The Clustering of Galaxies in the Completed SDSS-III Baryon … · 2019. 11. 3. · MNRAS 000,...

MNRAS 000, 000–000 (0000) Preprint 7 June 2021 Compiled using MNRAS LATEX style file v3.0

The Clustering of Galaxies in the Completed SDSS-IIIBaryon Oscillation Spectroscopic Survey: Cosmic Flowsand Cosmic Web from Luminous Red Galaxies

Metin Ata1?, Francisco-Shu Kitaura1,2,3†, Chia-Hsun Chuang4,1,Sergio Rodŕıguez-Torres4,5,6, Raul E. Angulo7, Simone Ferraro2,3, Patrick McDonald2,

Carlos Hernández Monteagudo7, Volker Müller1, Gustavo Yepes6, Falk Baumgarten1,

Florian Beutler8, Joel R. Brownstein9, Angela Burden10, Daniel J. Eisenstein11,

Hong Guo12, Shirley Ho13, Cameron McBride11, Mark Neyrinck14,

Matthew D. Olmstead15, Nikhil Padmanabhan10, Will J. Percival8, Francisco Prada4,5,16,

Graziano Rossi17, Ariel G. Sánchez18, David Schlegel3, Donald P. Schneider19,20,

Hee-Jong Seo21, Alina Streblyanska22, Jeremy Tinker23, Rita Tojeiro24,

Mariana Vargas-Magana25Affiliations are listed at the end of the paper

7 June 2021.

ABSTRACT

We present a Bayesian phase space reconstruction of the cosmic large-scale mat-ter density and velocity fields from the SDSS-III Baryon Oscillations SpectroscopicSurvey Data Release 12 (BOSS DR12) CMASS galaxy clustering catalogue. We relyon a given ΛCDM cosmology, a mesh resolution in the range of 6-10 h−1 Mpc, anda lognormal-Poisson model with a redshift dependent nonlinear bias. The bias pa-rameters are derived from the data and a general renormalised perturbation theoryapproach. We use combined Gibbs and Hamiltonian sampling, implemented in theargo code, to iteratively reconstruct the dark matter density field and the coher-ent peculiar velocities of individual galaxies, correcting hereby for coherent redshiftspace distortions (RSD). Our tests relying on accurate N -body based mock galaxycatalogues, show unbiased real space power spectra of the nonlinear density field upto k ∼ 0.2h Mpc−1, and vanishing quadrupoles down to ∼ 20h−1 Mpc. We alsodemonstrate that the nonlinear cosmic web can be obtained from the tidal field tensorbased on the Gaussian component of the reconstructed density field. We find thatthe reconstructed velocities have a statistical correlation coefficient compared to thetrue velocities of each individual lightcone mock galaxy of r ∼ 0.68 including about10% of satellite galaxies with virial motions (about r = 0.76 without satellites). Thepower spectra of the velocity divergence agree well with theoretical predictions up tok ∼ 0.2hMpc−1. This work will be especially useful to improve, e.g. BAO reconstruc-tions, kinematic Sunyaev-Zeldovich (kSZ), warm hot inter-galactic medium (thermalSZ or X-rays), and integrated Sachs-Wolfe (ISW) measurements, or environmentalstudies.

Key words: cosmology: theory – large-scale structure of the Universe – catalogues– galaxies: statistics – methods: numerical

? E-mail:[email protected]† E-mail:[email protected], Karl-Schwarzschild-fellow

1 INTRODUCTION

The large scale structure of the Universe is a key observableprobe to study cosmology. Galaxy redshift surveys providea three dimensional picture of the distribution of luminoustracers across the history of the Universe after cosmic dawn.

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However, they are affected by many systematic effects, suchas the survey geometry, radial selection functions, galaxybias, and redshift space distortions caused by the peculiarmotions of galaxies.

Many studies require reliable reconstructions of thelarge scale gravitational potential from which also the co-herent peculiar velocities can be derived. This is the caseof the integrated Sachs Wolfe effect (see e.g., Granett et al.2008; Ilić et al. 2013), the kinematic Sunyaev-Zeldovich ef-fect (see e.g., Planck Collaboration et al. 2016; Hernández-Monteagudo et al. 2015; Schaan et al. 2015), the cosmicflows (e.g. Watkins et al. 2009; Lavaux et al. 2010; Bran-chini et al. 2012; Courtois et al. 2012; Kitaura et al. 2012c;Heß & Kitaura 2016), or the baryon acoustic oscillations(BAO) reconstructions (see e.g. Eisenstein et al. 2007; Pad-manabhan et al. 2012; Anderson et al. 2014a; Ross et al.2015). Also environmental studies of galaxies demonstratedto benefit from accurate density and velocity reconstructions(see Nuza et al. 2014).

In addition, a number of works have suggested non-linear transformations, Gaussianising the density field toobtain improved cosmological constraints (Neyrinck et al.2009, 2011; Yu et al. 2011; Joachimi et al. 2011; Carron &Szapudi 2014; Simpson et al. 2016). Also, linearised densityfields can yield improved displacement and peculiar velocityfields (Kitaura & Angulo 2012; Kitaura et al. 2012b; Falcket al. 2012).

Nevertheless, all these studies are affected by redshiftspace distortions and the sparsity of the signal, which mustbe handled carefully (McCullagh et al. 2016). Indeed, Seljak(2012) has pointed out that if not properly modeled, non-linear transformations on density fields including redshiftspace distortions can lead to biased results. Such a carefulmodeling is one motivation for the current work.

The inferred galaxy line-of-sight position is a combina-tion of the so-called Hubble flow, i.e. their real distance, andtheir peculiar motion. The modifications produced by thiseffect are referred to as redshift space distortions (RSD).They can be used to constrain the nature of gravity andcosmological parameters (see e.g. Berlind et al. 2001; Zhanget al. 2007; Jain & Zhang 2008; Guzzo et al. 2008; Nesseris &Perivolaropoulos 2008; Song & Koyama 2009; Song & Perci-val 2009; Percival & White 2009; McDonald & Seljak 2009;White et al. 2009; Song et al. 2011; Zhao et al. 2010; Songet al. 2010, for recent studies). The measurement of RSDhave in fact become a common technique (Cole et al. 1995;Peacock et al. 2001; Percival et al. 2004; da Ângela et al.2008; Okumura et al. 2008; Guzzo et al. 2008; Blake et al.2011; Jennings et al. 2011; Kwan et al. 2012; Samushia et al.2012; Reid et al. 2012; Okumura et al. 2012; Chuang & Wang2013a,b; Chuang et al. 2013b,a; Samushia et al. 2013; Zhenget al. 2013; Blake et al. 2013; de la Torre et al. 2013; Beutleret al. 2014; Samushia et al. 2014; Sánchez et al. 2014; Belet al. 2014; Tojeiro et al. 2014; Okumura et al. 2014; Beutleret al. 2014; Wang 2014; Alam et al. 2015b). These studiesare usually based on the large-scale anisotropic clusteringdisplayed by the galaxy distribution in redshift space, al-though N -body based models for fitting the data to smallerscales have been presented in Reid et al. (2014); Guo et al.(2015a,b, 2016). A recent study suggested to measure thegrowth rate from density reconstructions (Granett et al.2015). However, instead of correcting redshift space distor-

tions, these were included in the power spectrum used torecover the density field in redshift space.

Different approaches have been proposed in the litera-ture to recover the peculiar velocity field from galaxy dis-tributions (Yahil et al. 1991; Gramann 1993; Zaroubi et al.1995; Fisher et al. 1995; Davis et al. 1996; Croft & Gaz-tanaga 1997; Monaco & Efstathiou 1999; Branchini et al.2002; Lavaux et al. 2008; Branchini et al. 2012; Wang et al.2012; Kitaura et al. 2012c; Heß & Kitaura 2016), based onvarious density-velocity relations (see Nusser et al. 1991;Bernardeau 1992; Chodorowski et al. 1998; Bernardeau et al.1999; Kudlicki et al. 2000; Mohayaee & Tully 2005; Bilicki &Chodorowski 2008; Jennings & Jennings 2015; Kitaura et al.2012b; Nadkarni-Ghosh & Singhal 2016).

The main objective of this paper is to perform a self-consistent inference analysis of the density and peculiarvelocity field on large scales accounting for all the abovementioned systematic effects (survey geometry, radial se-lection function, galaxy bias, RSD, non-Gaussian statistics,shot noise). We will rely on the lognormal-Poisson modelwithin the Bayesian framework (Kitaura et al. 2010) to in-fer the density field from the galaxy distribution. Lognormal-Poisson Bayesian inference performed independently on eachdensity cell reduces to a sufficient statistic characterizing thedensity field at the two-point level (Carron & Szapudi 2014),but including the density covariance matrix as done herecarries additional statistical power. Furthermore we will it-eratively solve for redshift space distortions relying on lineartheory (Kitaura et al. 2016b).

More complex priors describing the density field thanthe lognormal assumption can be used (Coles & Jones 1991),based on perturbation theory (Kitaura 2013; Jasche & Wan-delt 2013; Wang et al. 2013; Heß et al. 2013), or even onparticle mesh approaches (Wang et al. 2014). Also the like-lihood describing the statistical distribution of galaxies canbe improved modelling the deviation from Poissonity (Ataet al. 2015). Moreover, the relation between the density andthe peculiar velocity field could be more accurately mod-elled including tidal field tensors (Kitaura et al. 2012b). Inthis work, we want to focus however, on the simplest andmost efficient models, which permit us to make the leastassumptions with the smallest number of parameters. Weleave a more complex nonlinear analysis for future work. Infact we will see that with simple models we can recover thelarge scale density and peculiar velocity in the presence oflight-cone, survey mask, and selection function effects, witha given ΛCDM cosmology, having chosen the resolution atwhich our models apply (6-10 h−1 Mpc). The majority ofprevious Bayesian density field reconstructions applied togalaxy redshift surveys did not correct for the anisotropicredshift space distortions (see e.g. Erdogdu et al. 2004; Ki-taura et al. 2009; Jasche et al. 2010, 2015; Granett et al.2015). We aim at filling that gap in this work, and thinkthat the approach presented in this work could become stan-dard in the analysis of galaxy surveys due to its efficiency,simplicity, and its critical accuracy isotropizing the galaxydistribution while dealing with survey masks, selection func-tions and bias.

Ongoing and future surveys, such as the BOSS1 (White

1 http://www.sdss3.org/surveys/boss.php

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Cosmic Flows and Cosmic Web from LRGs 3

et al. 2011; Bolton et al. 2012; Alam et al. 2015a), eBOSS(Dawson et al. 2013), DESI2/BigBOSS (Schlegel et al. 2011),DES3 (Frieman & Dark Energy Survey Collaboration 2013),LSST 4 (LSST Dark Energy Science Collaboration 2012), J-PAS5 (Benitez et al. 2014), 4MOST6 (de Jong et al. 2012)or Euclid7 (Cimatti et al. 2009; Laureijs 2009), will requirespecial data analysis techniques, like the one presented here,to extract the maximum available cosmological information.

The structure of this paper is as follows: in Sec. 2 wepresent the main aspects of our reconstruction method andthe argo-code (Algorithm for Reconstructing the Galaxytraced Overdensities). We emphasize the challenges of deal-ing with a galaxy redshift survey including cosmic evolutionand the novel imporovements to this work. In Sec. 3 we de-scribe the BOSS CMASS DR12 data and the mock galaxycatalogues used in this study. In Sec. 4 we show and eval-uate the results of our application. We finally present theconclusions in Sec. 5.

2 METHOD

Our basic approach relies on an iterative Gibbs-samplingmethod, as proposed in Kitaura & Enßlin (2008); Kitauraet al. (2012a) and presented in more detail in Kitaura et al.(2016b). The first step samples linear density fields definedon a mesh δL with Nc cells compatible with the numbercounts on that mesh NG of the galaxy distribution in realspace {r}. The second step obtains the real space distribu-tion for each galaxy given its observed redshift space sobs

position required for the first step, from sampling the pecu-liar velocities {v (δL, fΩ)} (with the growth rate given byfΩ ≡ d logD(a)/d log a, and D(a) being the growth fac-tor for a scale factor a = 1/(1 + z) or redshift z), as-suming that the density field and the growth rate fΩ areknown. The Gibbs-sampling conditional probablity distribu-tion functions can be written as follows showing the quan-tities, linear densities δL and a set of galaxies in real space{r}, which are sampled from the corresponding conditionalPDFs:

δL x Pδ (δL|NG ({r}),w,CL ({pc}) , {bp}) , (1)

{r} x Pr({r}|{sobs}, {v (δL, fΩ)}

), (2)

which is equivalent to sample from the following joint prob-ability distribution function:

Pjoint(δL, {r}|{sobs},w,CL({pc}), {bp}, fΩ

). (3)

To account for the angular completeness (survey mask) andradial selection function, we need to compute the 3D com-pleteness w defined on the same mesh, as the density field(see e.g. Kitaura et al. 2009). Also we have to assume agiven covariance matrix CL ≡ 〈δ+L δL〉 (a Nc × Nc matrix),determined by a set of cosmological parameters {pc} within

2 http://desi.lbl.gov/3 http://www.darkenergysurvey.org4 http://www.lsst.org/lsst/5 http://j-pas.org/6 https://www.4most.eu/7 http://www.euclid-ec.org

a ΛCDM framework. We aim at recovering the dark mat-ter density field which governs the dynamics of galaxies.Since galaxies are biased tracers, we have to assume someparametrised model relating the density field to the galaxydensity field with a set of bias parameters {bp}.

After these probabilities reach their so-called stationarydistribution, the drawn samples are represetatives of the tar-get distribution. In the following we define Eqs. 1 and 2 indetail and describe our sampling strategy.

2.1 Density sampling

The posterior probability distribution of Eq. 1 is sampledusing a Hamiltonian Monte Carlo (HMC) technique (seeDuane et al. 1987). For a comprehensive review see Neal(2012). This technique has been applied in cosmology in anumber of works (see e.g. Taylor et al. 2008; Jasche & Ki-taura 2010; Jasche et al. 2010; Kitaura et al. 2012a,c; Ki-taura 2013; Wang et al. 2013, 2014; Ata et al. 2015; Jasche& Wandelt 2013). To apply this technique to our Bayesianreconstruction model, we need to define the posterior dis-tribution function through the product of a prior π (seeSec. 2.1.1) and a likelihood L (see Sec. 2.1.2) which up to anormalisation is given by

Pδ (δL|NG ({r}),w,C ({pc}) , {bp}) ∝ (4)π(δL|C ({pc}))× L(NG|ρobsG , {bp}) , (5)

with ρobsG being the expected number counts per volumeelement. The overall sampling strategy then is enclosed inSec. 2.1.3.

2.1.1 Lognormal Prior

As a prior we rely on the lognormal structure formationmodel introduced in Coles & Jones (1991). This model givesan accurate description of the matter statistics (of the cos-mic evolved density contrast δ ≡ ρ/ρ̄ − 1) on scales largerthan about 6-10 h−1 Mpc (see e.g. Kitaura et al. 2009). Insuch a model one considers that the logarithmically trans-formed density field δL is a good representation of the lineardensity field

δL ≡ log (1 + δ)− µ , (6)

with

µ ≡ 〈log (1 + δ)〉 , (7)

and is Gaussian distributed with zero mean and a givencovariance matrix CL

− lnπ(δL|CL ({pc})) =1

2δ+L C

−1L δL + c , (8)

with c being some normalisation constant of the prior. Thismodel yields, however, a poor description of the three-pointstatistics (see White et al. 2014; Chuang et al. 2015), and willhave a different mean field µ depending on the higher orderstatistics of the dark matter field. The mean field computedbased on the density field, as obtained from N -body simula-tions using the definition in Eq. 7, can strongly deviate fromthe theoretical prediction for lognormal fields µ = −σ2/2 de-pending on the resolution (with σ2 being the variance of thefield δL). In fact, if one expands the logarithm of the densityfield in a series with the first term being the linear density

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field followed by all the higher order terms δ+ (see Kitaura& Angulo 2012)

log(1 + δ) = δL + δ+ , (9)

one finds that the mean field depends on the order of theexpansion

µ ≡ 〈log(1 + δ)〉 = 〈δ+〉 . (10)

In practice, the data will determine the mean field µ. Inunobserved regions, the mean field should be given by thetheoretical lognormal value (µ = −σ2/2). In observed re-gions, the number density and completeness will determinethe value of the mean field. Since galaxy redshift surveyshave in general a varying completeness as a function of dis-tance, the assumption of a unique mean field can introducean artificial radial selection function. For this reason we sug-gest to follow Kitaura et al. (2012a) and iteratively samplethe mean field from the reconstructed linear density fieldassuming large enough volumes 〈δ〉 = 0 = 〈eδL+µ − 1〉, i.e.,µ = − ln(〈eδL〉). The assumption that volume averages ofthe linear and nonlinear density field vanish in the ensembleaverage, does not imply that this happens for the individ-ual reconstructions, which will be drawn from our posterioranalysis allowing for cosmic variance. We will consider, asa crucial novel contribution, individual redshift z and com-pleteness w bins

µ(z,w) = − ln(〈eδL〉(z,w)) . (11)

This can be expressed as an additional Gibbs-sampling step

µ(z,w) x Pµ(µ(z,w)|δL(r, z),w

). (12)

In this way we account for redshift and completeness depen-dent renormalised lognormal priors. In practice, since theevolution of the three point statistics can be considered tobe negligible within the covered redshift range for CMASSgalaxies (Kitaura et al. 2016a), we will perform the ensembleaverage only in completeness bins.

2.1.2 Likelihood and data model

The likelihood describes the data model. In our case theprobability to draw a particular number of galaxy countsNGi per cell i, given an expected number count per cellρobsGi , is modelled by the Poisson distribution function

− lnL(NG|ρobsG , {bp}) =Nc∑i

(−NGi ln ρobsGi + ρobsGi

)+ c , (13)

with Nc being the total number of cells of the mesh, andc being some normalisation constant of the likelihood. Thisexpectation value is connected to the underlying matter den-sity δi by the particular chosen bias model B(ρG|δ). In par-ticular, we rely on a power-law bias (linear in the log-densityfield) connecting the galaxy density field to the underlyingdark matter density ρG ∝ (1 + δ)b (de la Torre & Peacock2013). More complex biasing models can be found in theliterature (Fry & Gaztanaga 1993; Cen & Ostriker 1993;McDonald & Roy 2009; Kitaura et al. 2014; Neyrinck et al.2014; Ahn et al. 2015). In fact threshold bias can be very rel-evant to describe the three point statistics of the galaxy field(Kitaura et al. 2015, 2016a), and stochastic bias (Kitauraet al. 2014) is crucial to properly describe the clustering on

small scales. All these bias components have been investi-gated within a Bayesian framework in Ata et al. (2015). Wewill, however, focus in this work on the two-point statisticson large scales (k


to relate the galaxy correlation function in redshift space tothe matter real space correlation function

ξsG(z) = K(z) ξG(z)

= K(z) b2L(z)G2(z, zref) ξM(zref) . (20)

From the last two equations we find a quadratic expressionfor bL(z) for each redshift z

b2L(z) +2

3fΩ(z)bL(z) +

1

5f2Ω(z)−

(csL(z))2

G2(z, zref)= 0 , (21)

with only one positive solution, leaving the bias correctionfactor fb as a potential free parameter in our model (see therenormalised perturbation theory based derivation below)

bL(z) = −1

3fΩ(z) +

√− 4

45fΩ(z)2 + (csL(z))

2

(D(zref)

D(z)

)2. (22)

By coincidence, the bias measured in redshift space onlarge scales csL(z) = 1.84 ± 0.1 (with respect to the darkmatter power spectrum at redshift z = 0.57) is constant forCMASS galaxies across the considered redshift range (see,e.g., Rodŕıguez-Torres et al. 2015). Nevertheless, the (realspace) linear bias bL(z) is not, as it needs to precisely com-pensate for the growth of structures (growth factor) and theevolving growth rates, ranging between 2.00 and 2.30. Thenonlinear bias correction factor fb is expected to be less than“one”, since we are using the linear bias in the power-law.One can predict fb from renormalised perturbation theory,which in general, will be a function of redshift. Let us Taylorexpand our bias expression (Eq. 17) to third order

δg(zi) ≡ρgρ̄g

(zi)− 1 ' bL(zi)fb(zi)δ(zi) (23)

+1

2bL(zi)fb(zi)(bL(zi)fb(zi)− 1)

((δ(zi))

2 − σ2(zi))

+

1

3!bL(zi)fb(zi)(bL(zi)fb(zi)− 1)(bL(zi)fb(zi)− 2) (δ(zi))3 ,

with δ(zi) = G(zi, zref)δ(zref). The usual expression for theperturbatively expanded overdensity field to third order ig-noring nonlocal terms is given by

δg(zi) = cδ(zi)δ(zi)+1

2cδ2(zi)(δ

2(zi)−σ2(zi))+1

3!cδ3(zi)δ

3(zi) .

(24)

Correspondingly, one can show that the observed, renor-malised, linear bias is given by (see McDonald & Roy 2009)

bδ(zi) = cδ(zi) +34

21cδ2(zi)σ

2(zi) +1

2cδ3(zi)σ

2(zi) . (25)

By considering that in our case the observable linear biasis expected to be given by bL(zi) and identifying the coef-ficients {cδ = fbbL,cδ2 = fbbL(fbbL − 1),cδ3 = fbbL(fbbL −1)(fbbL−2)} from Eqs. 23 and 24 one can derive the follow-ing cubic equation for fb

f3b

(1

2b3L(zi)σ

2(zi)

)(26)

+f2b

(−3

2b2L(zi)σ

2(zi) +34

21b2L(zi)σ

2(zi)

)+fb bL(zi)

(1 +

(−34

21+ 1

)σ2(zi)

)− bL(zi) = 0 .

Let us consider the case of a cell resolution of 6.25h−1 Mpc.

The only real solutions for redshift z = 0.57 (G = 0.78) andbL = 2.1 ± 0.1, are fb = 0.62±0.01 including the variancefrom the nonlinear transformed field (σ2(δ) = 1.75), andfb = 0.71± 0.02 including the variance from the linear field(σ2(δL) = 0.91). This gives us a hint of the uncertainty inthe nonlinear expansion. Let us, hence, quote as the theo-retical prediction for the bias correction factor the averagebetween both mean values with the uncertainty given by thedifference between them fb = 0.66±0.1. These results showlittle variation (±0.01) across the redshift range (see §3.2).Leaving fb as a free parameter and sampling it to match thepower spectrum on large scales yields fb = 0.7 ± 0.05 (see§4). Although there is an additional uncertainty associatedto this measure, since the result depends on the particu-lar k mode range used in the goodness of fit. Therefore, onecan conclude that the theoretical predictions account for thenonlinear correction within the associated uncertainties.

2.1.3 Hamiltonian Monte Carlo of the linear density field

In this section we recap the Hamiltonian Monte Carlo sam-pling technique (HMC) to sample the matter density withinthe Bayesian framework. This technique requires the gradi-ents of the lognormal-Poisson model, as introduced in Ki-taura et al. (2010). The HMC technique was first appliedto this model with a linear bias in Jasche & Kitaura (2010)and later with more complex bias relations and likelihoodsin Ata et al. (2015). In this approach one defines a potentialenergy U(x), given by the negative logarithm of the poste-rior distribution function, and a kinetic energy K(p)

U(x) = − lnP(x) (27)H(x,p) = U(x) +K(p) , (28)

where the Hamiltonian H(x,p) is given by the sum of thepotential and the kinetic energy. In this formalism we use xas a pseudo spatial variable (in our case the linear densityfield δL) and p as the conjugate momentum. HMC requiresthe computation of the negative logarithm of Eq. 5 and itsderivatives with respect to the sampled quantity (the lineardensity field δL in our case). The kinetic energy term is con-structed on the nuisance parameters given by the momentap and mass variance M:

K(p) ≡ 12

∑ij

piM−1ij pj . (29)

The canonical distribution function defined by the Hamil-tonian (or the joint distribution function of the signal andmomenta) is then given by:

P (x,p) =1

ZHexp(−H(s,p))

=

[1

ZKexp(−K(p))

] [1

ZEexp(−U(x))

]= P (p)P (x) , (30)

with ZH , ZK and ZE being the partition functions so thatthe probability distribution functions are normalised to one.In particular, the normalisation of the Gaussian distribu-tion for the momenta is represented by the kinetic partitionfunction ZK . The Hamiltonian sampling technique does notrequire the terms which are independent of the configurationcoordinates as we will show below.

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From Eq. (30) it can be noticed that in case we havea method to sample from the joint distribution functionP (x,p), marginalizing over the momenta we can in fact,sample the posterior P (x).

The Hamiltonian dynamics provides such a method. Wecan define a dynamics on phase-space (positions and mo-menta) with the introduction of a time parameter t. TheHamiltonian equations of motion are given by:

dxidt

=∂H∂pi

=∑j

M−1ij pj , (31)

dpidt

= −∂H∂xi

= −∂U(x)∂xi

. (32)

To sample the posterior one has to solve these equationsfor randomly drawn momenta according to the kinetic termdefined by Eq. (29). This is done by drawing Gaussian sam-ples with a variance given by the mass M which can tune theefficiency of the sampler (see Jasche & Kitaura 2010). Werely on the Fourier formulation to capture the correlationfunction through the power spectrum and include some pre-conditioning diagonal matrices to speed up the algorithm.The marginalization over the momenta occurs by drawingnew momenta for each Hamiltonian step disregarding theones of the previous step.

It is not possible to follow the dynamics exactly, as onehas to use a discretized version of the equations of motion.It is convenient to use the leapfrog scheme which has theproperties of being time-reversible and conserve phase-spacevolume being necessary conditions to ensure ergodicity:

pi(t+

�

2

)= pi(t)−

�

2

∂U(x)

∂xl

∣∣∣∣xi(t)

, (33)

xi (t+ �) = xi(t) + �∑j

M−1ij pj(t+

�

2

), (34)

pi (t+ �) = pi(t+

�

2

)− �

2

∂U(x)

∂xl

∣∣∣∣xi(t+�)

. (35)

The dynamics of this system are followed for a period oftime ∆τ , with a value of � small enough to give acceptableerrors and for Nτ = ∆τ/� iterations. In practice � and Nτare randomly drawn from a uniform distribution to avoidresonant trajectories (see Neal 1993).

The solution of the equations of motion will move thesystem from an initial state (s,p) to a final state (s′,p′) aftereach sampling step. Although the Hamiltonian equations ofmotion are energy conserving, our approximate solution isnot. Moreover, the starting guess will not be drawn from thecorrect distribution and a burn-in phase will be needed. Forthese reasons a Metropolis-Hastings acceptance step has tobe introduced in which the new phase-space state (x′,p′) isaccepted with probability:

PA = min [1, exp(−δH)] , (36)with δH ≡ H(x′,p′)−H(x,p).

In particular, the required lognormal-Poisson gradientsfor the prior and likelihood including cosmic evolution aregiven by

− ∂∂δL

lnπ = C−1L δL , (37)

and

− ∂ lnL∂δL

|i =(−NGiρobsGi

+ 1

)· bL(z)fbG(z, zref)(1 + δi)

1 +G(z, zref)δiρobsGi , (38)

respectively. The linear density field is defined at the refer-ence redshift zref .

2.2 Velocity sampling

The peculiar motions of galaxies can be divided into twocategories: coherent flows (Kaiser 1987) and quasi-virialisedor dispersed velocities. While the former are very well con-strained by the large scale density field, the latter becomerelevant on smaller nonlinear scales (see e.g. Reid et al.2014). Thus, one can write the total velocity field as thesum of the curl-free coherent bulk flow, which can directlybe inferred from the large scale density field within lineartheory, and the dispersion term vdisp

v(r, z) = −fΩ(a)H(a) a∇∇−2δ(r, z) + vdisp , (39)

where H is the Hubble constant. A simple way of includingthe dispersion term is to randomly draw it from a Gaus-sian with a particular standard deviation. One may considerabout 50 km s−1 (see §4), the typical 1-σ uncertainty withinlinear theory (Kitaura et al. 2012b). More precise and sofisti-cated ways of dealing with quasi-virialised RSD are left forfuture work (see e.g. Heß et al. 2013; Jennings & Jennings2015; Kitaura et al. 2016b). Here we aim at focussing on thecoherent flows on the limit of vanishing dispersions (see §4for a comparison study with and w/o dispersion). In prac-tice we are restricting our study to resolutions in the rangebetween 6 and 10 h−1 Mpc, which yield robust results onlarge scales (see Kitaura et al. 2016b). Tidal field correctionscould be included in the model (see Kitaura et al. 2012b).Also one could try to get improved velocty reconstructionsfrom the linear component rather than from the nonlinearone as we do here (see Falck et al. 2012; Kitaura & Angulo2012). Nevertheless, there is a (nearly constant) bias fromthe lognormal transformation present in the linear densityfield, which we want to avoid to reduce the number of pa-rameters (see Neyrinck et al. 2009). The mapping betweenreal space and redshift space positions for each individualgalaxy is described by

rj+1 = sobs −

(v(rj , z

)· r̂

H(a) a

)r̂, (40)

where j and j+ 1 are two subsequent Gibbs-sampling itera-tions, and r̂ denotes the unit vector in line of sight direction.The peculiar velocity needs to be evaluated in real space,which requires an iterative sampling scheme. Each galaxyrequires in principle a peculiar velocity field computed atthat redshift, as the growth rate changes with redshift. Inpractice we construct a number of peculiar velocity fields de-fined on the same mesh but at different redshifts, i.e., fromdensity fields multiplied with the corresponding growth fac-tors and rates. Each galaxy will get a peculiar velocity fieldassigned interpolated to its position within the cell takenfrom the peculiar velocity mesh at the corresponding red-shift bin.

3 INPUT DATA

In this paper, we use N -body based mock galaxy cataloguesconstructed to match the clustering bias, survey mask, se-lection functions, and number densities of the BOSS DR12

MNRAS 000, 000–000 (0000)

Cosmic Flows and Cosmic Web from LRGs 7D

EC

RA

Figure 1. Left panel: Angular mask (right ascension RA vs declination DEC), ranging from zero to one, showing the completeness on

the sky of the SDSS-III BOSS DR12 survey. Right panel: Slice (in the x− y plane) of the 3D-projected angular mask on a volume of1250 h=1 Mpc side.

Figure 2. Radial selection function f(r) for a subvolume of theCMASS galaxy survey normalised to unity before and after RSD

corrections with argo. The mean is calculated by calculating f(r)

for 2000 reconstructions.

CMASS galaxies. This permits us to test our method, asboth real space and redshift space catalogues are known.Finally, we apply our analysis method to the BOSS DR12CMASS data. Let us describe the input galaxy cataloguesbelow.

3.1 BOSS DR12 galaxy catalogue

This work uses data from the Data Release DR12 (Alamet al. 2015a) of the Baryon Oscillation Spectroscopic Sur-vey (BOSS) (Eisenstein et al. 2011). The BOSS survey usesthe SDSS 2.5 meter telescope at Apache Point Observatory(Gunn et al. 2006) and the spectra are obtained using thedouble-armed BOSS spectrograph (Smee et al. 2013). Thedata are then reduced using the algorithms described in(Bolton et al. 2012). The target selection of the CMASS andLOWZ samples, together with the algorithms used to createlarge scale structure catalogues (the mksample code), arepresented in Reid et al. (2016).

We restrict this analysis to the CMASS sample of lu-minous red galaxies (LRGs), which is a complete sample,nearly constant in mass and volume limited between the

redshifts 0.43 ≤ z ≤ 0.7 (see Anderson et al. (2014b) fordetails of the targeting strategy).

3.2 Mock galaxy catalogues in real and redshiftspace

The mock galaxy catalogues used in this study were pre-sented in Rodŕıguez-Torres et al. (2015), and are ex-tracted from one of the BigMultiDark simulations8

(Klypin et al. 2016), which was performed using gadget-2 (Springel et al. 2005) with 3, 8403 particles on a volumeof (2.5h−1Gpc )3 assuming ΛCDM Planck cosmology with{ΩM = 0.307115,Ωb = 0.048206, σ8 = 0.8288, ns = 0.9611},and a Hubble constant (H0 = 100h km s

−1 Mpc−1) given byh = 0.6777. Haloes were defined based on the Bound DensityMaxima (BDM) halo finder (Klypin & Holtzman 1997).

They have been constructed based on the Halo Abun-dance Matching (HAM) technique to connect haloes togalaxies (Kravtsov et al. 2004; Neyrinck et al. 2004; Tasit-siomi et al. 2004; Vale & Ostriker 2004; Conroy et al. 2006;Kim et al. 2008; Guo et al. 2010; Wetzel & White 2010;Trujillo-Gomez et al. 2011; Nuza et al. 2013).

At first order HAM assumes a one-to-one correspon-dence between the luminosity or stellar and dynamicalmasses: galaxies with more stars are assigned to more mas-sive haloes or subhaloes. The luminosity in a red-band issometimes used instead of stellar mass. There should besome degree of stochasticity in the relation between stel-lar and dynamical masses due to deviations in the mergerhistory, angular momentum, halo concentration, and evenobservational errors (Tasitsiomi et al. 2004; Behroozi et al.2010; Trujillo-Gomez et al. 2011; Leauthaud et al. 2011).Therefore, we include a scatter in that relation necessary toaccurately fit the clustering of the BOSS data (Rodŕıguez-Torres et al. 2015).

8 http://www.multidark.org/MultiDark/

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http://www.multidark.org/MultiDark/

8 Ata, Kitaura et al.Pj

Pi Pi

P̂−

1

ic × 10−6

0.20

0.15

0.10

0.05

0.000.0 0.5 1.0 1.5 2.0

Figure 3. Convergence analysis of the Gibbs-Hamiltonian sampler. Left and middle panels: Power spectrum correlation matrix Rijof the first 1000 iterations of Argo with a mesh of 1283. Each entry of the matrix represents the correlation coefficient of the power

spectra Pi and Pj : Rij =Cij√Cii Cjj

, where Cij = 〈(Pi − 〈Pi〉)(Pj − 〈Pj〉)〉 is the covariance matrix. Left panel: correlation matrix for

all modes of the power spectrum, middle panel: correlation matrix for the lowest 30 modes, corresponding up to k = 0.2h−1 Mpc.Right panel: Potential scale reduction factor P̂ of the Gelman & Rubin (1992) test comparing the mean of variances of different chains

with the variance of the different chain means. The cell number ic of a 1283 mesh is plotted against the potential scale reduction factor

(P̂ − 1). Commonly a P̂ − 1 of less then 0.1 (blue line) is required to consider the chains to be converged at the target distribution.Here only two chains were compared, already showing that the majority of cells have converged. This result is already satisfactory, since

including more chains will increase the statistics and reduce the potential scale reduction factor, eventually showing that all cells haveconverged.

1.60

1.68

1.76

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1.92

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0.768

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0.756

0.752

0.748

0.744

0.740

0.052

0.056

0.060

0.064

0.068

0.072

0.076

Figure 4. Additional sampled quantities. Based on a light-cone mock catalog in redshift space with 6.25h−1 Mpc resolution: slices ofthickness ∼6 h−1 Mpc in the x− z plane of the 3D cubical mesh of side 1250 h−1 Mpc and 2003 cells for the following quantities: leftpanel: the linear real space bias bL multiplied with the nonlinear constant correction factor fb = 0.7, middle panel: the lognormal

mean field µ, and right panel: the galaxy number density normalisation γ.

4 RESULTS

In this section we present the results obtained with the argocode including the cosmic evolution treatment described inSec. 2 on the mock galaxy catalogues and finally on the data.Let us first describe the preparation of the data.

4.1 Preparation of the data

As explained in Sec. 2 our method requires the galaxynumber counts on a mesh. Therefore we need first to as-sume a fiducial cosmology (the same as the mock cataloguesdescribed in Sec. 3.2), and transform angular coordinates(right ascension α and declination δ) and redshifts into co-

moving Cartesian coordinates x, y, z

x = r cosα cos δ

y = r sinα cos δ

z = r sin δ ,

with the comoving distance given by

r =H0c

z∫0

dz′√ΩM(1 + z′)3 + ΩΛ

. (41)

With these transformations we can then grid the galaxies ona mesh and obtain the galaxy number count per cell NG.In paticular, we consider in our analysis cubical volumes of1250 h−1 Mpc side length with 2003 cells, and with the lower

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Figure 5. Slices of thickness ∼30 h−1 Mpc in the x− y plane of the 3D cubical mesh of side 1250 h−1 Mpc and 2003 cells, showing azoom-in region of 900 h−1 Mpc side for visual purposes. Left panel: the 3D completeness. Cosmic velocity fields with 6.25h−1 Mpcresolution with an additional Gaussian smoothing of the density and velocity field of 13h−1 Mpc smoothing radius based on (middlepanel:) a light-cone mock catalog in redshift space and on right panel: the BOSS DR12 data. The density of the stream lines corresponds

to the field strength of the flows, whereas the color of the stream lines indicates its velocity at a particular position. The colour code forthe density field is red for high and blue for low densities. A more quantitive comparison is shown in the figures below and in §4.3.

Figure 6. Based on a light-cone mock catalog in redshift space with 6.25h−1 Mpc resolution: slices in the x− y plane of (left panel:)the 3D completeness, and the x-component of the velocity field for middle panel: the averaged mock galaxy velocities per cell, and

right panel: one reconstructed velocity field sample with argo (compensating for completeness). The colour code for the density fieldis red for positive and blue for negative peculiar velocities. A more quantitive comparison is shown in Fig. 7.

left corner of the box at

xllc = −1500h−1Mpcyllc = −650h−1Mpczllc = 0h

−1Mpc .

4.1.1 Completeness: angular mask and radial selectionfunction

Furthermore, our data model requires the completeness ineach cell w (see Eq. 17). The first ingredient in our threedimensional completeness is the angular mask of a positionin the sky. The mask is provided as polygons with equalcompleteness (see left panel of Fig. 1, and the mangle soft-ware package, Hamilton & Tegmark 2004; Swanson et al.2008). As a first step in the 3D completeness calculation weproject the angular mask to 3D by throwing large numbersof sight lines evaluating the sky mask with mangle. The re-

sult of such a projection is shown on the right panel of Fig. 1.Next we need to define the radial selection function from thenumber density distribution as a function of redshift

f(r) ∝ 1r2

∆NG∆r

. (42)

normalised to one. In principle the radial selection func-tion should be evaulated in real space to avoid the so-called”Kaiser rocket” effect (Kaiser 1987; Nusser et al. 2014). Thisis only possible when the real space positions are recon-structed, as we do here. Obtaining the real space radial se-lection function can be expressed as an additional Gibbs-sampling step for iteration j + 1

f(r)j+1 x Pf(f(r)|{rj}

), (43)

for the set of recovered galaxy distances in the previousiteration {rj}. Once we have the radial selection functionwe can multiply it with the 3D projected angular mask toget the 3D completeness. The radial selection functions as

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r = 0.61

vargo

[kms−

1]

vmock [km s−1]

r = 0.68

〈vargo〉

[kms−

1]

vmock [km s−1]

r = 0.7

〈vargo〉 w>

0.5

[kms−

1]

vmock [km s−1]

r = 0.76

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%outl

iers

[s−

1km

]

vmock [km s−1]

r = 0.8

〈vargo〉 −

10%

outl

iers

[s−

1km

]

vmock [km s−1]

Figure 7. Velocity correlation taking one component of the velocity field for reconstructions with resolutions of dL = 6.25h−1 Mpc

with additional Gaussian smoothing of rS = 2h−1 Mpc. Upper left panel: for one reconstructed sample, upper right panel: for the

mean over 6000 reconstructed samples, lower left panel: same as upper right panel, but considering only galaxies with completeness

w > 0.5 (for about 209000 galaxies, ∼82% of the whole CMASS sample in the considered volume), lower middle panel: same as upperright panel, but excluding galaxies for which the difference in the velocity reconstruction eceeds |v| = 700 km s−1 (i.e., excluding about3.5% of the sample), and lower right panel: same as upper right panel, but excluding galaxies for which the difference in the velocity

reconstruction eceeds |v| = 500 km s−1 (i.e., excluding about 10% of the sample).

provided by the CMASS galaxy catalog in redshift spaceand the reconstructed real space one are shown in Fig. 2.The agreement between both is very good, being compati-ble within 2-σ throughout almost the entire redshift range.However, we see some tiny differences at distances wherethe selection function suffers strong gradients at the small-est distances, indicating that this approach could becomeimportant if such extreme cases happen more often.

4.2 Application to galaxy catalogues

In this section we present results from first testing themethod on light-cone mocks resembling the BOSS CMASSsurvey geometry, radial selection function, and galaxy biasfor which both the galaxy in real space and in redshift spaceare available; and second applying the same method to theBOSS DR12 data. We explore the scales between 6 and 10h−1 Mpc. In particular, we consider grids with 1283 and 2003

cells with cubical volumes of 1250 h−1 Mpc side. To obtain12000 density and peculiar velocity samples we need using8 cores for a mesh of 1283 (2003) cells about 200 (550) CPUhours with about


10-110-3

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Pθθ

(k)[M

pc3/h

3]

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linear matter PS

rsmooth = 1

rsmooth = 2

rsmooth = 3

rsmooth = 5 rsmooth = 7

10-1

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0.5

0.0

0.5

1.0

Pθθ

(k)/Pδδ

(k)

non-linear matter PS

linear matter PSrsmooth = 1

rsmooth = 2

rsmooth = 3

rsmooth = 5

rsmooth = 7

Fit 1σ

Figure 8. Upper panel: power spectrum of the scaled di-

vergence of the peculiar velocity field for different smoothing

scales for a typical realisation on a mesh of 2003 with resolutiondL = 6.25h

−1 Mpc. Lower panel: ratio with respect to the non-linear power spectrum from Heitmann et al. (2010). The shadedregion represents the theoretical fit for the velocity divergence

bias bv = e−(k/a)b

by Hahn et al. (2015) with the sigma region

being computed based on the largest uncertainty found on theparameters a and b. The wiggles are due to the more pronounced

baryon acoustic oscillations in the mean theoretical power spec-

trum than in the particular realisation used in this plot.

for our analysis for each setup (meshes of 1283 and 2003

for mocks and observations). We further demonstrate thatwe succeed in sampling from the posterior distribution func-tion estimated through the Gelman & Rubin (1992) test asshown on the right panel of Fig. 3 (for details, see appendixin Ata et al. 2015). The linear real space bias bL, one typ-ical sample of the lognormal mean field µ = 〈log(1 + δ)〉,and of the galaxy density normalisation γ, are shown inFig. 4. We find that it is crucial to sample the bias andthe mean fields on at least 5 bins to get accurate densityreconstructions free of radial selection biases. However, thereconstructions are robust against different redshift bins inγ. We find that the theoretical prediction for the mean fieldµ = −σ2/2 ' −0.760 for resolutions of 6.25h−1 Mpc iscompatible within 4% with our numerical sampling result.The technique presented in this work permits us to get pe-culiar velocity fields which are compensated for the surveygeometry and selection functions. This can be qualitativelyappreciated in Figs. 5 and 6. On a quantitative level, we findthat the velocities are unbiased with the true velocities (seeFig 7) for the case in which the density fields was smoothedwith a Gaussian kernel with radius of rS = 2h

−1 Mpc fora resolution of dL = 6.25h

−1 Mpc. We note that the maxi-mum a posteriori (MAP) solution, such as Wiener-filtering,will yield biased results, although for Wiener-filtering thevariance can be separately added to the MAP solution andsuch a bias is known (Zaroubi et al. 1995). The statisticalcorrelation coefficient we find is about 0.7 including about10% of satellite galaxies with virial motions, which is whatone finds for CMASS galaxies. This correlation can be con-siderably improved by excluding those satellite galaxies fromthe analysis. As a proxy we consider two cases. One exclud-

ing galaxies for which the velocity difference between trueand reconstructed exceeds 500 and 700 km s−1. The first oneremoves ∼10% of the galaxies, and the second one ∼3.5%.Since not all satellite galaxies will be outliers the answerwill be probably closer to the latter case, rasing the statis-tical correlation coefficient to about r = 0.76, which is aconsiderable improvement with respect to previous meth-ods. Although we are using only linear theory here, ourmethod includes a couple of ingredients which can explainthis improvement, such as being a self-consistent (iterative)method, yielding linearised density fields, for which the pixelwindow has been exactly solved (the counts in cells, i.e., thenearest grid point, are treated through the full Poisson like-lihood), and nonlinear bias has been taken into account. Thesmoothing scale could be considered another parameter ofour model. However, it can be derived from the velocity di-vergence power spectrum Pθθ with θ ≡ − 1fHa∇ · v priorto running any Markov chain, as opposed to the nonlinearbias parameter. In particular, one expects Pθθ to convergetowards the linear power spectrum in the transition to thenonlinear regime at about k ∼ 0.15−0.2h Mpc−1 (Jennings2012; Hahn et al. 2015). This is expected as the velocitydivergence is closer to the Gaussian field than the gravita-tionally evolved density field (see, e.g., Kitaura et al. 2012b).In fact while the density is enhanced in the potential wells,virialisation prevents galaxies from getting larger and largervelocities. As a consequence, the power spectrum of the ve-locity divergence is close to the linear density field in thequasi-linear regime, eventually being even more suppressedat high k values. Fig. 8 shows that such an agreement downto scales of k ∼ 0.2h Mpc−1 is indeed achieved for smooth-ing scales of about rS = 2h

−1 Mpc. In fact for a smoothingscale rS between 1 and 2 h

−1 Mpc one can potentially ob-tain unbiased results beyond k = 0.5h Mpc−1. While ourchains with 1283 were run with velocities derived from den-sity fields smoothed with rS = 7h

−1 Mpc, our reconstruc-tions with 2003 were run using rS = 2h

−1 Mpc. This varietyof smoothing scales serves us to test the robustness of the ve-locity reconstructions depending on this parameter. In factwe manage to recover the monopoles in real space down toscales of about k ∼ 0.2h Mpc−1 (see left panels in Fig. 9 and10, for the lognormal-Poisson and the lognormal-negative bi-nomial models, respectively). We have checked that the the-oretical prediction from renormalised perturbation theoryfor the bias correction parameter fb can be sampled as afree parameter yielding compatible results, fb = 0.70± 0.05vs fb = 0.66± 0.1 from theory when considering the first 30bins in the power spectrum, i.e., k ∼ 120h

−1 Mpc) between the recoveredand the true quadrupoles are due to the large empty volumewhich pushes the solution to be closer to zero than in the ac-tual mock catalogue. In fact, we showed in a previous paperthat one can recover with this method the quadrupole fea-tures of the particular realisation when considering completevolumes (Kitaura et al. 2016b). The results are consistentwhen comparing lower to higher resolution reconstructions

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0 20 40 60 80 100 120 140 160

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−50

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100

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)·r

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1σ


−1 MpcBOSS DR12

Figure 9. Left panels: power spectra of the reconstructed density fields δ(zref) on a mesh and right panels: quadrupoles of thegalaxy distribution {sobs} and {r} based on upper panels: a light-cone mock (including survey geometry) with dL = 9.76h−1 Mpc,middle panels: the BOSS DR12 data with dL = 9.76h

−1 Mpc, and lower panels: the BOSS DR12 on with dL = 6.25h−1 Mpc. Power

spectra show the mean (dashed blue line) over 6000 samples with 1 and 2 σ contours (light and dark blue shaded areas, respectively),as compared to the raw galaxy power spectrum (black solid line), the nonlinear (red solid line), and the linear power spectrum (green

solid line) assuming the fiducial cosmology. Quadrupole correlation functions show the mean (dashed blue line) over 6000 samples (10

spaced samples in intervals from 500 iterations covering 4000 Gibbs-iterations for quadrupoles to reduce computations) with 1 and 2σ contours (light and dark blue shaded areas, respectively), as compared to the raw galaxy power spectrum (black solid line), and thecorresponding computations for the catalogues in real (green line for mocks only) and redshift space (red line).

(middle to lower right panels). However, we see that theuncertainty (shaded regions) in the quadrupole increases inthe higher resolution case. This is expected as the coarsergrid smooths the peculiar velocities and tends to underes-timate them. We see that the error bars in the quadrupoleare also increased when using a negative binomial likelihood(Fig. 10). This is also expected as the deviation from Pois-sonity increases the variance in the density field (see Ataet al. 2015). The stochastic bias permits us to fit the powerspectrum towards high k-values. We note, however, that a

proper treatment of the small scales requires also a compres-sion of the fingers-of-god, which we do not focus on here. Infact, we used a stochastic parameter of 0.2 for the negativebinomial likelihood, which is far too low, as compared tothe expected values of about 0.4 for LRGs (Kitaura et al.2015). Therefore, this computation should only be regarded,as a qualitative test. In addition, we have run a reconstruc-tion chain including velocity dispersion, showing that thiswill also enhance the error bars in the quadrupole, how-ever yielding the same qualitative results as without that

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−50

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)·r

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with velocity dispersion

Poisson likelihood

dL = 9.76h−1 Mpc

BOSS DR12

Figure 10. Same as Fig. 9 but upper panels: for the negative binomial likelihood case including an excessive deviation from Poissonity(negative binomial parameter of 0.2) to fit the power spectrum towards high k-values without correcting for fingers-of-god, and lower

panels: including velocity dispersion.

term (see lower panels in Fig. 10). We observe a slightlyenhanced uncertainty in the monopole and quadrupole onlarge scales. A proper treatment of the velocity dispersionrequires, however, at least a density dependent dispersionterm, or even looking at the tidal field Eigenvalues (see Ki-taura et al. 2016b). This is however, computationally moreexpensive and requires a number of additional parameters.We thus leave such an effort for later work. The accuracyof the quadrupole reconstruction presented in this paper isalready far superior than in BAO reconstruction techniques(Vargas-Magaña et al. 2015; Burden et al. 2015), see in par-ticular, right panel in Fig. 8 in Kitaura et al. (2016a) showingthe quadrupole after BAO reconstruction for a set of mockMultidark-patchy BOSS DR12 CMASS catalogs very simi-lar to the ones used here. We note that while the monopolesof the dark matter field are trivially computed from the re-constructed samples on complete meshes, the computationof the quadrupoles of the galaxies needs more computationalefforts to account for survey geometry and radial selectionfunctions (see, e.g., Anderson et al. 2014a). Fig. 11 showsslices in the x − z plane of the galaxy number counts, thecompleteness, and the reconstructed density fields. One canclearly recognise prominent features in the data in the recon-structed density fields. It is remarkable however, how thesefeatures appear balanced without selection function effects,

in such reconstructions. Only when one computes the meanover many realisations, one can see that larger significance inthe reconstructions correlates with higher completeness val-ues. The vanishing structures in unobserved regions furtherdemonstrates the success in sampling from the posterior dis-tribution function. Fig. 12 shows that the lognormal fieldsare indeed reasonably Gaussian distributed in terms of theunivariate probablity distribution function. In fact the ab-solute skewness is reduced from about 6.4 to less than 0.03with means being always smaller than |〈δL〉| < 0.13 for dif-ferent completeness regions. As we will analyse below the3pt statistics does, however, not correspond to a Gaussianfield.

4.3 The cosmic web from lognormal-Poissonreconstructions

So far we have been reconstructing the linear componentof the density field in Eulerian space at a reference red-shift within the lognormal approximation. We can, however,get an estimate of the nonlinear cosmic web by performingstructure formation within a comoving framework, i.e., with-out including the displacement of structures, as our recon-structed linear density fields already reside at the final Eule-rian coordinates. One can use cosmological perturbation the-

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Figure 11. Based on a light-cone mock catalog in redshift space with 6.25h−1 Mpc resolution: slices of thickness ∼30 h−1 Mpc inthe x − z plane of the 3D cubical mesh of side 1250 h−1 Mpc and 2003 cells for the following quantities: upper left panel: the 3Dcompleteness or window function multiplied with a factor of 0.8 for visualisation purposes, upper right panel: the number counts percell in real space, middle left panel: one reconstructed linear logarithmic density sample of the lognormal-Poisson field, middle right

panel: same as middle left panel for the Zeldovich transformed density, lower left panel: the mean over the linear logarithmic densitysample over 6000 reconstructions of the lognormal-Poisson field, and lower right panel: same as lower left panel for the Zeldovich

transformed density.

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4 2 0 2 40.00

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0.30

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mean 0


Figure 13. Same as Fig. 11, but for the x− y plane.

the uncertainty on the density is larger in the voids and inthe unobserved regions (see lower panels in Fig. 15). In fact,the variance depicts the negative of the filamentary network.These density maps can be used for environmental studies(see, e.g., Nuza et al. 2014), or to study the warm hot inter-galactic medium in filaments, cross correlating them withtemperature maps of the cosmic microwave background (see,

e.g., Génova-Santos et al. 2015), which we leave for laterwork. They could be used as a reference for future applica-tions including reconstructions of the initial conditions (seee.g. Kitaura et al. 2012c; Jasche & Wandelt 2013; Kitaura2013; Wang et al. 2013; Heß et al. 2013; Wang et al. 2014).

MNRAS 000, 000–000 (0000)


Figure 14. Slices of the ensemble averaged Zeldovich transformed density field shown in Figs. 11 (left) and 13 (right) with another color

bar for visualisation purposes and the corresponding real space galaxy number count per cell overplotted in red.

Figure 15. Slices of the variance corresponding to the (left panel:) x− z plane, and (right panel:) x− y plane shown in Figs. 11 and13, respectively.

5 SUMMARY AND CONCLUSIONS

In this work, we have presented a Bayesian phase-space(density and velocity) reconstruction of the cosmic large-scale matter density and velocity field from the SDSS-IIIBaryon Oscillations Spectroscopic Survey Data Release 12(BOSS DR12) CMASS galaxy clustering catalogue. We havedemonstrated that very simple models can yield accurate re-sults on scales larger than k ∼ 0.2h−1 Mpc.

In particular we have used a set of simple assumptions.Let us list them here

• the statistical distribution of galaxies is described bythe lognormal-Poisson model,

• linear theory relates the peculiar velocity field to thedensity field,

• the volume is a fair sample, i.e. ensemble averages areequal to volume averages,

• cosmic evolution is modelled within linear theory withredshift dependent growth factors, growth rates, and bias,

• a power law bias, based on the linear bias multiplied bya correction factor, which can be derived from renormalisedperturbation theory, relates the galaxy expected numbercounts to the underlying density field.

MNRAS 000, 000–000 (0000)


This has permitted us to reduce the number of parametersand derive them consistently from the data, with a givensmoothing scale and a particular ΛCDM cosmological pa-rameter set.

We have included a number of novel aspects in the argocode extending it to account for cosmic evolution in thelinear regime. In particular, the Gibbs-scheme samples

• the density fields with a lognormal-Poisson model,• the mean fields of the lognormal renormalised priors for

different completeness values,• the number density normalisation at different redshift

bins,• the real space positions of galaxies from the recon-

structed peculiar velocity fields,• and the real space radial selection function from the

reconstructed real space positions of galaxies (accountingfor the “Kaiser-rocket” effect).

Our results show that we can get unbiased dark matterpower spectra up to k ∼ 0.2h Mpc−1, and unbiased isotropicquadrupoles down to scales of about 20 h−1 Mpc, being farsuperior to redshift space distortion corrections based ontraditional BAO reconstruction techniques which start todeviate at scales below 60 h−1 Mpc.

As a test case study we also analyse deviations of Pois-sonity in the likelihood, showing that the power in themonopole and the scatter in the quadrupoles is increasedtowards small scales.

The agreement between the reconstructions with mocksand BOSS data is remarkable. In fact, the identical algo-rithm with the same set-up and parameters were used forboth mocks and observations. This confirms that the cos-mological parameters used in this study are already close tothe true ones, the systematics are well under control, andgives further support to ΛCDM at least on scales of about0.01


the National Science Foundation, and the U.S. Depart-ment of Energy Office of Science. The SDSS-III web siteis http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical ResearchConsortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, theBrazilian Participation Group, Brookhaven National Lab-oratory, University of Cambridge, Carnegie Mellon Uni-versity, University of Florida, the French ParticipationGroup, the German Participation Group, Harvard Univer-sity, the Instituto de Astrofisica de Canarias, the MichiganState/Notre Dame/JINA Participation Group, Johns Hop-kins University, Lawrence Berkeley National Laboratory,Max Planck Institute for Astrophysics, Max Planck Insti-tute for Extraterrestrial Physics, New Mexico State Univer-sity, New York University, Ohio State University, Pennsyl-vania State University, University of Portsmouth, PrincetonUniversity, the Spanish Participation Group, University ofTokyo, University of Utah, Vanderbilt University, Universityof Virginia, University of Washington, and Yale University.

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