The cosmic web of the Local Universe: cosmic variance, matter content and its relation to galaxy morphology
Sebastian E. Nuza, Francisco-Shu Kitaura, Steffen Hess, Noam I. Libeskind, Volker Mueller
aa r X i v : . [ a s t r o - ph . C O ] A ug Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 30 September 2018 (MN L A TEX style file v2.2)
The cosmic web of the Local Universe: cosmic variance,matter content and its relation to galaxy morphology
Sebasti´an E. Nuza ⋆ , Francisco-Shu Kitaura † , Steffen Heß ‡ , Noam I. Libeskindand Volker M¨uller Leibniz-Institut f¨ur Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
ABSTRACT
We present, for the first time, a Local Universe (LU) characterization using high preci-sion constrained N -body simulations based on self-consistent phase-space reconstruc-tions of the large-scale structure in the Two-Micron All-Sky Galaxy Redshift Survey.We analyse whether we live in a special cosmic web environment by estimating cosmicvariance from a set of unconstrained ΛCDM simulations as a function of distance torandom observers. By computing volume and mass filling fractions for voids, sheets,filaments and knots, we find that the LU displays a typical scatter of about 1 σ at scales r & h − Mpc, in agreement with ΛCDM, converging to a fair unbiased sample whenconsidering spheres of about 60 h − Mpc radius. Additionally, we compute the mat-ter density profile of the LU and found a reasonable agreement with the estimatesof Karachentsev (2012) only when considering the contribution of dark haloes. Thisindicates that observational estimates may be biased towards low density values. As afirst application of our reconstruction, we investigate the likelihood of different galaxymorphological types to inhabit certain cosmic web environments. In particular, we findthat, irrespective of the method used to define the web, either based on the densityor the peculiar velocity field, elliptical galaxies show a clear tendency to preferentiallyreside in clusters as opposed to voids (up to a level of 5 . σ and 9 . σ respectively)and conversely for spiral galaxies (up to a level of 5 . σ and 5 . σ respectively). Thesefindings are compatible with previous works, albeit at higher confidence levels. Key words: cosmology: large-scale structure of the Universe – cosmology: theory –galaxies: general – methods: observational – fig: methods: numerical
The location of our place in the Universe was found to bequite particular in a number of studies. Our local environ-ment within a distance of about 50 Mpc appears to be lessdense than the expected mean density of the Universe (e.g.,Vennik 1984; Tully 1987; Magtesyan 1988; Bahcall et al.2000; Crook et al. 2007; Makarov & Karachentsev 2011;Karachentsev 2012). Additionally, a prominent low den-sity region, the so-called Local Void, is found to be lo-cated in our immediate vicinity (Tully & Fisher 1987). Atlarger distances, it has been found that most of the nearbyclusters lie on an approximately ring-like structure thatsurrounds our location. This feature led de Vaucouleurs(1953) to propose the super-galactic coordinate system in ⋆ E-mail: [email protected] † E-mail: [email protected], Karl-Schwarzschild-fellow ‡ E-mail: [email protected] the plane of this structure to enhance this curious pattern(de Vaucouleurs et al. 1976, 1991).After considering these peculiarities a natural questionarises: is our location in the Universe special? More specif-ically: how likely are the prevailing structures in our LocalUniverse (LU)? Are these local patterns capable of challeng-ing the ΛCDM cosmology? To answer these questions wefirst need to define those cosmic patterns in a quantitativeway. The distribution of galaxies as obtained by galaxy red-shift surveys, such as the Two-Micron All-Sky Galaxy Red-shift Survey (2MRS; Huchra et al. 2012) which we considerin this work, reveals the existence of an intricate networkof interconnected structures comprising a web of walls, fil-aments, galaxy clusters and voids (Zel’dovich 1970; Peebles1980; Bond et al. 1996). According to the picture introducedby Zel’dovich (1970) such a cosmic web naturally arises as aresult of gravitational collapse. Early numerical simulations(e.g., Doroshkevich et al. 1980; Davis et al. 1985) have con-firmed this scenario. These different environments are char- c (cid:13) Nuza et al.
Figure 1.
Mollview projection in Galactic coordinates of all galaxies in the 2MRS catalogue (red dots) at distances of 50 − h − Mpcgenerated using HEALPix (G´orski et al. 2005). The grey colour scale shows the DM density field of our best-correlated N -body recon-struction. The initial conditions of the reconstruction have been produced using the kigen -code (see Section 2.2). Note that the emptyregion around the Galactic plane corresponds to the Zone of Avoidance . acterized by their distinctive dynamical nature: voids areexpanding low-density regions whereas clusters are collaps-ing dense structures residing at the intersection of elongatedfilaments. Similarly, filaments are ‘chains’ of galaxies beingconstantly stretched across their major axes that are locatedwhere two-dimensional expanding sheets meet.Ever since the pioneering works of Fry & Peebles (1978)and White & Rees (1978) the hierarchical structure forma-tion paradigm has been established in cosmology. Accordingto this picture, structures are formed in a hierarchical pro-cess, in which smaller objects merge to form larger ones.Such a process will predominantly occur in higher densityenvironments, where mergers are more likely to happen.Numerical simulations have long since shown that mergersor tidal interactions can destroy galactic discs convertingspiral or irregular galaxies into elliptical and S0 galaxies(Toomre & Toomre 1972; Farouki & Shapiro 1981). There-fore, this mechanism can naturally generate a morphologi-cal segregation of galaxies as the density of the environmentincreases. In fact, such a morphology-density relation wasalready discovered by Oemler (1974) and Dressler (1980),showing that star-forming, disc-dominated galaxies prefer-entially reside in lower density regimes as opposed to ellip-tical ones.A large number of works have further investigatedthe relation between the environment and galaxy proper-ties in the LU, such as morphological type, stellar mass, (specific) star formation rate, colour and luminosity (e.g.,Balogh et al. 2001, 2004; Hogg et al. 2004; Kauffmann et al.2004; Tanaka et al. 2004; Blanton et al. 2005; Croton et al.2005; Alonso et al. 2006; Baldry et al. 2006; Mart´ınez et al.2006; Weinmann et al. 2006; Einasto et al. 2007; Park et al.2007; Ball et al. 2008; van der Wel 2008; Deng et al.2011; Tempel et al. 2011; Zandivarez & Mart´ınez 2011;Alonso et al. 2012; Wetzel et al. 2012; Lackner & Gunn2013). Most of these studies have focused on the high densityregime, which can relatively easily be determined throughthe local number density of galaxies. Additionally, otherauthors have focused in the classification of filamentary-like structures (e.g., Sousbie et al. 2008; Stoica et al. 2010;Sousbie et al. 2011; Smith et al. 2012; Beygu et al. 2013;Tempel et al. 2014) and voids (e.g., M¨uller et al. 2000;Hoyle & Vogeley 2004; Ceccarelli et al. 2006; Kreckel et al.2012; Lietzen et al. 2012; Pan et al. 2012; Sutter et al. 2012;Ceccarelli et al. 2013; Sutter et al. 2013; Tavasoli et al.2013) as a way to shed light on the galaxy formation processin these environments.Despite these efforts, a comprehensive study of the lo-cal environment with respect to the nonlinear cosmic web isstill missing. In this respect, some remarkable attempts havebeen presented in the literature. For instance, Lee & Lee(2008) used a LU reconstruction based on a linear Wienerfilter method that, however, tends to smooth out struc-tures at scales of the order of 10 h − Mpc (see Erdoˇgdu et al. c (cid:13) , 000–000 he cosmic web of the Local Universe N -body simulations with a group galaxy catalog. Alternatively,Aragon-Calvo (2012) performed an ensemble of randomlyseeded N -body simulations displaying the same large-scalestructure, but different small-scale perturbations, aiming atstudying the statistics of haloes as a function of environ-ment.In this context, it is mandatory to revise the environ-mental studies with more accurate reconstructions of thelarge-scale structure. We therefore extend such works us-ing the recently performed high precision constrained sim-ulations that correlate with 2MRS galaxies down to a fewMpc scales (Heß et al. 2013). These simulations are based onthe first self-consistent phase-space reconstruction methodof the primordial fluctuations corresponding to a set of mat-ter tracers (the kigen -code: Kitaura 2013). At the sametime, the unprecedented accuracy of our simulations, per-mit us to characterize the dark matter (DM) content withinthe LU and compare with observational estimates.In this work, we present a systematic study of the cos-mic web, as predicted both by random simulations and pre-cise reconstructions of the LU after studying the eigenval-ues of the tidal field tensor of the density field (Hahn et al.2007; Forero-Romero et al. 2009). As will be explained fur-ther, while classifying the web, we will use the informationcontained in the nonlinear and linear reconstructed densityfields to assess the robustness of the measurements. It isworth noting, however, that the linear overdensity will beestimated only by means of the nonlinear velocities of thereconstruction which are known to be more linear than thedensity field (see e.g., Zaroubi et al. 1999; Kitaura & Angulo2012 and references therein). Moreover, the reconstructedvelocity field yields a complementary environmental viewthat is based on the kinematics of the LU.In summary, the aim of the present work is twofold. Inthe first place, we aim at studying the LU matter contentas well as the impact of cosmic variance as a function ofdistance to the observer. In this way, we will be able to as-sess the scale at which our LU becomes a fair sample . To doso, we characterize the cosmic web of the LU by using highprecision constrained N -body simulations with two differ-ent approaches. In particular, we compute the volume andmass filling fractions (VFFs and MFFs, respectively) of dif-ferent cosmic web environments. Then, we compare thesestatistics with those corresponding to a set of unconstrainedrandomly seeded ΛCDM simulations. Secondly, we want tobenefit from the high level of accuracy of our reconstructionsto measure the correlation between galaxy morphology andtheir location within the cosmic web, presenting a first ap-plication of the LU density field estimation for the galaxypopulation inhabiting the nearby Universe.This paper is structured as follows. First, in Section 2,we introduce the dataset used to generate the ICs, thenwe briefly present the method applied to reconstruct thelarge-scale structure, as well as some details of our con-strained simulations. In Section 3 we compute the matter density parameter of the LU as a function of distance to theobserver and compare with recent observational estimates.In Section 4 we discuss the large-scale structure classifica-tion methods used to define the local web and the resultingstatistics in each case. In Section 5 we assess the cosmicvariance level of the LU in comparison with the expecta-tions of ΛCDM. In Section 6 we study the galaxy morphol-ogy/cosmic web correlation, for our web classifications, asa first application of the LU reconstruction. Finally, in Sec-tion 7, we present a summary and give our conclusions. Our study is based on the Two-Micron All-Sky Galaxy Red-shift Survey (2MRS) K s = 11 .
75 catalogue presented byHuchra et al. (2012). The unprecedented sky coverage (91%)and uniform completeness (97%) in the 2MRS galaxy cat-alogue are ideal to probe the characteristics of local struc-tures. Observations are only limited by the
Zone of Avoid-ance . We note that we could treat the mask in a self-consistent way within the reconstruction process (see e.g.,Jasche et al. 2010). However, the small number of galaxiesaffected by the mask, permits us to correct for this effectusing the data augmented catalog by Erdoˇgdu et al. (2006).This is done adding random galaxies drawn from the cor-responding adjacent strips of the survey (Yahil et al. 1991).We consider the volume within a box of 180 h − Mpc on aside with its centre located at the observer’s position. Asa result, our final sample comprises 31,017 galaxies, whichcorresponds to about 76% of the 2MRS survey. In this way,we are able to avoid the steep decrease of the radial selec-tion function at the edge of the survey containing less thanone quarter of the available galaxies. This also permits us tominimize the Kaiser-rocket effect (Branchini et al. 2012).
To obtain the full nonlinear phase-space distribution of aset of matter tracers in redshift space, we rely on the ki-gen -code (Kitaura 2013; Kitaura et al. 2012), which is thefirst self-consistent, phase-space, forward reconstruction ap-proach. Other remarkable forward approaches have been de-veloped (see Jasche & Wandelt 2013; Wang et al. 2013), al-though without including a self-consistent estimation of pe-culiar velocities, as we do here.The advantage of the kigen -code is manifold, as it in-cludes:(i) An accurate gravitational collapse model on Mpcscales combining second order LPT (2LPT) at large scaleswith the spherical collapse model at cluster scales (dubbed‘Augmented’ Lagrangian perturbation theory, hereafterALPT; Kitaura & Heß 2013) that suppress shell-crossingin the high-density regime and improves the description offilaments at lower densities (see also Neyrinck 2013).(ii) Nonlinear coherent and virialised redshift-spacedistortions built-in in the second step (likelihood compar-ison) of the kigen -code (see Heß et al. 2013), i.e. without c (cid:13) , 000–000 Nuza et al.
Figure 2.
DM overdensity map in a slice of 22 . h − Mpc centred on the position of the observer for our best LU reconstruction in realand redshift space (left and right panels respectively; see Heß et al. 2013). Units are in supergalactic coordinates. artificially compressing the Fingers-of-God.(iii) A nonlinear scale dependent bias in Lagrangian spaceto ensure that the reconstructed primordial density fieldsyield unbiased power spectra, as compared to the lineartheoretical model (this happens in the first step of ki-gen ). In particular, we use the exponential relation pro-posed by Cen & Ostriker (1993) that is able to modelthe nonlinear scale-dependent bias as it was shown inde la Torre & Peacock (2013). We note that the joint treat-ment of peculiar velocities (see (i)) and biasing in the initialconditions is essential to break the degeneracy present in thetwo-point statistics. N -body simulations In this work, we rely on constrained N -body simulationsbased on reconstructions performed with the kigen -code ap-plied to the 2MRS survey (Heß et al. 2013). In particular,we consider in our study a subset of 25 constrained sim-ulations displaying the largest cross-correlations with thegalaxy data. To estimate the degree of correlation betweenthe reconstructed DM density field ( δ DM ) and galaxy over-densities ( δ G ) we define the cross power-spectrum as XP ( k )[ δ DM , δ G ] ≡ h| ˆ δ DM ( k )ˆ δ G ( k ) |i p P DM ( k ) p P G ( k ) , (1)where the ensemble brackets denote angular averaging and P DM ( k ) and P G ( k ) are the associated power spectra. A cell-to-cell comparison of the logarithmic density fields in config-uration space shows a typical Pearson coefficient of 74% for acell width of 1 . h − Mpc. The high degree of correlation be-tween the 2MRS galaxy distribution and the reconstructeddensity field can be seen in Fig. 1. This plot shows a Mol- lview sky projection of all matter (grey scale) and galaxies(red dots) in our best-correlated real-space reconstruction ata distance of 50 − h − Mpc. A remarkable spatial coinci-dence between the galaxy tracers and the underlying densityfield can be observed.The difference between real and redshift space for ourLU reconstruction can be seen in Fig. 2 (left and right panelsrespectively). This figure demonstrates the squashing effectproduced along the line-of-sight as a result of the coherentpeculiar motions of galaxies. In this sense, it is important toperform the cosmic web analysis both in configuration andredshift space in order to check for differences in the results.These runs are complemented with an analogous set of DM-only simulations carried out using unconstrained randomphases for the ICs adopting the same cosmology and sim-ulation parameters. This set of unconstrained simulationswill serve to assess the level of cosmic variance of the LUreconstructions. The reader is referred to Heß et al. (2013)for further details.A flat ΛCDM model consistent with WMAP7 results(Komatsu et al. 2011) was considered in these simulations,i.e. with a matter density Ω M = 0 . Λ = 0 . b = 0 . h = 0 . σ = 0 .
807 and a scalar spectral in-dex n s = 0 . z = 100 and were evolved with the gadget-3 code (Springel2005; Springel et al. 2008) using 384 particles, which trans-lates into a DM particle mass of 7 . × h − M ⊙ , and acomoving gravitational softening of 15 h − kpc. Throughoutthis paper, we will use the density fields of the N -body sim-ulations obtained from their corresponding DM distributionusing a standard cloud-in-cell technique. The volume has c (cid:13) , 000–000 he cosmic web of the Local Universe Figure 3.
Matter density parameter as a function of radius forincreasingly larger spheres as obtained from our reconstructionof the LU. Shown are the results corresponding to the DM aswell as the halo contribution for virial halo mass cuts of M vir > and 5 × h − M ⊙ (black and blue lines respectively).Also shown are the observational results of Karachentsev (2012)which have been estimated from an updated local galaxy sample(circles). The solid circle with error bars stands for the integratedobservational result within 50 Mpc ( ≡ h − Mpc). been gridded with 128 cells thus giving a cell side length ofabout 1 . h − Mpc.DM haloes in the simulations are identified using the ahf code (Knollmann & Knebe 2009) as spherical overden-sities with a density 200 times above the critical densityof the universe. In this way, a halo sample with masses inthe range ∼ − h − M ⊙ was selected for our LU re-constructions (see Fig. 10 of Heß et al. 2013, for the massfunction of our constrained realisations). To test for reso-lution and cosmology effects, when haloes are considered,we also used a higher resolution reconstruction with 768 particles within the Planck cosmology (with Ω M = 0 . . × h − M ⊙ . Throughout the years there have been claims that the LUmay not be a fair sample of the Universe after studyingthe contribution of galaxy groups to its DM content. Typi-cally these estimates place the local matter density value inthe range Ω M , LU ≈ . − .
2, i.e. below the cosmologicalmean density by a factor of a few (see e.g. Karachentsev2012 and references therein). These results have been some-times interpreted as the consequence of an underlying ‘miss-ing DM problem’ that could potentially be in conflict withthe ΛCDM cosmological model. In particular, the recentstudy of Karachentsev (2012) presents an updated anal-ysis of a sample of local galaxies – that includes dwarfs,pairs, triplets and larger groups – favouring an estimate of Ω M , LU = 0 . ± .
02 for the volume contained within a dis-tance of 50 Mpc.The amount of DM as a function of volume in our re-constructions can be straightforwardly checked for consis-tency with observational results. This is shown in Fig. 3where the cumulative matter density parameter is calcu-lated as a function of distance to the observer for our best(real-space) N -body reconstruction. It can be seen that theDM density shows some modulation owing to the particularLU realisation to finally converge to the mean cosmologi-cal value at a distance of about 60 h − Mpc, i.e. approxi-mately 85 Mpc. The Local Void manifests itself as a slighttrough in the cumulative matter density for distances ofabout 30 − h − Mpc. This clearly indicates that the ob-served DM density estimates in the LU cannot be the resultof cosmic variance as the corresponding minimum value inour reconstruction is well above Ω M = 0 . M vir > and 5 × h − M ⊙ (black and blue lines respectively) forthe low (high) resolution N -body reconstruction. These cutsare roughly consistent with the minimum virial mass value ofgalaxy groups with N > . × h − M ⊙ (Makarov & Karachentsev 2011).In general, for the reconstructed halo samples, the predic-tions of both constrained simulations using different matterdensity parameters (Ω M = 0 .
272 and Ω M = 0 . r . h − Mpc). Additionally, the simulated halomatter density shows similar bumps to observations at dis-tances of about 10 − h − M ⊙ which are mainly owingto the presence of the Local Supercluster. At scales below10 h − Mpc, the mismatch between data and the matter den-sity of haloes is likely owing to a combination of the limitedresolution of our simulation and the uncertainties present inthe selection function used in the reconstruction.These results demonstrate that the ‘missing DM’ in theLU can be simply interpreted as being located outside mas-sive haloes, i.e. forming part of the cosmic web ‘field’ inagreement with the ΛCDM expectations. This interpreta-tion is strengthened by the results of Section 4.3. There itwill be shown that, irrespective of the cosmic web classifi-cation used, most of the mass is located outside the densestregions.
To carry out the cosmic web characterisation of the LU weapply to our reconstruction the tidal field tensor classifica- c (cid:13) , 000–000 Nuza et al.
Figure 4.
VFF (left panel) and MFF (right panel) of our best correlated LU reconstruction in real space as a function of thresholdeigenvalue for the T- and div V-web cosmic web classification methods. Different environments are indicated. tion proposed by Hahn et al. (2007). In addition, we alsoexplore the possibility of using the reconstructed nonlinearvelocity field to perform the classification. Throughout thispaper, we will refer to these approaches as the ‘T-web’ and‘div V-web’ respectively. These methods are implementedwithin the classic software package, which will be presentedin a forthcoming publication (Kitaura & Nuza, in prepara-tion).
We study the dynamics of matter by computing the eigen-values λ i ( i = 1 , ,
3) of the tidal field tensor T ij ≡ ∂ φ rec ∂x i ∂x j , (2)where φ rec is the gravitational potential of the reconstructednonlinear density field. By analysing the signature of theeigenvector ~λ = ( λ , λ , λ ) at a given spatial point it is then possible to distinguish between different dynamical be-haviours. For instance, collapsing (expanding) structures inall spatial dimensions will be classified as halo-like (void-like) regions. In a similar way, elongated-like (sheet-like)structures can be identified in the case of a 1-dimensional(2-dimensional) expansion. In what follows, we will desig-nate to these four cases as knots , voids , filaments and sheets respectively.However, not all positive (negative) eigenvalues will giverise to an actual collapse (expansion) along the correspond-ing axis in the inmediate future. Therefore, to achieve amore realistic correspondence between the tidal field ten-sor dynamical classification and the actual cosmic web wefollowed the approach of Forero-Romero et al. (2009): weclassify structures according to the number of eigenvalues( N λ ) above a certain threshold ( λ th ) which, in general, willbe nonzero. As a consequence, cells with N λ = 3 , , , c (cid:13) , 000–000 he cosmic web of the Local Universe S G Y [ h − M p c ] − S G Y [ h − M p c ] − S G Y [ h − M p c ] −
50 SGX [ h − Mpc] −
50 0 50 S G Y [ h − M p c ] − h − Mpc] −
50 0 50
Figure 5.
Cosmic web density field of our best-correlated LU reconstruction in real space as obtained with the T-web method using λ th = 0 . h − Mpc. The differentweb structures are shown in light colours. The resulting VFFs of the web elements are about 7.6%, 0.6%, 20.4% and 71.4% respectively(see Section 4.3). Units are in supergalactic coordinates.
As an alternative way of performing the cosmic web classifi-cation, we use the information of the reconstructed nonlin-ear velocities alone. By taking the divergence of the velocityfield we can infer the expected density distribution under thelinear approximation. Specifically, the linear density field isobtained by computing δ lin = − ∇ · v rec fHa , (3)where v rec is the reconstructed nonlinear velocity field, f isthe logarithmic derivative of the linear growth factor, H isthe Hubble constant and a is the expansion factor. Then, for computing the web, we use the corresponding linear poten-tial as an input of the tidal field tensor classification methoddiscussed above.It is worth noting that in this method – as well as in anyother classification derived from the velocity field – cautionmust be taken when considering knots, as shell crossing isknown to dramatically affect the gridded peculiar velocityfield (see e.g., Hoffman et al. 2012; Hahn et al. 2014). Fig. 4 shows the volume and mass filling fractions (VFFsand MFFs respectively) of the simulated LU as a function ofthreshold eigenvalue for the best-correlated LU reconstruc- c (cid:13) , 000–000 Nuza et al. tion of Heß et al. (2013) in real space as obtained from theT-web (upper panels) and div V-web (lower panels) classifi-cation methods. As expected, the relative fraction of occu-pied volume and mass of the different structures vary as thethreshold eigenvalue is increased. Interestingly, the VFFs forthe two classification methods studied show somewhat sim-ilar values for a given eigenvalue threshold. Nevertheless,differences can be large for λ th ≈
0. For larger thresholdstypical differences are of the order of 10 − To choose the effective threshold eigenvalue defining the cos-mic web, we rely on results of alternative classification meth-ods. In particular, our aim is to reproduce the VFF of voidsin a ΛCDM universe which is found to be considerably largerthan about 17%, i.e. the value obtained in the pioneeringclassification of Hahn et al. (2007). In general, such alter-native methods predict a void VFF in the range 70 − λ th = 0 . After applying the T-web classification with λ th = 0 . N -body reconstruction in configurationspace, we obtain VFFs of about 71 . . . . . . . . λ th = 0 . h − Mpc on a side that is centred in the observer’s po-sition.To compare the two classification methods consideredhere we use, for the div V-web method, an eigenvalue thresh-old of λ th = 0 .
8, aiming at approximately obtaining thesame VFF of voids as with the T-web. In this case, the re-sulting VFFs and MFFs are of about 66 . . . .
38% and 29 . . . .
4% for voids, sheets, fil-aments and knots respectively for the same LU reconstruc-tion. The small MFFs obtained for knots in this case clearlyindicates that the div V-web classification fails in character-ising the densest regions, as expected.
We compare the volume and mass statistics of the recon-structed cosmic web with those of the random simulationsto assess the cosmic variance in the LU. Therefore, the par-ticular method used to characterise the cosmic web is notrelevant, as soon as the same method is applied to bothsets of simulations. This can be seen in Figs. 6 and 7 forthe T- and div V-web classifications. The latter method isshown as inset plots to readily compare between the twoweb ‘finders’ in each case. The different panels show thecumulative VFFs (MFFs) of knots (upper-left panel), fil-aments (upper-right panel), sheets (lower-left panel) andvoids (lower-right panel) as a function of distance to theobserver, where the solid lines indicate the results for ourbest-correlated reconstruction in real space and the errorbars (standard deviation) have been estimated using ourensemble of 25 reconstructions. To construct these plots wespherically averaged the corresponding filling fractions fora given scale to simplify the description of the problem.The grey shaded regions indicate the 1 σ cosmic variancelevel of an ordinary ΛCDM universe, built by placing 1000‘observers’ at different locations within our random set ofunconstrained N -body simulations, whereas dotted lines in-dicate the mean value. We have checked that our results aremarginally affected if, instead, we compute these statisticsin redshift space, given the adopted resolution in our study.(Nevertheless, the impact of redshift-space distortions willbe further considered in Section 6.) The first conclusion wecan extract from these plots is that the LU is completelyconsistent with the expectations of ΛCDM, i.e. the mea-sured LU statistics are well within the predicted 1 σ fluctua-tions of the concordance model at most of the studied scales c (cid:13) , 000–000 he cosmic web of the Local Universe Figure 6.
Cumulative VFF of the best-correlated reconstructed LU as a function of distance from the centre of the box for differentenvironments. The cosmic web has been computed using the T-web method for a threshold of λ th = 0 .
9. The inset plots show theresults obtained using the div V-web classification with λ th = 0 .
8. The error bars show the standard deviation of our ensemble of 25reconstructions. The shaded regions represent the 1 σ cosmic variance fluctuations as obtained from unconstrained N -body simulationsby placing 1000 ‘observers’ at different locations within the box, whereas the dotted line stands for the mean value. and environments (despite of some departures for distancessmaller than ∼ h − Mpc where the selection function isless constrained). Irrespective of the classification, some re-markable features are evident for r & h − Mpc, i.e. as oneincreases the distance from the observer. At smaller scales,however, we found some differences between our two cosmicweb methods. Since at these scales the selection function isnot well constrained, in what follows, we will mainly focus onthe results at r & h − Mpc. In particular, for a sphere ofradius r = 20 h − Mpc the VFF of our reconstructed web isdominated by voids, occupying around 60% of the volume,followed by sheets and filaments, that comprise 30 − −
10% of the available space respectively. These val-ues represent a fluctuation of about 1 σ with respect to theexpected mean of a ΛCDM universe, which is about 70 − −
25% for sheets and 5 −
7% for filaments atthis scale. High density regions, as characterised by knots,occupy less than 1% of the volume, which is close to theexpectation for an unconstrained universe. Despite the factthat the LV is mainly populated by sheets/voids, our imme-diate vicinity shows a smaller fraction of void-like regions in comparison to the mean ΛCDM expectation. This be-haviour is, however, inverted for matter inside spheres withradius between 30 − h − Mpc, where the VFF is above(below) the mean expectation in the case of voids (sheets,filaments and knots) reaching a peak (dip) at a distance ofabout 40 h − Mpc. This indicates that, at that particularradius, the fraction of voids is higher than average, thuscontributing with more voids to the corresponding VFF.As expected, for increasingly larger spheres, the VFFs ofthe different cosmic web structures converge to the globalmean value of the whole reconstruction. This behaviour isalso observed for the MFF case, where we find results con-sistent with ΛCDM for radii greater than ∼ h − Mpc. Inthis case, however, the trends in the cumulative mass frac-tions present some differences between the two web ‘find-ers’. In fact, these discrepancies are expected as the cosmicweb resulting from different approaches do not completelyoverlap (e.g., Cautun et al. 2013). This is most noticeablefor knots owing to the field linearisation carried out in thediv V-web classification. However, this effect generates dif-ferences in other cosmic web structures as well. Therefore, c (cid:13) , 000–000 Nuza et al.
Figure 7.
Idem as Fig. 6 for the cumulative MFF. this fact prevents us from making strong conclusions aboutthe cumulative MFF profile in these environments. Never-theless, we can safely conclude that, irrespective of the clas-sification, the LU is in agreement with the expected ΛCDMfluctuations as our measured statistics are always containedwithin 1 σ at scales r & h − Mpc. In addition, we canstate that, when considering a volume with a radius of about60 h − Mpc, the LU becomes a fair sample thus convergingto the mean ΛCDM expectations.
As a first application of our detailed LU reconstructions weuse the 2MRS catalogue to correlate the position of galaxieswith the environmental information provided by the cosmicweb classifications computed in Section 4.3. To avoid anygridding effect all fields analysed here will be convolved witha Gaussian filter using a length of one cell. Nevertheless, wehave checked that the resulting trends and significances areslightly affected if no smoothing is used.We divide the observed galaxy sample in 4 broad cate-gories according to the morphological type assigned by the2MRS team, namely: ellipticals (E), lenticulars (S0), spirals (Sp) and irregulars (Irr). After selecting objects within thevolume of the reconstruction we end up with a sample of21,893 galaxies, out of which 8,367 correspond to ellipticals,1,323 to lenticulars, 11,903 to spirals and 300 to irregulars.To every galaxy, we assign an environment corresponding tothe cell where it is placed as determined by a given cosmicweb classification method. Following Lee & Lee (2008), wedefine what we call the ‘excess probability ratio’ as η ( T |E ) ≡ P ( T |E ) P ( G|E ) , (4)where G represents a random galaxy in the volume, T isthe galaxy type, E is the considered environment (knot, fil-ament, sheet or void) and P ( X |Y ) is the conditional prob-ability of X given Y . The interpretation of the excess prob-ability ratio is therefore rather simple: if galaxies of a giventype displayed some preference to be located in a particu-lar environment then the excess probability ratio has to be η ( T |E ) >
1, whereas η ( T |E ) < η ( T |E ) = 1 indicates no distinctionfrom the random distribution.To estimate the actual conditional probabilities we mea-sure galaxy number counts provided by the 2MRS cata-logue. Specifically, we approximate their values as P ( T |E ) ≈ c (cid:13) , 000–000 he cosmic web of the Local Universe N g ( T |E ) /N g ( T ) and P ( G|E ) ≈ N g ( E ) /N tot , where N g ( Z ) isthe number of galaxies satisfying the condition Z and N tot is the total number of galaxies in our sample. Therefore,our estimator for the excess probability presented in Eq. (4)results in η ( T |E ) ≈ N g ( T |E ) N tot N g ( T ) N g ( E ) . (5)The shot noise contribution of the number counts ismodelled assuming Poisson statistics thus allowing us to es-timate the magnitude of the associated errors. Fig. 8 shows the resulting excess probabilities of the differentgalaxy morphologies within the cosmic web for the redshift-space LU reconstruction. In this way, we can directly com-pare the reconstruction to observations. To compute the ex-cess probability signal we use both the T-web and div V-webclassification methods. In the first place, it can be seen thatthere is a global tendency for elliptical galaxies to preferen-tially reside in knots and in filaments rather than in sheetsand voids. This trend is inverted in the case of spiral andirregular galaxies. This can be more clearly seen in the T-web case whereas, when adopting the div V-web method todefine the web, the error bars tend to increase thus erasingthe signal corresponding to spirals. However, quite gener-ally, lenticular galaxies display no preference for a partic-ular environment irrespective of the classification method.These measurements provide the strength of the tendencyto inhabit/avoid a certain environment. According to theT-web classification, which provides the most significant de-tections, it can be seen that ellipticals are most easily foundin knots ( η = 1 . ± .
05) than in filaments ( η = 1 . ± . η = 0 . ± .
02) lessthan voids ( η = 0 . ± . η = 1 . ± .
03) than insheets ( η = 1 . ± .
02) whereas they tend to avoid fila-ments ( η = 0 . ± .
02) less than knots ( η = 0 . ± . . σ (1 . σ ),4 . σ (3 . σ ), 3 . σ (1 . σ ) and 9 . σ (4 . σ ) for knots, filaments,sheets and voids, respectively. Similarly, for spirals, we ob-tain significances of 5 . σ (0 . σ ), 4 . σ (3 . σ ), 2 . σ (1 σ ) and5 . σ (2 . σ ) for the corresponding environments. As an additional exercise we also computed the probabilityratios using the reconstructed real-space reconstruction asshown in Fig. 9. To compare our reconstruction with obser-vations, we transformed the observed data from redshift toreal space. This is, in general, not a trivial task although, us-ing our reconstructions, it is possible to establish a mappingbetween redshift and configuration space (see Fig. 2). Specif-ically, we use our set of constrained haloes in real and red-shift space to estimate the most likely real-space position ofthe corresponding observed 2MRS galaxies. In particular, for each galaxy, we search for the closest halo in redshift spaceusing our ensemble of cosmological simulations, and assignthe associated real-space position of the halo. As can be seenfrom Fig. 9, the excess probability correlations in real spacedisplay, in general, the same trends discussed before. How-ever, some new features are worth mentioning. In this case,the correlations show a smaller scatter, typically decreasingthe size of the error bars. Moreover, the η -morphology rela-tions derived using the T-web and the div V-web get moreconsistent with each other. This is mainly owing to the factthat working in configuration space reduces the artificialshell crossing caused by redshift-space distortions. For el-lipticals, in real space, the significance of the correspondingsignals – according to the T-web (div V-web) classification –are: 5 . σ (1 . σ ), 4 . σ (5 . σ ), 2 . σ (0 . σ ) and 9 . σ (7 . σ ) forknots, filaments, sheets and voids, respectively. Similarly,for spirals, we obtain significances of 5 . σ (1 . σ ), 3 . σ (5 σ ),2 . σ (0 . σ ) and 5 . σ (4 . σ ) for the corresponding environ-ments. Note that, in general, the results for the T-web anddiv V-web methods are now more similar to each other thanin the redshift-space case, as expected.Interestingly, for a given environment, the excess proba-bility displays a clear correlation as a function of galaxy typefrom ellipticals to irregulars, which provide insights to theprocess of galaxy formation and the build up of the Hubblesequence. The resulting slope in the η -morphology relationturns out to be positive in the sheet and void cases whileit gets negative for filaments and knots. These trends areconsistent with the idea of a continuous transition from ir-regular/spiral to elliptical morphology. The direction of thetransition is suggested by the gradual increase of the slopeas one follows the sequence knots–filaments–sheets–voids,i.e. from the more to the less dense environments. This indi-cates that spheroidal-like systems are most probably formedin high-density regions as a result of mergers between irreg-ular/spiral objects. Conversely, the lower merger probabilityin less dense environments permits the latter to retain theirmorphology. In this work we have presented a characterisation of thematter content of the LU including a cosmic web analysisand an environmental-galaxy morphology study based on aconstrained cosmological simulation. This is only possibleowing to the high-precision of the reconstructed initial con-ditions performed by the Bayesian self-consistent methodof Kitaura (2013). Our reconstruction covers a volume of180 h − Mpc and it was produced using the spatial infor-mation provided by 31,107 2MRS galaxies (Heß et al. 2013).To characterise the cosmic web of the reconstruction we haveapplied the tidal field tensor classification method on thenonlinear (T-web) and linear (div V-web) density fields sep-arately (see Section 4). The latter has been computed usingthe reconstructed nonlinear velocity field within the lineartheory approximation.To estimate the expected cosmic variance level we useda suite of unconstrained N -body simulations with the sameparameter setting as in our reconstructions. In this way,we were able to assess the consistency level of the specificLU realisation with the ΛCDM cosmology. To that end we c (cid:13) , 000–000 Nuza et al.
Figure 8. ‘Excess probability’ η ( τ, ǫ ) in redshift space for a galaxy of a given morphological type τ (i.e., E: elliptical; S0: lenticular; Sp:spiral; Irr: irregular) to inhabit a particular environment as indicated in the panels. Results are presented for the T-web (solid circles)and div V-web (open circles) classification methods using a threshold of λ th = 0 . . σ (2 σ ) levels. have measured the corresponding VFFs and MFFs of voids,sheets, filaments and knots of the LU and compared themwith those resulting from N -body universes with randominitial seeds. This work brings up a number of novel aspectsthat are important to remark: X We use a state-of-the-art reconstruction of the non-linear density and peculiar velocity fields based on highprecision constrained simulations that reach an accuracy ofabout 3 Mpc. X Based on such a reconstruction we present the firstfully nonlinear cosmic web classification of the LU using thetidal field tensor method on the reconstructed density field. X For the first time, we use the reconstructed nonlinearpeculiar velocity field to characterise basic statistics of theLU cosmic web as well as to perform a study on the relationbetween galaxy morphology and environment.In what follows we summarise the main findings of ourwork: • We measured the DM density profile in the LU byusing our N -body reconstruction. As a result, we found that the so-called ‘missing DM problem’ can be simplyinterpreted as the consequence of ignoring matter locatedoutside haloes above a given mass threshold. In fact, ifwe select simulated haloes with a similar virial mass cutas in the observations of Karachentsev (2012) (see alsoMakarov & Karachentsev 2011) we are able to reproducethe observed value within 50 Mpc (Ω M , LU ≈ .
1) and theshape of the profile (see Fig. 3). Instead, if we considerall matter present in the cosmic web, the matter densityreadily converges to the mean universal value. This suggeststhat the observationally-derived low matter density valuescannot be the result of cosmic variance alone. • Both the reconstructed VFF and MFF statistics,computed using increasingly larger spheres centred on theobserver’s location, display, in general, a nice agreementwith the expectations of ΛCDM. In particular, for ascale-radius larger than r & h − Mpc the deviations forall web environments are within 1 σ of the expected cosmicvariance. In particular, when considering a sphere with aradius of about 60 h − Mpc, these statistics converge tothe mean value of the random ΛCDM realisations, thusrepresenting a fair sample . • We have measured the tendency of 2MRS galaxies c (cid:13) , 000–000 he cosmic web of the Local Universe Figure 9.
Idem as Fig. 8 but in real space. A mapping between the real- and redshift-space reconstruction has been used to assign themost likely real-space position to every 2MRS galaxy (see text). to inhabit/avoid different environments as a function ofmorphology quantified in terms of the excess probabilityratio η . In agreement with previous work, we found thatelliptical systems are prone to be found in higher densityregions, which we identify with knots and filaments in theweb, rather than in sheets and voids. The opposite is truefor spirals. In general, we found that the η -morphologycorrelations are best defined in real space where artificialshell crossing effects caused by redshift-space distortionsare not present. • In particular, if the T-web classification is adoptedto define the cosmic web in our real-space reconstruction,elliptical galaxies show a clear signal ( η = 1 . ± .
05) topreferentially reside in clusters, at a 5 . σ level, as opposedto sheets ( η = 0 . ± .
03) and voids ( η = 0 . ± . . σ and 9 . σ , respectively. Interestingly, we alsofound that elliptical galaxies show an excess probability of η = 1 . ± .
03 in filaments that represents a 4 . σ detection.However, filaments in the tidal field classification are notvery well defined in the vicinity of high density regions, asit is shown in Forero-Romero et al. (2009) (their Fig. 1).Therefore, the latter result should be taken with caution.For spiral galaxies, we found a tendency to reside in voids( η = 1 . ± .
03) and sheets ( η = 1 . ± .
02) at a 5 . σ and2 . σ , as opposed to filaments ( η = 0 . ± .
02) and knots( η = 0 . ± . . σ and 5 . σ level, respectively. If, instead, we use the div V-web classification to define theweb we found that we can reproduce globally the sametrends. • Irrespectively of the classification adopted to definethe web we found that, in general, lenticular (S0) galaxiesin the 2MRS catalogue do not show a preference for anyparticular environment. Irregulars, on the contrary, showsimilar trends as in the spiral case, although at much lowersignificance as a result of the small galaxy number in oursample.As a final remark, we would like to note that our re-sults concerning the validity of ΛCDM have to be taken asa consistency check since the constrained simulations wereperformed assuming that very same model. It is neverthe-less remarkable that our constrained simulations are ableto provide an explanation to the properties of the LU byeither cosmic variance or observational biases without in-voking a paradigm shift in cosmology. In this regard, wewant to emphasise that the phases of the primordial fluctu-ations in our reconstructions – which determine the locationof peaks and troughs – are purely constrained by the data.This work demonstrates that high precision constrained sim-ulations can indeed help to characterise the properties ofthe LU in different environments. Therefore, we anticipatea large number of applications in which galaxy formation c (cid:13) , 000–000 Nuza et al. can be tested by directly cross-correlating observations tosimulations including the full phase-space information.
ACKNOWLEDGMENTS
The authors acknowledge the anonymous referee for a con-structive report that helped to improve this paper. SEN andFSK also thank Marius Cautun and Peter Creasey for usefulcomments on the manuscript. SEN, VM and SH acknowl-edge support by the Deutsche Forschungsgemeinschaft un-der the grants NU 332/2-1, MU1020 16-1 and GO563/21-1.NIL is also supported by the Deutsche Forschungsgemein-schaft. The simulations analysed in this work have been per-formed at NIC (J¨ulich, Germany).
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