Mass accretion rates and multi-scale halo environment in cold and warm dark matter cosmologies
MMNRAS , 1–16 (0000) Preprint 6 January 2021 Compiled using MNRAS L A TEX style file v3.0
Mass accretion rates and multi-scale halo environment incold and warm dark matter cosmologies
Payaswinee Dhoke (cid:63) , Aseem Paranjape † Dharampeth M. P. Deo Memorial Science College, North Ambazari Road, Nagpur 440033, India Inter-University Centre for Astronomy & Astrophysics, Ganeshkhind, Post Bag 4, Pune 411007, India draft
ABSTRACT
We study the evolving environment dependence of mass accretion by dark haloes insimulations of cold and warm dark matter (CDM and WDM) cosmologies. The latterallows us to probe the nature of halo growth at scales below the WDM half-mode mass,which form an extreme regime of nonlinear collisionless dynamics and offer an excellenttest-bed for ideas relating to hierarchical growth. As environmental proxies, we usethe local halo-centric matter density δ and tidal anisotropy α , as well as large-scalehalo bias b . Our analysis, while reproducing known trends for environment-dependentaccretion in CDM, as well as the comparison between accretion in CDM and WDM,reveals several interesting new features. As expected from excursion set models, WDMhaloes have higher specific accretion rates, dominated by the accretion of diffuse mass,as compared to CDM haloes. For low-mass WDM haloes, we find that the environment-dependence of both diffuse mass accretion as well as accretion by mergers is almostfully explained by α . For the other cases, δ plays at least a comparable role. We detect,for the first time, a significant and evolving assembly bias due to diffuse mass accretionfor low-mass CDM and WDM haloes (after excluding splashback objects), with a z = 0strength higher than with almost all known secondary variables and largely explainedby α . Our results place constraints on semi-analytical merger tree algorithms, which inturn could affect the predictions of galaxy evolution models based on them. Key words: cosmology: theory, dark matter, large-scale structure of the Universe –methods: numerical
The growth of gravitationally bound haloes of collisionlesscold dark matter (CDM) through accretion and mergers isone of the primary physical processes of interest in the hi-erarchical structure formation paradigm. The stochasticityinherent in the initial cosmological seed fluctuations, coupledwith the nonlinearity of gravitational evolution of a colli-sionless fluid, renders this problem analytically challenging,although considerable insights may be gained through simpli-fied models of structure formation (Bond et al. 1991; Lacey& Cole 1993; Musso & Sheth 2014). Since dark haloes areexpected to be the cradles of galaxy formation and evolution(White & Rees 1978), understanding the evolving nature ofhalo mass accretion and its dependence on the local andlarge-scale environment of haloes is expected to yield impor-tant clues into corresponding correlations in the observeddistribution and evolution of galaxies in the Universe.The understanding that accretion rates in the late Uni-verse are sensitive to the shape of the initial power spectrum (cid:63)
E-mail: [email protected] † E-mail: [email protected] (Lacey & Cole 1993, see also below) suggests a useful toolfor investigating the nature of mass accretion, in the formof CDM-like power spectra which are suppressed at smallscales (large k ), mimicking the effects of a thermally pro-duced warm dark matter (WDM) particle. The steep cut-offin power in such models creates an extreme situation wherehaloes forming close to the corresponding mass scale experi-ence dramatically enhanced growth, and haloes below thisscale simply do not exist (e.g., Angulo et al. 2013). Simu-lations performed with WDM-like power spectra, therefore,offer ideal test-beds for the environment dependence of massaccretion: any physical model that purports to explain thenature of mass accretion in CDM must also do so for WDM,since the physics of a self-gravitating collisionless fluid is com-mon to both (Hahn & Paranjape 2014). Of course, WDMmodels are physically interesting in their own right, from thepoint of view of small-scale challenges for the CDM frame-work (see Bullock & Boylan-Kolchin 2017, for a review),although this is not the focus of the present work.There has been considerable work to date studying massaccretion by haloes via mergers and smooth (or diffuse)accretion (Fakhouri & Ma 2010; Genel et al. 2010; Bensonet al. 2013; Elahi et al. 2014), as well as its dependence on halo © a r X i v : . [ a s t r o - ph . GA ] J a n Dhoke & Paranjape environment (defined typically in terms of halo-centric darkmatter density using different smoothing schemes, see Genelet al. 2010; Fakhouri & Ma 2009, 2010; Maulbetsch et al.2007; Borzyszkowski et al. 2017; Lee et al. 2017, see also Moreet al. 2015). The overall understanding that has emergedfrom these studies, regardless of the exact choice of definitionof halo-centric density, is that denser environments tend topromote mergers while underdense environments are moreconducive to diffuse accretion. There are also stark differencesbetween accretion in CDM and WDM, particularly at lowmass, as expected from the discussion above, with low-massWDM haloes accreting rapidly and primarily through diffuseaccretion.A related line of study is that of halo assembly bias or secondary bias, i.e., the correlation (at fixed halo mass)between secondary halo properties other than mass and halo-centric density (or bias) measured at cosmological scales(Sheth & Tormen 2004; Gao et al. 2005). Although assemblybias has been studied over a wide range of halo mass andredshift using many choices of secondary variables such asage (Sheth & Tormen 2004; Gao et al. 2005; Jing et al. 2007),concentration (Wechsler et al. 2006; Angulo et al. 2008),shape (Faltenbacher & White 2010; van Daalen et al. 2012),angular momentum or spin (Gao & White 2007) and veloc-ity dispersion structure (Faltenbacher & White 2010), thecorrelation between halo bias and mass accretion rate hasthus far been limited to massive objects at z = 0 (Lazeyraset al. 2017). Moreover, recent work on assembly bias at z = 0has revealed the importance of the local tidal environment ofhaloes (Hahn et al. 2009; Borzyszkowski et al. 2017; Paran-jape et al. 2018) in explaining the assembly bias of manysecondary variables, including concentration, spin, shape andvelocity dispersion structure over a wide range of halo mass(Ramakrishnan et al. 2019, see also Han et al. 2019). It istherefore very interesting to ask whether the halo tidal envi-ronment plays a similar role in explaining any assembly biastrends with mass accretion.In this work, we perform a detailed study of the natureof the environment dependence of mass accretion by haloes,segregated into contributions due to mergers and diffusemass, as a function of redshift and for a range of halo masses.We will do so using N -body simulations of both CDM andWDM cosmologies; as mentioned above, the latter will allowus to better resolve the multi-scale environment dependenceof mass accretion due to its sensitivity to the shape of theinitial matter power spectrum. We will connect these resultsto the assembly bias literature by performing, for the firsttime, a comprehensive study of assembly bias due to massaccretion across cosmic time, along with its connection tothe evolving local halo environment, for both CDM andWDM. Our results reproduce previously observed trendswhile extending these to new local environmental variablessuch as the tidal anisotropy, and are expected to place usefulconstraints on semi-analytical models of halo formation andgrowth (e.g., Somerville & Kolatt 1999; Parkinson et al. 2008;Jiang & van den Bosch 2014), which in turn form the bedrockof several semi-analytical models of galaxy formation andevolution (e.g., Benson & Bower 2010; Benson 2012; Barausse2012; Dayal et al. 2014; Birrer et al. 2014; Somerville et al.2015; Yung et al. 2019).The paper is organised as follows. We describe our simu-lations and analysis techniques in section 2. We present ourresults along with a discussion in section 3, with section 3.1focusing on environment-independent evolution, sections 3.2 and 3.3 focusing on trends with local environment and sec-tion 3.4 devoted to assembly bias. We discuss some of thesetrends in the framework of the excursion set approach insection 4 and conclude with a summary of our main resultsin section 5. Throughout, the base-10 logarithm is denotedby ‘log’ and the natural logarithm by ‘ln’. Here we describe our simulations and analysis tools for iden-tifying haloes and measuring their accretion rates and localas well as large-scale environments.
We have used N -body simulations of CDM and WDM per-formed using the tree-PM code gadget-2 (Springel 2005) in a cubic, periodic box of comoving length 150 h − Mpc sam-pled with 1024 particles, corresponding to a particle mass m p = 2 . × h − M (cid:12) , with a 2048 PM grid and comov-ing force resolution 4 . h − kpc corresponding to 1 /
30 of theLagrangian inter-particle spacing.The CDM transfer function T cdm ( k ) for generating initialconditions was computed using the code camb (Lewis et al.2000) with a spatially flat ΛCDM cosmology having totalmatter density parameter Ω m = 0 . b = 0 . H = 100 h kms − Mpc − with h = 0 .
7, primordial scalar spectral index n s = 0 . h − Mpc, σ = 0 . Wilkin-son Microwave Anisotropy Probe (WMAP7, Komatsu et al.2011).For the WDM model, we additionally suppress small-scale power in the linear transfer function as appropriate forthe free-streaming of a thermally produced WDM particlewith mass m dm = 0 . T wdm ( k ) = T cdm ( k ) (cid:2) αk ) µ (cid:3) − /µ , (1)with µ = 1 .
12 and α ≡ . (cid:18) Ω m . (cid:19) . (cid:18) h . (cid:19) . (cid:16) m dm (cid:17) − . h − Mpc . (2)The resulting “half-mode” mass-scale (c.f., e.g., Schneideret al. 2012) of M hm (cid:39) × h − M (cid:12) is resolved with ∼ Although a WDM particle with m dm = 0 . http://camb.info The collisionless WDM fluid is assumed to be in the perfectlycold limit, i.e., we ignore the small thermal velocity dispersion of areal WDM fluid. This is expected to be accurate at the epochs ofour interest, well after perturbations have been suppressed belowthe largest free-streaming scale in linear theory (Angulo et al.2013). MNRAS000
12 and α ≡ . (cid:18) Ω m . (cid:19) . (cid:18) h . (cid:19) . (cid:16) m dm (cid:17) − . h − Mpc . (2)The resulting “half-mode” mass-scale (c.f., e.g., Schneideret al. 2012) of M hm (cid:39) × h − M (cid:12) is resolved with ∼ Although a WDM particle with m dm = 0 . http://camb.info The collisionless WDM fluid is assumed to be in the perfectlycold limit, i.e., we ignore the small thermal velocity dispersion of areal WDM fluid. This is expected to be accurate at the epochs ofour interest, well after perturbations have been suppressed belowthe largest free-streaming scale in linear theory (Angulo et al.2013). MNRAS000 , 1–16 (0000) ass accretion rates and environment and its environment dependence in the absence of small-scaleperturbations.Initial conditions for both CDM and WDM were gener-ated at z = 99 with the same random seed using 2 nd orderLagrangian perturbation theory (Scoccimarro 1998) with thecode music (Hahn & Abel 2011). Snapshots were storedstarting from z = 12 to z = 0 at time intervals equallyspaced in the scale factor with ∆ a = 0 . Haloes were identified using the code rockstar (Behrooziet al. 2013a) which implements a Friends-of-Friends algo-rithm in 6-dimensional phase space and provides informationon gravitationally bound haloes as well as their substructure.Merger trees were constructed using the 201 snapshots in eachsimulation (CDM and WDM) using the code consistent-trees (Behroozi et al. 2013b). Since we wish to focus onthe accretion rates of well-resolved haloes in this work, weexclude all subhaloes and further consider only those ob-jects for which the virial energy ratio η = 2 T / | U | satisfies0 . < η < .
5, which mitigates the effects of unrelaxedobjects and numerical artefacts (Bett et al. 2007).We further exclude splashback objects, which spatiallymimic genuine haloes at the redshift of interest but havepassed through a larger host in the past and hence arephysically closer to subhaloes. We do so using the valueof the ‘first accretion scale’ reported by consistent-trees ,which records the earliest epoch at which the main progenitorof a given descendant passed through the virial radius of alarger object; we exclude all descendants whose first accretionscale is earlier than the epoch of interest. For completeness,we report that splashback haloes at z = 0 form (cid:46)
6% (4 . (cid:46) .
7% of high-mass objects in either cosmology (seebelow for the corresponding mass ranges), with the fractionsbecoming vanishingly small at high redshifts.Throughout, we will quote halo and progenitor massesusing the mass definition M vir as reported by rockstar .This corresponds to the bound mass contained in a halo-centric sphere of virial radius R vir which encloses a densityequal to ∆ vir times the critical density of the Universe, where∆ vir is the spherical collapse overdensity and is taken fromthe fitting function provided by Bryan & Norman (1998) (at z = 0, ∆ vir (cid:39)
98 for our cosmology, while ∆ vir → π (cid:39) z during matter domination). Additionally, we usethe rockstar values of M to infer the halo-centric radius R which encloses a density equal to 200 times the meandensity of the Universe at each redshift; this will be usefulwhen characterising local halo environments below. http://hpc.iucaa.in https://bitbucket.org/gfcstanford/rockstar https://bitbucket.org/pbehroozi/consistent-trees To avoid effects of numerical precision in comparing scale factorvalues, we define splashbacks as objects whose first accretion epochis more than 100 Myr in the past of the epoch of interest. Wehave checked that varying this threshold by factor 2 on either sidemakes no difference to our results.
Figure 1. Halo mass function:
Dashed (solid) lines show thehalo mass function in CDM (WDM) at z = 0 . z = 1 . M vir . The horizon-tal axis starts at M vir = 1 . × h − M (cid:12) , corresponding to a 60particle cut which we use for identifying progenitors. The first ver-tical dotted line from the left indicates M vir = 1 . × h − M (cid:12) or 500 particles, which we use as the threshold for identifyingdescendants at any redshift (see text for other criteria applied inselecting a clean halo sample). The low-mass and high-mass binswe use are indicated in blue and red colours, respectively, demar-cated by the remaining vertical lines and with the correspondingranges mentioned in the labels. Highlights:
In the WDM massfunction, we clearly see the suppression of halo counts below thehalf-mode mass M hm (cid:39) × h − M (cid:12) , and (at low redshift) thespurious upturn at very low masses due to numerical artefacts. Figure 1 shows the mass functions for haloes in CDM(dashed) and WDM (solid) at z = 0 and z = 1 .
5. The verticallines demarcate the mass bins we will use below; the lowestmass threshold corresponds to 500 particles and the massfunctions in the low-mass (high-mass) bin we employ arecoloured blue (red). The CDM mass function shows the well-studied power law rise towards low masses and exponentialdecline at high masses (Press & Schechter 1974; Bond et al.1991; Sheth & Tormen 1999; Tinker et al. 2008).The WDM mass function is identical to the CDM oneat high masses for each redshift, but turns over around thehalf-mode mass (see above), leading to a suppression of haloabundances at low masses as expected from the lack of small-scale perturbations (Hahn & Paranjape 2014; see also Bensonet al. 2013; Schneider et al. 2013). At masses smaller than ∼ . h − M (cid:12) , however, we see an up-turn in the z = 0WDM mass function, which is a well-known consequenceof numerical artefacts in the N -body technique applied toinitial conditions with suppressed small-scale power (Wang& White 2007; Angulo et al. 2013; Lovell et al. 2014; Agarwal& Corasaniti 2015). Note that the virial cut of 0 . < η < . initial proto-patches MNRAS , 1–16 (0000)
Dhoke & Paranjape from which these objects evolve (Lovell et al. 2014), whichwe have not performed here.Our choice of the lowest mass bin for descendant haloesis sufficiently far above the mass scale where spurious objectsstart becoming numerically relevant. We have checked this byapplying the correction suggested by Schneider et al. (2013)by fitting a power law to the mass function below the up-turnscale (not displayed) and subtracting it from the measuredmass function; the result agrees with the measured massfunction in the mass bins of interest at better than ∼
2% atall redshifts. Spurious objects therefore do not contribute toour chosen populations of descendant haloes at any redshift.However, when identifying progenitor haloes in the mergertrees, we will employ a lower mass cut of 60 particles (theleft edge of the horizontal axis in figure 1). This affects thequantification of mass accretion rates in the WDM case atlow redshifts, by artificially enhancing the accretion dueto mergers and correspondingly decreasing the accretion ofdiffuse mass. We return to this point below.
For the analysis below, we use haloes in the two mass binsdiscussed above, all of which are well-resolved with more than500 particles. We used merger trees generated by consistent-trees to find all the progenitors of a given halo within theprevious 2 dynamical times at any chosen redshift. Thevalues of the dynamical time T dyn and the redshift interval∆ z ≡ z i − z f corresponding to 2 T dyn at different redshifts aregiven in table 1. Our snapshot resolution of ∆ a (cid:39) .
005 (seeabove) provides us with excellent sampling of the required∆ z at all the redshifts we probe (see table 1).Throughout, we retain progenitors that are resolvedwith more than 60 particles ( M vir > . × h − M (cid:12) ).This leads to a different dynamic range in progenitor-to-descendant mass ratio χ for the low-mass ( χ (cid:38) .
04) andhigh-mass bin ( χ (cid:38) × − ). To the extent that accretionrates are self-similar with mass (Fillmore & Goldreich 1984;Bertschinger 1985), this would lead to accretion by ‘mergers’being systematically enhanced in high-mass haloes simplybecause relatively low-mass objects are counted as progen-itors as compared to the low-mass bin. Provided accretionrates are accurately estimated without double-counting dueto (artificial) fragmentation, however, the contribution togenuine mergers from very low mass ratios (say χ < − )should be small (Genel et al. 2010). consistent-trees im-proves the consistency of progenitor/descendant assignmentacross time steps by solving differential equations for theexpected locations of potential descendants (Behroozi et al.2013b), so that effects of artificial fragmentation are reduced.Also, since we define accretion rates over 2 T dyn , any residualeffects of instantaneous fragmentation should typically beaveraged over. Moreover, environmental trends have beenfound to be relatively insensitive to whether progenitors aredefined using a fixed minimum particle count or minimum χ (Fakhouri & Ma 2010). We therefore proceed with ouranalysis using a fixed threshold of 60 particles on progenitormass and comment below on analyses using other choices. We have repeated our main analysis for accretion rates calcu-lated over 1 dynamical time and find qualitatively similar butnoisier results. We therefore focus on results using accretion ratesover 2 dynamical times.
Table 1.
Value of dynamical time T dyn in Gyr, the redshift interval∆ z corresponding to 2 T dyn in the past, and the correspondingnumber of simulation snapshots N snap tracked, at each redshiftstudied in this work. z T dyn (Gyr) 3.14 2.55 2.08 1.67 1.46 1.08 0.83∆ z (2 T dyn ) 0.68 0.80 0.95 1.11 1.20 1.50 2.23 N snap
89 70 57 47 42 34 32
We follow Fakhouri & Ma (2010) and divide the massaccreted by a halo into two parts: (i) via other resolved haloesas mergers and (ii) in the form of diffuse mass, which includesbound structures below the chosen mass resolution, actuallyunbound particles which may have been tidally stripped fromother objects, or genuinely diffuse mass which has never beenpart of a bound structure earlier. In practice, we calculatethe mass accreted in a given time interval z i > z > z f due tomergers as the total mass of all progenitors except the mainprogenitor. Correspondingly, diffuse mass accreted in thesame time interval is the difference between the descendanthalo’s mass and the total mass of all the progenitors. Finally,the total mass accreted in this time interval is the sum ofthe masses accreted via mergers and diffuse accretion, i.e.,the difference between the descendant halo’s mass and themass of the main progenitor. Symbolically, if we define M : virial mass of descendant halo at z = z f , M j : virial mass of the j th progenitor halo at z = z i , M : virial mass of the main progenitor at z = z i ,then the dimensionless specific accretion rates Γ mer (merger),Γ dif (diffuse) and Γ tot (total) – where Γ ∼ d ln M/ d ln a (e.g.,Diemer & Kravtsov 2014) – can be written asΓ mer ≡ (cid:16)(cid:80) j ≥ M j (cid:17) M ( z i − z f ) × (1 + z f ) , Γ dif ≡ (cid:16) M − (cid:80) j ≥ M j (cid:17) M ( z i − z f ) × (1 + z f ) , Γ tot ≡ ( M − M ) M ( z i − z f ) × (1 + z f ) . (3)We have chosen a convention such that positive values of therates correspond to increase in mass from z i to z f .Using the progenitors selected as above, we computedΓ mer , Γ dif and Γ tot within the previous 2 dynamical timesat seven redshifts z = 0 .
0, 0.25, 0.5, 0.8, 1.0, 1.5 and 2.0 forthe CDM and WDM simulations. These rates are foundto show distributions similar to those shown in Fakhouri& Ma (2010). Figure 2 shows the normalised probabilitydistributions for Γ dif (left panel) and Γ mer (right panel) for z = 0 haloes in the low-mass (blue) and high-mass (red) bin,for CDM (dashed lines) and WDM (solid lines). Arrows inthe left panel (with identical colour-coding and line styles)indicate the corresponding median values of Γ dif . These As a check, we also explicitly calculated Γ tot over 100 My,finding that these are within ∼
10% of the values reported by consistent-trees . The differences are likely due to differentchoices of progenitor mass resolution between our values and thedefault settings of consistent-trees , and are not expected toalter our conclusions qualitatively. MNRAS000
10% of the values reported by consistent-trees . The differences are likely due to differentchoices of progenitor mass resolution between our values and thedefault settings of consistent-trees , and are not expected toalter our conclusions qualitatively. MNRAS000 , 1–16 (0000) ass accretion rates and environment Figure 2. Accretion rate distributions:
Histograms in the left (right) panel show the distributions of the specific mass accretionrates Γ dif (Γ mer ) due to diffuse mass (mergers) at z = 0 (see equations 3). Dashed (solid) lines show results for CDM (WDM) haloes,with the red (blue) curves showing results for the high-mass (low-mass) bin. The legend in the left panel indicates the number of haloesthat contributed to each mass bin, while the arrows mark the median values of Γ dif for the respective distributions. Highlights:
Low-massWDM haloes have the highest Γ dif and lowest Γ mer values on average. arrows show that the median Γ dif is the highest for WDMlow-mass haloes. For the high-mass bin, CDM and WDMhaloes show very similar distributions of the two accretionrates. Γ mer is always positive except if there were no recentmergers, in which case Γ mer = 0 exactly. Γ dif can be negativeif the total mass of progenitors is more than the mass of thedescendant halo, which can happen due to tidal stripping; thisis evidently a significant effect only for low-mass CDM haloes.Below, we will investigate in detail the redshift evolution andenvironmental trends of both Γ dif and Γ mer .As mentioned earlier, the dominant presence of spuri-ous objects in the WDM simulation at low redshifts andmasses (cid:46) . h − M (cid:12) affects the measurement of WDMaccretion rates. In particular, Γ mer is overestimated and Γ dif correspondingly underestimated due to the fact that massis locked up in spuriously identified objects. However, the amount of mass in these low-mass spurious objects is notdramatically large. By integrating the measured and power-law-corrected (see section 2.2) mass functions, weighted bymass, over the range 60 m p ≤ M vir ≤ m p , we find thatthe contribution of spurious objects in this range is (cid:39) z = 0 and falls to (cid:46)
4% by z = 1 .
5. Moreover, wewill see below that accretion due to mergers is subdominantin WDM in any case. We therefore tentatively conclude that,although the WDM merger accretion rates are likely to besystematically overestimated due to the presence of spuri-ous objects, accounting for this spurious contribution is notexpected to alter any of our qualitative conclusions.To assess the effects of the potential caveats discussedabove, namely, the effect of a fixed particle threshold versus athreshold in mass ratio for defining progenitors, and the effectof spurious WDM haloes, we have repeated our entire analysisusing two different selection criteria. In the first modification,we replaced the fixed cut of >
60 particles per progenitor witha lower threshold of >
30 particles, which should emphasizethe effects of spurious WDM haloes. In the second, we used the 60 particle threshold for progenitors of the low-massdescendants, but a threshold of M vir > . × h − M (cid:12) for progenitors of high-mass descendants. This corresponds to χ (cid:39) .
08 with respect to the median descendant mass in thehigh-mass bin, which is the same as the ratio between the 60particle threshold and the median mass of the low-mass bin,and should emphasize any differences between environmentaltrends due to the different progenitor selection. We find that all our qualitative results remain unchanged in each case ,with only a few minor differences which we comment on later.This indicates that (a) the environmental trends we studyare indeed insensitive to the choice of progenitor thresholdmass and (b) spurious haloes in WDM do not affect theinference of the trends. All our results below will be quotedfor the (more conservative) fixed threshold of >
60 particlesper progenitor for each mass bin.
We quantify the local environment of haloes using scalarsconstructed from the smoothed halo-centric tidal tensor T ij ( x ) = ∂ i ∂ j Ψ R ( x ), where the Newtonian potential Ψ R satisfies the normalised Poisson equation ∇ Ψ R = δ R with δ R being the halo-centric dark matter overdensity Gaussian-smoothed on comoving scale R . In practice, for any smooth-ing scale R , we invert the Poisson equation in Fourier spaceusing cloud-in-cell (CIC) interpolation for the unsmootheddensity on a 512 grid. Namely, we Fourier transform theCIC overdensity to obtain δ ( k ), using which the tidal tensoris the inverse Fourier transform of ( k i k j /k ) δ ( k )e − k R / .As the two scalars of choice, we use the overdensity δ R itself, and the tidal anisotropy α R introduced by Paranjapeet al. (2018). If the eigenvalues of T ij are denoted λ ≤ λ ≤ λ , then we have δ R = λ + λ + λ (4) MNRAS , 1–16 (0000)
Dhoke & Paranjape
Figure 3. Local tidal anisotropy:
Normalised distributions of the tidal anisotropy α (equation 5) for low-mass (left panel) andhigh-mass (right panel) haloes in CDM (dashed curves) and WDM (solid curves) at four redshifts between 0 ≤ z ≤ . α = 0 .
5, with higher (lower) values corresponding to filamentary (node-like) environments.
Highlights:
Distributions for low-mass WDM haloes are systematically narrower than their CDM counterparts. and α R = (cid:113) q R / (1 + δ R ) (5)where q R is the tidal shear (Heavens & Peacock 1988; Catelan& Theuns 1996) defined as, q R = 12 [( λ − λ ) + ( λ − λ ) + ( λ − λ ) ] . (6)For the reasons discussed by Paranjape et al. (2018) andRamakrishnan et al. (2019), we define the local halo environ-ment at scales R = 4 R / √ α ≡ α R and δ ≡ δ R .A common approach to classifying cosmic web environ-ment is by counting the signs of the eigenvalues of the tidaltensor or density Hessian defined at some fixed smoothingscale; with the tidal tensor we would have (e.g., Hahn et al.2007) λ > λ < λ > λ < λ > λ < α and δ provides an alternative, continuous definition of halo envi-ronment adapted to the halo size. These variables, althoughcorrelated, are not completely degenerate (Ramakrishnanet al. 2019, see also below). Such a continuous measure ofenvironment is much better suited for studies of large-scaleenvironmental correlations or assembly bias (Paranjape et al.2018; Ramakrishnan et al. 2019) as well as environmentaltrends in galaxy evolution (Zjupa et al. 2020).Physically, small values of the adaptively defined α cor-respond to isotropic environments (nodes), while large valuescorrespond to very anisotropic environments (filaments). Asdiscussed by Paranjape et al. (2018) and Paranjape (2020),the value α = 0 . α at four redshiftsfor CDM (dashed) and WDM (solid) haloes in the low-mass (left panel) and high-mass bin (right panel) . The distributionof α at fixed redshift depends on halo mass, with massivehaloes having preferentially lower values of α and low-masshaloes spanning a wide range of α . Haloes at higher redshiftsalso have preferentially lower values (as well as narrowerdistributions) of α , which is possibly mainly a consequenceof mass resolution (since haloes at fixed mass are rarer inthe past), but might also be partially reflecting a genuineevolution of the local cosmic web environment of haloes.Figure 4 is formatted identically to figure 3 and shows thecorresponding distributions of δ . At fixed redshift, we seethat the distributions for low-mass haloes have wider tailsthan for high-mass haloes, at both low and high density. Atfixed mass, high-redshift haloes span substantially narrowerranges of δ than those at lower redshift. For both α and δ , theresults for WDM and CDM at each redshift are very similarto each other for high-mass haloes, while the distributionsfor low-mass WDM haloes are narrower than their CDMcounterparts (with the median δ also being systematicallyhigher for WDM). This extends the results of Paranjape et al.(2018), who studied CDM haloes at z = 0, to significantlyhigher redshift as well as WDM cosmologies. As an indicator of the large-scale halo environment, we es-timate the linear bias b for each halo using the techniqueoutlined by Paranjape et al. (2018) and Paranjape & Alam(2020). This is essentially an object-by-object version of theusual cross-correlation definition of bias in Fourier space– P hm ( k ) /P mm ( k ) – averaged over low- k modes using theweights discussed by Paranjape & Alam (2020), where P hm MNRAS000
Distributions for low-mass WDM haloes are systematically narrower than their CDM counterparts. and α R = (cid:113) q R / (1 + δ R ) (5)where q R is the tidal shear (Heavens & Peacock 1988; Catelan& Theuns 1996) defined as, q R = 12 [( λ − λ ) + ( λ − λ ) + ( λ − λ ) ] . (6)For the reasons discussed by Paranjape et al. (2018) andRamakrishnan et al. (2019), we define the local halo environ-ment at scales R = 4 R / √ α ≡ α R and δ ≡ δ R .A common approach to classifying cosmic web environ-ment is by counting the signs of the eigenvalues of the tidaltensor or density Hessian defined at some fixed smoothingscale; with the tidal tensor we would have (e.g., Hahn et al.2007) λ > λ < λ > λ < λ > λ < α and δ provides an alternative, continuous definition of halo envi-ronment adapted to the halo size. These variables, althoughcorrelated, are not completely degenerate (Ramakrishnanet al. 2019, see also below). Such a continuous measure ofenvironment is much better suited for studies of large-scaleenvironmental correlations or assembly bias (Paranjape et al.2018; Ramakrishnan et al. 2019) as well as environmentaltrends in galaxy evolution (Zjupa et al. 2020).Physically, small values of the adaptively defined α cor-respond to isotropic environments (nodes), while large valuescorrespond to very anisotropic environments (filaments). Asdiscussed by Paranjape et al. (2018) and Paranjape (2020),the value α = 0 . α at four redshiftsfor CDM (dashed) and WDM (solid) haloes in the low-mass (left panel) and high-mass bin (right panel) . The distributionof α at fixed redshift depends on halo mass, with massivehaloes having preferentially lower values of α and low-masshaloes spanning a wide range of α . Haloes at higher redshiftsalso have preferentially lower values (as well as narrowerdistributions) of α , which is possibly mainly a consequenceof mass resolution (since haloes at fixed mass are rarer inthe past), but might also be partially reflecting a genuineevolution of the local cosmic web environment of haloes.Figure 4 is formatted identically to figure 3 and shows thecorresponding distributions of δ . At fixed redshift, we seethat the distributions for low-mass haloes have wider tailsthan for high-mass haloes, at both low and high density. Atfixed mass, high-redshift haloes span substantially narrowerranges of δ than those at lower redshift. For both α and δ , theresults for WDM and CDM at each redshift are very similarto each other for high-mass haloes, while the distributionsfor low-mass WDM haloes are narrower than their CDMcounterparts (with the median δ also being systematicallyhigher for WDM). This extends the results of Paranjape et al.(2018), who studied CDM haloes at z = 0, to significantlyhigher redshift as well as WDM cosmologies. As an indicator of the large-scale halo environment, we es-timate the linear bias b for each halo using the techniqueoutlined by Paranjape et al. (2018) and Paranjape & Alam(2020). This is essentially an object-by-object version of theusual cross-correlation definition of bias in Fourier space– P hm ( k ) /P mm ( k ) – averaged over low- k modes using theweights discussed by Paranjape & Alam (2020), where P hm MNRAS000 , 1–16 (0000) ass accretion rates and environment Figure 4. Local overdensity:
Same as figure 3, showing results for halo-centric local overdensity δ (equation 4). Highlights:
As with α ,the distributions of δ for low-mass WDM haloes are systematically narrower than their CDM counterparts, and also have higher medianvalues. and P mm are the halo-matter cross power spectrum andmatter auto-power spectrum, respectively.The box size of 150 h − Mpc leads to small-volume sys-tematic effects in the absolute value of b measured for eachobject, due to missing long-wavelength modes. Here, however,we are only interested in the correlation between b and otherhalo properties like accretion rates and halo environment.As demonstrated by Ramakrishnan et al. (2019), these cor-relations are relatively insensitive to volume effects and wetherefore expect our results to be robust to such systematics. We now turn to our main results, starting with theenvironment-independent evolution of mass accretion, fol-lowed by trends with local environment and finally assemblybias.
Figure 5 shows the evolving fraction f no merge of haloes whichdo not accrete any mass through mergers in the last 2 dy-namical times (i.e., haloes without any secondary progenitorlarger than the 60 particle threshold, having Γ mer = 0), forCDM and WDM. High-mass CDM haloes essentially alwayshad mergers, except at very low redshifts where a smallfraction have evolved with Γ mer = 0. At fixed mass, thevalue of f no merge is always larger in WDM than in CDM.This can be easily understood as being due to the lack ofsmall-scale structure and hence fewer low-mass bound struc-tures in WDM. In fact, we see that haloes in the low-massWDM bin accrete mainly diffuse mass at all redshifts, with f no merge (cid:38) . z . This is consistent with correspondingresults from Elahi et al. (2014).Figure 6 shows the evolution of the median Γ dif andΓ mer ( left and middle panels , respectively) with the totalaccretion rate Γ tot shown in the right panel . Error bars on themeasurements were calculated by bootstrap sampling: at each Figure 5. Mergers versus diffuse accretion:
Evolution withredshift of the fraction f no merge of haloes with zero merger ratein the previous 2 dynamical times. Results for the low-mass (high-mass) bin are shown in blue (red), with the dashed (solid) curvesshowing results for CDM (WDM). Highlights:
Low-mass WDMhaloes accrete mainly diffuse mass at all epochs, while essentiallyevery high-mass CDM halo has accreted some mass through merg-ers in the previous 2 dynamical times, except at very low redshift. redshift and for each mass bin, the accretion rate data aresampled with repetition a number of times and the standarddeviation in the median values of each sample gives the valueof error. We see that WDM high-mass haloes have nearlythe same total accretion rate as CDM high-mass haloes, atall epochs (consistent with the earlier results by Knebe et al.2002; Benson et al. 2013). WDM low-mass haloes, on theother hand, have higher total accretion rates than, both,
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Figure 6. Accretion history:
Evolution with redshift of the median values of the specific accretion rates Γ dif (left panel) , Γ mer (middlepanel) and Γ tot (right panel) , for low-mass (blue) and high-mass (red) haloes in CDM (dashed) and WDM (solid). Error bars werecalculated by bootstrap sampling at each redshift.
Highlights: Γ dif for low-mass WDM haloes is the highest of all categories. In thehigh-mass bin, although Γ tot is nearly identical between CDM and WDM, the relative contribution to this total from mergers versusdiffuse accretion is higher in CDM than in WDM. CDM low-mass haloes and also WDM (and CDM) high-masshaloes.Upon splitting the accretion rate between mergers anddiffuse accretion, we see that diffuse accretion dominates theaccretion budget at all redshifts, in each mass bin for bothCDM and WDM. Comparing low-mass and high-mass resultsfor diffuse accretion, in CDM we see that low-mass haloeshave lower Γ dif than high-mass haloes, while the opposite istrue for WDM (see also figure 2). For accretion by mergers,on the other hand, high-mass haloes have higher Γ mer thanlow-mass haloes in both CDM and WDM.And, interestingly, while the high-mass total accretionrates in CDM and WDM are nearly identical, the relative con-tribution to this total from mergers versus diffuse accretionis higher in CDM than in WDM.These results are all consistent with an overall picture inwhich the arrested growth of small-scale structure in WDMinhibits mass accretion through mergers, as compared toCDM. The fact that differences between WDM and CDMaccretion rates are dramatic at low masses is not surprising,considering that our low-mass bin is at slightly smaller massthan the half-mode mass for our WDM cosmology (see sec-tion 4 for analytical insights into this behaviour). The factthat high-mass haloes also show differences between WDMand CDM can be attributed to the difference in availablesubstructure for such haloes in the two cosmologies. In thefollowing sections, we explore the relation between theseaccretion rates and the local environment of haloes.
To start with, we focus on whether the environmental vari-ables α and δ evolve differently on average for haloes that addmass with and without mergers, in both CDM and WDMcosmologies. Figure 7 shows the median α (left panel) and δ (right panel) for haloes having zero and non-zero mergerrates in the low-mass (blue, cyan) and high-mass (red, or-ange) bins. The solid (dashed) lines show results for WDM(CDM). Error bars on the measurements were calculated by bootstrap sampling at each redshift. Several interestingtrends are apparent, as we discuss next.Overall, we see that the tidal environments of objects atany fixed mass are typically more anisotropic at later times,while most local densities evolve moderately or decrease atlater times. This is broadly consistent with the growth of thecosmic web such that a larger fraction of haloes at fixed massfind themselves in filamentary or sheet-like environments atlate times.Comparing the trends for low-mass and high-mass ob-jects, we see for CDM that high-mass objects (dashed, warmcolours) are in relatively denser and more isotropic environ-ments than low-mass objects (dashed, cool colours), at anyredshift. This is fully consistent with the standard hierar-chical picture in which massive haloes are more clusteredand dominate their tidal environments more than low-masshaloes.In WDM, while massive haloes do live in more isotropicenvironments than their low-mass counterparts (solid warmversus solid cool in left panel), unlike CDM, now the low-mass haloes are in denser environments than high-mass ones(same in right panel, see also figure 4). These low-mass WDMenvironments are also more overdense than the correspondinglow-mass CDM environments. We return to this point later.We focus next on the high-mass bin where one might ex-pect similarities between CDM and WDM haloes (althoughsee section 3.1). Indeed, we see that haloes with mergers oc-cupy nearly identical median density and tidal environmentsat all redshifts in CDM and WDM (compare the dashedversus solid orange curves in each panel). And, within er-rors, this is also true for haloes without recent mergers (redcurves).These high-mass environments tend to be the mostisotropic of all (c.f. figure 3) while having intermediate den-sities, at almost any redshift. Thus, most of the differencesbetween high-mass CDM and WDM haloes are related to the We have also checked that the median b values for varioushalo categories are nearly identical between CDM and WDM, andconform to the usual expectation of low-mass objects being lessbiased than high-mass ones. MNRAS000
To start with, we focus on whether the environmental vari-ables α and δ evolve differently on average for haloes that addmass with and without mergers, in both CDM and WDMcosmologies. Figure 7 shows the median α (left panel) and δ (right panel) for haloes having zero and non-zero mergerrates in the low-mass (blue, cyan) and high-mass (red, or-ange) bins. The solid (dashed) lines show results for WDM(CDM). Error bars on the measurements were calculated by bootstrap sampling at each redshift. Several interestingtrends are apparent, as we discuss next.Overall, we see that the tidal environments of objects atany fixed mass are typically more anisotropic at later times,while most local densities evolve moderately or decrease atlater times. This is broadly consistent with the growth of thecosmic web such that a larger fraction of haloes at fixed massfind themselves in filamentary or sheet-like environments atlate times.Comparing the trends for low-mass and high-mass ob-jects, we see for CDM that high-mass objects (dashed, warmcolours) are in relatively denser and more isotropic environ-ments than low-mass objects (dashed, cool colours), at anyredshift. This is fully consistent with the standard hierar-chical picture in which massive haloes are more clusteredand dominate their tidal environments more than low-masshaloes.In WDM, while massive haloes do live in more isotropicenvironments than their low-mass counterparts (solid warmversus solid cool in left panel), unlike CDM, now the low-mass haloes are in denser environments than high-mass ones(same in right panel, see also figure 4). These low-mass WDMenvironments are also more overdense than the correspondinglow-mass CDM environments. We return to this point later.We focus next on the high-mass bin where one might ex-pect similarities between CDM and WDM haloes (althoughsee section 3.1). Indeed, we see that haloes with mergers oc-cupy nearly identical median density and tidal environmentsat all redshifts in CDM and WDM (compare the dashedversus solid orange curves in each panel). And, within er-rors, this is also true for haloes without recent mergers (redcurves).These high-mass environments tend to be the mostisotropic of all (c.f. figure 3) while having intermediate den-sities, at almost any redshift. Thus, most of the differencesbetween high-mass CDM and WDM haloes are related to the We have also checked that the median b values for varioushalo categories are nearly identical between CDM and WDM, andconform to the usual expectation of low-mass objects being lessbiased than high-mass ones. MNRAS000 , 1–16 (0000) ass accretion rates and environment Figure 7. Local environments with and without mergers:
Evolution of median local tidal anisotropy α (left panel) and localoverdensity δ (right panel) of haloes selected to have grown without mergers (blue, red) and with mergers (cyan, orange) in the previous2 dynamical times, for the low-mass (cooler colours) and high-mass (warmer colours) bin. Results for CDM (WDM) are shown usingdashed (solid) curves. Error bars were calculated by bootstrap sampling at each redshift. Highlights:
While most of these trends conformto expectations based on the preponderance of small-scale bound structures in CDM as compared to WDM, two trends are noteworthy:(i) low-mass CDM haloes growing without mergers are in slightly more anisotropic environments compared to other categories and (ii)low-mass WDM haloes are in denser environments than other categories (see also figure 4). lack of substructure and corresponding dominance of diffuseaccretion in WDM seen in figure 6.In the low-mass bin , the results are more nuanced, withseveral differences between zero and non-zero merger envi-ronments in CDM and WDM which can be summarized asfollows: • The environments of low-mass CDM haloes with andwithout mergers have similar densities at all redshifts (dashedcyan versus dashed blue in the right panel), but are system-atically more anisotropic at low redshift for haloes withoutmergers at any redshift (same in the left panel). We discussthis further below. • In contrast, low-mass WDM haloes without mergers(solid blue) live in systematically less dense and more isotropic environments than those with mergers (solid cyan). This issensible, since isolated, isotropic environments would allowdiffuse accretion to dominate over mergers. • Low-mass WDM haloes without mergers (solid blue) livein more dense environments (with similar tidal anisotropy)than their CDM counterparts (dashed blue). This is consis-tent with an enhancement of diffuse accretion in WDM; thelack of substructure in WDM compared to CDM means thatenvironments conducive to purely diffuse accretion (i.e., norecent mergers) can be denser in WDM than in CDM. • In contrast, low-mass WDM haloes with mergers (solidcyan) live in substantially more dense and more anisotropic environments than their CDM counterparts (dashed cyan).We discuss this below.These low-mass results can be mostly understood keep-ing in mind the preponderance of small objects in WDM ascompared to CDM, as well as the facts that a higher localtidal anisotropy would typically inhibit the accretion of dif-fuse mass (e.g., by redirecting local flows towards nodes; seeBorzyszkowski et al. 2017), while higher densities would en-hance mergers. Some of these trends, however, deserve more careful consideration. For example, the CDM haloes withoutmergers are in more anisotropic environments than thosewith mergers, contrary to what is suggested by the argumentabove. Similarly, the higher anisotropy of low-mass WDMhaloes with mergers as compared to their CDM counterpartsis intriguing as well. On the other hand, the higher density of low-mass WDM haloes without mergers as compared toall other categories might be understood as being due to thesubstantially decreased occurrence of mergers in the smoothWDM cosmic web, with only the highest density environ-ments being capable of sustaining merger events. In the nextsection, we will explore in more detail the correlations be-tween environment and mass accretion rate, which will shedfurther light on the nature and origin of these trends.
To quantify the dependence of mass accretion rates on envi-ronment, in this section we study the evolving correlationsbetween our environmental proxies α and δ , and the massaccretion rates due to diffuse accretion (Γ dif ) and mergers(Γ mer ).Following Ramakrishnan et al. (2019), we calculateSpearman rank correlation coefficients between the envi-ronmental variables and accretion rates; namely, in eachmass bin and for every redshift, we calculate the coefficients α ↔ Γ dif , δ ↔ Γ dif , α ↔ Γ mer , δ ↔ Γ mer , as well as thecoefficient α ↔ δ for reference (which was also studied byRamakrishnan et al. 2019, for z = 0 CDM haloes). In orderto assess which, if any, of the two environmental proxies ismore important, we also compute conditional coefficients .For stochastic variables a , b and c , the coefficient between a and b , conditioned on c , is given by( a ↔ b ) | c = ( a ↔ b ) − ( a ↔ c )( b ↔ c ) . (7) MNRAS , 1–16 (0000) Dhoke & Paranjape
If the variables were Gaussian distributed with a joint dis-tribution that had the structure p ( a, b, c ) = p ( a | c ) p ( b | c ) p ( c ),then the variables a and b would be conditionally indepen-dent and we would have ( a ↔ b ) | c = 0 although ( a ↔ b )need not be zero. We point the reader to Ramakrishnan et al.(2019) for a detailed justification for constructing conditionalcoefficients using equation (7) with Spearman rank coeffi-cients, even for variables that are not Gaussian distributed,as is the case here (see also below).Figure 8 (figure 9) shows the correlation coefficientsfor CDM (WDM). In each case, in the left (right) panel we display correlations between the environment and Γ dif (Γ mer ) as a function of redshift. Additionally, the left panel in each case also shows the evolving correlation ( α ↔ δ ).Results for the low-mass (high-mass) bin are displayed usingcool (warm) colours. The solid curves show the primarycorrelations such as ( α ↔ Γ mer ) while the dotted curvesshow conditional coefficients such as ( α ↔ Γ mer ) | δ . Errorbars on the measurements were calculated by bootstrapsampling at each redshift. This represents the first detailedstudy in the literature, of the evolving correlation betweenlocal environment and mass accretion rates in either CDMor WDM cosmologies.We see that α and δ are always positively correlatedfor both CDM and WDM, with a strength that is relativelyindependent of halo mass and that only weakly depends onredshift and dark matter type. This extends the results ofRamakrishnan et al. (2019) (who studied CDM haloes at z = 0) to significantly higher redshift, as well as to WDMcosmologies. The remaining correlations, which depend onlyweakly on redshift, can be summarized as follows. • High-mass bin: – For both CDM and WDM, we see a strong negativecorrelation ( δ ↔ Γ dif ), and a weaker but significant positivecorrelation ( δ ↔ Γ mer ), at all redshifts (solid orange).This is consistent with the expectation that overdenseenvironments enhance mergers while underdense regionsallow for more diffuse accretion (Fakhouri & Ma 2010).– The corresponding primary correlations ( α ↔ Γ dif )and ( α ↔ Γ mer ) (solid red) are all negative, with mag-nitudes comparable to those in the case of δ , for bothCDM and WDM. Interestingly, however, the conditional coefficient ( α ↔ Γ dif ) | δ (dotted red in left panels) for bothCDM and WDM almost vanishes at all z , while the cor-responding conditional coefficient ( α ↔ Γ mer ) | δ does notvanish (dotted red in right panels). Thus, for high-masshaloes, α and δ are equally important for accretion viamergers, while δ almost completely explains the environ-ment dependence of diffuse accretion. • Low-mass bin: – In low-mass CDM haloes, Γ mer shows almost no cor-relation with either α or δ at any redshift, consistent with For this correlation analysis, we only evaluate Γ mer using thosehaloes that had mergers in the previous 2 dynamical times, i.e.,objects having Γ mer (cid:54) = 0. Γ dif on the other hand, is evaluatedusing all haloes in the bin. When using a mass threshold M vir > . × h − M (cid:12) fordefining progenitors (see section 2.3), the high-mass negative ( α ↔ Γ mer ) correlation becomes weaker, and in fact slightly positive atlow redshift. This is the only qualitative difference we find betweendifferent progenitor definitions. similar weak correlations seen previously using other envi-ronmental proxies (c.f. figure 4 of Fakhouri & Ma 2010).– The situation changes in WDM: we now see positive correlations ( α ↔ Γ mer ) and ( δ ↔ Γ mer ), comparable instrength to each other and increasing at higher redshift.Moreover, at least at low redshift, we see that the condi-tional coefficient ( δ ↔ Γ mer ) | α (dotted cyan) is close tozero. In fact, this is the case within errors at all z (cid:46) . Thus, α mostly accounts for the environment dependenceof Γ mer at nearly all redshifts for low-mass WDM haloes,with Γ mer being enhanced in more anisotropic tidal envi-ronments. – For Γ dif in low-mass CDM haloes, both ( δ ↔ Γ dif )and ( α ↔ Γ dif ) are negative, with comparable magnitudes(solid cyan and blue). Also, neither of the conditionalcoefficients ( α ↔ Γ dif ) | δ and ( δ ↔ Γ dif ) | α is close to zero(dotted cyan and blue). Thus, both α and δ play a role forΓ dif , consistent with the expectation that diffuse accretionshould be more efficient far from crowded regions, i.e. inunderdense, isotropic environments in CDM.– For Γ dif in low-mass WDM haloes, on the otherhand, we see a stronger ( α ↔ Γ dif ) anticorrelation (solidblue), which almost completely explains the ( δ ↔ Γ dif )(solid cyan) anticorrelation, seen as the near-vanishing of( δ ↔ Γ dif ) | α (dotted cyan) at all redshifts. Thus, the envi-ronment dependence of Γ dif in low-mass haloes is related toboth α and δ for CDM and almost completely explained by α for WDM. The dominant role played by the tidal environ-ment for low-mass WDM haloes is consistent with previoussimulation results (Angulo et al. 2013) which indicate thatthese objects form in a well-established tidal environmentwhich can then further affect their mass accretion.The nature of the environmental correlations discussedabove reveals a complex interplay between mass accretionrates and local environment. While this is perhaps not sur-prising, considering the intimate connection between massaccretion in the halo outskirts and the internal structure ofhaloes (e.g., Diemer & Kravtsov 2014; More et al. 2015), ourresults above show several interesting aspects. In section 3.2,for example, we saw that low-mass WDM haloes with merg-ers live in systematically more anisotropic environments thantheir CDM counterparts, a trend consistent with the correla-tions seen above in which ( α ↔ Γ mer ) is positive for WDMbut negative for CDM. This is counter-intuitive, consider-ing that accretion due to mergers is enhanced in overdense,isotropic environments in CDM, and one might expect thistrend to be even more pronounced in WDM which has lesssmall-scale structure available in all environments. Moreover,the small magnitude of ( δ ↔ Γ mer ) | α for these low-massWDM haloes shows that α , in fact, dominates the environ-mental trends for Γ mer . This connection between mergersand the local tidal environment of low-mass WDM haloesdeserves further study.Another puzzling result from section 3.2 was that theenvironments of low-mass CDM haloes without mergers aremore anisotropic than of those with mergers. In this caseas well, the correlation results above do not particularlyclarify the situation, since they are consistent with Γ dif beingenhanced in underdense, isotropic environments (with both δ and α playing comparable roles) and Γ mer being nearlyuncorrelated with environment. We have checked that thesetrends are not restricted to the median α alone; rather, the MNRAS000
If the variables were Gaussian distributed with a joint dis-tribution that had the structure p ( a, b, c ) = p ( a | c ) p ( b | c ) p ( c ),then the variables a and b would be conditionally indepen-dent and we would have ( a ↔ b ) | c = 0 although ( a ↔ b )need not be zero. We point the reader to Ramakrishnan et al.(2019) for a detailed justification for constructing conditionalcoefficients using equation (7) with Spearman rank coeffi-cients, even for variables that are not Gaussian distributed,as is the case here (see also below).Figure 8 (figure 9) shows the correlation coefficientsfor CDM (WDM). In each case, in the left (right) panel we display correlations between the environment and Γ dif (Γ mer ) as a function of redshift. Additionally, the left panel in each case also shows the evolving correlation ( α ↔ δ ).Results for the low-mass (high-mass) bin are displayed usingcool (warm) colours. The solid curves show the primarycorrelations such as ( α ↔ Γ mer ) while the dotted curvesshow conditional coefficients such as ( α ↔ Γ mer ) | δ . Errorbars on the measurements were calculated by bootstrapsampling at each redshift. This represents the first detailedstudy in the literature, of the evolving correlation betweenlocal environment and mass accretion rates in either CDMor WDM cosmologies.We see that α and δ are always positively correlatedfor both CDM and WDM, with a strength that is relativelyindependent of halo mass and that only weakly depends onredshift and dark matter type. This extends the results ofRamakrishnan et al. (2019) (who studied CDM haloes at z = 0) to significantly higher redshift, as well as to WDMcosmologies. The remaining correlations, which depend onlyweakly on redshift, can be summarized as follows. • High-mass bin: – For both CDM and WDM, we see a strong negativecorrelation ( δ ↔ Γ dif ), and a weaker but significant positivecorrelation ( δ ↔ Γ mer ), at all redshifts (solid orange).This is consistent with the expectation that overdenseenvironments enhance mergers while underdense regionsallow for more diffuse accretion (Fakhouri & Ma 2010).– The corresponding primary correlations ( α ↔ Γ dif )and ( α ↔ Γ mer ) (solid red) are all negative, with mag-nitudes comparable to those in the case of δ , for bothCDM and WDM. Interestingly, however, the conditional coefficient ( α ↔ Γ dif ) | δ (dotted red in left panels) for bothCDM and WDM almost vanishes at all z , while the cor-responding conditional coefficient ( α ↔ Γ mer ) | δ does notvanish (dotted red in right panels). Thus, for high-masshaloes, α and δ are equally important for accretion viamergers, while δ almost completely explains the environ-ment dependence of diffuse accretion. • Low-mass bin: – In low-mass CDM haloes, Γ mer shows almost no cor-relation with either α or δ at any redshift, consistent with For this correlation analysis, we only evaluate Γ mer using thosehaloes that had mergers in the previous 2 dynamical times, i.e.,objects having Γ mer (cid:54) = 0. Γ dif on the other hand, is evaluatedusing all haloes in the bin. When using a mass threshold M vir > . × h − M (cid:12) fordefining progenitors (see section 2.3), the high-mass negative ( α ↔ Γ mer ) correlation becomes weaker, and in fact slightly positive atlow redshift. This is the only qualitative difference we find betweendifferent progenitor definitions. similar weak correlations seen previously using other envi-ronmental proxies (c.f. figure 4 of Fakhouri & Ma 2010).– The situation changes in WDM: we now see positive correlations ( α ↔ Γ mer ) and ( δ ↔ Γ mer ), comparable instrength to each other and increasing at higher redshift.Moreover, at least at low redshift, we see that the condi-tional coefficient ( δ ↔ Γ mer ) | α (dotted cyan) is close tozero. In fact, this is the case within errors at all z (cid:46) . Thus, α mostly accounts for the environment dependenceof Γ mer at nearly all redshifts for low-mass WDM haloes,with Γ mer being enhanced in more anisotropic tidal envi-ronments. – For Γ dif in low-mass CDM haloes, both ( δ ↔ Γ dif )and ( α ↔ Γ dif ) are negative, with comparable magnitudes(solid cyan and blue). Also, neither of the conditionalcoefficients ( α ↔ Γ dif ) | δ and ( δ ↔ Γ dif ) | α is close to zero(dotted cyan and blue). Thus, both α and δ play a role forΓ dif , consistent with the expectation that diffuse accretionshould be more efficient far from crowded regions, i.e. inunderdense, isotropic environments in CDM.– For Γ dif in low-mass WDM haloes, on the otherhand, we see a stronger ( α ↔ Γ dif ) anticorrelation (solidblue), which almost completely explains the ( δ ↔ Γ dif )(solid cyan) anticorrelation, seen as the near-vanishing of( δ ↔ Γ dif ) | α (dotted cyan) at all redshifts. Thus, the envi-ronment dependence of Γ dif in low-mass haloes is related toboth α and δ for CDM and almost completely explained by α for WDM. The dominant role played by the tidal environ-ment for low-mass WDM haloes is consistent with previoussimulation results (Angulo et al. 2013) which indicate thatthese objects form in a well-established tidal environmentwhich can then further affect their mass accretion.The nature of the environmental correlations discussedabove reveals a complex interplay between mass accretionrates and local environment. While this is perhaps not sur-prising, considering the intimate connection between massaccretion in the halo outskirts and the internal structure ofhaloes (e.g., Diemer & Kravtsov 2014; More et al. 2015), ourresults above show several interesting aspects. In section 3.2,for example, we saw that low-mass WDM haloes with merg-ers live in systematically more anisotropic environments thantheir CDM counterparts, a trend consistent with the correla-tions seen above in which ( α ↔ Γ mer ) is positive for WDMbut negative for CDM. This is counter-intuitive, consider-ing that accretion due to mergers is enhanced in overdense,isotropic environments in CDM, and one might expect thistrend to be even more pronounced in WDM which has lesssmall-scale structure available in all environments. Moreover,the small magnitude of ( δ ↔ Γ mer ) | α for these low-massWDM haloes shows that α , in fact, dominates the environ-mental trends for Γ mer . This connection between mergersand the local tidal environment of low-mass WDM haloesdeserves further study.Another puzzling result from section 3.2 was that theenvironments of low-mass CDM haloes without mergers aremore anisotropic than of those with mergers. In this caseas well, the correlation results above do not particularlyclarify the situation, since they are consistent with Γ dif beingenhanced in underdense, isotropic environments (with both δ and α playing comparable roles) and Γ mer being nearlyuncorrelated with environment. We have checked that thesetrends are not restricted to the median α alone; rather, the MNRAS000 , 1–16 (0000) ass accretion rates and environment Figure 8. Correlations between accretion rates and local environment (CDM haloes):
Spearman rank correlation coefficientsbetween local environmental variables α , δ and specific accretion rates Γ dif (left panel) and Γ mer (right panel) , as a function of redshift.In each panel, cool (warm) colours indicate results for low-mass (high-mass) haloes. Solid curves show the correlations (Γ ↔ α ) (red, blue)and (Γ ↔ δ ) (orange, cyan) and, in the left panel , ( α ↔ δ ) (brown, dark blue). Dotted curves show conditional coefficients (Γ ↔ α ) | δ and(Γ ↔ δ ) | α , calculated using equation (7). Highlights: (i) There are significant anti-correlations (Γ dif ↔ δ ) and (Γ dif ↔ α ) in both massbins (left panel, solid curves). (ii) Γ mer correlates positively (negatively) with δ ( α ) in the high-mass bin (right panel, solid orange andred), and does not correlate with either in the low-mass bin (right panel, solid cyan and blue). (iii) The conditional coefficient (Γ dif ↔ α ) | δ for high-mass haloes (left panel, dotted red) almost vanishes at all z , indicating that δ almost fully explains the environment dependenceof diffuse accretion in high-mass CDM haloes. Figure 9.
Same as figure 8, for
WDM haloes . Highlights: (i) The coefficient (Γ mer ↔ α ) for low-mass haloes is, surprisingly, positive(right panel, solid blue) and the corresponding conditional coefficient (Γ mer ↔ δ ) | α (right panel, dotted cyan) is close to zero. (ii) Theconditional coefficient (Γ dif ↔ δ ) | α for low-mass haloes nearly vanishes (left panel, dotted cyan), even though the primary correlation(Γ dif ↔ δ ) (solid cyan) is significantly non-zero. The same is true for the conditional coefficient (Γ dif ↔ α ) | δ for high-mass haloes (leftpanel, dotted red). Thus, α ( δ ) almost completely explains the environment dependence of diffuse accretion for low-mass (high-mass)WDM haloes. entire distributions of α in such haloes are shifted relativeto each other.As a check on systematic errors due to our use of corre-lation coefficients and equation (7), we repeated the analysisby measuring the median accretion rates Γ dif and Γ mer innarrow bins of α and δ . The results, although noisy, are fullyconsistent with the conclusions above. In this section, we explore the nature of the correlationsbetween mass accretion rate and the large-scale halo environ-ment (characterised by halo bias b ; see section 2.5) in fixedmass bins, also known as halo assembly bias or secondarybias.Previous studies have largely focused on secondary halo MNRAS , 1–16 (0000) Dhoke & Paranjape variables such as age, concentration, shape, angular momen-tum and velocity dispersion structure (see the Introductionfor references). To our knowledge, the only previous workthat studied the assembly bias of mass accretion rates wasby Lazeyras et al. (2017), who focused on the total massaccretion rates of high-mass CDM haloes at z = 0. Ouranalysis below therefore substantially extends these resultsin terms of halo mass, redshift range and dark matter type.We define the assembly bias of mass accretion as the twocorrelations ( b ↔ Γ dif ) and ( b ↔ Γ mer ) measured usingSpearman rank correlation coefficients in different mass binsfor all redshifts. The results of Ramakrishnan et al. (2019)indicate that, at z = 0 in CDM, the assembly bias of eachof the internal properties halo concentration, spin, shapeand velocity dispersion can be mostly attributed to twofundamental correlations, one between b and α and theother between α and the halo internal property. Consideringthat there are relatively strong correlations between the massaccretion rates Γ mer , Γ dif and the local environmental proxies α and δ (see figures 8 and 9), it is interesting to ask (a) whatis the overall strength of assembly bias with mass accretionrate in comparison to other internal halo properties, and (b)what role, if any, do the local environmental proxies play inexplaining these trends?Figure 10 (figure 11) shows the results for CDM (WDM)haloes. As with figures 8 and 9, the left (right) panels showresults for Γ dif (Γ mer ), with the solid (dotted) curves show-ing primary (conditional) coefficients. Additionally, the left(right) panels show the correlation ( b ↔ α ) (( b ↔ δ )) forcomparison.Similarly to the results for ( α ↔ δ ) in figures 8 and 9,we see that the correlations ( b ↔ α ) and ( b ↔ δ ) aresignificantly positive, very similar across halo mass as well asbetween CDM and WDM, and evolve moderately between z = 2 and z = 0. The ( b ↔ α ) correlation at any redshift is ∼
50% larger than the corresponding ( b ↔ δ ) correlation, forboth CDM and WDM. This extends the z = 0 CDM resultsof Paranjape et al. (2018) and Ramakrishnan et al. (2019)to significantly higher redshifts and to WDM cosmologies.In the high-mass bin , the correlations ( b ↔ Γ dif ) and( b ↔ Γ mer ) are relatively weak at nearly all redshifts for bothCDM and WDM. The exception is at the highest redshift z = 2 for WDM, where ( b ↔ Γ dif ) is negative while ( b ↔ Γ mer ) is positive, but with large errors. Overall, therefore,we conclude that high-mass objects in CDM and WDM donot show significant assembly bias for either mass accretionrate.In the low-mass bin , we see more interesting results.Although the ( b ↔ Γ mer ) correlation is essentially zeroacross all redshifts for both CDM and WDM, the sameis not true for the ( b ↔ Γ dif ) correlation, which remainssignificantly negative at nearly all redshifts, in both CDMand WDM. In fact, the strength of the correlation at z =0, namely | ( b ↔ Γ dif ) | (cid:38) .
05 (0 .
1) for CDM (WDM), isamongst the strongest seen for any secondary variable atthis halo mass (see, e.g., the middle panel of figure 2 inRamakrishnan et al. 2019). The magnitude of the correlationis nearly constant with redshift for CDM while showing anincreasing trend with time for WDM.
Thus, the assembly biasof low-mass accretion rates, especially at low redshift, is acomparatively large effect, entirely driven by diffuse accretion.
More interestingly, the conditional coefficient ( b ↔ Γ dif ) | α for low-mass haloes is close to zero at all redshifts,for both CDM and WDM. (For CDM, the coefficient ( b ↔ Γ dif ) | δ also has a small magnitude, but is not as close to zeroas ( b ↔ Γ dif ) | α .) Thus, the local tidal anisotropy plays a key role in ex-plaining the accretion rate assembly bias of both CDM andWDM haloes , with local overdensity also playing a significantrole in CDM. These findings are consistent with previous re-sults in the literature (e.g., Dalal et al. 2008; Hahn et al. 2009;Borzyszkowski et al. 2017) which show that low-mass haloeswhich are in the vicinity of massive haloes, and are hencehighly clustered at large-scales while residing in anisotropiclocal tidal environments, show suppressed (diffuse) accretionrates.
In this section, we discuss the numerical results presentedabove in the language of the excursion set approach (Bondet al. 1991; Lacey & Cole 1993, 1994; Bond & Myers 1996).Below, we briefly recapitulate the basic concepts underlyingthe excursion set approach and its modern variants, beforediscussing our results in this context.
In the excursion set framework, halo abundances, large-scaleclustering and accretion rates are estimated by identifyingand counting likely locations of virialisation identified inthe initial stochastic density field (linearly extrapolated topresent epoch). A key ingredient in the traditional excursionset approach is a choice of collapse threshold or ‘barrier’ B which must be crossed by random walks in the linearlyextrapolated density field δ R as a function of Lagrangiansmoothing scale R . The barrier B is usually adopted fromspherical (Gunn & Gott 1972) or ellipsoidal (Sheth et al.2001) collapse models, and depends on the redshift z ofinterest and possibly additional stochastic variables (suchas those related to tidal effects in the ellipsoidal model, seeCastorina et al. 2016).The statistical properties of the random walks, on theother hand, are determined by the choice of initial matterpower spectrum and a smoothing filter, which fixes the func-tional relation σ ( m ) where σ is the variance of the linearlyextrapolated density contrast smoothed on a Lagrangianscale corresponding to mass m ∼ R . If a random walkfirst upcrosses the barrier B ( z ) at scale R ( m ) (starting from R → ∞ or σ → m is declared to format redshift z . While early work on the subject focused on afilter that is sharp in Fourier space (for reasons of analyticalsimplicity), later developments have shown how to efficientlyanalyze the effects of more realistic filters that are compactin real space and lead to random walks with correlated stepsas R is varied (Musso & Sheth 2012; see also Bond et al. 1991;Zentner 2007; Paranjape et al. 2012). Finally, the recognitionthat the sites of virialisation are special (Sheth et al. 2001) –e.g., peaks in the linearly extrapolated density field (Bardeenet al. 1986) – leads to an ‘excursion set peaks’ approach(ESP, Paranjape & Sheth 2012; Appel & Jones 1990) whichhas been shown to agree with simulated halo mass functionsand large-scale clustering in CDM cosmologies at the ∼ MNRAS000
In the excursion set framework, halo abundances, large-scaleclustering and accretion rates are estimated by identifyingand counting likely locations of virialisation identified inthe initial stochastic density field (linearly extrapolated topresent epoch). A key ingredient in the traditional excursionset approach is a choice of collapse threshold or ‘barrier’ B which must be crossed by random walks in the linearlyextrapolated density field δ R as a function of Lagrangiansmoothing scale R . The barrier B is usually adopted fromspherical (Gunn & Gott 1972) or ellipsoidal (Sheth et al.2001) collapse models, and depends on the redshift z ofinterest and possibly additional stochastic variables (suchas those related to tidal effects in the ellipsoidal model, seeCastorina et al. 2016).The statistical properties of the random walks, on theother hand, are determined by the choice of initial matterpower spectrum and a smoothing filter, which fixes the func-tional relation σ ( m ) where σ is the variance of the linearlyextrapolated density contrast smoothed on a Lagrangianscale corresponding to mass m ∼ R . If a random walkfirst upcrosses the barrier B ( z ) at scale R ( m ) (starting from R → ∞ or σ → m is declared to format redshift z . While early work on the subject focused on afilter that is sharp in Fourier space (for reasons of analyticalsimplicity), later developments have shown how to efficientlyanalyze the effects of more realistic filters that are compactin real space and lead to random walks with correlated stepsas R is varied (Musso & Sheth 2012; see also Bond et al. 1991;Zentner 2007; Paranjape et al. 2012). Finally, the recognitionthat the sites of virialisation are special (Sheth et al. 2001) –e.g., peaks in the linearly extrapolated density field (Bardeenet al. 1986) – leads to an ‘excursion set peaks’ approach(ESP, Paranjape & Sheth 2012; Appel & Jones 1990) whichhas been shown to agree with simulated halo mass functionsand large-scale clustering in CDM cosmologies at the ∼ MNRAS000 , 1–16 (0000) ass accretion rates and environment Figure 10. Accretion rate assembly bias (CDM haloes):
Spearman rank correlation coefficients between large-scale halo bias b and specific accretion rates Γ dif (left panel) and Γ mer (right panel) for low-mass (blue) and high-mass (red) CDM haloes, as a function ofredshift. Solid curves show the correlations (Γ ↔ b ) and, in the left [right] panel , green curves show ( b ↔ α ) [( b ↔ δ )], with light (dark)green for high-mass (low-mass) haloes. Dashed and dotted curves respectively show conditional coefficients (Γ ↔ b ) | α and (Γ ↔ b ) | δ ,calculated using equation (7). Error bars were estimated using bootstrap sampling at each redshift. Highlights:
There is a significantnegative correlation (Γ dif ↔ b ) for low-mass haloes at all redshifts (left panel, solid blue), which is largely explained by the (Γ dif ↔ α )correlation (conditional coefficient shown by the dashed blue curve is close to zero). Figure 11.
Same as figure 10, for
WDM haloes . Highlights:
There is a significant negative correlation (Γ dif ↔ b ) for low-mass haloes(left panel, solid blue), which increases in magnitude at low redshift, is stronger than the corresponding correlation for CDM haloes and ismostly explained by the (Γ dif ↔ α ) correlation (conditional coefficient shown by the dashed blue curve is close to zero). HP14), who demonstrated that the suppression of small-scale power in these models is an excellent diagnostic tool fortesting the assumptions underlying the excursion set (peaks)approach. In particular, HP14 argued that a single-barrierframework is insufficient to properly explain halo abundancesdue to small but systematic errors in the ellipsoidal collapsemodel (Monaco 1999; Giocoli et al. 2007; Ludlow et al. 2014),which become dramatically amplified in WDM models ascompared to CDM. Here, however, we are interested in theprediction (Lacey & Cole 1993, 1994) that mass accretionrates are intimately connected with the slope of the initialpower spectrum near the scales of interest, with steeper spectra leading to higher accretion rates. The consequencesof the latter effect for WDM can be understood withoutdelving into the details of any specific excursion set modeland only depend on the correlated nature of the randomwalks, as we discuss next.
In the following, in addition to the barrier B for linearlyextrapolated density fluctuations and the σ ( m ) relation men-tioned above, we will need two more quantities from theexcursion set lexicon. The first is the proto-halo ‘significance’ MNRAS , 1–16 (0000) Dhoke & Paranjape ν ( m, z ) = δ c ( z ) /σ ( m ), where δ c ( z ) ∝ /D ( z ) is the collapsethreshold from the spherical model, with D ( z ) being the lin-ear theory growth factor. The second is the typical peak curva-ture (cid:104) x | ν (cid:105) at scale ν ( m, z ), where x = −∇ δ R / (cid:112) Var( ∇ δ R )(see figure 6 of Bardeen et al. 1986).As mentioned above, accretion rates in CDM and WDMare expected to be very different due to the comparativesteepness of the initial matter power spectrum in WDMaround the half-mode mass scale. To see what this implies,we must mainly keep in mind that the σ ( m ) relation becomes‘stretched out’ for a truncated power spectrum such as WDM,with σ ( m ) → a constant below the half-mode mass (see,e.g., figure 2 of HP14). In CDM, on the other hand, σ ( m )continues to increase down to very small masses. A simplecalculation now shows that the diffuse mass accretion ratecan be written as (e.g., Lazeyras et al. 2017)Γ = d ln M d ln a = 1 σ B/ d σ d B/ d ln a d ln σ/ d ln M ≈ σ γ (cid:104) x | ν (cid:105) | d B/ d ln a || d ln σ/ d ln M | , (8)where the first line is a manipulation of variables and thesecond line relates the slope at barrier crossing, d B/ d σ , tothe peak curvature (cid:104) x | ν (cid:105) (Musso & Sheth 2012; Paranjape& Sheth 2012), with γ being a spectral variable of orderunity in both CDM and WDM ( γ (cid:39) . (cid:104) x | ν (cid:105) is large, σ is small and | d ln σ/ d ln M | is finite. At masses smaller thanthe half-mode mass for WDM, (cid:104) x | ν (cid:105) is smaller than at highmasses, σ is constant and | d ln σ/ d ln M | approaches zero.This last feature makes the accretion rates very high. Atlow masses for CDM, on the other hand, (cid:104) x | ν (cid:105) is similar tothat in WDM, σ is significantly larger than at high massesand | d ln σ/ d ln M | is finite, so that low-mass CDM accretionrates remain small. This qualitatively explains the overalltrends seen in the right panel of figure 6.Quantitatively, in our high-mass bin at z = 0, forboth CDM and WDM we have σ (cid:46) . (cid:104) x | ν (cid:105) (cid:38) | d ln σ/ d ln M | (cid:46) .
2. Assuming that the time dependenceof the barrier is completely determined by δ c ( z ), we alsohave | d B/ d ln a | ∼ z = 0. Equation (8) then predictshigh-mass accretion rates of Γ ∼ . z = 0. In the low-mass bin, for CDM, we have σ ∼ . (cid:104) x | ν (cid:105) ∼ . | d ln σ/ d ln M | (cid:46) .
15. For WDM, on the other hand, while σ and (cid:104) x | ν (cid:105) are not very different from CDM for our cosmol-ogy and choice of mass bin, we have | d ln σ/ d ln M | (cid:46) . z = 0 are predicted to be about a factor 2 higherfor our WDM low-mass bin than the corresponding CDMvalues, which agrees with the trend seen in the right panelof figure 6. The values predicted, Γ ∼ . .
0) for CDM(WDM), are higher than the measured ones, which is per-haps not surprising considering the approximate nature ofour calculation. The discussion of environmental trends for mass ac-cretion rates requires a prediction of the cross-correlationbetween the slopes of random walks at barrier crossing and We have also ignored the issue of mass reassignment discussedby HP14, in which the σ ( m ) relation must effectively be modifiedfor collapsed objects, with the effect becoming more prominent inWDM at scales substantially smaller than the half-mode mass. the values attained by these walks at larger smoothing scales.This is conceptually easiest for the density environment,which is the natural variable used in excursion set calcula-tions. Lazeyras et al. (2017) have applied such arguments toshow that an overall assembly bias trend, in which slowlyaccreting haloes are strongly clustered as compared to rapidaccretors, i.e. a negative correlation ( b ↔ Γ), is a natu-ral prediction of the excursion set peaks framework. Thedistinction between diffuse accretion and that via mergers,on the other hand, requires a higher level of sophisticationin the simultaneous prediction of continuous and discreteaccretion rates, an aspect which excursion set models haveonly recently begun to explore (Musso & Sheth 2014). Andthe trends we have noted with tidal anisotropy α , especiallythe result that α largely explains low-mass assembly biastrends with mass accretion, are currently not predictableby any excursion set (peaks) model that we are aware of(although see Castorina et al. 2016, for some initial steps inthis direction). We therefore defer a discussion of analyticalpredictions for the environment dependence of mass accre-tion to future work, where we hope to develop a versatileexcursion set framework that can address these issues. We have investigated the evolving correlations between massaccretion rates and environment in N -body simulations ofCDM and WDM cosmologies. The latter are characterisedby a strong suppression of small-scale power (we deliberatelychose a somewhat extreme case with m dm = 0 . ≥ z ≥ mer ) and diffuse mass (Γ dif ; see equa-tions 3) can be understood in terms of the lack of small-scalestructure below the half-mode mass scale in WDM, whichaffects not only the low-mass haloes but also the accretionhistories of massive objects. We summarize these below andalso highlight a few puzzling findings that deserve furtherstudy. Environment-independent trends: • Specific accretion rates of low-mass CDM haloes arelower than those of high-mass CDM haloes, while low-massWDM haloes have higher accretion rates than their high-masscounterparts (figure 6). This is a straightforward prediction ofthe excursion set approach with correlated steps (section 4). • Mass accretion in WDM haloes is dominated by Γ dif ,consistent with previous work (Benson et al. 2013; Elahi et al.2014). Γ dif (Γ mer ) in WDM haloes is also higher (lower) than MNRAS000
0) for CDM(WDM), are higher than the measured ones, which is per-haps not surprising considering the approximate nature ofour calculation. The discussion of environmental trends for mass ac-cretion rates requires a prediction of the cross-correlationbetween the slopes of random walks at barrier crossing and We have also ignored the issue of mass reassignment discussedby HP14, in which the σ ( m ) relation must effectively be modifiedfor collapsed objects, with the effect becoming more prominent inWDM at scales substantially smaller than the half-mode mass. the values attained by these walks at larger smoothing scales.This is conceptually easiest for the density environment,which is the natural variable used in excursion set calcula-tions. Lazeyras et al. (2017) have applied such arguments toshow that an overall assembly bias trend, in which slowlyaccreting haloes are strongly clustered as compared to rapidaccretors, i.e. a negative correlation ( b ↔ Γ), is a natu-ral prediction of the excursion set peaks framework. Thedistinction between diffuse accretion and that via mergers,on the other hand, requires a higher level of sophisticationin the simultaneous prediction of continuous and discreteaccretion rates, an aspect which excursion set models haveonly recently begun to explore (Musso & Sheth 2014). Andthe trends we have noted with tidal anisotropy α , especiallythe result that α largely explains low-mass assembly biastrends with mass accretion, are currently not predictableby any excursion set (peaks) model that we are aware of(although see Castorina et al. 2016, for some initial steps inthis direction). We therefore defer a discussion of analyticalpredictions for the environment dependence of mass accre-tion to future work, where we hope to develop a versatileexcursion set framework that can address these issues. We have investigated the evolving correlations between massaccretion rates and environment in N -body simulations ofCDM and WDM cosmologies. The latter are characterisedby a strong suppression of small-scale power (we deliberatelychose a somewhat extreme case with m dm = 0 . ≥ z ≥ mer ) and diffuse mass (Γ dif ; see equa-tions 3) can be understood in terms of the lack of small-scalestructure below the half-mode mass scale in WDM, whichaffects not only the low-mass haloes but also the accretionhistories of massive objects. We summarize these below andalso highlight a few puzzling findings that deserve furtherstudy. Environment-independent trends: • Specific accretion rates of low-mass CDM haloes arelower than those of high-mass CDM haloes, while low-massWDM haloes have higher accretion rates than their high-masscounterparts (figure 6). This is a straightforward prediction ofthe excursion set approach with correlated steps (section 4). • Mass accretion in WDM haloes is dominated by Γ dif ,consistent with previous work (Benson et al. 2013; Elahi et al.2014). Γ dif (Γ mer ) in WDM haloes is also higher (lower) than MNRAS000 , 1–16 (0000) ass accretion rates and environment that in CDM haloes of the same mass, as expected from thelack of small-scale bound structures in WDM. Trends with local environment: • The evolving median local density and tidal anisotropyof haloes with and without mergers (figure 7) are also largelyconsistent with expectations based on the lack of small-scale structure in WDM. The only exceptions, which deservefurther study, are– the environments of low-mass WDM haloes are denserthan those of all other categories, and– the environments of low-mass CDM haloes withoutmergers are more anisotropic than of those with mergers. • The correlations between environment and accretionrates in figures 8 and 9 add further detail to these results.For both CDM and WDM, at all redshifts in the high-massbin , – there is a strong negative correlation between δ andΓ dif , a weaker positive correlation between δ and Γ mer ,and comparable negative correlations between α and bothΓ dif and Γ mer ,– while α and δ are equally important for Γ mer , δ almostcompletely explains the environment dependence of Γ dif . • In the low-mass bin at all z , for CDM haloes, α and δ areequally important in explaining the environment dependenceof Γ dif and show no correlation with Γ mer . In WDM, on theother hand, the environment dependence of both Γ mer andΓ dif is almost fully explained by α . In fact, the correlation( α ↔ Γ mer ) is positive , which is counter-intuitive and deservesfurther study. In summary, at all z , the local tidal anisotropy α plays acompletely dominant role for both Γ mer and Γ dif in low-massWDM haloes, and a bigger role than the local density δ for Γ mer in high-mass haloes in both CDM and WDM. In contrast, δ plays the dominant role for Γ dif in high-mass CDM haloes. In the other cases, δ and α play equally important roles. Assembly bias:
We defined assembly bias using the Spearman rank correla-tion coefficients ( b ↔ Γ dif ) and ( b ↔ Γ mer ) in each massbin (c.f., Ramakrishnan et al. 2019). We find that the fol-lowing holds for both CDM and WDM haloes (figures 10and 11): • ( b ↔ α ) is always higher than ( b ↔ δ ) at any z . Thisextends the z = 0 CDM results of Paranjape et al. (2018) andRamakrishnan et al. (2019) to higher redshifts and WDMcosmologies. • We detect assembly bias in the low-mass bin, drivenentirely by Γ dif , with a strength ( b ↔ Γ dif ) at z = 0 that isthe highest of nearly all known secondary variables for CDMand is higher in WDM than in CDM (solid blue curves in theleft panels of figures 10 and 11, compare the middle panel offigure 2 in Ramakrishnan et al. 2019). In the high-mass bin,there is no significant assembly bias at any but the highestredshifts we study. • The tidal anisotropy α plays a dominant role in ex-plaining the low-mass assembly bias, especially for WDMhaloes.Our results place important constraints on (semi-) an-alytical excursion set models of halo formation and growth and, by extension, on galaxy evolution models built uponsuch approximate techniques (see the Introduction for refer-ences). As argued by HP14, any such model which purportsto explain environmental trends of evolving haloes must logi-cally work equally well for CDM and WDM, since the physicsof collisionless self-gravitating systems is common to both.As we have seen, however, the suppression of small-scalepower in WDM leads to an intricate dependence of the massaccretion on local (and large-scale) halo environment, partic-ularly at masses smaller than the WDM half-mode mass. Aswith the mass function at these scales discussed by HP14,producing an accurate model of mass accretion with thecorrect environment dependence is likely to reveal interestingfeatures of collisionless dynamics in the shell-crossed regime.It will be very interesting to confront our results above withexcursion set models of mergers and mass accretion (Lacey& Cole 1994; Mitra et al. 2011; Musso & Sheth 2014), as wellas excursion set-inspired semi-analytical algorithms tuned toreproduce CDM results (Somerville & Kolatt 1999; Parkin-son et al. 2008; Jiang & van den Bosch 2014). We leave thisto future work. ACKNOWLEDGEMENTS
We thank Oliver Hahn and Sujatha Ramakrishnan for usefuldiscussions. PD thanks IUCAA for hospitality and workingfacilities, Akhilesh Peshwe (Principal, DMPDM Science Col-lege) for his kind support, Isha Pahwa for useful discussionsand Dhairyashil Jagadale for constant support and discussion.The research of AP is supported by the Associateship Schemeof ICTP, Trieste and the Ramanujan Fellowship awarded bythe Department of Science and Technology, Government ofIndia. This work used the open source computing packagesNumPy (Van Der Walt et al. 2011), SciPy (Virtanen et al.2020) and the plotting software Veusz. We gratefullyacknowledge the use of high performance computing facilitiesat IUCAA, Pune.
DATA AVAILABILITY
The data underlying this work will be shared upon reasonablerequest to the authors.
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