MNRAS 000, 1–19 (2020) Preprint 29 October 2020 Compiled using MNRAS LATEX style file v3.0

BIRTH of the COSMOS Field: Primordial and Evolved DensityReconstructions During Cosmic High Noon

Metin Ata1★, Francisco-Shu Kitaura2,3, Khee-Gan Lee1, Brian C. Lemaux4,Daichi Kashino5, Olga Cucciati6, Mónica Hernández-Sánchez2,3 & Oliver Le Fèvre71Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), WPI,The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan2Instituto de Astrofísica de Canarias (IAC), Calle Vía Lactea s/n, 38200, La Laguna, Tenerife, Spain3Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206, La Laguna, Tenerife, Spain4Department of Physics, University of California, Davis, One Shields Ave., Davis, CA 95616, USA5Department of Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, CH-8093 Zürich, Switzerland6INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, 40129 Bologna, Italy7Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACTThis work presents the first comprehensive study of structure formation at the peak epoch ofcosmic star formation over 1.4 ≤ 𝑧 ≤ 3.6 in the COSMOS field, including the most massivehigh redshift galaxy proto-clusters at that era.We apply the extended COSMIC BIRTH algorithmto account for a multi-tracer and multi-survey Bayesian analysis at Lagrangian initial cosmictimes. Combining the data of five different spectroscopic redshift surveys (zCOSMOS-deep,VUDS, MOSDEF, ZFIRE, and FMOS-COSMOS), we show that the corresponding unbiasedprimordial density fields can be inferred, if a proper survey completeness computation from theparent photometric catalogs, and a precise treatment of the non-linear and non-local evolutionon the light-cone is taken into account, including (i) gravitational matter displacements, (ii)peculiar velocities, and (iii) galaxy bias. The reconstructions reveal a holistic view on theknown proto-clusters in the COSMOS field and the growth of the cosmic web towards lowerredshifts. The inferred distant dark matter density fields concurrently with other probes liketomographic reconstructions of the intergalactic medium will explore the interplay of gas anddark matter and are ideally suited to study structure formation at high redshifts in the light ofupcoming deep surveys.

Key words: cosmology: large-scale structure of Universe – cosmology: theory – galaxies:high-redshift – surveys

1 INTRODUCTION

Our current standard cosmological model predicts a hierarchicalclustering of subsequently merging small structures to greater ones(see White & Rees 1978; Fry & Peebles 1978) up to the formationof the largest galaxy super-clusters observed in our present localUniverse (e.g. Tully et al. 2014, with a total mass of ∼ 1017𝑀�).The formation history and the exploration of the underlying physicalphenomena of galaxy clustering over cosmic history remains animportant question (see the pioneering work of Kauffmann et al.1999, and references therein).

Furthermore, progenitors of galaxy clusters and super-clustersand their halos are key probes to understand early structure forma-tion (Cohn & White 2008; Gao et al. 2018) and can be used toconstrain a particular dark matter model (Bode et al. 2001). Also,

★ E-mail: metin.ata@ipmu.jp, Kavli IPMU Fellow

the analysis of galaxy clusters allow us to study galaxy formation,test cosmological parameters (Allen et al. 2011) and constrain non-standard cosmological models (Kravtsov & Borgani 2012; CostanziAlunno Cerbolini et al. 2013).

In particular, the range of 2 . 𝑧 . 3 marks the peak epochof star formation in the Universe, frequently referred to as “Cos-mic High Noon” (Somerville & Davé 2015). The processes drivingstar formation of galaxies and clusters have been studied in previ-ous works, showing a nontrivial relation of star formation and thegalaxies’ environmental density (Cooper et al. 2008; Koyama et al.2013; Kawinwanichakij et al. 2017; Muldrew et al. 2018; Ji et al.2018). Moreover, the quenching of star formation within massivegalaxy clusters has been found (Cooper et al. 2008), suggestingon average a passive evolution of galaxy clusters during the last≈ 10 Gyr and studying an environmental dependency (see Lemauxet al. 2012; Belfiore et al. 2017; Tomczak et al. 2019; Lemaux et al.2019). Therefore, an accurate description of the dark matter density

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distribution at these redshifts will potentially help to understandthe interconnection of star formation (see e.g. Behroozi et al. 2013;Madau & Dickinson 2014; Shi et al. 2019) and the location withinthe cosmic web (e.g. Bond et al. 1996). Detailed analyses of highredshift structures have been accessible via numerical simulations(e.g. Watson et al. 2013; Henriques et al. 2014; Chiang et al. 2017),studying their formation history and the importance of the environ-ment.

While wide field galaxy surveys like SDSS-III Baryon Oscilla-tion Spectroscopic Survey (BOSS) (Alam et al. 2017) and 2MASS(Skrutskie et al. 2006) have played an important role in spatiallymapping the large-scale structure of galaxies and quasars (see e.g.Ata et al. 2018), these have focused on scales of the baryon acousticoscillations (BAOs) (Eisenstein et al. 2005; Percival et al. 2007).Some endeavours have been done to map the structure formationdown to megaparcec scales over to smaller footprints in so-calledpencil-beams1, such as ALMAdeep fields (Casey et al. 2018) reach-ing out to 𝑧 ∼ 1.5 − 2.5. However, no spectroscopic surveys haveabundantly mapped the large-scale density distribution down tomegaparcec scales beyond 𝑧 > 1.5, although multiple projects aimto push beyond this boundary in the near future, such as DESI (Leviet al. 2013), EUCLID (Amendola et al. 2018), PFS (Takada et al.2014), 4MOST (de Jong et al. 2012), and MOONS (Cirasuolo et al.2014). For this reason we are still missing observational valida-tion of quasi-linear structure formation at about 𝑧 & 1.5, beforenon-linearities start dominating the emerging cosmic web.

Over nearly two decades, the Cosmic Evolution Survey (COS-MOS) (see latest version Laigle et al. 2016) (Capak et al. 2007;Scoville et al. 2007) has been a major ongoing effort to photometri-cally observe galaxies across a sufficiently wide footprint to resolvetransverse large-scale structure, while simultaneously having suffi-cient depth to probe the entire span of cosmic history from the LocalUniverse right into (and eventually beyond) the Epoch of Reioniza-tion (e.g. Scoville et al. 2013). Observations within COSMOS haveidentified large aggregations of galaxies at high redshifts (Yuanet al. 2014), i.e. galaxy proto-clusters, that have been extensivelystudied over a large redshift range at Cosmic Noon (Diener et al.2013, 2015; Chiang et al. 2015; Casey et al. 2015; Wang et al. 2016;Lee et al. 2016; Cucciati et al. 2018; Lemaux et al. 2018; Darvishet al. 2020). These overdense structures have been proposed to bepossible progenitors of the Coma-like galaxy clusters, potentiallyassuming total masses of ∼ (1 − 2) × 1015𝑀� at 𝑧 = 0 (Lee et al.2016; Cucciati et al. 2018; Lemaux et al. 2018; Darvish et al. 2020).The COSMOS field is therefore ideally suited to study early struc-ture formation and the evolution of galaxy proto-clusters into themature structures we observe at the current epoch.

However, some of the aforementioned pioneering studies havetypically been based on individual spectroscopic surveys in theCOSMOS field and also not taking selection criteria into account.Therefore, a consistent analysis of the confirmed structures com-bining the multiple deep galaxy spectroscopic surveys over a suffi-ciently large redshift range is still missing. This has led to a hetero-geneous view of structures that may or may not co-evolve, e.g. thereported overdensities at 𝑧 ≈ 2.4−2.5 that arewithin∼ 100 ℎ−1Mpcof each other (Diener et al. 2015; Chiang et al. 2015; Casey et al.2015; Wang et al. 2016). Moreover, analysis of the overdensitieshave typically been carried out by comparing with analogous struc-tures in 𝑁-body simulations rather than direct analysis of the ob-

1 Pencil-beam surveys do not statistically resolve the transverse large-scalestructures at ∼ 10 ℎ−1Mpc scales.

served structures, which leads to greater uncertainties due to thediversity of structures with similar aggregate properties in the sim-ulations (e.g. aperture mass, velocity dispersion etc). This situationcompounds an attempt to compare the results of these findings.

Our goal in thiswork is to recover the darkmatter density distri-bution in the COSMOS field during Cosmic Noon (1.4 ≤ 𝑧 ≤ 3.6),jointly constrained from different spectroscopic galaxy surveys, re-vealing a consistent reconstruction of all structures within the field.We infer the Gaussian density field at redshift 𝑧 = 100 and the den-sity field corresponding to the observed redshift over the range of1.4 ≤ 𝑧 ≤ 3.6, which we refer to as ‘initial’ and ‘final’ conditions,respectively. Over the last two decades, several density reconstruc-tionmethods have been proposed in literature, starting from iterativemethods (Zaroubi et al. 1995;Wang et al. 2009; Kitaura et al. 2009),over non-linear ones using, e.g., a lognormal prior and a Poissonlikelihood (Kitaura et al. 2010), to more realistic structure forma-tion models, such as Lagrangian perturbation theory (LPT) (Kitaura2013; Jasche & Wandelt 2013; Wang et al. 2013; Schmittfull et al.2017; Hada&Eisenstein 2018; Patrick Bos et al. 2019; Kitaura et al.2019; Lavaux et al. 2019) or Particle-Mesh based codes (Wang et al.2014; Jasche & Lavaux 2019). Recent reconstruction approachesalso aim to infer initial and final conditions from the observationsof absorption lines from the Intergalactic medium (see e.g. Kitauraet al. 2012a; Horowitz et al. 2019; Porqueres et al. 2019).

In this work we use five spectroscopic galaxy surveys in theCOSMOS field, (i) zCOSMOS-deep (Lilly et al. 2009), (ii) VIMOSUltra Deep Survey (VUDS, Le Fèvre et al. 2015), (iii) MOS-FIRE Deep Evolution Field (MOSDEF, Kriek et al. 2015), (iv)KECK/MOSFIRE Spectroscopic Survey of Galaxies in Rich Envi-ronments (ZFIRE, Nanayakkara et al. 2016), and the (v) FMOS-COSMOS survey (Silverman et al. 2015; Kashino et al. 2019) tojointly reconstruct the initial dark matter density field with the COS-MIC BIRTHmethod (Kitaura et al. 2019), which is ideally suited forthis purpose as it deals for the first time with light-cone evolutioneffects beyond the Zel’dovich approximation. Considering thesesurveys we firstly develop a formalism to combine them within ourBayesian reconstruction framework. Secondly, we calculate the se-lection functions of each survey to estimate the completeness ofthe observations as a function of the angular and radial dimensions.Thirdly, we study the galaxy bias beyond passive evolution as afunction of redshift.

The inferred density fields are ideally suited to comparewith the intergalactic medium (IGM) absorption maps, from the𝑧 ∼ 2 − 2.5 CLAMATO (COSMOS Ly𝛼 Mapping And Tomo-graphic Observations) survey (Lee et al. 2014a,b, 2016, 2018), andthe 𝑧 ∼ 2.2 − 2.8 LATIS (Ly𝛼 Tomography IMACS Survey) (New-man et al. 2020), based on theLyman-𝛼 forest tomography technique(Pichon et al. 2001) and thus will provide insights into the relation-ship of matter clustering and the properties of the IGM. Other futureapplications include a direct study of galaxy properties as a func-tion of underlying matter density and a full constrained 𝑁-Bodysimulation starting from the inferred initial conditions.

This article is structured as follows. In Section §2 we give abrief introduction to the recently developed initial density pertur-bations reconstructions algorithm COSMIC BIRTH. We discuss thechallenges to reconstruct the density perturbations in the COSMOSfield, and present the necessary extensions to our algorithm. In Sec-tion §3 we give details of the surveys used in this work, how weselected the data and how we constructed the selection functions.In Section §4 we give details of our large-scale bias computationand how we determined the bias evolution with increasing redshift.In Section §5 we describe the application of COSMIC BIRTH to the

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BIRTH of the COSMOS Field 3

surveys in the COSMOS field, present the inferred density fieldsand provide additional diagnostics to validate the COSMIC BIRTHreconstruction algorithm. Finally, we give a summary in Section §6,discuss our results and forecast future efforts based on this work.

Throughout this paper we use a fiducial flatΛCDM cosmologywith a set of cosmological parameters {𝑝c} of

{𝑝c} = {ΩM = 0.31,ΩΛ = 0.69, 𝜎8 = 0.82, 𝑛𝑠 = 0.96, ℎ = 0.68} (1)

concertedly chosen with the parameters in Lee et al. (2018). Alldistances are given in comoving ℎ−1Mpc units.

2 COSMIC BIRTH-ALGORITHM

Density field reconstructions from observed galaxy distributionsare an ongoing effort in cosmological science. The aim of thiswork is to infer the initial density fluctuations 𝛿(𝒒) at Lagrangiancoordinated 𝒒 given galaxy positions that have been observed inEulerian redshift-space 𝒔. These two frames can be connected if thepeculiar velocity 𝒗r (𝒒) and displacement 𝝍(𝒒) fields are known, as

𝒒 = 𝒔 − 𝒗r (𝒒) − 𝝍(𝒒) . (2)

Equation 2 shows that the mapping problem is analytically ill-defined, as we need a priori knowledge of the Lagrangian coor-dinates 𝒒 at which the displacement field and the peculiar velocityfield (see Kitaura et al. 2016) are evaluated. Thus, an iterative so-lution is required to solve this problem, as proposed in pioneeringworks (Yahil et al. 1991; Monaco & Efstathiou 1999). This classof iterative mapping schemes were further developed in Kitaura &Angulo (2012); Kitaura et al. (2012b); Kitaura (2013); Heß et al.(2013); Kitaura et al. (2019). One of the key ingredients to achievehigher accuracy on small scales consists of introducing a Bayesianinference framework in the initial conditions reconstruction whichtakes into account the likelihood of the dark matter tracers. In thisway, the typical additional Gaussian smoothing can be avoided andthe number counts of objects within the mesh in a given voxel res-olution can be correctly treated (Kitaura & Enßlin 2008; Kitauraet al. 2010; Jasche & Kitaura 2010). This allows for high precisionon a few Mpc scales (see Kitaura et al. 2012c; Nuza et al. 2014).

COSMIC BIRTH (Kitaura et al. 2019) is a Bayesian inferenceframework, using a nested Gibbs-sampling scheme to infer the ini-tial density perturbations 𝜹(𝒒) on a regular cubical mesh grid with𝑁C voxels. It maps the galaxy distribution represented by its Carte-sian (for a given set of cosmological parameters) Eulerian redshift-space positions {𝒔obs} to Lagrangian real-space coordinates {𝒒}expressed as number counts of galaxies per voxel on a regular mesh:𝑁G (𝒒). For efficiency the displacement field is computed relying onAugmented Lagrangian perturbation theory (Kitaura & Hess 2013)and so effectively solving Equation 2 (see Section 2.1).

In thiswaywe have a precise description of the action of gravitywithin ΛCDM on Mpc scales at redshifts larger than one (Neyrinck2013). To compute the displacement 𝝍(𝒒) and velocity fields 𝒗r (𝒒)from the initial density field 𝜹(𝒒) we apply Hamilton Monte-Carlo(HMC) sampling (see Duane et al. 1987; Jasche & Kitaura 2010;Neal 1993, 2011) with a bias description 𝑩(𝜹(𝒒)) and the galaxycounts on the mesh grid 𝑵G. This is key to perform a forwardmodelling, as we do not obtain the displacement field from thedensity field defined in Eulerian space, done by inverse approaches(see e.g. Eisenstein et al. 2007). We account for selection effectsof the galaxy survey data, i.e. the survey geometry, the angularand the radial selection functions, within a response operatorR (seeSection 3.2 for details). The COSMIC BIRTH code uses a particularly

efficient fourth order leap-frog implementation (see Hernández-Sánchez et al. 2019) to solve the Hamiltonian equations of motion.We recap the gravity model in Section 2.1 following a descriptionof the probabilistic model in more detail in Section 2.2 and thenexpand the calculations in Section 2.4 to combine multiple surveys.

2.1 Gravity Model

In this section we recapitulate the analytical model used within theCOSMIC BIRTH code to compute the gravitational evolution of thecosmic density field.

We rely on augmented Lagrangian Perturbation Theory(ALPT) to simulate structure formation (details can be found inKitaura & Hess 2013). In this approximation the displacement field𝝍(𝒒), mapping a distribution of dark matter particles at initial La-grangian positions 𝒒 to the final Eulerian positions 𝒙(𝑧) at redshift 𝑧(𝒙(𝑧) = 𝒒+𝝍(𝒒)), is split into a long-range𝝍L (𝒒) and a short-rangecomponent 𝝍S (𝒒), i.e. 𝝍(𝒒) = 𝝍L (𝒒) + 𝝍S (𝒒). The long-rangecomponent is computed with second order Lagrangian PerturbationTheory (2LPT) 𝝍2LPT(for details on 2LPT see Bouchet et al. 1992,1995; Catelan 1995). The resulting displacement field is convolutedwith a kernelK:𝝍L (𝒒) = K(𝒒, 𝑟S)∗𝝍2LPT (𝒒), given by aGaussianfilter K(𝒒, 𝑟S) = exp (−|𝒒 |2/(2𝑟2S)), with 𝑟S being the smoothingradius. The short-range component is modelled with the sphericalcollapse approximation 𝝍SC (𝒒) (see Bernardeau 1994; Mohayaeeet al. 2006; Neyrinck 2013). The resulting ALPT displacement fieldfrom combining the long and the short range components given by:

𝝍ALPT (𝒒) = K(𝒒, 𝑟S) ∗ 𝝍2LPT (𝒒) + (1 − K(𝒒, 𝑟S)) ∗ 𝝍SC (𝒒) (3)is used to move a set of homogenously distributed particles fromLagrangian initial conditions to the Eulerian final ones. We thengrid the particles following a clouds-in-cell scheme and phase spacemapping (Abel et al. 2012; Hahn et al. 2013) to produce a smoothdensity field 𝜹(𝒓). Some improvements can be obtained preventingvoids within larger collapsing regions, which essentially extendsthese regions towards moderate underdensities (seemusclemethodin Neyrinck 2016). This approach requires about eight additionalconvolutions being about twice as expensive, as the approach usedhere. Moreover, we have checked that the improvement providedby including muscle is not perceptible when using grids with cellresolutions of the order ∼ ℎ−1Mpc.

The mapping between Eulerian real space 𝒙(𝑧) and redshiftspace 𝒔(𝑧) is given by: 𝒔(𝑧) = 𝒓 (𝑧)+𝒗𝑟 (𝑧), with 𝒗𝑟 ≡ (𝒗 · �̂�) �̂�/(𝐻𝑎);where �̂� is the unit sight line vector, 𝐻 the Hubble constant, 𝑎 thescale factor, and 𝒗 = 𝒗(𝒙) the 3-d velocity field interpolated at theposition of each halo in Eulerian-space 𝒓 using the displacementfield 𝝍ALPT (𝒒). We split the peculiar velocity field into a coherent𝒗coh and a (quasi) virialized component 𝒗𝜎 : 𝒗 = 𝒗coh + 𝒗𝜎 . Thecoherent peculiar velocity field is computed in Lagrangian-spacefrom the linear Gaussian field 𝛿 (1) (𝒒) using the ALPT formulationconsistently with the displacement field (see Equation 3):

𝒗cohALPT (𝒒) = K(𝒒, 𝑟S) ∗ 𝒗2LPT (𝒒) + (1 − K(𝒒, 𝑟S)) ∗ 𝒗SC (𝒒) , (4)with 𝒗2LPT (𝒒) being the second order and 𝒗SC (𝒒) being the spher-ical collapse component (for details see Kitaura et al. 2014). Weuse the high correlation between the local density field and thevelocity dispersion to model the displacement due to (quasi) virial-ized motions. Effectively, we sample a Gaussian distribution func-tion (G) with a dispersion (see also Ata et al. 2017) given by𝜎𝑣 ∝ (1 + 𝛿 (𝒓))𝛾 . Consequently we assume,𝒗𝜎𝑟 ≡ (𝒗𝜎 · �̂�) �̂�/(𝐻𝑎) x G

(𝑔 × (1 + 𝛿(𝒓))𝛾

)�̂� . (5)

MNRAS 000, 1–19 (2020)

4 Metin Ata et al.

For the Gaussian streaming model see Reid & White (2011), fornon-Gaussian models see e.g. Tinker (2007). In closely virialisedsystems the kinetic energy approximately equals the gravitationalpotential and a Keplerian law predicts 𝛾 close to 0.5, leaving onlythe proportionality constant 𝑔 as a free parameter in the model.We leave a detailed investigation of the impact of redshift spacedistortions for future work.

2.2 Probabilistic Model

We use a Bayesian framework to draw samples of the den-sity field 𝜹(𝒒) from a posterior probability density P(𝜹(𝒒) |𝑵G (𝒒), 𝑩(𝜹(𝒒)),R(𝒒)). The posterior itself is a product of a priorand a likelihood function which we will describe in more detailin the following. We express the expectation value of galaxies pervoxel 𝝀𝑖 = 〈𝑵G (𝒒)〉𝑖2 for all voxels 𝑖 ∈ [1 . . . 𝑁C] as:

𝜆(𝒒)𝑖 = 〈𝑵G (𝒒)〉𝑖 = 𝑓�̄� 𝑖 (𝒒)∑︁𝑘

𝑅𝑖𝑘 (𝒒)𝐵𝑘 (𝜹(𝒒)) , (6)

with the normalisation factor 𝒇 �̄� (𝒒) = �̄�/〈𝑩(𝜹(𝒒))〉 ensuring theright galaxy number density �̄� , as given by each survey. In our case,the response function R is limited to the completeness 𝑤𝑖 in eachcell 𝑖 as described in Section 3.2: 𝑅𝑖 𝑗 (𝒒) = 𝑅𝑖𝑖𝛿K𝑖 𝑗 . We then relatethe expected number counts of galaxies per voxel to the actuallyobserved number counts 𝑵G with a Poisson likelihood (Kitaura &Enßlin 2008; Kitaura et al. 2010) model:

L(𝑵G | 𝝀(𝒒)) =∏𝑖

𝜆𝑖 (𝒒)𝑁G𝑖 exp (−𝜆𝑖 (𝒒))𝑁G𝑖!

, (7)

where 𝝀(𝒒) = 𝝀(𝑓�̄� (𝒒), 𝑩(𝜹(𝒒)),R(𝒒)

). This is an adequate as-

sumption for tracers with a vanishing small scale clustering, whichotherwise become sources of super-Poissonity (e.g. Peebles 1980)that can be modelled with a negative-binomial likelihood (Kitauraet al. 2014; Neyrinck et al. 2014; Ata et al. 2015). However, sincethe tracers are mapped to Lagrangian space at very high redshifts(e.g.: 𝑧 = 100), a deviation from Poissonity becomes insignificantexcept for the most massive galaxies (Modi et al. 2017; Abidi &Baldauf 2018; Schmittfull et al. 2019), which is not the case for ourgalaxy samples (Laigle et al. 2016) (see Section 3).

The above mentioned conditions are ideal to describe the mat-ter distribution with a lognormal prior towards high redshifts (Coles& Jones 1991). Its derivation is precisely based on a comovingframework at initial cosmic times, before shell crossing occurs.We apply a logarithmic transformation, which further linearises thedensity field (Neyrinck et al. 2009) as:

𝜹L (𝒒) = log(1 + 𝜹(𝒒)) − 𝝁 , (8)

with 𝝁 = 〈log(1 + 𝜹)〉 = − log(〈e𝜹L

〉)(Kitaura et al. 2012a).

Since we consider early cosmic times, the overdensity field haslittle power |𝛿 | � 1. Hence, the logarithmic transformation ensurespositive densities 𝝆 (with 𝜹 = 𝝆/�̄� − 1 = exp(𝜹L + 𝝁) − 1), whilewe can model the prior 𝜋(𝜹(𝒒))3 for the linear density field 𝜹L bya Gaussian distribution with zero mean

𝜋(𝜹(𝒒) | CL (𝒒)) =1√︁

(2𝜋)𝑁C det(CL (𝒒))exp

(−12𝜹†L (𝒒)C

−1L (𝒒)𝜹L (𝒒)

), (9)

2 The expectation value is given by the ensemble average: 〈𝑿 〉.3 Note that the prior is actually a function of the linearised density field𝜹L (𝒒) , which is in turn a function of the original density field 𝜹 (𝒒) .

where CL (𝒒) =〈𝜹†L (𝒒)𝜹L (𝒒)

〉is the covariance matrix of the lin-

earised density fields, which depends on the cosmological param-eters {𝑝c}. Finally, we can express the posterior of 𝜹(𝒒) throughBayes theorem as:

P(𝜹(𝒒) | 𝑵G (𝒒), 𝑩(𝜹(𝒒)),R(𝒒), {𝑝c}) ∝𝜋 (𝜹(𝒒) | CL ({𝑝c})) × L(𝑵G ((𝒒)) | 𝝀(𝒒),R(𝒒)) , (10)

where the normalisation given by the evidence is not necessarywithin HMC. Table 1 summarises the main quantities that are sam-pled and how they are connected to each other.

2.2.1 Galaxy Bias Description

Finally, we need to specify the connection between the likelihoodand the prior through the bias relation 𝑩(𝜹). In the COSMIC BIRTHframework, non-local bias is described through the displacement.The split-background bias (Kaiser 1984; Bardeen et al. 1986), whichis necessary in Eulerian space particularly for massive galaxies (seee.g. Kitaura et al. 2015), however, becomes negligible when ho-mogenising the galaxy distribution mapping it to Lagrangian space.Hence, we can assume a power-law Lagrangian bias as discussed inKitaura et al. (2019):

𝑩(𝜹(𝒒)) = (1 + 𝜹(𝒒))𝑏 (𝑧𝑞) 𝑓𝑏 (𝑧𝑞) , (11)

where 𝑧𝑞 is the redshift at which the Lagrangian coordinates areevaluated (for this study 𝑧𝑞 = 100), 𝑏 the linear large-scale biasand 𝑓𝑏 the non-linear correction factor of our bias description (Ataet al. 2017). The correction factor 𝑓𝑏 (𝑧𝑞) can be determined itera-tively as presented in Kitaura et al. (2019), ensuring that 𝑏 exactlycorresponds the large-scale linear bias. By using a power-law biaswe ensure that the density field is positive, since otherwise any biasless than one can potentially cause negative densities at voxels with𝛿 close to −1.

The advantage of this bias description is that the only free pa-rameter of our method is reduced to the large-scale bias at Eulerianspace, which can be connected to Lagrangian space through passiveevolution (Nusser & Davis 1994; Fry 1996):

𝑏(𝑧𝑞) = (𝑏(𝑧) − 1)𝐷 (𝑧)𝐷 (𝑧𝑞)

+ 1 , (12)

including the linear growth factor 𝐷 (𝑧) . We will show in Section4.2 how we describe the large-scale bias evolution for the employedgalaxy catalogs.

2.3 Summary of the Algorithm

In summary, the joint probability distributions of all the above men-tioned variables can be expressed within a Gibbs-sampling schemebased on the corresponding conditional probabilities:

𝛿(𝒒) x P𝛿 (𝜹(𝒒) | 𝑵G (𝒒), 𝑩(𝜹(𝒒)),R(𝒒),CL (𝒒))

{𝒓} x P𝑟({𝒓}|{𝒔obs}, 𝒗r (𝒒),M𝑣

),

{𝒒} x P𝑞({𝒒}|{𝒓},𝝍(𝒒),M𝜓

),

R(𝒒) x P𝑅(R(𝒒) |R(𝒔),𝝍(𝒒),M𝜓

),

𝑩(𝜹(𝒒)) x P𝐵(𝑩(𝜹(𝒒)) |𝑩(𝜹(𝒔)),𝝍(𝒒),M𝜓

), (13)

where the functional dependency of 𝒒 and 𝒔 stand for Lagrangianreal-space, and Eulerian redshift-space coordinates, respectively.The curved left arrows stand for the sampling process.M𝑣 andM𝜓represent the models describing peculiar motions and displacementfields.

MNRAS 000, 1–19 (2020)

BIRTH of the COSMOS Field 5

Inferred quantity Parent quantity Connected via

{𝒓} Eulerian real-space {𝒔obs} Eulerian redshift-space 𝒗r (𝒒) Peculiar velocity

{𝒒} Lagrangian real-space {𝒓} Eulerian real-space 𝝍 (𝒒) Displacement field

R(𝒒) Lagrangian response function R(𝒔) Eulerian response function 𝝍 (𝒒) , 𝒗r (𝒒)

𝝀 (𝒒) Galaxy number expectation 𝑵G (𝒒) Galaxy number counts 𝒇 �̄� (𝒒) , 𝑩 (𝜹 (𝒒)) , R(𝒒) (see caption)

𝑓𝑏 (𝑧) Bias correction 𝑏 (𝑧) Linear bias 𝑩 (𝜹 (𝒒)) , 𝑵G (𝒒) ,K(𝑟S) (see caption)

𝜹 (𝒒) Lagrangian density {𝒒} Lagrangian real-space P(𝜹 (𝒒) | 𝑵G (𝒒) , 𝑩 (𝜹 (𝒒)) , R(𝒒) , CL (𝒒)) (see caption)

Table 1. Inferred quantities of COSMIC BIRTH: The Eulerian real-space positions {𝒓} are inferred from the observed redshift-space positions {𝒔obs} viathe peculiar velocity 𝒗r. The Lagrangian positions {𝒒} are calculated from the Eulerian ones by applying the displacement field 𝝍 (𝒒) . The same mappingis used to calculate the response operator R(𝒒) in Lagrangian space. The expectation value of galaxy number counts 𝝀 is estimated from the galaxycounts 𝑵G (𝒒) , connected by the normalisation factor 𝒇 �̄� (𝒒) , the bias model 𝑩 (𝜹 (𝒒)) and the response function R(𝒒)) . We sample the initial densitydensity field 𝜹 (𝒒) at Lagrangian coordinates 𝒒 from the posterior probability function P(𝜹 (𝒒) | 𝑵G (𝒒) , 𝑩 (𝜹 (𝒒)) , R(𝒒) , CL (𝒒)) . The connection quantities𝒗r ,𝝍 (𝒒) , 𝑩 (𝜹 (𝒒)) , R(𝒒) , CL (𝒒) depend on a set of cosmological parameters {𝑝c }. 𝑓𝑏 is the bias correction term, derived from the large-scale linear bias 𝑏and a smoothing kernel K with radius 𝑟S. Note that, 𝝀 (𝒒) , 𝑵G (𝒒) , 𝑩 (𝜹 (𝒒)) , 𝒇 �̄� (𝒒) , and 𝜹 (𝒒) are arrays of scalar quantities of 𝑁C entries, while 𝒗r (𝒒) ,𝝍 (𝒒) are arrays of three-dimensional vector quantities of 𝑁C entries. The quantities R(𝒒) and CL (𝒒) are matrix operators of 𝑁C × 𝑁C dimensionality.

2.4 Multi-tracer Formalism in Lagrangian-space

In this study, we aim at combining the data of five spectroscopic sur-veys in the COSMOSfield that share spatially overlapping footprintsand similar redshift distributions, however relying on different ob-serving strategies (for more details see Section 3). Thus merging thesurveys into a single catalogue by a pre-processing step, i.e. addingthe different catalogues into one data set, represents a difficult task,which in general cannot be accomplished without making a seriesof simplifying assumptions, e.g. neglecting the different selectioncriteria.

Some of the previous pioneering Bayesian inference studieshave already applied multi-tracer treatment, however, all performedin Eulerian space, and without taking into account separate surveyfootprints within the Bayesian scheme (Jasche et al. 2015; Granettet al. 2015), or separate footprints from the same (super)-set ofcatalogs (Jasche & Lavaux 2017; Jasche & Lavaux 2019).

In this work, we aim at performing joint analysis of entirelydifferent surveys within the Bayesian framework. According to Sec-tion 2.2, we perform the reconstructions in Lagrangian space (in ourcase corresponding to a redshift of 𝑧 = 100), where gravitationalinteractions have not yet introduced mode couplings of density per-turbations. Therefore, mapping the galaxies to Lagrangian spacebefore the density sampling step (see Table 1) is gradually reducingthe covariance of the different surveys and homogenizes the galaxyfields. This enables us to treat each survey as a distinct component,avoiding mixed terms in the likelihood expression. Therefore, spa-tial overlap in Lagrangian space does not represent a problem, aslong as we make sure that no galaxy is multiply counted among thedifferent surveys.

In this way, we are able to combine different catalogues (in-dexed with superscript 𝑘) taking into account their distinct surveyselection functions R𝑘 (𝒒), number densities 𝑵𝑘G (𝒒), and galaxybias functions 𝑩𝑘 (𝜹(𝒒)).

The Eulerian to Lagrangian mapping of COSMIC BIRTH,shown in Equation 2, can lead to a change of the redshift bin ofa tracer. Consequently, the bias of this tracer will not coincide withthe bias of the redshift bin at its new location. This can be takeninto account by keeping track of galaxies staying at a redshift bin,or jumping from one redshift bin to another, which causes a “biasmixing” implemented in the COSMIC BIRTH code (see Section 3 in

Kitaura et al. 2019). While this effect is negligible when interpolat-ing the bias within the redshift bins, it has the advantage that a multi-tracer treatment is already implemented in this framework. We canthen extend COSMIC BIRTH to perform a full Bayesian multi-tracer& multi-survey analysis to address the challenges of this work fol-lowing the calculations presented in e.g. Ata et al. (2015, AppendixA) to express the corresponding posterior Pmulti in Lagrangian co-ordinates and combine the surveys with their specific likelihoods toconstruct the combined likelihood Lmulti by:

Lmulti(𝑵G (𝒒) |𝝀

(𝑓�̄� (𝒒), 𝑩(𝜹(𝒒)),R(𝒒)

) )∝ (14)∏

𝑘

L (𝑘)(𝑵

(𝑘)G (𝒒) |𝝀

(𝑘)(𝑓(𝑘)�̄�

(𝒒), 𝑩 (𝑘) (𝜹(𝒒)),R(𝑘) (𝒒))),

where index 𝑘 ∈ [1 . . . 𝑁S] denotes the different surveys. Accord-ingly, we need to generalise Equation 6 and distinguish between thesurveys in the reconstructed volume:

𝝀 (𝑘) =〈𝑵

(𝑘)G (𝒒)

〉= 𝑓 (𝑘)

�̄�(𝒒)R(𝑘) (𝒒)𝑩 (𝑘) (𝜹(𝒒)) . (15)

In Hamiltonian sampling, one seeks to draw samples of the potentialenergy termU of the Hamiltonian, that is linked to the posterior inEquation 10 via

U = − lnP . (16)

Thus, the multi-tracer & multi-survey posterior Pmulti writes for anumber of 𝑁S surveys as:

− lnPmulti(𝜹(𝒒) |𝝀 (1) (𝒒), 𝝀 (2) (𝒒), . . . , 𝝀 (𝑁s) (𝒒)

)= (17)

𝑐 − ln 𝜋(𝜹(𝒒) | CL (𝒒))

− lnL (1)(𝑵

(1)G |𝝀

(1)(𝑓(1)�̄�

(𝒒), 𝑩 (1) (𝜹(𝒒)),R(1) (𝒒)))

− lnL (2)(𝑵

(2)G |𝝀

(2)(𝑓(2)�̄�

(𝒒), 𝑩 (2) (𝜹(𝒒)),R(2) (𝒒)))

.

.

.

− lnL (𝑁S)(𝑵

(𝑁S)G |𝝀

(𝑁S)(𝑓(𝑁S)�̄�

(𝒒), 𝑩 (𝑁S) (𝜹(𝒒)),R(𝑁S) (𝒒))),

where the constant 𝑐 does not depend on 𝜹(𝒒).

MNRAS 000, 1–19 (2020)

6 Metin Ata et al.

Survey 𝑁Obj 𝑧 range Parent catalog

zCOSMOS-deep 3544 1.4 ≤ 𝑧 ≤ 3.6(*) COSMOS

VUDS 1822 1.4 ≤ 𝑧 ≤ 3.6(*) COSMOS

MOSDEF 401 1.4 ≤ 𝑧 ≤ 3.6(*) 3D-HST

ZFIRE 149 2.0 ≤ 𝑧 ≤ 2.2 ZFOURGE

FMOS-COSMOS 587 1.4 ≤ 𝑧 ≤ 1.7 COSMOS

Table 2. Summary of five surveys used for this study. 𝑁Obj resemblesthe number of galaxies that we use after applying spectroscopic qualitycriteria and removing duplicates. (*) zCOSMOS-deep, VUDS &MOSDEFexceed the redshift range of our reconstructions. MOSDEF observes theredshift range in intervals of 1.37 ≤ 𝑧 ≤ 1.70 , 2.09 ≤ 𝑧 ≤ 2.61, and2.95 ≤ 𝑧 ≤ 3.80. In the case of FMOS-COSMOS, about half of thegalaxies in the central footprint are used in this work.

3 SURVEY DATA & COMPLETENESS

Within the COSMOS field, several spectroscopic surveys have beenundertaken (see e.g. Hasinger et al. 2018, for a summary), focusingmainly on star forming galaxies at high redshifts. This work usesdata fromfive different surveys, summarised in Table 2.We describeeach survey in more detail below and afterwards explain our methodto estimate the corresponding survey completenesses.

3.1 Surveys in the COSMOS Field

Let us briefly recap the main characteristics of the different spec-troscopic galaxy surveys considered in this work.

• zCOSMOS-deepThe zCOSMOS-deep survey is the high-redshift component ofthe zCOSMOS spectroscopic survey, which covers the central1 deg2 of the zCOSMOS footprint (Lilly et al. 2007, Lilly et alin prep.) using the VIMOS spectrograph (LeFevre et al. 2003) onthe VLT. The targets were chosen from the then-current versionof the multi-colour photometric COSMOS catalog (Capak et al.2007). To isolate galaxies at redshifts of 𝑧 > 1.5, several selectioncriteria were applied. In particular, a selection in the (U-B)/(V-R)colour-colour plane, called “UBR” selection (Steidel et al. 2004)was combined with the “BzK” selection (Daddi et al. 2004). Forboth the “UGR” and “BzK” selections, an additional selection of22.5 < 𝐵AB < 25.0 and a deep 𝐾-band imaging reaching down to𝐾AB ∼ 23.5 were applied. Our analysis is done on a tentative ver-sion of the zCOSMOS-deep catalog that has been used for the bulkof previous works that employ zCOSMOS-deep data. A refined ver-sion of this catalogwill be available in the future (Lilly et al. in prep).

• VUDSThe VIMOS Ultra Deep Survey (Le Fèvre et al. 2015), hereafterVUDS, is another spectroscopic survey carried out on the VIMOSspectrograph that was partly operated in the COSMOS field, butconsiderably deeper than zCOSMOS-deep with up to ∼ 3× largerintegration times. VUDS peaks in number density at 𝑧 ∼ 3 andwas designed to explore multiple questions, including those relatedto the formation rates of stars and merging of galaxies during theperiod of time when galaxies were most active. Another majorsuccess was to identify and characterize galaxy protoclusters at𝑧 ∼ 2 − 4 (e.g. Cucciati et al. 2014; Lemaux et al. 2018; Cucciatiet al. 2018). The selection is based on photometric redshift

selections (Ilbert et al. 2013), with a small fraction of galaxiesselected using the Lyman-break technique (Steidel et al. 1996).VUDS and zCOSMOS are based on different versions of COSMOSparent catalog, so the astrometry was not identical for all sourcesbetween the two surveys. Because of the different catalogs used, weemployed amatching radius of 0.1−0.2′′ to identify true duplicates.

• MOSDEFThe MOSFIRE Deep Evolution Field (MOSDEF) Survey (Krieket al. 2015) is partly taken in the COSMOS field, separated intothree redshift intervals at 1.37 ≤ 𝑧 ≤ 1.70, 2.09 ≤ 𝑧 ≤ 2.61, and2.95 ≤ 𝑧 ≤ 3.80, down to fixed 𝐻AB magnitudes of 24.0, 24.5, and25.0 for each interval. The MOSFIRE spectrosgraph (McLean et al.2012) on the Keck-I telescope was used to obtain near infra-redemission line redshifts of the targeted galaxies. The targets wereselected from the photometric and grism 3D-HST (Brammeret al. 2012) data, applying photometric redshift and magnituderequirements to the parent catalog and show a spectroscopicsuccess rate of around ∼ 80%.

• ZFIREThe KECK/MOSFIRE Spectroscopic Survey of Galaxies inRich Environments at 𝑧 ∼ 2 (ZFIRE) (Nanayakkara et al. 2016)using the MOSFIRE spectrosgraph was partly taken in theCOSMOS and Hubble Ultra Deep Survey (UDS) field (Beckwithet al. 2006). For the COSMOS field the targets were 𝐾-bandselected from the photometric parent FourStar Galaxy EvolutionSurvey (ZFOURGE) catalog (Straatman et al. 2016), requiringa photometric redshift 2.0 . 𝑧phot . 2.2. ZFIRE was designedto observe primarily the 𝑧 = 2.095 galaxy cluster (Yuan et al. 2014).

• FMOS-COSMOSThe FMOS-COSMOS survey (e.g. Silverman et al. 2015; Kashinoet al. 2019) (hereafter FMOS) used the Fibre Multi-Object Spec-trograph (Kimura et al. 2010) at the Subaru Telescope, observingstar forming galaxies at redshifts 𝑧 ∼ 1.6 in the near infra-red.The targets were selected from the COSMOS photometric catalogwithin a redshift range of 1.4 . 𝑧phot . 1.7, additionally applyinglimits on the UltraVISTA K-band magnitude limit and the 𝐻𝛼 fluxpredicted from SED fitting. FMOS objects are pre-selected with se-cure photometric redshift using the filters available in the COSMOSphotometric survey. In this analysis we utilize FMOS observationsin the range of 149.8 ≤ R.A. ≤ 150.4 and 1.8 ≤ DEC ≤ 2.5.

In summary, the surveys apply different colour, magnitude and pho-tometric redshift pre-selection cuts to the parent photometric cat-alogs to efficiently select high redshift targets for spectroscopy. Inthe case of galaxies that were spectroscopically observed in morethan one of these surveys, we kept the one with better redshift qual-ity, based on the signal-to-noise ratio of the observed spectra (seeSection 3.2.1). From the redshift distributions of the surveys shownin Figure 1 we can see that the number density of galaxies peaks at2 ≤ 𝑧 ≤ 2.5, mainly contributed by the zCOSMOS-deep survey.

The footprints (galaxy distribution in angular coordinates) ofeach survey are shown in Figure 2. From this figure we can recog-nise that the edges of zCOSMOS-deep, represented in blue, havebeen observed with only a single pointing, thus having a lower num-ber density, while in the central area the pointings are overlapping,allowing for a denser targeting. Regarding the VUDS survey, repre-sented in orange, Figure 2 shows the gaps in between the quadrantsof an individual VIMOS pointing, whereas within one quadrant thetarget sampling is very homogeneous. Focusing on the MOSDEF

MNRAS 000, 1–19 (2020)

BIRTH of the COSMOS Field 7

1.0 1.5 2.0 2.5 3.0 3.5 4.0z

100

200

300

400

500

dN/dz

zCOSMOS-deep

VUDS

MOSDEF

ZFIRE

FMOS

Used range

Figure 1.Radial distribution functions for the five considered surveys shownas function of redshift with a bin width of Δ𝑧 = 0.16. The two grey dashedvertical lines signpost the considered redshift range 1.4 ≤ 𝑧 ≤ 3.6 in thiswork.

and ZFIRE surveys depicted in green and red, respectively, one cansee that they cover smaller areas in the central part of the COS-MOS field. The FMOS survey, represented in magenta, shows asparser targeting compared to the rest of the surveys considered inthis study. While the three surveys MOSDEF, ZFIRE, and FMOSlie withing the footprints of zCOSMOS-deep and VUDS, the latterones only partly overlap. VUDS shares 86% angular coverage withzCOSMOS-deep, which corresponds to 69%VUDS coverage of thezCOSMOS-deep footprint.

3.2 Survey Completeness Estimation

The response function R (see Table 1, Section 2) represents theefficiency how each voxel 𝑖 has been observed as compared to thenumber of possible targets (see Figure 3 for details of the selec-tion stages). In a Bayesian analysis, the likelihood function shouldtherefore account for the uncertainty of the galaxy number counts𝑵G (𝒒) as a function of the completeness, shown in Equation 6.The response function is calculated from an angular and a radialcomponent, R𝛼 and R𝑟 , respectively, as they can be independentlycalculated:

R = R𝛼 · R𝑟 . (18)

Both components are calculated on the reconstructed mesh gridand then multiplied for each voxel. It is practical to consider themseparately, since the angular part is not subject to redshift space dis-tortions, contrary to the radial part. Following Kitaura et al. (2019),we can compute the angular response operator in Eulerian spaceR𝛼 (𝒔) = R𝛼 (𝒓) once, and in each Gibbs-sampling iteration mapit to Lagrangian space R𝛼 (𝒒) through the displacement field (seeTable 1). The radial response function can be trivially computedfrom the distribution of large-scale tracers in Lagrangian real spacecoordinates {𝒒} in each Gibbs-sampling iteration, and multipliedaccording to Equation 18. Let us thus focus in detail on the compu-tation of the angular completeness.

149.6 149.8 150.0 150.2 150.4 150.6R.A.

1.8

2.0

2.2

2.4

2.6

DE

C

zCOSMOS-deep

VUDS

MOSDEF

ZFIRE

FMOS

Figure 2. Survey footprints: Spatial distribution of the observed galaxiesin the right ascension (R.A.)-declination (DEC) plane. We follow the samecolour coding as in Figure 1. At a redshift of 𝑧 = 2.5, the maximumseparation ΔDEC = 0.91◦ corresponds to a comoving transverse distance of𝑑ΔDEC = 64.8 ℎ−1Mpc, while for ΔR.A. = 1.02◦ the maximum separationis 𝑑ΔR.A. = 70.2 ℎ−1Mpc assuming the cosmological parameters in Section1.

3.2.1 Angular completeness

In the following we adopt the terminology of the VIMOS relatedsurveys (e.g. VVDS,VIPERS) (see e.g. Ilbert et al. 2005; Zucca et al.2009; Guzzo et al. 2014; Granett et al. 2015; Scodeggio et al. 2018)to estimate the spatial completeness in the reconstructed volume.

To compute the angular completeness, we need to distinguishbetween four selection stages, starting from the photometric parentcatalog and yielding the final spectroscopic survey, shown in Figure3. In the first stage, observers apply photometric selection criteriaon top of the parent photometric catalog (I) to select galaxies withcertain properties (e.g. star forming, redshift range etc). The result-ing catalog consists of photometric targets (II), leading to a secondstage, as shown in Figure 3. The transition from the photometricparent (I) to the photometric target catalog (II) is called ColourSampling Rate (CSR) and accounts for the colour-colour, colour-magnitude and photometric redshift selections. A fraction of thephotometric targets (II) is chosen for spectroscopy, which we callSpectroscopic Parent (III). The ratio of (II) and (III) is called TargetSampling Rate (TSR). Many factors have an influence on the finalselection of spectroscopic galaxies such as the spatial arrangementof the slits/fibres, the conditions during the observations etc. Thismeans, that not all the spectra taken for individual galaxies can betranslated into a reliable redshifts. We call the ratio of the spec-troscopic parents (III) to the final selection (IV) the SpectroscopicSuccess Rate (SSR).

The selection estimation approach consists on reproducing thetargeting strategy of each survey to precisely estimate the ratio fromthe parent photometric catalog to the final spectroscopic galaxies’

MNRAS 000, 1–19 (2020)

8 Metin Ata et al.

(I) Photo-metric Parent

Colour-magnitude, photo-z selectionColour Sampling Rate (CSR)

(II) Photomtet-ric Target

Assign slits, fibers to targetsTarget Sampling Rate (TSR)

(III) Spectro-scopic Parent

Quality selection of redshiftsSpectroscopic Success Rate (SSR)

(IV) FinalSelection

Spectroscopic galaxy sam-ple used for this analysis

Figure 3. Flowchart describing the process to obtain the final spectroscopiccatalog, starting from the parent photometric survey. The arrow symbol onthe left represents the name of the survey. The descriptive text on the rightdenotes the selection/operation that is undertaken upon the survey, leadingto the next stage. We use an adapted terminology of the VVDS/VIMOScollaboration: Stage (I) to (II) is called Colour Sampling Rate (CSR), (II)to (III) Target Sampling Rate (TSR), and finally (III) to (IV) SpectroscopicSuccess Rate (SSR).

selection. This has been neglected in previous studies based on theCOSMOS field (see Section §5 and references therein).

Let us describe the different selection steps in more detail:

• Colour Sampling Rate (CSR)One can make the robust assumption that the photometric pre-selections applied on the parent photometric catalog (I) are constantover the footprint of each survey. In such a case, the CSR will onlyresult in an overall normalization factor of the number density ofgalaxies (see Pezzotta et al. 2017), but not influence the angulardependent clustering. Therefore, we can safely absorb this factorinto the radial selection factor.

• Target Sampling Rate (TSR)First, we reproduce the photometric pre-selection criteria. Then webuild the ratio of the photometric targets 𝑁II with the number of thespectroscopic parents 𝑁III, regardless of the quality of the spectra,TSR = 𝑁III/𝑁II. We construct a mesh grid 80 × 80 cells overthe R.A.-DEC plane (Figure 2), resulting in a resolution of about0.9 arcmin per cell. We note that the reconstructions are computedwith a comoving resolution of 𝑑R = 2 ℎ−1Mpc (see Section 5). Thiscorresponds to a angular aperture of 𝜗 = 1.5′ at a redshift of 𝑧 = 3.6and thus coarser than the angular completeness resolution. We setthe value of the selection function in between the VUDS quadrantsand outside the borders of the surveys to zero.

• Spectroscopic Sampling Rate (SSR)Finally, we select a subset 𝑁IV of galaxies from the spectroscopicparent sample, that have high redshift accuracies, as described fol-lows. For the zCOSMOS-deep and VUDS surveys we demandredshift flags of ≥ 2.For the MOSDEF survey we apply quality

1.6

1.9

2.2

2.5

2.8

DE

C

zCOSMOS-deep VUDS

1.6

1.9

2.2

2.5

2.8

DE

C

MOSDEF

149.5 149.8 150.1 150.4 150.7R.A.

ZFIRE

149.5 149.8 150.1 150.4 150.7R.A.

1.6

1.9

2.2

2.5

2.8

DE

C

FMOS

0.2

0.4

0.6

0.8

1.0

Figure 4. Final angular completenessmask𝑤𝑘𝛼 for all five surveys computedfrom TSR and SSR in the R.A.-DEC plane normalized to unity.

flags of = 3, which corresponds to a redshift confidence of > 95%and a minimum signal-to-noise ratio of 2 ≤ 𝑆/𝑁 ≤ 3. For theZFIRE survey we apply quality flag of 2, which corresponds to a𝑆/𝑁 ≥ 5 and |𝑧spec−𝑧phot | ≤ 0.2. For the FMOS surveywe demand3 ≤ 𝑆/𝑁 ≤ 5, which translates into a redshift quality flag of ≥ 2.The SSR is then calculated as SSR = 𝑁IV/𝑁III.

The resulting angular selection masks

𝑤𝑘𝛼 = TSR𝑘 × SSR𝑘 , (19)

in the R.A-DEC plane are shown in Figure 4 for each survey 𝑘 .In the case of the zCOSMOS-deep survey the higher targeting ratein the center of the survey footprint, caused by several overlappingpointingswith theVIMOS spectrograph, can be appreciated. For theVUDS survey on the contrary, each pointing is unique, and hence,the areas do not overlap. This results in thin stripes between thequadrants of the pointings in the footprint, showing the inter-CCDgaps of ∼ 2 arcmin in the VIMOS focal plane.

Once we have computed the angular mask, we need to projectit into three dimensions, as required by the COSMIC BIRTH code.

In particular, we project each R.A.-DEC bin value of 𝑤𝑘𝛼 intoour cubical mesh grid on which we perform the reconstructions.As the resolution of 𝑤𝑘𝛼 is higher compared to the reconstructionmesh grid, we average over each sight at a voxel 𝑖 and thus obtainthe angular completeness 𝑅𝑘

𝑖𝑖for all 𝑘 surveys and cell 𝑖. Figure

MNRAS 000, 1–19 (2020)

BIRTH of the COSMOS Field 9

1.52.02.53.03.5Redshift z

1.0

2.2

3.4

DE

C[d

eg]

Eulerian Completeness Rα(s)

0.0 0.2 0.4 0.6 0.8 1.0

1.52.02.53.03.5Redshift z

1.0

2.2

3.4

DE

C[d

eg]

Lagrangian Completeness Rα(q)

0.0 0.2 0.4 0.6 0.8 1.0

Figure 5. Single slice plot of the projected window function R𝛼 for the VUDS survey into the reconstructed volume, normalized to unity, showing 0 fornon observed regions and 1 for maximum completeness. The top panel represents the window calculated directly from the observations in Eulerian space 𝒔,showing the effects of the gaps in between the VUDS quadrants (compare Figure 4), emphasizing the importance of an accurate selection function handlingfor this work. The bottom panel presents the same slice in Lagrangian space 𝒒. The whole declination angle on the 𝑌 -axis corresponds to 200 ℎ−1Mpc incomoving distance.

5 exemplary shows the projection of the VUDS angular selectionmask into the reconstructed volume. The data preparation and de-tails of the coordinate system are explained in Section 5.1. The toppanel of Figure 5 shows the stripes and the corresponding gaps inbetween the pointings of the survey in the Eulerian frameR𝛼 (𝒔), inaccordance with Figures 2 and 4. The VUDS inter-CCD gaps trans-late into empty regions of 2.3 ℎ−1Mpc at redshifts of 𝑧 = 3.6. Thetranslation of the angular response operator to Lagrangian spaceR𝛼 (𝒒) through the action of gravity causes a deformation in thesurvey window, which is represented in the lower panel of Figure5. This means, that unobserved angular regions in Eulerian space,might have been effectively partially observed in Lagrangian space.In turn, the deformation of the survey window can also cause thatregions are effectively unobserved in Lagrangian space, that wereobserved in Eulerian space (see Figure 5 e.g. around redshift 𝑧 ∼ 2).As expected, we can see that the deformation is stronger towardslower redshifts, where the growth of structures is more evolved.

4 LARGE-SCALE GALAXY BIAS

The COSMIC BIRTH algorithm accounts for stochastic and non-linear Lagrangian bias, while non-local Eulerian bias is modelledthrough the displacement field connecting Eulerian to Lagrangianspace (see Section 2 and for further details in Kitaura et al. 2019).The only free parameter is the large-scale Eulerian bias (Kaiser1984), which needs to be determined from observations or sim-ulations. We rely on detailed bias studies of highly star forminggalaxies due to the nature of the surveys considered in this study(see Section 4.1).

4.1 Bias Studies in Simulations and Observations

The various galaxy properties are in general correlated with theirclustering behaviour, and hence, are indicators of how they trace theunderlying dark matter density field. More massive and luminous

galaxies, such as luminous red galaxies (LRGs) for instance (seee.g. Alam et al. 2017) show a strong clustering, tracing mainly thepeaks of the density field (Kitaura et al. 2015). These galaxies arepassively evolving, showing low stellar formation activity and oldstellar populations.

Apart from LRGs, star forming galaxies can be identified withphotometric techniques (see e.g. Daddi et al. 2004) and emissionline spectroscopy, frequently called emission line galaxies (ELGs)in literature. The galaxy bias of highly star forming galaxies, suchas [O II], [O III], H𝛼 detected galaxies (see e.g. Delubac et al.2017; Kaasinen et al. 2017), UV emitting Lyman-break galaxies(Kollmeier et al. 2003) and Lyman-𝛼 emitters (LAEs, see e.g. Ken-nicutt (1998)) have been extensively studied in the literature.

Unlike LRGs that trace only the densest peaks of the densityfield, ELGs can populate the density field at nearly the whole rangeof overdensities. Therefore, their bias is close to unity at low red-shifts (Favole et al. 2016). However, in this work we consider higherredshifts, when the dark matter field was less evolved, displayingweaker density perturbations. Hence, these types of galaxies displayan increasing bias towards high redshifts (Okada et al. 2016; Guoet al. 2019).

Studies based on numerical simulations have shown the biasas a function of redshift and star formation rate, comparing thegalaxy and matter density fields (see Table 1 in Chiang et al. 2013).Within the VUDS survey a bias measurement for the redshift rangeof 2 ≤ 𝑧 ≤ 5 has been accomplished in Durkalec et al. (2015,2018) using two-point clustering analysis. Similar analysis has beenperformed for the FMOS-COSMOS survey, studying the projectedcorrelation function and from there estimating the bias at a medianredshift of 𝑧 = 1.58 (Kashino et al. 2017).

4.2 Large-Scale Bias Evolution

In the following we explain our strategy to compute the large-scalebias, as required to perform the darkmatter reconstructions through-out the redshift range of 1.4 ≤ 𝑧 ≤ 3.6.

MNRAS 000, 1–19 (2020)

10 Metin Ata et al.

1.5 2.0 2.5 3.0 3.5 4.0Redshift z

1.5

2.0

2.5

3.0

3.5

Bia

sb(z)

Passive Evolution

Fit r = 0.6, b(1.5) = 1.9

Passive Evolution 1σ

Fit 1σ

Chiang et al (2013), Simulation

Durkalec et al (2015)

Cochrane et al (2017)

Kashino et al (2017)

Figure 6. Large-scale galaxy bias 𝑏 (𝑧) as a function of redshift 𝑧. The bluesolid line and the corresponding blue shaded error band show the best fit forthe passive evolution model of Equation 12 with 𝑏 (𝑧 = 1.5) = 1.9 ± 0.33.The solid orange line and the orange shaded error band show the fittingresult assuming Equation 20 with the parameters 𝑟 (𝑧 = 1.5) = 0.6 and𝑏 (𝑧 = 1.5) = 1.9 ± 0.33. The circles denote previous bias measurementswhereas the red triangles show values derived from numerical simulations.

Given that the galaxy populations from the 5 considered sur-veys are similar, we assume as a null hypothesis, that galaxies sharethe same bias at a given redshift, and that their bias passively evolvesthrough Equation 12 (equivalent to a perfect correlation 𝑟 (𝑧2) = 1 inEquation 20). Then, we choose a narrow redshift range for which thedeviation from passive evolution is expected to be negligible. In par-ticular we select the range 1.4 . 𝑧 . 1.8 embedded in a rectangularvolume that extends from 2875 ℎ−1Mpc ≤ 𝑑L ≤ 3387 ℎ−1Mpcin line-of-sight distance. We perform a series of COSMIC BIRTHruns with the data set in this volume with a varying bias at 𝑧 = 1.5from 1 to 2.5 (the general set-up of the COSMIC BIRTH reconstruc-tions is described in Section 5.1). From these runs we find thatthe reconstructed primordial matter density shows unbiased powerspectra w.r.t. the theoretical linear one for 𝑏 ≈ 1.9 ± 0.3 at redshift𝑧 = 1.5 being conservative (and the rest of bias values given bypassive evolution). This is in good agreement with the findings ofCochrane et al. (2017, see Table 4, where they obtain 𝑏 = 1.78+0.06−0.08at 𝑧 = 1.48), and also roughly compatible with the clustering analy-sis of the FMOS-COSMOS survey (Kashino et al. 2017, see Table3, finding 𝑏 = 2.440.380.32 at median redshift 𝑧 = 1.59) within theestimated uncertainties.

The passive evolutionmodel (Nusser&Davis 1994; Fry 1996),based on our low redshift bias measurement, evolves to higher red-shifts according to the blue solid line including the large associateduncertainties represented by the shaded light blue area in Figure 6.From this figure we can assume that there is a trend in the bias mea-surements towards higher bias values with increasing redshift thanwhat is predicted by passive evolution, although within the largeuncertainties the passive evolution model is still nearly compatiblewith the data points.

In such a case the redshift evolution of the galaxy populationscan be modelled. This can be achieved by introducing a correla-tion coefficient 𝑟 between the dark matter and the galaxy densityfield as a function of redshift (Tegmark & Peebles 1998). As theselection criteria of the surveys do not significantly change with ob-served distance, we expect a moderate enhancement of the galaxybias towards higher redshifts beyond the one described by passiveevolution. Thus, we write:

𝑏(𝑧2) =

√︂(1 − 𝐷 (𝑧2)

𝐷 (𝑧1)

)2− 2𝑟 (𝑧1)

(1 − 𝐷 (𝑧2)

𝐷 (𝑧1)

)𝑏(𝑧1) + 𝑏2 (𝑧1)(

𝐷 (𝑧2)𝐷 (𝑧1)

) (20)for two redshifts 𝑧1 and 𝑧2 with 𝑧1 < 𝑧2, a correlation coefficient𝑟 (𝑧1), and the linear growth function 𝐷 (𝑧) given by

𝐷 (𝑧) = 𝐻 (𝑧)𝐻0

∞∫𝑧

dz′

𝐻3 (𝑧′)

/ ∞∫0

dz′

𝐻3 (𝑧′), (21)

normalized to unity at redshift zero 𝐷 (𝑧 = 0) = 1. The evolution ofthe bias and the correlation coefficient are coupled as:

𝑟 (𝑧1) =((1 −

(𝐷 (𝑧2)𝐷 (𝑧1)

))+ 𝑟 (𝑧2)

(𝐷 (𝑧2)𝐷 (𝑧1)

)𝑏(𝑧2)

) /𝑏(𝑧1) . (22)

A perfect correlation of 𝑟 (𝑧2) = 1 at an earlier redshift 𝑧2 willremain like that for all times, whereas 𝑟 (𝑧1) always tends towards1, regardless of its initial value, according to Equation 22. In thecase of a perfect correlation 𝑟 (𝑧2) = 1, Equation 20 equals Equation12 (described in Section 2) and no change of the galaxy populationis expected along redshift. However, 𝑟 (𝑧2) < 1 implies a varyingcorrelation coefficient, effectively describing a cosmic evolution ofthe galaxy distribution, which may be caused by galaxy formationor evolution (Tegmark & Peebles 1998).

On the other hand, a series of COSMIC BIRTH runs disfavour2𝜎 deviations from the upper bias limits quoted in the literature,as they lead to unreasonable biased dark matter reconstructions(see Appendix A). We note, that the selection function can lead toan excess of power in the two point statistics on large scales, andthereby higher bias values can be inferred (see e.g. Thomas et al.2011). We therefore investigate the theoretical predictions for thebias evolution in simulations. As a result, we find that the galaxysamples considered in this work cover the stellarmass range of𝑀∗ =109.5𝑀� to 𝑀∗ = 1010.5𝑀� , peaking at 𝑀∗ ∼ 109.8𝑀� (Lamauxet al. in prep.). According to this finding, we obtain the data pointsrepresented in red upwards pointing triangles in Figure 6 (see Table1 in Chiang et al. 2013). We find that these simulation based dataare in agreement with the observational measurements, howeverfavour slightly lower bias values. In the spirit of being conservative,and avoid bias ranges which can be affected by selection effects,we include the simulation (Chiang et al. 2013) and observational(Cochrane et al. 2017; Kashino et al. 2017; Durkalec et al. 2015,2018) data points in a least squares fit. The resulting bias evolutionmodel is represented in solid red with the uncertainty given by thelight red shaded area in Figure 6.

According to this bias study, we have found some moderateevidence (given the hitherto large uncertainties due to the smallvolumes covered by high redshift galaxy surveys) for a cosmic evo-lution of the galaxy bias beyond passive evolution, hinting towardsongoing galaxy formation and merging processes at these redshifts.A coefficient of 𝑟 (𝑧 = 1.5) = 0.6 corresponds to 𝑟 (𝑧 = 3.6) = 0.42(see Equation 22). This apparently tiny variation, reduces the ten-sion with observationally constrained biases at redshift 𝑧 > 1.5.

MNRAS 000, 1–19 (2020)

BIRTH of the COSMOS Field 11

However, a proper verification of a deviation from passive evolu-tion requires a deeper study, extending the runs we performed at lowredshift to higher ones. The available data at this stage might notbe sufficient to make stronger claims and we leave a more detailedinvestigation of the bias evolution to a forthcoming work.

The resulting large-scale bias calculations presented in thissection can be fed into the COSMIC BIRTH code to produce unbiaseddark matter reconstructions, as shown in Section 5.

5 COSMIC BIRTH APPLIED TO THE COSMOS FIELD

In this section we present the application of COSMIC BIRTH code(see Kitaura et al. 2019, and Section 2) to the spectroscopic surveysin the COSMOS field (see Section 3). Previous pioneering COS-MOS density field estimates were based on photometric redshifts(Kovac et al. 2010) and used tessellations of the observed galaxyfields (Scoville et al. 2013; Darvish et al. 2015), but did not accountfor the selection function (Amara et al. 2012; Smolčić et al. 2017),or focused on individual high density peaks (e.g. Wang et al. 2016).Therefore, this work represents the first comprehensive study to ad-dress all the above mentioned issues, taking into account structureformation, selection functions, redshift-dependent bias descriptionsand redshift-space distortions within a forward Bayesian analysis.

5.1 Setup of the Reconstructions

The COSMIC BIRTH code in its current version performs calcula-tions on cubical regular meshes in comoving Cartesian coordinatesand uses the corresponding galaxy positions in redshift-space onthe light-cone with their corresponding bias obtained from galaxycatalogues as input source.

First we assume in this study a ΛCDM model with cosmo-logical parameters defined in Equation 1. Then we select the inputcatalogues comprehending 5 different redshift surveys, as describedin Section 3.1. The large-scale bias as a function of redshift is ob-tained from the data itself and further constrained according tosome previous studies, as explained in Section 4.2. From this theconnection to Lagrangian bias including a nonlinear and nonlo-cal treatment is internally computed, as explained in Section 2.2.1.With the given cosmology we can translate the angular and redshiftcoordinates for each galaxy into comoving Cartesian coordinates(𝑥, 𝑦, 𝑧). The corresponding geometry and angular completeness toeach survey is computed, as explained in detail in Section 3.2.1.While the COSMIC BIRTH code does not assume the plane paral-lel approximation at any step, this approximation is nearly fulfilledgiven the large distances to the galaxies of the considered redshiftrange and the narrow angular coverage (see Figure 1). We take ad-vantage of that for visualisation purposes, and choose a coordinatetransformation so that the centre of the zCOSMOS-deep survey isaligned to DEC = 0◦ and R.A. = 180◦, which makes the 𝑌 -axisapproximately coincide with the declination angle (for angles closeto zero, as in this case) and the 𝑋-axis with the redshift 𝑧 (see e.g.Figures 5,11). The final results are presented in the original co-ordinate system. The reconstructions comprise the galaxies of thefive mentioned surveys within a redshift range of 1.4 ≤ 𝑧 ≤ 3.6,which translates into a comoving distance of 𝑑Box = 1898 ℎ−1Mpcalong the line-of-sight. For computational reasons we have split thereconstruction into four cubical volumes making sure that a largeenough volume (in terms of mode-coupling) is taken in each caseof 512 ℎ−1Mpc side length (Sorce et al. 2016). This resulted in amesh grid resolution resolution of 2 ℎ−1Mpc using meshes of 2563

10−3

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101

P(k

)[h−

3M

pc3

]

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10−1 100

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0.60.81.01.2

P/P

Th

eory

0

10

20

30

40

50

60

Figure 7. Power spectra of the inferred initial density field 𝛿 (𝑞) at 𝑧 = 100shown colour coded for the first 60HMC samples for the sub-volume rangingfrom 1.7 ≤ 𝑧 ≤ 2.2. In the bottom panel the ratio of the power spectra andthe theoretical prediction are shown. Convergence is roughly achieved afterthe ∼ 40th samples.

voxels. We placed the four volumes successively along the line-of-sight considering an overlapping region of 50 ℎ−1Mpc on each side.This is an adequate choice, acknowledging that the galaxy-galaxycorrelation function drops steeply at scales larger than 20 ℎ−1Mpc(see e.g. Anderson et al. 2012). However, boundary effects (e.g.velocity correlations) at the transition region of the sub-volumesare the primary source of uncertainty in the reconstructions, whichwill be further analyzed in forthcoming works. Additionally, we didnot place galaxies in a buffering zone of 25 ℎ−1Mpc at the edgesof each volume, which we took into account within the radial se-lection function accordingly. This means, that the line-of-sight dataregion is 5-7 times larger than in transverse directions for each sub-volume (see Figure 2). Thus, the full reconstructions extends from2875 ℎ−1Mpc to 4773 ℎ−1Mpc in comoving line-of-sight distance.We note, that COSMIC BIRTH code takes light-cone evolution intoaccount within each reconstructed volume. We choose 8 redshiftbins for each sub-volume (see Figure 4 in Kitaura et al. 2019).

5.2 Numerical Assessment & Convergence

Our numerical tests showed that the COSMIC BIRTH reconstructionsare insensitive in terms of the power spectrum to deviations fromup to 30% from our best large-scale bias evolution estimate shownin Figure 6. We note, that this already excludes the high end of biasvalues allowed within 2𝜎 confidence levels by the observations (seediscussion in Section 4.2). The convergence behaviour of the powerspectra from the Lagrangian density fields (for the best large-scalebias model), starting from a perfectly homogeneous density field𝜹(𝒒) = 0 is shown in Figure 7. The reconstructions show unbi-ased power spectra already after the 40th iteration step. Up to thisiteration we consider the chain to be in the burn-in phase and notrepresenting the target distribution of which we aim at drawing sam-ples from. Furthermore, the top panel of Figure 8 shows the meanpower spectrum of the initial density field 𝛿(𝑞) at 𝑧 = 100 aver-aged over 6000 samples, illustrating the 1𝜎 standard deviation as agrey band. The bottom panel shows the corresponding ratios withthe theoretical linear ΛCDM prediction. The reconstructed density

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12 Metin Ata et al.

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)[h−

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Theory P z=100th (k)

Mean birth

1σ birth

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k[hMpc−1]

0.00.51.01.52.0

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eory

Figure 8. Mean power spectrum of the inferred initial density field 𝛿 (𝑞)shown for the sub-volume ranging from 1.7 ≤ 𝑧 ≤ 2.2. The red dashed linerepresents the mean power spectrum averaged over 6000 individual HMCrealisations with 1𝜎 standard deviation shown as a grey band. In the bottompanel we show the ratio of the mean power spectrum and its uncertaintyband with the theoretical ΛCDM prediction.

0 50 100 150 200Iteration Distance n

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Cor

rela

tion

Len

gthC(δ j

)n

200 400 600 800 1000δj

Figure 9.Correlation length C(𝛿 𝑗 )𝑛 of 1000 randomly chosen density fieldvoxels 𝛿 𝑗 , where the completeness is 𝑤𝑗 > 0. We compute the correlationlength over a sample of 𝑁 = 6500 HMC realization with an iteration length𝑛 ∈ [0...200] shown for the sub-volume ranging from 1.7 ≤ 𝑧 ≤ 2.2. Thecolour code indicates the correlation length for a particular 𝛿 𝑗 while theblack solid line represents the average over all density voxels. We considerC(𝛿 𝑗 )𝑛 < 0.1 to be uncorrelated, which our Gibbs-sampling chain dropsbelow after 𝑛 = 40 iterations. Thus, each 40th realization is an independentsample.

fields show unbiased power spectra over all scales, confirming anaccurate bias treatment. The effect of a mishandled bias in our studyis shown in Appendix A. We note that sampling the large-scale biasfrom sparse data sets as considered herewith such a low data volumefraction is going to be inaccurate, although the Bayesian frameworkallows for it (see Granett et al. 2015; Jasche & Lavaux 2017).

Once the cosmology has been chosen, and the data input de-

0.0

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1.0

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F

Lagrangian density at z = 100

−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00δ

0.0

0.5

1.0

PD

F

Eulerian light-cone density 1.7 ≤ z ≤ 2.2

Figure 10. Histogram of the inferred density fields normalized to unity for−1 ≤ 𝛿 ≤ 1. The top panel shows a histogram of the density at initialconditions (Lagrangian coordinates) at 𝑧 = 100. The bottom panel shows ahistogram of the light-cone density field in Eulerian coordinates in the rangeof 1.7 ≤ 𝑧 ≤ 2.2. The Lagrangian density on the top panel shows a closelyGaussian distribution with a negligible skewness of 𝑠 = 0.001, while theEulerian density on the bottom panel presents a highly skewed distributionwith 𝑠 = 5.373.

fined, there is only one free parameter in the COSMIC BIRTH code,which is the large-scale bias of the different populations. As shownin Section 4, there is a confidence region for the large-scale bias as afunction of redshift for the galaxies considered in this study. Giventhe the low volume filling fraction of the data region with respect tothe entire volume, which needs to be considered to keep the mode-coupling effects from large-scale modes low (see e.g. Sato et al.2009; Takada & Hu 2013), the question arises whether the recon-structions are sensitive to bias. To validate the independence of theHMC samples, we calculate the correlation length for the inferreddensity field. The correlation length C(𝛿 𝑗 )𝑛 for a particular densityvoxel 𝛿 𝑗 at an iteration distance 𝑛 over 𝑁 samples is given by:

C(𝛿 𝑗 )𝑛 =1

𝑁 − 𝑛

𝑁−𝑛∑︁𝑖=0

(𝛿𝑖𝑗− 〈𝛿 𝑗 〉)√︃𝜎2 (𝛿 𝑗 )

(𝛿𝑖+𝑛𝑗

− 〈𝛿 𝑗 〉)√︃𝜎2 (𝛿 𝑗 )

, (23)

where 〈𝛿 𝑗 〉 = 1𝑁∑𝑖 𝛿

𝑖𝑗is the mean of the density voxel 𝛿 𝑗 over 𝑁

samples and 𝜎2 (𝛿 𝑗 ) = 1𝑁∑𝑖

(𝛿𝑖𝑗− 〈𝛿 𝑗 〉

)2the corresponding vari-

ance. We show the correlation length in Figure 9 for 1000 randomlychosen density voxels 𝛿 𝑗 , with 𝑗 ∈ [1...1000] in the data region ofour reconstructed volume. This demonstrates that we draw indepen-dent samples each ∼40th Gibbs-sampling iteration.

To quantify the Lagrangian to Eulerian mapping using ALPTwithin COSMIC BIRTH, we show the density distribution of theinitial and final density fields in Figure 10. On the same scaleon the x-axis the density field value 𝛿 is plotted against the nor-malized probability distribution. We can see that the density fieldin Lagrangian space follows closely a Gaussian distribution withmean peaked at zero, and very small variance and skewness,𝜇 ≈ 0, 𝜎2 = 0.001, 𝑠 = 0.001. The histogram for the Euleriandensity field also has a mean value close to zero, 𝜇 ≈ 0, howevershows a variance of 𝜎2 = 0.95, and a skewness of 𝑠 = 5.373. Thisis induced by gravity over cosmic time scales and is in excellentagreement to previous findings studied in detail in Neyrinck (2013),comparing the displacement fields of second order Lagrangian per-turbation theory (2LPT) and 𝑁-body simulations at different red-

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BIRTH of the COSMOS Field 13

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5

Figure 11. COSMIC BIRTH reconstructed density field 1+𝜹 shown as slices in line-of-sight and declination coordinates with thickness of 6 ℎ−1Mpc. Phase-spacemapping with the corresponding tetra-hedra tesselation has been performed as mentioned in Section 2. The whole declination angle on the𝑌 -axis correspondsto 200 ℎ−1Mpc in comoving coordinates. From top to bottom the two upper slice plots, panels 1 and 2, show firstly a single realization the Lagrangian initialdensity field 1 + 𝛿 (𝒒) at 𝑧 = 100 and secondly a light-cone realization at Eulerian redshift-space 1 + 𝜹 (𝒔) with the individual galaxy positions plotted on top.Panels 3 and 4 represent the mean distributions of the inferred initial Lagrangian 〈1 + 𝜹 (𝒒) 〉 at 𝑧 = 100 and final Eulerian 〈1 + 𝜹 (𝒔) 〉 density fields averagedover 6000 HMC realizations, respectively, where on top of the Eulerian density field the galaxy positions are plotted with black dots. Finally, we show in panel5 the signal-to-noise ratio 𝜇/𝜎 of the Eulerian density field with the galaxy positions plotted on top.

shifts. Especially at redshifts 𝑧 ≥ 1 the 2LPT displacements andthe 𝑁-body results are in very good agreement and thus representa very reasonable choice for this study (even more so, since we useALPT).

5.3 Density Inference Results

The resulting reconstructions corresponding to our best large-scalebias evolution estimation are shown in Figure 11 as slice plots. On

the 𝑋-axis we show the line-of-sight distance4 as redshift 𝑧, and onthe 𝑌 -axis we show the declination (DEC). The slices are shownwith a thickness of 6 ℎ−1Mpc (averaging over three neighbouringcells). The corresponding convergence behaviour, power spectra,

4 We note that the line-of-sight distance corresponds to the redshift of theobservations only for the light-cones. For the initial density fields 𝜹 (𝒒) at𝑧 = 100, 𝑋 -axis only shows a distance measure.

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14 Metin Ata et al.

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1004× 10−1 6× 10−1 2× 100

Figure 12. Tomographic slices of the reconstructed matter density field in the declination-redshift plane. On the left we show the angular footprint of the 5surveys in the same colour code as Figure 2 with a grey line showing the R.A. thickness of the corresponding slice. On the right the mean light-cone densityreconstructions are shown.

and matter statistics was discussed in Section 5.2 and shown inFigures 7–10.

We find homogeneously distributed initial cosmic densityfields (see first panel of Figure 11 for individual reconstructionsof 1 + 𝜹(𝒒)). This is further demonstrated in the statistical matterdistribution shown in the upper panel of Figure 10) with negligibleskewness and kurtosis values, as expected for a Gaussian densityfield. Furthermore, the top panel of Figure 11 shows no redshiftevolution. This is in contrast with the second upper panel, in whichthe Eulerian density for the corresponding reconstruction 1 + 𝜹(𝒔)is shown. In fact, the corresponding matter statistics shown in thelower panel of Figure 10 shows a highly non-Gaussian distribution.Further inspection of the second panel of Figure 11, shows an in-crease of the matter fluctuations towards low redshifts, particularly

enhanced through the appearance of large cosmic voids, depicted indark blue. The corresponding ensemble averages (over 6000 Gibbs-sampling iterations) to panels 1 and 2 are shown in the panels 3 and4, 〈1 + 𝜹(𝒒)〉 and 〈1 + 𝜹(𝒔)〉, respectively. In particular, the aver-age over the ensemble of Eulerian density fields (i.e., the expecteddark matter field from a Bayesian calculation) shows a high corre-lation with the galaxy field, as it should happen, when the initialcosmic density field is accurately recovered. One can also appreci-ate the vanishing fluctuations in regions with low completeness, inconcordance with a Bayesian analysis (see for a comparison, as anexample the completeness of VUDS depicted in Figure 5). Panel5 shows the signal-to-noise ratio obtained through the ratio of themean density over its standard deviation 𝜇/𝜎. As expected, we finda higher signal-to-noise ratio in the data region. Interestingly, cos-

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BIRTH of the COSMOS Field 15

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2.32.352.42.452.5Redshift z

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Figure 13. Density plots for a zoomed region. On the left panel the galaxy density 1 + K (𝑟S) ∗ 𝛿G (𝒔) is shown convoluted with a Gaussian smoothing kernelK(𝑟S) with smoothing radius of 𝑟S = 2 ℎ−1Mpc, followed by the mean density field in initial (middle panel) 1 + 𝛿 (𝒒) and final (right panel) conditions1 + 𝛿 (𝒔) for the same slice as shown in Figure 11. Overplotted with black dots are shown the corresponding galaxy positions.

mic voids are particularly prominent in this measure. The missingdata or equivalently, empty window function regions increase theuncertainty of the corresponding density cells and thus the varianceof the density field (see Equation 6). This would be reflected in thesignal-to-noise ratio, shown in the bottom panel of Figure 11. Wehowever see no significant decrease in the signal-to-noise ratio atthe gaps of the VUDS survey.

To further assess the matter distribution in the reconstructions,we show tomographic slice plots for various R.A. ranges (greyvertical stripes) in Figure 12, showing the survey footprints on theleft-hand side and the corresponding density reconstructions next toit on the right. This permits us to evaluate the extension of the proto-clusters and cosmic voids. A comparison between the Lagrangianand Eulerian ensemble average reconstructions (middle panels inFigure 11) shows higher density fluctuations in Eulerian space,and small displacements of the center of mass of the proto-clusterstowards high redshifts. Focusing on the lowest redshift regions onecan observe stronger variations of the shape of the proto-clusters.

To further investigate this, we zoom into a prominent proto-cluster region with 2.28 ≤ 𝑧 ≤ 2.51 and 1.6◦ ≤ DEC ≤ 2.9◦(see Figure 13). This comparison shows three high density proto-clusters at 𝑧 ∼ 2.45, 𝑧 ∼ 2.4 and 𝑧 ∼ 2.3 growing throughoutthe cosmological timescale starting from 𝑧 = 100 down to theirlight-cone redshifts. At first glance, the proto-clusters seem to growin place, with negligible displacements, in accordance with linearperturbation theory, describing the growth of perturbation fixed incomoving frame between the initial redshift 𝑧𝑞 to the final redshift𝑧 𝑓 with 𝛿(𝒓, 𝑧 𝑓 ) = 𝛿(𝒓, 𝑧𝑞) 𝐷 (𝑧 𝑓 )/𝐷 (𝑧𝑞), where 𝐷 (𝑧) is the lin-ear growth factor. Linear theory from 𝑧𝑞 = 100 to the redshift ofour observed light-cone (∼ 𝑧 𝑓 = 2.3) predicts a growth by a fac-tor of approximately 30, which is consistent with the colour bar ofthe first and second panel of Figure 11. However, a more detailedinspection reveals a complex non-spherical accumulation of masswhen comparing the middle and the right panels. This can be wellappreciated when comparing to the Gaussian smoothed galaxy fieldshown on the left panel. We also find from this comparison howthe two proto-clusters on the left, which appear as separate entitiesin the left panel, are actually connected, most likely having a darkmatter bridge in between their respective galaxy distributions. It isalso interesting to see the accumulation of matter through the actionof gravity from a ring-like shape cloud to a spherical overdensityregion in the region centred at 𝑧 ∼ 2.38 and DEC ∼ 1.9◦, compar-ing the Lagrangian and Eulerian density fields. We also focus onthe prominent 𝑧 ∼ 2.1 proto-cluster and show how this region isreconstructed within different individual realizations in Appendix

B. This Bayesian analysis taking the completeness into account,also reveals regions that are more likely to be true cosmic voids. Inparticular, the right panel shows a deep void towards higher decli-nation angles in the redshift range 2.3 < 𝑧 < 2.4, which feeds theoverdensity peaks on its left and right. This is in agreement withthe cosmic void initially discovered from the three dimensional Ly𝛼forest tomography (Krolewski et al. 2018). Another void region isforming directly under the proto-clusters at 𝑧 ∼ 2.45 and 𝑧 ∼ 2.4.We will provide a more detailed analysis of the various structuresin the reconstructed density field in a subsequent paper.

6 CONCLUSIONS AND DISCUSSION

In this work we presented for the first time a comprehensive multi-survey reconstruction effort of the primordial and evolved densityfields with the COSMIC BIRTH algorithm performed in the COS-MOS field during the epoch of Cosmic Noon (1.4 ≤ 𝑧 ≤ 3.6). Toour knowledge, this is the ever attempted large-scale structure anal-ysis of its kind at high redshift, probing the quasi-linear regime ofgravitational structure formation, before non-linear shell-crossingstarted dominating the emerging cosmic web.

We combined the data from five spectroscopic galaxy surveysin the COSMOS field, which have partially overlapping footprints,and do not share a common observing strategy. Therefore, we had tomake a special effort in estimating the angular selection function foreach survey, based on the the individual targeting strategies for theparent photometric catalogs, which was missing for these surveys.

Also, we applied for the first time a multi-tracer and multi-survey likelihood formalism in Lagrangian space within a Bayesianinference framework. This allowed us to combine surveys with dif-ferent selection functions, galaxy bias, number densities, and red-shift ranges, connected through the underlying dark matter distribu-tion. Although earlier works have presented reconstructions usingmultiple galaxy populations, they however shared a unique surveygeometry. Therefore, our new method is more general and has awider range of possible applications. Since we expect a correla-tion between the spatial distribution of the galaxy catalogues, acovariance term would in principle be necessary when different ob-servations are combined within one joint likelihood analysis. Webypass this problem by firstly mapping all tracers to Lagrangiancoordinates. This allows us to assume Poisson likelihoods, sinceonly in the homogeneous epoch before gravity coupled separatedspatial regions, identical and independently distributed large-scalestructure tracers can be assumed.

MNRAS 000, 1–19 (2020)

16 Metin Ata et al.

Despite of the large number of surveys, the COSMIC BIRTHcode showed an efficient performance, converging within about 40iterations, and showing low correlation lengths of about 40 Gibbs-sampling iterations.

The resulting reconstructions reveal for the first time a holisticview on the matter density field and its primordial fluctuationsjointly inferred from five spectroscopic surveys.

We also revised the bias of star forming galaxies towards highredshifts, finding some moderate evidence for a stronger evolutionthan the one described by passive evolution.

The inferred density fields have a large number of potential ap-plications. In particular, we have found several high density regionsacross the whole reconstructed volume. We successfully recon-structed a number of observationally known proto-cluster regionspreviously reported by Cucciati et al. (2014); Chiang et al. (2015);Diener et al. (2015); Casey et al. (2015); Lee et al. (2016); Wanget al. (2016); Nanayakkara et al. (2016); Darvish et al. (2020). How-ever, these previous proto-cluster studies were typically performedusing individual galaxy surveys directly on the observed galaxypositions in redshift-space. Major improvements have been donecombining two surveys and statistically sampling the redshifts intwo-dimensional slices including a Voronoi tessellation to estimatethe galaxy density field (Cucciati et al. 2018). Still, the mass esti-mates are prone to projection effects due to their peculiar velocities(Kaiser 1984, 1987), and also assuming a velocity dispersion mea-sure which is not valid for non-virialised objects at high redshifts.In a subsequent paper, we will carry out a more detailed analysis ofthese structures. Since we have reconstructed the initial conditionsof the COSMOS volume, we will be able to run constrained 𝑁-bodysimulations based on the inferred initial conditions, enabling us tostudy the full cosmic evolution of the proto-clusters in detail (Ataet al. in prep.). This will, for exampl