800853 CMASS galaxies and 361775 LOWZ galaxies.The effective redshifts of the sample are z = 0.59 and z = 0.32respectively. The details of generating this sample are described inReid et al. (2016).

2.2 The Planck Data

Planck (Tauber et al. 2010; Planck Collaboration I 2011) is the thirdgeneration space mission, following COBE and WMAP, to measurethe anisotropy of the CMB. It observed the sky in nine frequencybands covering the range 30–857 GHz with high sensitivity and an-gular resolutions from 31’ to 5’. The Low Frequency Instrument(LFI; Bersanelli et al. 2010; Mennella et al. 2011) covers the bandscentred at 30, 44, and 70 GHz using pseudo-correlation radiometersdetectors, while the High Frequency Instrument (HFI; Planck HFICore Team 2011) covers the 100, 143, 217, 353, 545, and 857 GHzbands with bolometers. Polarisation is measured in all but the high-est two bands (Leahy et al. 2010; Rosset et al. 2010). In this paper,we used the 2015 Planck release (Planck Collaboration I 2015),which included the full mission maps and associated data products.

4 http://www.sdss3.org/

c© 0000 RAS, MNRAS 000, 000–000


2.3 The Mock Galaxy Catalogues

We use 2000 BOSS DR12 MultiDark-PATCHY (MD-PATCHY)mock galaxy catalogues (Kitaura et al. 2015b) for validating ourmethodology and estimating the covariance matrix in this study.These mock catalogues were constructed using a similar proceduredescribed in Rodrıguez-Torres et al. 2015 where they constructedthe BOSS DR12 lightcone mock catalogues using the MultiDarkN -body simulations. However, instead of using N -body simula-tions, the 2000 MD-PATCHY mocks catalogues were constructedusing the PATCHY approximate simulations. These mocks are pro-duced using ten boxes at different redshifts that are created with thePATCHY-code (Kitaura et al. 2014). The PATCHY-code is com-posed of two parts: 1) computing approximate dark matter densityfield; and 2) populating galaxies from dark matter density field withthe biasing model. The dark matter density field is estimated us-ing Augmented Lagrangian Perturbation Theory (ALPT; Kitaura& Hess (2013)) which combines the second order perturbation the-ory (2LPT; e.g. see Buchert (1994); Bouchet et al. (1995); Cate-lan (1995)) and spherical collapse approximation (see Bernardeau(1994); Mohayaee et al. (2006); Neyrinck (2013)). The biasingmodel includes deterministic bias and stochastic bias (see Kitauraet al. (2014, 2015) for details). The velocity field is constructedbased on the displacement field of dark matter particles. The mod-eling of finger-of-god has also been taken into account statistically.The mocks match the clustering of the galaxy catalogues for eachredshift bin (see Kitaura et al. (2015b) for details) and have beenused in recent galaxy clustering studies (Cuesta et al. 2015; Gil-Marın et al. 2015a,b; Rodrıguez-Torres et al. 2015; Slepian et al.2015) and void clustering studies (Kitaura et al. 2015a; Liang et al.2015). They are also used in Alam et al. (2016) (BOSS collabo-ration paper for final data release) and its companion papers (thispaper and Ross et al. (2016); Vargas-Magana et al. (2016); Beut-ler et al. (2016a); Satpathy et al. (2016); Beutler et al. (2016b);Sanchez et al. (2016a); Grieb et al. (2016); Sanchez et al. (2016b);Chuang et al. (2016); Slepian et al. (2016a,b); Salazar-Albornozet al. (2016); Zhao et al. (2016); Wang et al. (2016)

3 METHODOLOGY

We develop a new methodology to extract the cosmological con-straints from the joint data set of the Planck CMB data and BOSSgalaxy clustering measurements fitting the LSS data with parametercombinations defining the key cosmological dependencies, whileincluding CMB constraints to simultaneously constrain other pa-rameters. This means that we can define constraints that can sub-sequently be used to constrain a wide-range of Dark Energy mod-els. Similar approaches have been applied to these data separately.Our work is the first to investigate how in detail this joint analysisshould be performed.

3.1 Likelihood from BOSS galaxy clustering

In this section, we describe the steps to compute the likelihood fromthe BOSS galaxy clustering.

3.1.1 Measure Multipoles of the Two-Point Correlation Function

We convert the measured redshifts of the BOSS CMASS andLOWZ galaxies to comoving distances by assuming a fiducialmodel, i.e., flat ΛCDM with Ωm = 0.307115 and h = 0.6777

which is the same model adopted for constructing the mockcatalogues (see Kitaura et al. (2015b) ). To compute the two-dimensional two-point correlation function, we use the two-pointcorrelation function estimator given by Landy & Szalay (1993):

ξ(s, µ) =DD(s, µ)− 2DR(s, µ) +RR(s, µ)

RR(s, µ), (1)

where s is the separation of a pair of objects and µ is the cosineof the angle between the directions between the line of sight (LOS)and the line connecting the pair the objects. DD, DR, and RR rep-resent the normalized data-data, data-random, and random-randompair counts, respectively, for a given distance range. The LOS isdefined as the direction from the observer to the centre of a galaxypair. Our bin size is ∆s = 1h−1Mpc and ∆µ = 0.01. The Landyand Szalay estimator has minimal variance for a Poisson process.Random data are generated with the same radial and angular selec-tion functions as the real data. One can reduce the shot noise due torandom data by increasing the amount of random data. The num-ber of random data we use is about 50 times that of the real data.While calculating the pair counts, we assign to each data point aradial weight of 1/[1+n(z) ·Pw], where n(z) is the radial numberdensity and Pw = 1 · 104 h−3Mpc3 (see Feldman et al. 1994).

The traditional multipoles of the two-point correlation func-tion, in redshift space, are defined by

ξl(s) ≡ 2l + 1

2

∫ 1

−1

dµ ξ(s, µ)Pl(µ),

where Pl(µ) is the Legendre Polynomial (l =0 and 2 here). Weintegrate over a spherical shell with radius s, while actual measure-ments of ξ(s, µ) are done in discrete bins. To compare the measuredξ(s, µ) and our theoretical model, the last integral in Eq.(2) shouldbe converted into a sum,

ξl(s) ≡

∑s−∆s

2<s′<s+ ∆s

2

∑06µ61

(2l + 1)ξ(s′, µ)Pl(µ)

Number of bins used in the numerator, (2)

where ∆s = 5 h−1Mpc in this work.Fig.1 shows the monopole (ξ0) and quadrupole (ξ2) measured

from the BOSS CMASS and LOWZ galaxy sample compared withthe best fit theoretical models.

We are using the scale range s = 40 − 180h−1Mpc and thebin size is 5 h−1Mpc. The data points from the multipoles in thescale range considered are combined to form a vector, X , i.e.,

X = ξ(1)0 , ξ

(2)0 , ..., ξ

(N)0 ; ξ

(1)2 , ξ

(2)2 , ..., ξ

(N)2 ; ..., (3)

where N is the number of data points in each measured multipole;here N = 28. The length of the data vector X depends on thenumber of multipoles used.

3.1.2 Theoretical Two-Point Correlation Function

Following Chuang et al. (2016), companion paper, we use twomodels to compute the likelihood of the galaxy clustering mea-surements. One is a fast model which is used to narrow down theparameters space scanned; the other is a slower model which isused to calibrate the results from the fast model.

Fast model: The fast model we use is the two-dimensionaldewiggle model explained in Chuang et al. (2016), companion pa-per. The theoretical model can be constructed by first and higherorder perturbation theory. We first adopt the cold dark matter model

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-100

-80

-60

-40

-20

0

20

40

60

80

40 60 80 100 120 140 160 180

s2 ξ 0,2

(s)

s (Mpc/h)

Monopole 0.15 < z < 0.43Monopole best fit model

Quadrupole 0.15 < z < 0.43Quadrupole best fit model

-150

-100

-50

0

50

100

40 60 80 100 120 140 160 180

s2 ξ 0,2

(s)

s (Mpc/h)

Monopole 0.43 < z < 0.75Monopole best fit model

Quadrupole 0.43 < z < 0.75Quadrupole best fit model

Figure 1. Left panel: measurement of monopole and quadrupole of the correlation function from the BOSS DR12 LOWZ galaxy sample within 0.15 < z <

0.43 compared to the best fited theoretical models (solid lines). Right panel: measurement of effective monopole and quadrupole of the correlation functionfrom the BOSS DR12 CMASS galaxy sample within 0.43 < z < 0.75 compared to the best fitted theoretical models (solid lines). The error bars are thesquare roots of the diagonal elements of the covariance matrix. In this study, our fitting scale ranges are 40h−1Mpc < s < 180h−1Mpc; the bin size is5h−1Mpc.

and the simplest inflation model (adiabatic initial condition). Com-puting the linear matter power spectra, Plin(k), by using CAMB(Code for Anisotropies in the Microwave Background, Lewis et al.2000) we can decomposed it into two parts:

Plin(k) = Pnw(k) + P linBAO(k), (4)

where Pnw(k) is the “no-wiggle” power spectrum calculated usingEq.(29) from Eisenstein & Hu (1998) and P linBAO(k) is the “wig-gled” part defined by previous Eq. (4). Nonlinear damping effectof the “wiggled” part, in redshift space, is approximated followingEisenstein et al. (2007) by

PnlBAO(k, µk) = P linBAO(k) · exp

(− k2

2k2?

[1 + µ2k(2f + f2)]

),

(5)where µk is the cosine of the angle between k and the LOS, f is thegrowth rate, and k? is computed following Crocce & Scoccimarro(2006) and Matsubara (2008) by

k? =

[1

3π2

∫Plin(k)dk

]−1/2

. (6)

Thus dewiggled power spectrum is

Pdw(k, µk) = Pnw(k) + PnlBAO(k, µk). (7)

We include the linear redshift distortion as follows (reference(Kaiser 1987)),

P sg (k, µk) = b2(1 + βµ2k)2Pdw(k, µk), (8)

where b is the linear galaxy bias and β is the linear redshift distor-tion parameter.

To compute the theoretical two-point correlation function,ξ(s, µ), we Fourier transform the non-linear power spectrumP sg (k, µk) by using Legendre polynomial expansions and one-dimensional integral convolutions as introduced in Chuang & Wang(2013a).

We times calibration functions to the fast model by

ξcal0 (s) = (1− e−ss1 + e

−(ss2

)2

)ξ0(s), (9)

ξcal2 (s) = (1− e−ss3 + e

−(ss4

)2

)ξ2(s), (10)

so that it mimics the slow model presented bellow. We find thecalibration parameters, s1 = 12, s2 = 14, s3 = 20, and s4 = 27,by comparing the fast and slow models by visual inspection. Itis not critical to find the best form of calibration function and itsparameters as the model will be callibrated later when performingimportance sampling with slow model.

Slow model: The slower but accurate model we use is ”Gaus-sian streaming model” described in Reid & White (2011). Themodel assumes that the pairwise velocity probability distributionfunction is Gaussian and can be used to relate real space cluster-ing and pairwise velocity statistics of halos to their clustering inredshift space by

1 + ξsg(rσ, rπ) =

∫ [1 + ξrg(r)

]e−[rπ−y−µv12(r)]2/2σ2

12(r,µ) dy√2πσ2

12(r, µ),

(11)where rσ and rπ are the redshift space transverse and LOS dis-tances between two objects with respect to the observer, y is thereal space LOS pair separation, µ = y/r, ξr

g is the real spacegalaxy correlation function, v12(r) is the average infall velocity ofgalaxies separated by real-space distance r, and σ2

12(r, µ) is the rmsdispersion of the pairwise velocity between two galaxies separatedwith transverse (LOS) real space separation rσ (y). ξr

g(r), v12(r)and σ2

12(r, µ) are computed in the framework of Lagrangian (ξr)and standard perturbation theories (v12, σ2

12).For large scales, only one nuisance parameter is necessary

to describe the clustering of a sample of halos or galaxies in thismodel: b1L = b − 1, the first-order Lagrangian host halo biasin real space. In this study, we consider relative large scales (i.e.

c© 0000 RAS, MNRAS 000, 000–000


40 < s < 180h−1Mpc), so that we do not include σ2FoG, to model

a velocity dispersion accounting for small-scale motions of halosand galaxies. Further details of the model, its numerical implemen-tation, and its accuracy can be found in Reid & White (2011).

3.1.3 Covariance Matrix

We use the 2000 mock catalogues created by Kitaura et al. 2015bfor the BOSS DR12 CMASS and LOWZ galaxy sample to estimatethe covariance matrix of the observed correlation function. We cal-culate the multipoles of the correlation functions of the mock cata-logues and construct the covariance matrix as

Cij =1

(N − 1)(1−D)

N∑k=1

(Xi −Xki )(Xj −Xk

j ), (12)

where

D =Nb + 1

N − 1, (13)

N is the number of the mock catalogues, Nb is the number of databins, Xm is the mean of the mth element of the vector from themock catalogue multipoles, and Xk

m is the value in the mth ele-ments of the vector from the kth mock catalogue multipoles. Thedata vector X is defined by Eq.(3). We also include the correction,D, introduced by Hartlap et al. (2007).

3.1.4 Compute Likelihood from Galaxy Clustering

The likelihood is taken to be proportional to exp(−χ2/2) (B.P.1992), with χ2 given by

χ2 ≡NX∑i,j=1

[Xth,i −Xobs,i]C−1ij [Xth,j −Xobs,j ] (14)

where NX is the length of the vector used, Xth is the vector fromthe theoretical model, and Xobs is the vector from the observeddata.

As explained in Chuang & Wang (2012), instead of recalcu-lating the observed correlation function while computing for differ-ent models, we rescale the theoretical correlation function to avoidrendering the χ2 values arbitrary. This approach can be consideredas an application of Alcock-Paczynski effect (Alcock & Paczynski1979). The rescaled theoretical correlation function is computed by

T−1(ξth(σ, π)) = ξth

(DA(z)

DfidA (z)

σ,Hfid(z)

H(z)π

), (15)

where ξth is the theoretical model computed in Sec. 3.1.2. Here,DA(z) and H(z) would be the input parameters and Dfid

A (z) andHfid(z) are 990.20Mpc, 80.16 km s−1 Mpc−1 at z = 0.32(LOWZ) and 1409.26Mpc, 94.09 km s−1 Mpc−1 at z = 0.59(CMASS). Then, χ2 can be rewritten as

χ2 ≡NX∑i,j=1

T−1Xth,i −Xfid

obs,i

C−1fid,ij ·

·T−1Xth,j −Xfid

obs,j

; (16)

where T−1Xth is the vector computed by eq. (2) from the rescaledtheoretical correlation function, eq. (15). Xfid

obs is the vector fromobserved data measured with the fiducial model (see Chuang &Wang 2012 for more details regarding the rescaling method).

-3000

-2000

-1000

0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500

Cl

l

Planck15 TTbest fit model for TT

Planck15 TE times 20best fit model for TE

Planck15 EE times 50best fit model for EE

Figure 2. Angular power spectrum of temperature and polarization mea-surement from Planck data and their best fits from our double probe analy-sis.

3.2 Likelihood from Planck CMB data

Our CMB data set consists of the Planck 2015 measurements(Planck Collaboration I 2015; Planck Collaboration XIII 2015).The reference likelihood code (Planck Collaboration XI 2015) wasdownloaded from the Planck Legacy Archive5. Here we combinethe Plik baseline likelihood for high multipoles (30 6 ` 6 2500)using the TT, TE and EE power spectra, and the Planck low-` mul-tipole likelihood in the range 2 6 ` 6 29 (hereafter lowTEB). Wealso include the new Planck 2015 lensing likelihood (Planck Col-laboration XV 2015), constructed from the measurements of thepower spectrum of the lensing potential (hereafter referred as ”lens-ing”). We using the Planck lensing likelihood, the Alens parameteris always set to 1 (Planck Collaboration XIII 2015).

3.3 Markov Chain Monte-Carlo Likelihood Analysis

3.3.1 basic procedure

We perform Markov Chain Monte Carlo (MCMC) likelihood anal-yses using CosmoMC (Lewis & Bridle 2002; Lewis 2013). Thefiducial parameter space that we explore spans the parameter setof Ωch2, Ωbh

2, ns, log(As), θ, τ , Ωk, w,H(z), DA(z), β(z),bσ8(z), b(z). The quantities Ωc and Ωb are the cold dark mat-ter and baryon density fractions, ns is the power-law index ofthe primordial matter power spectrum, Ωk is the curvature den-sity fraction, w is the equation state of dark energy, h is the di-mensionless Hubble constant (H0 = 100h km s−1Mpc−1), andσ8(z) is the normalization of the power spectrum. Note that, withthe joint data set (Planck + BOSS), the only parameter which isnot well constrained is b(z). We apply a flat prior of (1, 3) onit. The linear redshift distortion parameter can be expressed asβ(z) = f(z)/b. Thus, one can derive f(z)σ8(z) from the mea-sured β(z) and bσ8(z).

5 PLA: http://pla.esac.esa.int/

c© 0000 RAS, MNRAS 000, 000–000

http://pla.esac.esa.int/


3.3.2 Generate Markov chains with fast model

We first use the fast model (2D dewiggle model) to compute thelikelihood, Lfast and generate the Markov chains. The MonteCarlo analysis will go through many random steps keeping orthrowing the computed points parameter space according to theMarkov likelihood algorithm. Eventually, it will provide the chainsof parameter points with high likelihood describing the constraintsto our model.

3.3.3 Calibrate the likelihood using slow model

Once we have the fast model generated chains, we modify theweight of each point by

Wnew =WoldLslowLfast

, (17)

where Lslow and Lfast are the likelihood for given point of inputparameters in the chains. We save time by computing only the”important” points without computing the likelihood of the oneswhich will not be included in the first place. The methodology isknow as ”Importance sampling”. However, the typical Importancesampling method is to add likelihood of some additional data setto the given chains, but in this study, we replace the likelihood of adata set.

4 DOUBLE PROBE RESULTS

The 2-point statistic of galaxy clustering can be summarizedby Ωmh2, H(z), DA(z), f(z)σ8(z) (e.g. Chuang & Wang(2013a)). In some studies, Ωmh

2 was not included since a strongprior had been applied. Instead of using H(z) and DA(z), oneuses the derived parameters H(z)rs/rs,fid and DA(z)rs,fid/rsto summarize the cosmological information since these two quan-tities are basically uncorrelated to Ωmh

2, where rs is the soundhorizon at the redshift of the drag epoch and rs,fid is the rs of thefiducial cosmology. In this study, Ωmh

2 is well constrained by thejoint data set but we still use H(z)rs/rs,fid and DA(z)rs,fid/rsbecause they have tighter constraints.

Wang & Mukherjee (2007) showed that CMB shift parameters(la, R), together with Ωbh

2, provide an efficient and intuitive sum-mary of CMB data as far as dark energy constraints are concerned.It is equivalent to replace Ωbh

2 with z∗, the redshift to the photon-decoupling surface (Wang 2009). The CMB shift parameters aredefined as (Wang & Mukherjee 2007):

R ≡√

ΩmH20 r(z∗), (18)

la ≡ πr(z∗)/rs(z∗), (19)

and z? is the redshift to the photon-decoupling surface given byCAMB (Lewis et al. 2000)

The angular comoving distance to an object at redshift z isgiven by:

r(z) = cH−10 |Ωk|−1/2sinn[|Ωk|1/2 Γ(z)], (20)

which has simple relation with the angular diameter distanceDA(z) = r(z)/(1 + z).

In additional to the shift parameters, we include also the scalarindex and amplitude of the power law primordial fluctuation ns andAs to summarize the CMB information.

From the measured parameters Ωch2, Ωbh2, ns, log(As),

fσ8(0.59) 0.510± 0.047H(0.59)rs/rs,fid 97.9± 3.1

DA(0.59)rs,fid/rs 1422± 25

fσ8(0.32) 0.431± 0.063H(0.32)rs/rs,fid 79.1± 3.3

DA(0.32)rs,fid/rs 956± 27

R 1.7430± 0.0080la 301.70± 0.15

Ωbh2 0.02233± 0.00025

ns 0.9690± 0.0066ln(1010As) 3.040± 0.036

Ωk −0.003± 0.006

Table 1. Fiducial result of the double-probe approach. The units of H(z)and DA(z) are km s−1 Mpc−1 and Mpc.

θ, τ , Ωk, w,H(z), DA(z), β(z), bσ8(z), b(z), we derive theparameters R, la, Ωbh

2, ns, log(1010As), Ωk, H(z)rs/rs,fid,DA(z)rs,fid/rs, f(z)σ8(z) to summarize the joint data set ofPlanck and BOSS galaxy sample. Table 1 and 2 show the measuredvalues and their normalized covariance. A normalized covariancematrix is defined by

Nij =Cij√CiiCjj

, (21)

where Cij is the covariance matrix.To conveniently compare with other measurements using

CMASS sample within 0.43 < z < 0.7 (we are using0.43 < z < 0.75), we extrapolated our measurements atz = 0.57: H(0.57)rs/rs,fid = 96.7 ± 3.1 km s−1 Mpc−1 andDA(0.57)rs,fid/rs = 1405 ± 25Mpc (see Table 9 of Alam et al.2016).

5 CONSTRAIN PARAMETERS OF GIVEN DARKENERGY MODELS WITH DOUBLE-PROBE RESULTS

In this section, we describe the steps to combine our results withother data sets assuming some dark energy models. For a givenmodel and cosmological parameters, one can compute R, la,Ωbh

2, ns, log(1010As), Ωk, H(z)rs/rs,fid, DA(z)rs,fid/rs,f(z)σ8(z). one can take the covariance matrices, Mij,CMB+galaxy,of these 12 parameters (galaxy sample are divided in two redshiftbins). Then, χ2

CMB+galaxy can be computed by

χ2CMB+galaxy = ∆CMB+galaxyM

−1ij,CMB+galaxy∆CMB+galaxy, (22)

where

∆CMB+galaxy =

fσ8(0.59)−0.510

H(0.59)rs/rs,fid−97.9

DA(0.59)rs,fid/rs−1422

fσ8(0.32)−0.431

H(0.32)rs/rs,fid−79.1

DA(0.32)rs,fid/rs−956

R−1.7430

la−301.70

Ωbh2−0.02233

ns−0.9690

ln(1010As)−3.040

Ωk−0.003

, (23)

where the angular diameter distance DA(z) is given by:

DA(z) = (1 + z)cH−10 |Ωk|−1/2sinn[|Ωk|1/2 Γ(z)], (24)

c© 0000 RAS, MNRAS 000, 000–000


R la Ωbh2 ns ln(1010As) fσ8(0.59)

H(0.59)rs,fid/rs

DA(0.59)

rs/rs,fidfσ8(0.32)

H(0.32)rs,fid/rs

DA(0.32)

rs/rs,fidΩk

R 1.0000 0.6534 -0.7271 -0.8787 -0.0352 -0.0620 -0.1675 -0.0059 -0.0237 -0.0271 0.0027 0.6349la 0.6534 1.0000 -0.5212 -0.5770 -0.0651 -0.1067 -0.1957 0.0017 0.0073 0.0174 -0.0211 0.4329

Ωbh2 -0.7271 -0.5212 1.0000 0.6633 0.1175 0.0525 0.0822 0.0333 0.1373 0.0566 0.0321 -0.4070

ns -0.8787 -0.5770 0.6633 1.0000 0.0808 0.0381 0.1648 -0.0003 0.0285 0.0510 0.0303 -0.5547ln(1010As) -0.0352 -0.0651 0.1175 0.0808 1.0000 0.0034 0.0391 0.0175 -0.0066 0.0020 0.0516 0.5915fσ8(0.59) -0.0620 -0.1067 0.0525 0.0381 0.0034 1.0000 0.7153 0.6172 0.1531 0.1535 -0.0333 0.0252

H(0.59)rs/rs,fid -0.1675 -0.1957 0.0822 0.1648 0.0391 0.7153 1.0000 0.4168 0.0447 0.0968 -0.0388 -0.0959DA(0.59)rs,fid/rs -0.0059 0.0017 0.0333 -0.0003 0.0175 0.6172 0.4168 1.0000 0.0209 -0.0319 -0.0839 0.0038

fσ8(0.32) -0.0237 0.0073 0.1373 0.0285 -0.0066 0.1531 0.0447 0.0209 1.0000 0.6581 0.5250 0.1142H(0.32)rs/rs,fid -0.0271 0.0174 0.0566 0.0510 0.0020 0.1535 0.0968 -0.0319 0.6581 1.0000 0.3168 0.1165DA(0.32)rs,fid/rs 0.0027 -0.0211 0.0321 0.0303 0.0516 -0.0333 -0.0388 -0.0839 0.5250 0.3168 1.0000 0.0835

Ωk 0.6349 0.4329 -0.4070 -0.5547 0.5915 0.0252 -0.0959 0.0038 0.1142 0.1165 0.0835 1.0000

Table 2. Normalized covariance matrix of the fiducial result from the double-probe approach.

where Γ(z) =

∫ z

0

dz′

E(z′), and E(z) = H(z)/H0,

and sinn(x) = sin(x), x, sinh(x) for Ωk < 0, Ωk = 0, andΩk > 0 respectively; and the expansion rate the universe H(z) isgiven by

H(z) =

H0

√Ωm(1 + z)3 + Ωr(1 + z)4 + Ωk(1 + z)2 + ΩXX(z), (25)

where Ωm + Ωr + Ωk + ΩX = 1, and the dark energy densityfunction X(z) is defined as

X(z) ≡ ρX(z)

ρX(0). (26)

f is defined in relation to the linear growth factorD(τ) in the usualway as

f =d lnD(τ)

d ln a=

1

Hd lnD(τ)

dτ, (27)

where D is the growing solution to the second order differentialequation writen in comoving coordinates

d2D(τ)

dτ2+HdD(τ)

dτ=

3

2Ωm(τ)H2(τ)D(τ). (28)

We will be writing σ(z,R) as:

σ2(z,R) =1

(2π)3

∫d3kW 2(kR)P (k, z) (29)

with

W (kR) =3

(kR)3[sin(kR)− kR cos(kR)] (30)

being the top-hat window function. Thus

σ8(z) = σ(z,R = 8Mpc/h). (31)

In this way, one just need to compute linear theory to get χ2CMB+galaxy

to reproduce and combine CMB plus galaxy information. Theseequations assume no impact from massive neutrinos, mainly work-ing for the cases of massless or approximately massless neutrinos.When including neutrino species with a given mass one needs tosolve the full Boltzmann hierarchy as shown in Ma & Bertschinger(1995); Lewis & Challinor (2002).

Table 3 lists the constraints on the parameters of different darkenergy models obtained using our double-probe measurements.The results show no tension with the flat ΛCDM cosmologicalparadigm.

6 FULL-LIKELIHOOD ANALYSIS FIXING DARKENERGY MODELS

To validate our double-probe methodology, we perform the full-likelihood MCMC analyses with fixing dark energy models. Themain difference of this approach comparing our double-probe anal-ysis is that it has been given a dark energy model at first place.Opposite to the double probe approach, one cannot use the resultsfrom the full-likelihood analysis to derive the constraints for the pa-rameters of other dark energy models. Since the dark energy modelis fixed, the quantities, H(z), DA(z), β(z), bσ8(z), would bedetermined by the input parameters, Ωch2, Ωbh

2, ns, log(As),θ, τ , Ωk, w, as shown in Eq. 24, 25, 27, and 31. We show theresults in Table 4. In Fig. 3, 4 and 5, we compare these resultswith our double-probe approach and the single-probe approach(Chuang et al. (2016); companion paper). We find very good agree-ment among these three approaches. Note that deriving the darkenergy model constraints from our double-probe measurements ismuch faster than the full run. For example, using the same ma-chine, it takes ∼ 2.5 hours to obtain the constraints for ΛCDM us-ing double-probe measurements, but takes 6 days to reach similarconvergence for the full likelihood MCMC analysis (slower with afactor of 60).

Up to this point we have introduced two methodologies forextracting cosmological information, the double-probe method anda full likelihood analysis. Moreover, we are comparing these resultswith a third methodology already introduced in Chuang et al. 2016also called single-probe analysis combined with CMB. We showhere motivations for the use of each of them:

• Double-probe: Joint fit to LSS data and CMB constrainingthe full set of cosmological parameters without the need of extraknowledge on the priors. This methodology allow us to test on theprior information content assumed by other probes and give us thetool to have a dark energy independent measurements from LSSand CMB combined.• Full fit: Fit of cosmological parameter set to LSS and CMB

data, requiring an assumption of a dark energy model (i.e. not go-ing through DA, H and fσ8 as intermediate parameters) from thebeginning. This methodology provides a tool to check the informa-tion content of the data and we take it to be the answer to recoverfrom other methodologies as it does not have extra assumptionsappart from the dark energy model.• Single-probe+CMB: Likelihoods are determined from the

BOSS measurements of DArfids /rs, Hrs/rfids , fσ8, Ωmh2 to-

gether with Planck data. This methodology provides, in its firststep, measurements of large scale structure independent of CMB

c© 0000 RAS, MNRAS 000, 000–000


Ωm H0 σ8 Ωk w or w0 waΛCDM 0.304± 0.009 68.2± 0.7 0.806± 0.014 0 −1 0

oΛCDM 0.303± 0.010 68.6± 0.9 0.810± 0.015 0.002± 0.003 −1 0

wCDM 0.299± 0.013 69.0± 1.5 0.815± 0.020 0 −1.04± 0.06 0owCDM 0.302± 0.014 68.7± 1.5 0.811± 0.021 0.002± 0.003 −1.00± 0.07 0

w0waCDM 0.313± 0.020 67.6± 2.0 0.817± 0.016 0 −0.84± 0.22 −0.66± 0.68

ow0waCDM 0.313± 0.020 67.6± 2.2 0.815± 0.016 0.000± 0.004 −0.85± 0.24 −0.61± 0.80

Table 3. Constraints on cosmological parameters obtained by using our results assuming dark energy models (see Sec. 5).

data, thus showing as a good tool to test possible tensions betweendata sets.

7 MEASUREMENTS OF NEUTRINO MASS

In this section, we will focus on measuring the sum of the neu-trino mass Σmν using different methodologies described in previ-ous sections. First, we repeat the double-probe analysis describedin Sec. 3.3 with an additional free parameter, Σmν , and presentthe constraints on cosmological parameters. Second, we repeat theMCMC analysis with the full likelihood of joint data set describedin Sec. 6 and find that the full shape measurement of the monopoleof the galaxy 2-point correlation function introduces some detec-tion of neutrino mass. However, since the monopole measurementis sensitive to the observational systematics, we provide another setof cosmological constraints by removing the full shape informa-tion. Third, we also obtain the constraint of Σmν using the singleprobe measurement provided by Chuang et al. (companion paper).

7.1 measuring neutrino mass using double probe

Note first that for the study ofmν , we replaceR =√

ΩmH20 r(z∗)

with Ωbch2 = Ωbh

2 + Ωch2 (e.g. see Aubourg et al. (2015)),

since R depends directly on Ων . Thus, we use the following set ofparameters from the double probe analysis while measuring neu-trino mass, Ωbch2, la, Ωbh

2, ns, log(As), Ωk, H(z), DA(z),f(z)σ8(z).

We repeat the analysis described in Sec. 3.3, but here we setΣmν , to be free instead of setting it to 0.06 eV. The results areshown in Table 5 and 6.

As described in Sec. 5, one can constrain the parameters ofgiven dark energy models using Table 5 and 6. Table 7 presentsthe cosmological parameter constraints assuming some simple darkenergy models. Figure 6 shows the probability density for Σmν fordifferent dark energy models. Our measurements of Σmν usingdouble probe approach are consistent with zero. The upper limit(68% confidence level) varys from 0.1 to 0.35 eV depending ondark energy model.

In addition, we also derive the cosmological constraints byusing the results with fixed Σmν , i.e. Table 1 and 2 with R re-placed by Ωbch

2. Different from Table 3 (see Sec. 5), we includeΣmν as one of the parameters to be constrained. The results areshown in Table 8. We find that the results are very similar to Ta-ble 7, which showing our double probe measurements are insensi-tive to the Σmν assumption. Fig. 7 shows this point in a clear wayby comparing the 2D contours when including a covariance matrixvarying Σmν (using Table 5 and 6) or fixing Σmν (using Table 1and 2). We see that they lie on top of each other. Moreover, Fig.7 also exhibit the constraint given by fσ8 on the Σmν and w pa-rameters. We find the constraint on w become tighter while that inΣmν stays the same when including the fσ8 constraint. This is a

fσ8(0.59) 0.495± 0.051

H(0.59)rs/rs,fid 97.5± 3.2

DA(0.59)rs,fid/rs 1419± 27fσ8(0.32) 0.431± 0.066

H(0.32)rs/rs,fid 78.9± 3.6

DA(0.32)rs,fid/rs 964± 26Ωbch

2 0.1413± 0.0022

la 301.75± 0.14

Ωbh2 0.02209± 0.00025

ns 0.9639± 0.0068

ln(1010As) 3.062± 0.040Ωk −0.009± 0.006

Table 5. Results of double-probe analysis obtained with varying Σmν . Theunits of H(z) and DA(z) are km s−1 Mpc−1 and Mpc (see Sec. 7.1).

good news for future experiments as their power on the neutrinoconstraint would not highly rely on the growth rate measurementswhich are more sensitive to the observational systematics.

Furthermore, we have also checked the impact of adding su-pernovae Ia (SNIa) data, dubbed Joint Light-curve Analysis (JLA)(Betoule et al. 2014) and find that the upper limit of Σmν de-crease because SNIa breaks the degeneracy of the constraint fromPlanck+BOSS (see Fig. 8). In this way, we can get tighter con-straints on the upper limit by including SNIa data.

7.2 measuring neutrino mass using full likelihood analysis

We perform the same full MCMC analysis using the joint-full-likelihood of Planck and BOSS data as described in Sec. 6 to obtainthe cosmological parameter constraints including Σmν . Table 9presents the results. We show also the probability density for Σmν

in Fig. 9. We find more than 2 σ detection of non zero Σmν as-suming the models without fixingw to be -1. However, we find thatthe detection actually mainly comes from the monopole of galaxycorrelation function which is sensitive to some observational sys-tematics, e.g. see Ross et al. (2012); Chuang et al. (2013b). Fig. 10shows that the Σmν detection decreases when adding a polynomialto remove the full shape information of monopole. To be conserva-tive, we run again the full MCMC analysis to obtain the constrainton Σmν without including the full shape information and the re-sults are presented in Table 10. The probability density for Σmν

is shown in Fig. 11. One can see the detections of Σmν decrease.In addition, the upper limits in Fig. 11 are lower than Fig. 6 whichare expected. Since we do not include the parameter Σmν whensummarising the information of double probe, the Σmν constraintfrom Planck is lost.

Table 11 displays the constraints measured when allowing theCMB lensing amplitude parameter AL to vary. Fig. 13 shows thePlanck data shifts Σmν measurement to higher values allowinga higher detection from the combined data analysis when allow-ing AL free. Thus, we find again ∼ 2 σ detection even without

c© 0000 RAS, MNRAS 000, 000–000


Ωm H0 σ8 Ωk w or w0 waΛCDM 0.305± 0.008 68.0± 0.6 0.812± 0.009 0 −1 0

oΛCDM 0.300± 0.009 68.6± 1.0 0.816± 0.010 0.001± 0.003 −1 0

wCDM 0.298± 0.015 68.8± 1.8 0.818± 0.017 0 −1.02± 0.07 0owCDM 0.298± 0.017 68.8± 1.8 0.818± 0.018 0.001± 0.003 −1.01± 0.08 0

w0waCDM 0.311± 0.022 67.4± 2.3 0.808± 0.020 0 −0.85± 0.23 −0.51± 0.67

ow0waCDM 0.309± 0.025 67.8± 3.0 0.810± 0.024 0.000± 0.004 −0.86± 0.26 −0.50± 0.73

Table 4. Constraints on cosmological parameters from full-likelihood MCMC analysis of the joint data set (see Sec. 6).

0.275 0.300 0.325 0.350

Ωm

66.0

67.5

69.0

70.5

72.0

H0

Planck

Single Probe

Double Probe

Full Run

0.25 0.30 0.35 0.40

Ωm

0.030

0.015

0.000

0.015

ΩK

Planck

Single Probe

Double Probe

Full Run

Figure 3. Left panel: 2D marginalized contours for 68% and 95% confidence levels for Ωm and H0 (ΛCDM model assumed) from Planck-only (gray),derived using double probe measurements (blue), full -likelihood analysis with joint data (red; labeled as ”Full Run”), and Planck+single probe measurements(green). Right panel: 2D marginalized contours for 68% and 95% confidence level for Ωm and Ωk (oΛCDM model assumed). One can see that the latterthree measurements are consistent with each other.

accounting for the full shape of the monopole from the correlationfunction.

7.3 measure neutrino mass using measurements from singleprobe analysis

We use the single probe measurement provided by Chuang et al.(companion paper) combining with Planck (fixing AL = 1) andobtain the constraint of Σmν . Table 12 shows the cosmological pa-rameter constraints including Σmν for different dark energy mod-els. The probability densities for Σmν are shown in Fig. 14. Onecan see that it is consistent with Fig. 11. We have checked that therewould be some detection of neutrino mass while allowing AL to befree as seen in the case of full-likelihood analysis (see Sec. 7.2).

Fig. 15 presents the comparison between the three differentmethodologies. The three approaches agree very well with somesubtle differences. One can see that the constraint on Σmν fromthe double probe approach is weaker which is expected. The dif-ference comes from the fact that we do not include Σmν into oursummarized set of parameters, so that information from Planck islost. On the other hand, both single probe and full-likelihood anal-ysis include full Planck information and their measurements arevery similar.

7.4 combination with supernovae type Ia data

We combine our measurements using the full likelihood approachwith those from supernovae Ia (SNIa) data, Joint Light-curve Anal-ysis (JLA) (Betoule et al. 2014). As seen in Fig. 8, SN data breakssome degeneracies providing tighter constraints on Σmν . Resultscan be found in table 13 and Fig. 16) for the case of fixing AL = 1and table 14 and Fig. 17) for the case of varying AL. When addingSN1a data, we get tighter upper limits, e.g. Σmν < 0.12 againstΣmν < 0.14 in ΛCDM with AL = 1. We point out that the con-straints we obtained are still not sufficient to distinguish betweennormal and inverted hierarchy.

8 SUMMARY

In this work we have studied and compared three different ways ofextracting cosmological information from the combined data setsof Planck2015 and BOSS final data release (DR12) having greatcare in avoiding imposing priors on cosmological parameters whencombining these data.

First, we have extracted the dark-energy-model-independentcosmological constraints from the joint data sets of Baryon Oscilla-tion Spectroscopic Survey (BOSS) galaxy sample and Planck cos-

c© 0000 RAS, MNRAS 000, 000–000


0.2 0.3 0.4 0.5

Ωm

2.0

1.6

1.2

0.8

w

Planck

Single Probe

Double Probe

Full Run

0.04 0.02 0.00 0.02

ΩK

2.0

1.5

1.0

0.5

w

Planck

Single Probe

Double Probe

Full Run

Figure 4. Left panel: 2D marginalized contours for 68% and 95% confidence level for Ωm and w (wCDM model assumed) from Planck-only (gray),derived using double probe measurements (blue), full -likelihood analysis with joint data (red; labeled as ”Full Run”), and Planck+single probe measurements(green). Right panel: 2D marginalized contours for 68% and 95% confidence level for Ωk and w (owCDM model assumed). One can see that the latter threemeasurements are consistent with each other.

2.0 1.5 1.0 0.5

w0

2

1

0

1

2

wa

Planck

Single Probe

Double Probe

Full Run

0.04 0.02 0.00

ΩK

2.4

1.8

1.2

0.6

w0

Planck

Single Probe

Double Probe

Full Run

Figure 5. Left panel: 2D marginalized contours for 68% and 95% confidence level for w0 and wa (w0waCDM model assumed) from Planck-only (gray),derived using double probe measurements (blue), full -likelihood analysis with joint data (red; labeled as ”Full Run”), and Planck+single probe measurements(green). Right panel: 2D marginalized contours for 68% and 95% confidence level for Ωk and w0 (ow0waCDM model assumed). One can see that the latterthree measurements are consistent with each other.

mic microwave background (CMB) measurement. We measure themean values and covariance matrix of R, la, Ωbh

2, ns, log(As),Ωk, H(z), DA(z), f(z)σ8(z), which give an efficient summaryof Planck data and 2-point statistics from BOSS galaxy sample (seeTable 1). We called this methodology as ”double-probe” approach

since it combines two data sets to minimize the priors needed forthe cosmological parameters. We found that double probe measure-ments are insensitive to the assumption of neutrino mass (fixed ornot). But, the parameter R should be replaced by Ωbch

2 while hav-ing Σmν to be free.

c© 0000 RAS, MNRAS 000, 000–000


Ωbch2 la Ωbh

2 ns ln(1010As) fσ8(0.59)H(0.59)rs,fid/rs

DA(0.59)

rs/rs,fidfσ8(0.32)

H(0.32)rs,fid/rs

DA(0.32)

rs/rs,fidΩk

Ωbch2 1.0000 0.4607 -0.6377 -0.8376 0.0145 0.0075 0.0536 0.0672 -0.0870 0.0317 0.0049 0.3794

la 0.4607 1.0000 -0.4977 -0.5042 -0.0470 0.0201 -0.0525 0.0043 -0.0216 0.0765 0.0912 0.2919Ωbh

2 -0.6377 -0.4977 1.0000 0.7188 -0.0241 -0.0016 -0.0625 -0.0879 0.0692 0.0299 0.0149 -0.2708ns -0.8376 -0.5042 0.7188 1.0000 0.0475 -0.0131 -0.0591 -0.0499 0.0717 0.0268 -0.0686 -0.2894

ln(1010As) 0.0145 -0.0470 -0.0241 0.0475 1.0000 0.0095 -0.0352 -0.0065 0.0773 0.0225 0.0053 0.5576fσ8(0.59) 0.0075 0.0201 -0.0016 -0.0131 0.0095 1.0000 0.6546 0.5223 0.2427 0.2074 0.0634 0.1538

H(0.59)rs/rs,fid 0.0536 -0.0525 -0.0625 -0.0591 -0.0352 0.6546 1.0000 0.3777 0.0586 0.0615 0.0015 -0.0025DA(0.59)rs,fid/rs 0.0672 0.0043 -0.0879 -0.0499 -0.0065 0.5223 0.3777 1.0000 -0.0598 0.0272 -0.0474 -0.0578

fσ8(0.32) -0.0870 -0.0216 0.0692 0.0717 0.0773 0.2427 0.0586 -0.0598 1.0000 0.6531 0.4819 0.1487H(0.32)rs/rs,fid 0.0317 0.0765 0.0299 0.0268 0.0225 0.2074 0.0615 0.0272 0.6531 1.0000 0.1686 0.1165DA(0.32)rs,fid/rs 0.0049 0.0912 0.0149 -0.0686 0.0053 0.0634 0.0015 -0.0474 0.4819 0.1686 1.0000 0.0049

Ωk 0.3794 0.2919 -0.2708 -0.2894 0.5576 0.1538 -0.0025 -0.0578 0.1487 0.1165 0.0049 1.0000

Table 6. Correlation matrix of the double-probe measurements obtained with varying Σmν (corresponding to Table 5; see Sec. 7.1).

Ωm H0 σ8 Ωk w or w0 wa Σmµ(eV)ΛCDM 0.310± 0.010 67.6± 0.8 0.828± 0.019 0 −1 0 < 0.10 (< 0.22)

oΛCDM 0.310± 0.011 67.8± 1.0 0.828± 0.020 0.002± 0.003 −1 0 < 0.13 (< 0.27)

wCDM 0.296± 0.016 69.6± 1.9 0.824± 0.027 0 −1.11± 0.10 0 < 0.26 (< 0.52)owCDM 0.297± 0.017 69.8± 2.2 0.816± 0.033 0.001± 0.004 −1.13± 0.12 0 < 0.35 (< 0.75)

w0waCDM 0.312± 0.024 68.1± 2.6 0.812± 0.030 0 −0.88± 0.24 −0.89± 0.75 < 0.32 (< 0.60)

ow0waCDM 0.310± 0.026 68.3± 3.3 0.809± 0.034 −0.001± 0.004 −0.91± 0.29 −0.83± 0.87 < 0.31 (< 0.78)

Table 7. Constraints on cosmological parameters obtained by using the double-probe measurements presented in Table 5 and 6 assuming dark energy models.We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrino masses in the parentheses. The units ofH0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.1 and Fig. 6).

0.0 0.4 0.8 1.2 1.6

Σmν

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oCDM

wCDM

owCDM

w0waCDM

ow0waCDM

Figure 6. Probability density for Σmν from double-probe measurementsusing the covariance matrix with free parameter Σmν (see Sec. 7.1 andTable 7).

Second, we performed the full-likelihood-analysis from thejoint data set of Planck and BOSS assuming some simple darkenergy models. By comparing these results with the ones fromdouble-probe approach, we have demonstrated that the double-probe approach provides robust cosmological parameter constraintswhich can be conveniently used to study dark energy models. Us-ing our results, we obtain Ωm = 0.304± 0.009, H0 = 68.2± 0.7,

0.0 0.2 0.4 0.6 0.8

Σmν

1.75

1.50

1.25

1.00

w

fix mν, no fσ8

vary mν, no fσ8

fix mν

vary mν

Figure 7. Comparison of 2D contours for 68% and 95% confidence levelon Σmν and w from the double probe methodology using covariance ma-trix from first step varying and fixing neutrinos. One can see that the con-straints are insensitive to the assumption of Σmν . We also show the resultsfrom double probe measurement excluding f(z)σ8(z). One can see thatf(z)σ8(z) improve the constraint on w but not Σmν .

and σ8 = 0.806 ± 0.014 assuming ΛCDM; Ωk = 0.002 ± 0.003assuming oCDM; w = −1.04 ± 0.06 assuming wCDM; Ωk =0.002 ± 0.003 and w = −1.00 ± 0.07 assuming owCDM; andw0 = −0.84±0.22 andwa = −0.66±0.68 assumingw0waCDM.The results show no tension with the flat ΛCDM cosmologicalparadigm. Note that deriving the dark energy model constraints

c© 0000 RAS, MNRAS 000, 000–000



oΛCDM 0.307± 0.010 68.2± 0.9 0.796± 0.021 0.003± 0.003 −1 0 < 0.19 (< 0.37)

wCDM 0.295± 0.014 69.5± 1.8 0.798± 0.023 0 −1.10± 0.10 0 < 0.27 (< 0.53)owCDM 0.296± 0.015 70.1± 2.3 0.781± 0.033 0.003± 0.004 −1.13± 0.14 0 < 0.45 (< 0.91)

w0waCDM 0.307± 0.020 68.5± 2.3 0.782± 0.028 0 −0.92± 0.22 −0.77± 0.73 < 0.39 (< 0.63)

ow0waCDM 0.302± 0.021 69.4± 2.8 0.775± 0.034 0.002± 0.004 −1.01± 0.28 −0.53± 0.88 < 0.47 (< 0.93)

Table 8. Constraints on cosmological parameters obtained by using our double-probe measurements obtained with fixed Σmν assuming dark energy models.We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrino masses in the parentheses. The units ofH0

and Σmν are km s−1 Mpc−1 and eV respectively. One can see that the results are very similar to Table 7, which showing our double probe measurementsare insensitive to the Σmν assumption


oΛCDM 0.313± 0.013 67.9± 1.1 0.792± 0.020 0.004± 0.004 −1 0 0.25+0.13−0.17 (< 0.49)

wCDM 0.293± 0.016 70.1± 2.0 0.808± 0.019 0 −1.15± 0.11 0 0.30+0.17−0.14 (< 0.52)

owCDM 0.299± 0.019 70.0± 2.4 0.795± 0.021 0.004± 0.004 −1.14± 0.13 0 0.40+0.17−0.17

(+0.34−0.33

)w0waCDM 0.316± 0.023 67.8± 2.5 0.785± 0.023 0 −0.87± 0.23 −0.96± 0.68 0.36+0.17

−0.15

(+0.26−0.29

)ow0waCDM 0.313± 0.026 68.4± 2.8 0.787± 0.027 0.002± 0.004 −0.91± 0.26 −0.82± 0.77 0.39+0.15

−0.15

(+0.32−0.32

)Table 9. Constraints on cosmological parameters from the full-likelihood-analysis of the joint data set. Σmν is one of the parameters to be constrained. Planckdata includes lensing with AL = 1. The overall shape information of the monopole of the correlation function from the BOSS galaxy clustering is included.We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrino masses in the parentheses. The units ofH0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.2 and Fig. 9).

0.0 0.6 1.2 1.8 2.4

Σmν

3.0

2.5

2.0

1.5

1.0

w

Planck

JLA

Double Probe

Double + JLA

Figure 8. 2D marginalized contours for 68% and 95% confidence level forw and Σmν (wCDM model assumed) from Planck-only (gray), double-probe (blue), JLA (green), and double probe + JLA (red).

from our double-probe measurements is much faster than the fullrun. For example, it takes ∼ 2.5 hours to obtain the constraintsfor ΛCDM using double-probe measurements, but takes 6 days toreach similar convergence for the full MCMC run (slower with afactor of 60).

We have extended our study to measure the sum of neutrinomass using different methodologies including double probe anal-ysis (introduced in this study), full-likelihood analysis, and sin-

0.0 0.3 0.6 0.9 1.2

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Pro

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1] ΛCDM

oΛCDM

wCDM

owCDM

w0waCDM

ow0waCDM

Figure 9. Probability density for Σmν from the full-likelihood-analysis ofthe joint data set. Σmν is one of the parameters to be constrained. Planckdata including lensing with AL = 1. The overall shape information of themonopole of the correlation function from the BOSS galaxy clustering isincluded (see Sec. 7.2 and Table 9).

gle probe analysis. We found that double probe has weaker con-straint on the neutrino mass since it does not include the constrain-ing power on the neutrino mass from Planck data. While includinglensing information, we have performed the analyses with vary-

c© 0000 RAS, MNRAS 000, 000–000



oΛCDM 0.310± 0.012 67.9± 1.0 0.805± 0.017 0.002± 0.003 −1 0 < 0.18 (< 0.36)

wCDM 0.296± 0.017 69.6± 2.1 0.818± 0.021 0 −1.11± 0.11 0 < 0.25 (< 0.42)owCDM 0.300± 0.019 69.1± 2.2 0.813± 0.021 0.001± 0.004 −1.08± 0.12 0 < 0.21 (< 0.43)

w0waCDM 0.312± 0.027 68.2± 3.1 0.803± 0.028 0 −0.91± 0.27 −0.70± 0.79 < 0.33 (< 0.49)

ow0waCDM 0.311± 0.025 68.0± 2.7 0.803± 0.026 0.000± 0.004 −0.92± 0.25 −0.59± 0.78 < 0.28 (< 0.45)

Table 10. Constraints on cosmological parameters from the full-likelihood-analysis of the joint data set. Σmν is one of the parameters to be constrained.Planck data includes lensing with AL = 1. The overall shape information of the monopole of the correlation function from the BOSS galaxy clustering isremoved with a polynomial function. We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrinomasses in the parentheses. The units of H0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.2 and Fig. 11).

Ωm H0 σ8 Ωk w or w0 wa Σmµ(eV) ALΛCDM 0.308± 0.011 67.7± 0.9 0.782± 0.026 0 −1 0 0.17+0.08

−0.13 (< 0.34) 1.07± 0.06

oΛCDM 0.314± 0.013 67.9± 1.0 0.752± 0.037 0.005± 0.004 −1 0 0.34+0.17−0.22 (< 0.66) 1.12± 0.07

wCDM 0.290± 0.019 70.4± 2.5 0.781± 0.032 0 −1.16± 0.14 0 0.33+0.16−0.18 (< 0.60) 1.10± 0.07

owCDM 0.300± 0.023 69.8± 2.8 0.754± 0.041 0.005± 0.005 −1.11± 0.15 0 0.44+0.23−0.22 (< 0.81) 1.13± 0.07

w0waCDM 0.292± 0.031 70.4± 3.9 0.781± 0.037 0 −1.15± 0.34 −0.09± 0.94 0.32+0.18−0.20 (< 0.61) 1.10± 0.06

ow0waCDM 0.292± 0.030 70.8± 3.7 0.763± 0.044 0.004± 0.005 −1.18± 0.32 0.11± 0.94 0.42+0.22−0.22 (< 0.77) 1.14± 0.09

Table 11. Constraints on cosmological parameters from the full-likelihood-analysis from the joint data set. Both Σmν and AL are the parameters to beconstrained. The overall shape information of the monopole of the correlation function from the BOSS galaxy clustering is removed with a polynomialfunction. We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrino masses in the parentheses. Theunits of H0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.2 and Fig. 12).

0.0 0.2 0.4 0.6 0.8

Σmν

1.4

1.2

1.0

w

Full Run nopol

Full Run pol

Figure 10. 2D marginalized contours for 68% and 95% confidence level forw and Σmν (wCDM model assumed) from Planck+BOSS. The blue con-tours are from full-likelihood-analysis without using a polynomial functionto remove the overall shape information of monopole; the red contours arefrom the analysis removing overall shape information with a polynomialfunction. One can see that the overall shape information shift the Σmν to alarger value.

ing AL or fixing AL = 1. We found that varying AL would shiftthe Σmν to a larger value. From the full-likelihood analysis withvarying AL, we obtained Σmν = 0.17+0.08

−0.13 assuming ΛCDM;Σmν = 0.34+0.17

−0.22 assuming oΛCDM; Σmν = 0.33+0.16−0.18 as-

0.0 0.2 0.4 0.6 0.8

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Pro

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oCDM

wCDM

owCDM

w0waCDM

ow0waCDM

Figure 11. Probability density for Σmν from the full-likelihood-analysis ofthe joint data set. Σmν is one of the parameters to be constrained. Planckdata includes lensing with AL = 1. The overall shape information of themonopole of the correlation function from the BOSS galaxy clustering isremoved with a polynomial function (see Sec. 7.2 and Table 10).

suming wCDM; Σmν = 0.44+0.23−0.22 assuming owCDM. We found

∼ 2σ detection of Σmν when allowing w and Ωk to be free.In addition, when performing the full-likelihood analysis, we

found that the overall shape of correlation function contributed tothe detection of neutrino mass significantly. However, since we do

c© 0000 RAS, MNRAS 000, 000–000



oΛCDM 0.313± 0.011 67.6± 0.9 0.804± 0.016 0.002± 0.004 −1 0 < 0.19 (< 0.37)

wCDM 0.303± 0.014 68.7± 1.7 0.812± 0.017 0 −1.08± 0.09 0 < 0.24 (< 0.42)owCDM 0.305± 0.014 68.6± 1.6 0.809± 0.018 0.001± 0.004 −1.06± 0.10 0 < 0.25 (< 0.48)

w0waCDM 0.314± 0.021 67.8± 2.2 0.800± 0.022 0 −0.91± 0.22 −0.70± 0.75 0.26+0.13−0.18 (< 0.51)

ow0waCDM 0.315± 0.020 67.6± 2.1 0.799± 0.022 −0.001± 0.004 −0.89± 0.21 −0.77± 0.74 0.24+0.08−0.22 (< 0.55)

Table 12. The cosmological constraints including total mass of neutrinos from the single probe measurements provided by Chuang et al. 2016 (companionpaper) combining with Planck data assuming different dark energy models. We show 68% 1-D marginalized constraints for all the parameters. We provide also95% constraints for the neutrino masses in the parentheses. The units of H0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.2 and Fig.14).


oΛCDM 0.309± 0.010 67.9± 0.9 0.807± 0.016 0.001± 0.004 −1 0 < 0.17 (< 0.33)

wCDM 0.305± 0.012 68.2± 1.2 0.812± 0.016 0 −1.04± 0.05 0 < 0.17 (< 0.33)owCDM 0.307± 0.013 68.3± 1.4 0.808± 0.019 0.001± 0.004 −1.03± 0.06 0 < 0.20 (< 0.43)

w0waCDM 0.309± 0.014 68.2± 1.3 0.807± 0.019 0 −0.92± 0.12 −0.64± 0.56 < 0.26 (< 0.43)ow0waCDM 0.310± 0.013 68.0± 1.3 0.803± 0.019 0.000± 0.004 −0.91± 0.11 −0.63± 0.59 < 0.27 (< 0.46)

Table 13. Constraints on cosmological parameters from the full-likelihood-analysis of the joint (Planck and BOSS dr12) and JLA data sets assuming variableΣmν . Planck data includes lensing withAL = 1. The overall shape information of the monopole of the correlation function from the BOSS galaxy clusteringis removed with a polynomial function. We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrinomasses in the parentheses. The units of H0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.2 and Fig. 16).

0.0 0.3 0.6 0.9 1.2

Σmν

Pro

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oCDM

wCDM

owCDM

w0waCDM

ow0waCDM

Figure 12. Probability density for Σmν from full-likelihood-analysis fromthe joint data set. Both Σmν and AL are the parameters to be constrained.The overall shape information of the monopole of the correlation functionfrom the BOSS galaxy clustering is removed with a polynomial function(see Sec. 7.2 and Table 11). One can see that the maximum of Σmν in-creases comparing to the cases with fixing AL = 1 (see Fig. 11).

not have high confidence on the overall shape because of the poten-tial observational systematics, we removed the overall shape infor-mation to be conservative. The numbers provided above have beenobtained without the overall shape information. Our study haveshown that one should be cautious to the impact of observational

0.0 0.3 0.6 0.9 1.2

Σmν

3.0

2.4

1.8

1.2

0.6

w

Planck var AlPlanck Al = 1

Full Run var AlFull Run Al = 1

Figure 13. 2D marginalized contours for 68% and 95% confidence levelfor w and Σmν (wCDM model assumed) from full run methodology andPlanck only for different lensing information used. Gray contours and greencontours are from Planck only with varying AL and fixing AL = 1 respec-tively; the blue contours and the red contours are from Planck+BOSS withvarying AL and fixing AL = 1 respectively using full-likelihood-analysis.One can see that Σmν shift to a large value when varyingAL for both datacombinations.

systematics when constraining neutrino mass using the large scalestructure measurements.

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Ωm H0 σ8 Ωk w or w0 wa Σmµ(eV) ALΛCDM 0.307± 0.010 67.8± 0.8 0.784± 0.026 0 −1 0 0.15+0.06

−0.13 (< 0.32) 1.07± 0.06

oΛCDM 0.311± 0.013 68.0± 1.1 0.755± 0.037 0.005± 0.005 −1 0 0.32+0.16−0.23 (< 0.63) 1.12± 0.08

wCDM 0.306± 0.012 68.2± 1.2 0.779± 0.030 0 −1.04± 0.06 0 0.21+0.09−0.18 (< 0.44) 1.08± 0.07

owCDM 0.310± 0.012 68.5± 1.3 0.748± 0.038 0.006± 0.004 −1.04± 0.06 0 0.40+0.19−0.25 (< 0.76) 1.13± 0.08

w0waCDM 0.310± 0.013 68.1± 1.2 0.769± 0.035 0 −0.93± 0.12 −0.70± 0.61 0.33+0.16−0.18 (< 0.61) 1.09± 0.07

ow0waCDM 0.310± 0.016 68.5± 1.6 0.756± 0.037 0.004± 0.005 −0.97± 0.14 −0.41± 0.67 0.38+0.20−0.27 (< 0.74) 1.12± 0.08

Table 14. Constraints on cosmological parameters from the full-likelihood-analysis of the joint (Planck and BOSS dr12) and JLA data sets assuming variableΣmν . Planck data includes lensing varying AL. The overall shape information of the monopole of the correlation function from the BOSS galaxy clusteringis removed with a polynomial function. We show 68% 1-D marginalized constraints for all the parameters. We provide also 95% constraints for the neutrinomasses in the parentheses. The units of H0 and Σmν are km s−1 Mpc−1 and eV respectively (see Sec. 7.2 and Fig. 17).

0.0 0.2 0.4 0.6 0.8

Σmν

Pro

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oΛCDM

wCDM

owCDM

w0waCDM

ow0waCDM

Figure 14. Probability density for Σmν from the single probe measure-ments provided by Chuang et al. 2016 (companion paper) combining withPlanck data (with fixing AL = 1). All the measurements are consistentwith Σmν = 0 (see Sec. 7.2 and Table 12).

ACKNOWLEDGEMENT

M.P.I. would like to thank Denis Tramonte and Rafael Rebolofor useful discussions. M.P.I. and C.C. thanks David Hogg, Sav-vas Nesseris, and Yun Wang for useful discussions. C.C. andF.P. acknowledge support from the Spanish MICINNs Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064and AYA2010-21231-C02-01 grant. C.C. was also supported by theComunidad de Madrid under grant HEPHACOS S2009/ESP-1473.C.C. was supported as a MultiDark fellow. M.P.I. acknowledgessupport from MINECO under the grant AYA2012-39702-C02-01.G.R. is supported by the National Research Foundation of Ko-rea (NRF) through NRF-SGER 2014055950 funded by the KoreanMinistry of Education, Science and Technology (MoEST), and bythe faculty research fund of Sejong University in 2016.

We acknowledge the use of the CURIE supercomputer at TresGrand Centre de calcul du CEA in France through the French par-ticipation into the PRACE research infrastructure, the SuperMUCsupercomputer at Leibniz Supercomputing Centre of the Bavarian

0.0 0.2 0.4 0.6 0.8

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1.4

1.2

1.0

wDouble Probe

Full Run

Single Probe

Figure 15. Comparison of 2D contours for 68% and 95% confidence levelon Σmν and w from the double probe, single probe, and full-likelihood-analysis approaches. One can see that the constraint on Σmν from the dou-ble probe approach is weaker which is expected. The difference comes fromthe fact that we do not include Σmν into our summarized set of parameters,so information from Planck is lost.

Academy of Science in Germany, the TEIDE-HPC (High Perfor-mance Computing) supercomputer in Spain, and the Hydra clusterat Instituto de Fısica Teorica, (UAM/CSIC) in Spain.

Funding for SDSS-III has been provided by the Alfred P.Sloan Foundation, the Participating Institutions, the National Sci-ence Foundation, and the U.S. Department of Energy Office of Sci-ence. The SDSS-III web site is http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consor-tium for the Participating Institutions of the SDSS-III Collabora-tion including the University of Arizona, the Brazilian ParticipationGroup, Brookhaven National Laboratory, Carnegie Mellon Univer-sity, University of Florida, the French Participation Group, the Ger-man Participation Group, Harvard University, the Instituto de As-trofisica de Canarias, the Michigan State/Notre Dame/JINA Par-ticipation Group, Johns Hopkins University, Lawrence BerkeleyNational Laboratory, Max Planck Institute for Astrophysics, MaxPlanck Institute for Extraterrestrial Physics, New Mexico StateUniversity, New York University, Ohio State University, Pennsyl-vania State University, University of Portsmouth, Princeton Uni-

c© 0000 RAS, MNRAS 000, 000–000


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Figure 16. Probability density for Σmν from the full likelihood analysismeasurement for joint and JLA data sets. We assume lensing likelihoodwith fixed AL = 1. All the measurements are consistent with Σmν = 0(see Sec. 7.2 and Table 13).

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Figure 17. Probability density for Σmν from the full likelihood analysismeasurement for joint and JLA data sets. We assume lensing likelihoodwith variable AL (see Sec. 7.2 and Table 14).

versity, the Spanish Participation Group, University of Tokyo, Uni-versity of Utah, Vanderbilt University, University of Virginia, Uni-versity of Washington, and Yale University.

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