Halo mass and weak galaxy-galaxy lensing profiles in rescaled cosmological N -body simulations
MMNRAS , 1–26 (2018) Preprint September 13, 2018 Compiled using MNRAS L A TEX style file v3.0
Halo mass and weak galaxy-galaxy lensing profiles inrescaled cosmological N -body simulations Malin Renneby , ? , Stefan Hilbert , and Ra´ul E. Angulo Excellence Cluster Universe, Boltzmannstraße 2, 85748 Garching bei M¨unchen, Germany Ludwig-Maximilians-Universit¨at, Fakult¨at f¨ur Physik, Universit¨ats-Sternwarte, Scheinerstraße 1, 81679 M¨unchen, Germany Centro de Estudios de F´ısica del Cosmos de Arag´on, Plaza de San Juan 1, 44001 Teruel, Spain
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We investigate 3D density and weak lensing profiles of dark matter haloes predictedby a cosmology-rescaling algorithm for N -body simulations. We extend the rescal-ing method of Angulo & White (2010) and Angulo & Hilbert (2015) to improve itsperformance on intra-halo scales by using models for the concentration-mass-redshiftrelation based on excursion set theory. The accuracy of the method is tested with nu-merical simulations carried out with different cosmological parameters. We find thatpredictions for median density profiles are more accurate than ∼ for haloes withmasses of . − . h − M (cid:12) for radii . < r/r m < . , and for cosmologieswith Ω m ∈ [0 . , . and σ ∈ [0 . , . . For larger radii, . < r/r m < , theaccuracy degrades to ∼
20 % , due to inaccurate modelling of the cosmological andredshift dependence of the splashback radius. For changes in cosmology allowed bycurrent data, the residuals decrease to (cid:46) up to scales twice the virial radius. Weillustrate the usefulness of the method by estimating the mean halo mass of a mockgalaxy group sample. We find that the algorithm’s accuracy is sufficient for currentdata. Improvements in the algorithm, particularly in the modelling of baryons, arelikely required for interpreting future (dark energy task force stage IV) experiments.
Key words: galaxies: haloes – gravitational lensing: weak – cosmology: theory –methods: numerical
The relation between galaxies and their dark matter haloesis of great interest not only for the study of galaxy evolu-tion, but also for precision cosmology. To fully exploit fu-ture large-scale structure measurements requires a thoroughquantitative understanding of the connection between galax-ies as visible tracers of cosmic structure and the predomi-nantly dark cosmic web. One of the most sensitive probes toconstrain this relation is galaxy-galaxy lensing (GGL).GGL quantifies the relationship between galaxies andthe dark matter density field through the cross-correlationof the observed shapes of distant galaxies and the positionsof foreground galaxies. These foreground galaxies, togetherwith their surrounding dark matter haloes, act as gravita-tional lenses since the associated gravity induces a differ-ential deflection of light from the background sources (e.g.Bartelmann & Schneider 2001). Typically, the resulting im-age distortions are small. However, the effect can be mea-sured statistically by considering a large number of systems. ? E-mail: [email protected]
Since its first detection by Brainerd et al. (1996), GGLhas become well understood in terms of statistical andsystematic uncertainties. Recent GGL observations reportsignal-to-noise ratios ∼ (Viola et al. 2015). The avail-able data will increase substantially from ongoing and up-coming surveys such as the Dark Energy Survey (DES), theKilo Degree Survey (KiDS), the Hyper Suprime-Cam Sub-aru Strategic Survey (HSC), the Large Synoptic Survey Tele-scope (LSST) survey, and the Euclid mission. This createsnew challenges for GGL theoretical modelling.Two of the most widely-used frameworks to interpretGGL measurements are halo-occupation distribution (HOD)models (e.g. Peacock & Smith 2000; Seljak 2000; Berlind &Weinberg 2002; Cooray & Sheth 2002; Leauthaud et al. 2011,2012; Zu & Mandelbaum 2015) and (sub-)halo abundancematching (SHAM) techniques (Kravtsov et al. 2004; Tasit-siomi et al. 2004; Vale & Ostriker 2006; Conroy et al. 2006;Conroy & Wechsler 2009; Moster et al. 2010; Behroozi et al.2010). There are however hints that there may be aspectspoorly understood for certain galaxy samples (Leauthaudet al. 2017). This might be a product of shortcomings ofand/or simplifications in these models. For instance, effects c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J u l Renneby, Hilbert, & Angulo such as assembly bias, the non-gravitational physics inducedby baryons, and the overall dependence on cosmological pa-rameters are difficult to incorporate accurately.A more faithful description of GGL might be con-structed from a joint numerical treatment of galaxy for-mation and the evolution of the density field. In recentyears, elaborate modelling of the baryonic gas physics hasbecome feasible in hydrodynamical simulations such as Il-lustris (Vogelsberger et al. 2014a,b; Genel et al. 2014) andEagle (Schaye et al. 2015; Crain et al. 2015) in sufficientlylarge volumes to allow for a direct comparison with GGLobservations (Leauthaud et al. 2017; Velliscig et al. 2017).A complementary approach is to employ semi-analyticalmodels (SAMs) of galaxy formation (White & Frenk 1991;Kauffmann et al. 1999; Springel et al. 2001; Bower et al.2006; De Lucia & Blaizot 2007; Guo et al. 2011; Henriqueset al. 2013, 2015) together with gravity-only simulations.In this approach, halo merger trees extracted from N -bodysimulations are populated with galaxies whose physical pro-cesses, such as cooling, star formation, and feedback, aretracked by a set of coupled differential equations. This al-lows for self-consistent and physically-motivated predictionsfor the galaxy population and the respective dark matter,which can then be used to compute the expected weak lens-ing signal for various lens galaxy samples (e.g. Hilbert et al.2009; Hilbert & White 2010; Pastor Mira et al. 2011; Saghihaet al. 2012; Gillis et al. 2013; Schrabback et al. 2015; Wanget al. 2016; Saghiha et al. 2017).The computationally cost of carrying out numerical sim-ulations over many different cosmological parameters is cur-rently prohibitively expensive. A way to alleviate this chal-lenge is to carry out a small number of high-quality simu-lations which could then be manipulated to mimic differentbackground cosmologies. This idea was originally broughtforth by Angulo & White (2010), henceforth AW10. Theirmethod is to rescale the time and length units such that thevariance of the linear matter field in the rescaled fiducial andtarget simulations match over a range of scales relevant forhalo formation. In Angulo & Hilbert (2015), hereafter AH15,an additional requirement on a matched linear growth his-tory was introduced, which improved the accuracy of pre-dictions for shear correlations functions.Despite the improvements, the rescaling method stillproduced noticeable biases in the internal structure of darkmatter haloes, owing to different formation times in the fidu-cial and target cosmologies. In this paper, we propose an en-hancement to the original algorithm by taking advantage ofrecent theory developments in predicting the concentration-mass relation of dark matter haloes by Ludlow et al. (2016),henceforth L16. We then investigate if the updated rescalingalgorithm can capture the small and intermediate scales ofthe cosmic web interpretable by GGL.This paper is organised as follows: In Section 2, we re-cap the key ingredients of our rescaling algorithm. Detailson the simulations, halo samples, and summary statistics fortesting the algorithm are described in Section 3. We presentthe results using the original as well as our updated scalingpredictions in Section 4. We discuss our results and theirimplications, e.g. for the estimation of lens masses and pre-dictions for concentration biases, in Section 5. We summariseour main findings in Section 6. In this section we present the main aspects of our scaling al-gorithm. We briefly recap the AW10 and AH15 algorithm inSection 2.1. In Section 2.2 and 2.3 we define halo concentra-tions and how they transform under rescaling. In Section 2.4,we summarise the model of L16, which will be employedlater in the paper. Throughout the paper we use comovingcoordinates and densities.
For the details of the rescaling algorithm, we refer to AW10and AH15. Here we note that it determines a length rescal-ing factor α and a redshift z ∗ in the fiducial cosmology tomatch to a redshift z in the target cosmology based on (i)the difference in the variance σ of the linear matter fieldbetween two smoothing lengths determined by the rangeof halo masses one would like to emulate and (ii) the dif-ference in growth history. Letting primed symbols denotequantities in the target cosmology, comoving positions x andsimulation particle masses m p in the fiducial simulation arerescaled as x [ Mpc /h ] x (cid:2) Mpc /h (cid:3) = α x [ Mpc /h ] , (1) m p [ M (cid:12) /h ] m p (cid:2) M (cid:12) /h (cid:3) = α Ω m Ω m h h m p [ M (cid:12) /h ]= β m m p [ M (cid:12) /h ] . (2)Here, Ω m denotes the cosmic mean matter density (in unitsof the critical density) and H = 100 h km/s/Mpc is theHubble constant. The comoving matter density ρ m thentransforms as: ρ m ρ m = α − β m ρ m . (3)The simulation box length and redshift change to: L → L = αL, (4) z → z , z (cid:54) z ∗ , z (cid:54) z , (5)where higher redshifts are acquired through the lineargrowth factor relation, D ( z ) = D ( z ) /D ( z ∗ ) · D ( z ) . (6)The growth constraint from AH15 is implementedthrough a comparison of a range of scale factors a around thevalue a ∗ in the (unscaled) fiducial cosmology correspondingto the best redshift fit z ∗ of the target simulation at z = 0 fora range of proposed scaling options ( α, z ∗ ) with the growthhistory of the target simulation. In AW10, the last stepof the algorithm involves a large-scale structure correctionto account for the differences in the primordial linear powerspectrum between the fiducial and target cosmologies, whichamounts to moving the particles with respect to one anotherto reach a better agreement with the positions in the tar-get simulation. Since this analysis focuses on the non-linearregime where this correction translates to an almost uniformdisplacement, we neglect this correction. As the snapshotoutput of an N -body simulation usually is discrete in time,the closest match to ( α, z ∗ ) is selected. The best relative weight on emulating the variance vs. thegrowth for a given observable is still an open question.MNRAS000
For the details of the rescaling algorithm, we refer to AW10and AH15. Here we note that it determines a length rescal-ing factor α and a redshift z ∗ in the fiducial cosmology tomatch to a redshift z in the target cosmology based on (i)the difference in the variance σ of the linear matter fieldbetween two smoothing lengths determined by the rangeof halo masses one would like to emulate and (ii) the dif-ference in growth history. Letting primed symbols denotequantities in the target cosmology, comoving positions x andsimulation particle masses m p in the fiducial simulation arerescaled as x [ Mpc /h ] x (cid:2) Mpc /h (cid:3) = α x [ Mpc /h ] , (1) m p [ M (cid:12) /h ] m p (cid:2) M (cid:12) /h (cid:3) = α Ω m Ω m h h m p [ M (cid:12) /h ]= β m m p [ M (cid:12) /h ] . (2)Here, Ω m denotes the cosmic mean matter density (in unitsof the critical density) and H = 100 h km/s/Mpc is theHubble constant. The comoving matter density ρ m thentransforms as: ρ m ρ m = α − β m ρ m . (3)The simulation box length and redshift change to: L → L = αL, (4) z → z , z (cid:54) z ∗ , z (cid:54) z , (5)where higher redshifts are acquired through the lineargrowth factor relation, D ( z ) = D ( z ) /D ( z ∗ ) · D ( z ) . (6)The growth constraint from AH15 is implementedthrough a comparison of a range of scale factors a around thevalue a ∗ in the (unscaled) fiducial cosmology correspondingto the best redshift fit z ∗ of the target simulation at z = 0 fora range of proposed scaling options ( α, z ∗ ) with the growthhistory of the target simulation. In AW10, the last stepof the algorithm involves a large-scale structure correctionto account for the differences in the primordial linear powerspectrum between the fiducial and target cosmologies, whichamounts to moving the particles with respect to one anotherto reach a better agreement with the positions in the tar-get simulation. Since this analysis focuses on the non-linearregime where this correction translates to an almost uniformdisplacement, we neglect this correction. As the snapshotoutput of an N -body simulation usually is discrete in time,the closest match to ( α, z ∗ ) is selected. The best relative weight on emulating the variance vs. thegrowth for a given observable is still an open question.MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations The chief advantage of the algorithm is that all quan-tities are calculated in the linear regime, wherein we eitherhave explicit predictions or adequate fits for a range of dif-ferent cosmologies. This allows for a fast evaluation ( (cid:54) on a contemporary laptop).
As a model for comoving matter density profiles of haloes,we consider the NFW profile (Navarro et al. 1996, 1997): ρ NFW ( r ) = ρ crit ( z ) δ c ( r/r s )(1 + r/r s ) . (7)Here, δ c denotes the characteristic density of the halo, r s its scale radius, and ρ crit ( z ) the comoving critical density athalo redshift z . For a spatially flat universe with cold darkmatter (CDM) and a cosmological constant Λ , ρ crit ( z ) =3 H (8 π G) − E ( z ) (1 + z ) − , where G is the gravitationalconstant, and E ( z ) = Ω m (1 + z ) + (1 − Ω m ) .For a given overdensity threshold ∆ , one may definethe halo radius r ∆ c as the radius at which the mean inte-rior density is ∆ × ρ crit ( z ) . The halo concentration c ∆ c isthen defined by c ∆ c = r ∆ c /r s with the associated halo mass M ∆ c = ∆(4 / πr c ρ crit ( z ) and the characteristic density δ c δ c = ∆3 c c ln(1 + c ∆ c ) − c ∆ c / (1 + c ∆ c ) . (8)We also consider as halo radius r ∆ m , at which the halo’smean interior density is ∆ times the cosmic mean. The asso-ciated halo concentration c ∆ m = r ∆ m /r s , and the halo mass M ∆ m = ∆(4 / πr m Ω m ρ crit (0) .In addition, we also model the density field with Einastoprofiles (Einasto 1965): ρ Einasto ( r ) = ρ s exp (cid:16) − α h(cid:16) rr s (cid:17) α − i(cid:17) , (9)where α denotes a profile shape parameter, r s the scaleradius, and ρ s is a density normalisation parameter. Theshape parameter is connected to the local average densityin the initial field, encompassing the peak curvature (Gaoet al. 2008; Ludlow & Angulo 2017). Following L16, we fix α = 0 . . The halo scale radii r s transform under rescaling as r s r s = αr s . NFW halo radii r ∆ m , masses M ∆ m , and concen-trations c ∆ m based on halo overdensities relative to the cos-mic mean density also follow simple transformation rules: r ∆ m r ∆ m = αr ∆ m , M ∆ m M ∆ m = β m r ∆ m , and c ∆ m c ∆ m = c ∆ m .The rescaling transformation laws for NFW profilequantities based on overdensities relative to the critical den-sity are more involved. Applying Eq. (3) to the NFW profiledefinition Eq. (7), we find for the characteristic densities: δ c ρ crit ( z ) = Ω m Ω m (cid:18) H H (cid:19) δ c ρ crit ( z ) . (10)Thus, the concentration c ∆ c transforms as c ∆ c c ∆ c , (11) with c ∆ c given by the (numerical) solution to δ c ( c ∆ c ) = Ω m Ω m (1 + z ) (1 + z ) E ( z ) E ( z ) δ c ( c ∆ c ) . (12)The halo mass M ∆ c then transforms according to M ∆ c M ∆ c = β c M ∆ c , (13)with β c = (cid:18) c ∆ c c ∆ c (cid:19) · α · (cid:18) H H (cid:19) E ( z ) E ( z ) (1 + z ) (1 + z ) , (14)and c ∆ c as the numerical solution to Eq. (12). As a rangeof c ∆ c values could correspond to a given M ∆ c , this meansthat the rank order of M ∆ c is not invariant under rescaling.One may also use M ∆ m = (cid:16) c ∆ m c ∆ c (cid:17) Ω m (1 + z ) E ( z ) M ∆ c , (15)to first convert M ∆ c to M ∆ m , then rescale M ∆ m to M ∆ m ,and then convert back to M ∆ c . We show how to rescaleEinasto concentrations in Appendix D. We focus on what excursion sets (Press & Schechter 1974;Bond et al. 1991) predict for the concentration of haloes(Lacey & Cole 1993). One approach for CDM has been to tiethe concentration to the mass accretion history of the halo(e.g. Ludlow et al. 2014; Correa et al. 2015). However, thisis not suitable for warm dark matter (WDM) models wherethe concentration-mass relation is non-monotonic despitethe different accretion histories of low and high mass haloes.Revisiting the original NFW argument (Navarro et al. 1996,1997), it was proposed that the characteristic density of thehalo δ c is an imprint of the critical density of the Universe atan appropriate collapse redshift, when progenitors exceedinga fraction f of the final virial halo mass constituted half ofthis mass. L16 argued that choosing the mean density h ρ s i inside the scale radius r s to be proportional to the criticaldensity of the Universe at the collapse redshift (instead of δ c ) and letting the mass inside the scale radius M s definethe characteristic collapsed mass (instead of the virial mass)yields a better agreement for CDM and WDM. This relationthen takes the form M s = 4 π r s h ρ s i = 4 π r s · C · ρ crit ( z s ) , (16)where C is a proportionality constant and z s the collapseredshift. According to excursion sets (Lacey & Cole 1993),the collapsed mass fraction is given by M s ( f, z ) M ∆ c = erfc δ sc ( z s ) − δ sc ( z ) √ · p σ ( fM ∆ c ) − σ ( M ∆ c ) ! , (17)where M ∆ c is the final mass at z , σ ( M ) the variance of thelinear density field on scales equivalent to the mass M , and δ sc ( z ) a linear barrier height δ sc ( z ) = δ sc ( z ) /D ( z ) , wherethe linear growth is normalised such that D ( z ) = 1 , andthe linear density threshold satisfies δ sc ( z ) = δ sc ( z = 0) ≈ . corresponding to spherical collapse at redshift z =0 . Combining this with Eq. (16) and an assumed densityprofile, this system of three equations yields numerical fits MNRAS , 1–26 (2018)
Renneby, Hilbert, & Angulo for the c ( M, z ) -relation. The best-fits for the two constantswere determined to be f = 0 . and C = 650 . We neglectthe mild cosmological and redshift dependences of δ sc ( z ) inthis study.In L16 this relation was found to fit the median c ( M, z ) -relation estimated with Einasto profiles for relaxed haloes(see Section 3.2) for the same simulations that we are usingin this paper (see Section 3.1) with the M ∆ c mass definitionwith ∆ = 200 . We thus calculate the c ( M, z ) -relation withEq. (16) and Eq. (17), assuming an NFW profile Eq. (7),with z = z ∗ and z in the fiducial and target simulations,respectively, then adapt the relations for M ∆ m and c ∆ m . In this section we present details of our adopted method-ology to test the performance of the scaling algorithm. InSection 3.1, we describe our fiducial simulation along withfive others carried out adopting significantly different cos-mologies. We discuss the construction of halo samples inSection 3.2. In Section 3.3, we define the differential excesssurface mass density profiles and provide details about howto measure them, as well as halo concentrations in our sim-ulations.
This study is conducted with several N -body simulationsemploying GADGET-2 (Springel 2005) with parti-cles. The fiducial simulation spans a (250 h − Mpc ) comov-ing volume, uses a softening length of l s = 5 h − kpc, andhas particle masses m p = 8 . × h − M (cid:12) . It assumes aflat Λ CDM cosmology with a cosmological constant energydensity parameter Ω Λ = 1 − Ω m = 0 . , a matter densityparameter Ω m = Ω cdm + Ω b = 0 . , baryon density parame-ter Ω b = 0 . , Hubble constant H = 100 h km s − Mpc − with h = 0 . , matter power spectrum normalisation σ =0 . , and spectral index n s = 1 . The cosmological parame-ters and force and mass resolution are identical to those ofthe Millennium simulation (Springel et al. 2005).We rescale the fiducial simulation to cosmologies withdifferent values for Ω m and σ . We then compare theserescaled simulations to simulations carried out directly as-suming the target cosmologies. These ‘direct’ and ‘rescaled’simulations have initial conditions with identical phases. Thesoftening lengths, box sizes, and particle masses in these di-rect simulations have been chosen to match those in therescaled simulations. Details are provided in Table 1 (theother configurations and parameters are the same as in thefiducial run).Though the rescaling algorithm captures non-linearstructure evolution, it cannot arbitrarily adapt to differentgrowth histories. As dark energy becomes more important at To achieve internal consistency for a spherical collapse model, C = 400 would have been the preferred value, but C = 650 pro-duced better fits. This inconsistency primarily affects high masshaloes, which are rare in our simulations. Moreover, we limit thepossible length scale factors to α ∈ [0 . , in Eq. (1). For thecosmological parameters in this study, this ensures that β m M ∆ m remains in the range of validity. Table 1.
Simulation configurations (fiducial cosmology in thefirst row) with their values of Ω m and σ listed. The scale factors α from Eq. (4) are obtained by dividing the box lengths L withthe first column entry. The softening lengths are set as α × l s for the direct simulations with α = 1 for the fiducial run. Theparticle masses m p are calculated using Eq. (2). The rescalingredshifts z ∗ of the fiducial cosmology’s snapshots are listed in thelast column. Ω m σ L (cid:2) h − Mpc (cid:3) m p (cid:2) h − M (cid:12) (cid:3) z ∗ lower redshifts, the growth and expansion histories of differ-ent Λ CDM cosmologies deviate in different manners from anEinstein-de-Sitter evolution. Thus, we expect the inaccuracyof the scaling to grow with cosmic time. For this reason, wefocus on structures at redshift z = 0 to obtain a conservativeestimate on the accuracy of the scaling method. Finally, notethat the rescaling parameters ( α, z ∗ ) are identified followingAW10 and AH15 for scales corresponding to halo masses inthe range − h − M (cid:12) . Haloes in the simulations are first identified using a friends-of-friends (FOF) algorithm (Davis et al. 1985) with a linkinglength of 0.2 times the mean particle separation. The FOFhaloes are then processed with
SUBFIND (Springel et al.2001), employing the same settings as for the MXXL simula-tion (Angulo et al. 2012), to identify self-bound structures,possibly returning a main subhalo and further self-boundsubhaloes.We will mostly consider halo samples defined by their(rescaled) M m mass. However, in some cases we will alsoconsider halo samples that only include matched haloes indirect-rescaled pairs of simulations. Following AW10, weidentify as match candidate for each halo in the direct simu-lation the halo in the rescaled simulation with the most par-ticles with ids matching those of the direct simulation’s halo.We repeat the process with the simulations’ roles swapped,and consider a haloes matched if they are each others matchcandidates.Note that the most accurate rescaling approach wouldbe to transform individual simulation particles and then re-run the group finding algorithm. However, this is compu-tationally expensive, and similarly accurate results can beobtained by directly rescaling the halo catalogue, as shownby Ruiz et al. (2011) (see also Mead & Peacock 2014a,b),which is the procedure we adopt here; we rescale the posi-tion and mass of each snapshot particle but keep the fiducialhalo catalogue and rescale it accordingly.Unrelaxed haloes are poorly described by NFW pro-files, and their best fit concentrations tend to be lowerthan those of relaxed systems (Neto et al. 2007). To testfor this in our results, in some cases we will consider sam-ples of haloes that satisfy two criteria. The first criterion MNRAS000
SUBFIND (Springel et al.2001), employing the same settings as for the MXXL simula-tion (Angulo et al. 2012), to identify self-bound structures,possibly returning a main subhalo and further self-boundsubhaloes.We will mostly consider halo samples defined by their(rescaled) M m mass. However, in some cases we will alsoconsider halo samples that only include matched haloes indirect-rescaled pairs of simulations. Following AW10, weidentify as match candidate for each halo in the direct simu-lation the halo in the rescaled simulation with the most par-ticles with ids matching those of the direct simulation’s halo.We repeat the process with the simulations’ roles swapped,and consider a haloes matched if they are each others matchcandidates.Note that the most accurate rescaling approach wouldbe to transform individual simulation particles and then re-run the group finding algorithm. However, this is compu-tationally expensive, and similarly accurate results can beobtained by directly rescaling the halo catalogue, as shownby Ruiz et al. (2011) (see also Mead & Peacock 2014a,b),which is the procedure we adopt here; we rescale the posi-tion and mass of each snapshot particle but keep the fiducialhalo catalogue and rescale it accordingly.Unrelaxed haloes are poorly described by NFW pro-files, and their best fit concentrations tend to be lowerthan those of relaxed systems (Neto et al. 2007). To testfor this in our results, in some cases we will consider sam-ples of haloes that satisfy two criteria. The first criterion MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations is based on the offset between the centre-of-mass r CM andthe gravitational potential minimum r pot relative to thehalo radius r (Thomas et al. 2001; Macci`o et al. 2007;Neto et al. 2007) d off = | r pot − r CM | /r . We considerhaloes relaxed if d off < . . The second criterion is a sub-structure threshold (Neto et al. 2007; Ludlow et al. 2012), f sub = M sub /M < . , where M sub is the mass of allbound particles in the subhaloes apart from the main haloidentified by the substructure finder.These criteria lead to similar results as imposing the d off cut and a dynamical age criterion, t (cid:62) . t cross (Jiang &van den Bosch 2016; Ludlow et al. 2016) curtailing the al-lowed accretion of the main progenitor w.r.t. its crossingtime t cross = 2 r /V , as they exclude recent mergers ofstructures with similar mass. With the M m mass defini-tion , the geometric cuts on f sub and d off are trivially invari-ant under the rescaling mapping . This invariance does nothold for other dynamical relaxation criteria such as boundson the virial ratio η = 2 K/ | U | (e.g. Cole & Lacey 1996) orthe spin parameter λ (e.g. Bett et al. 2007). We measure the spatial cross-correlation between the haloand matter fields in our simulations to obtain mass profilesin 3D and 2D. In 3D, we consider spherically averaged ra-dial matter density profiles for haloes as a function of halomass. As analytic approximations to these profiles we con-sider NFW profiles Eq. (7) and Einasto profiles Eq. (9).The 3D density field is not readily available in thereal Universe. However, galaxy-galaxy lensing can be usedto probe the cross-correlation between galaxies and mat-ter. Assuming statistical isotropy, this cross-correlation ξ gm ( | r − r | ) = h δ g ( r ) δ m ( r − r ) i between the total over-density of matter δ m and the overdensity of lens galaxies δ g at comoving positions r and r , respectively, is related tothe mean projected surface mass overdensity Σ at projectedcomoving transverse distance r through Σ( r ) = ¯ ρ Z d l ξ gm ( p r + l ) , (18)with ¯ ρ as the mean comoving density. The differential excesssurface mass density ∆Σ( r ) then reads ∆Σ( r ) = ¯Σ( (cid:54) r ) − Σ( r ) , (19) However, a dynamical timescale cut also discriminates againsthaloes at maximum contraction following a massive merger, whichare still present in our subsample. Given β c in Eq. (14), the cuts w.r.t. M c are not rescalinginvariant. Since the measured concentrations are influenced bythese cuts (Neto et al. 2007), a recursive rescaling fitting schemeis required to find the passing haloes in the target cosmology. provided we ignore implicit relations, e.g. redshift evolutionwhich affects f sub (e.g. van den Bosch et al. 2005) If the simulation’s softening length l s α s and α vel ≈ α forthe velocities whose transform is given in AW10 then η η ’ Ω m / Ω m ( H /H ) η with the potential U given in Springel et al.(2005). Since U and T have different transform prefactors, map-ping λ λ is non-trivial. In AW10, λ was comparable for the haloes in the direct andrescaled simulation snapshots, hinting at similar internal dynam-ical states, whereas the halo concentrations estimated from veloc-ities displayed a systematic bias. where ¯Σ( (cid:54) r ) = 1 πr Z r d r πr Σ( r ) , (20)denotes the mean projected surface mass overdensity insidea circular aperture with radius r . ∆Σ can be estimated fromthe tangential shear ¯ γ t = ∆Σ / Σ crit induced by lens galax-ies at redshift z d in images of source galaxies at redshift z s > z d (Miralda-Escud´e 1991; Squires & Kaiser 1996; Wil-son et al. 2001), where Σ crit = Σ crit ( z d , z s ) denotes the co-moving critical surface mass density for lenses at redshift z d and sources at redshift z s . Hence, the tangential shearof background galaxies provides information on the matterdistribution around foreground galaxies.As analytical models, we consider NFW lenses (Wright& Brainerd 2000; Baltz et al. 2009). The lensing expressionsare acquired by integrating the NFW density profile Eq. (7)along the line-of-sight. Expressed in terms of the dimension-less ratio x = r/r s , the projected surface mass density at aradius x is then acquired through Σ( x ) = 2 r s Z ∞ d l ρ NFW ( p l + x ) , (21)whereas ∆Σ is given by Eq. (19). We restrict the comparisonto scales (cid:46) the halo virial radii and leave modelling of thelarge scales for future studies. We do not model the lenseswith Einasto profiles as those are similar to NFW lenses(Retana-Montenegro et al. 2012; Sereno et al. 2016).Operationally, we compute 3D radial halo profiles ρ andprojected radial profiles Σ by binning all particles in spher-ical and cylindrical shells, respectively, around the recordedhalo centres given by the positions of their most boundparticles. To moderate triaxiality (e.g. Jing & Suto 2002)and other deviations from azimuthal symmetry, we projectthe cylinders along the three principal simulation box axesand let the mean signal describe the halo sample, effectivelytripling our sample size. For the rescaled simulation, the pro-files are computed after applying the adequate rescaling toensure matching bin boundaries.In order to assess the errors due to the limited volume,we bootstrap resample (e.g. Efron 1979) the haloes in eachmass bin with 100 realisations to estimate the variance. For ∆Σ we calculate 100 realisations per axis.We consider halo samples selected by mass with 0.1 dexwidth above h − M (cid:12) to approximately . h − M (cid:12) where we record twenty haloes per bin. For the halo massfunction we show the result in 0.05 dex bins. For the 3D pro-files, we follow Neto et al. (2007), where the matter densityprofiles were estimated using 32 log -equidistant bins between r c and log ( r/r c ) = − . where we replace r c with r m . To suppress the impact of outliers on the 3D profilefits, we use the median particle count per spherical shell asinput, unless otherwise specified. We then minimise the dif-ference in ln ρ between the measured median profile and theanalytic profile to determine the best fit parameters. We alsopresent concentration estimates for individual haloes fromthe separate particle counts. To investigate the transition We ignore differences between halo density and overdensity pro-files, since these do not affect the differential excess surface massdensity ∆Σ .MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo regime between the 1-halo and 2-halo terms, we bin the par-ticles in 64 log -equidistant bins for . r m < r < r m .GGL profiles for each mass-selected halo sample are ob-tained through Eq. (19), with the projected profiles com-puted by binning the particles in 40 log -equidistant bins inthe h − kpc − h − Mpc range. The average GGL profilesare fitted by analytical profiles Eq. (21) minimising χ = N r X i =1 r i [∆Σ data ( r i ) − ∆Σ NFW ( r i ; r m , c m )] , (22)w.r.t. r m and c m . The radial weights ∝ r are obser-vationally motivated, as the shape noise error on the sig-nal scales with the number density of background galaxies,which is proportional to the area of the projected cylinderassuming a constant source density. In observations, maskingand blending of background galaxies by foreground galaxiesbecomes a major systematic as one approaches the centralgalaxy (Viola et al. 2015), which motivates the lower cutoff. In this section we quantify the performance of the scaling al-gorithm and present alternatives to further improve it. Wefirst focus on the halo mass functions (Section 4.1), the 3Ddensity profiles (Section 4.2), and the differential excess sur-face mass density profiles (Section 4.3) for the original algo-rithm. The accuracy of the rescaling for the concentration-mass relation is quantified and compared to the theoreti-cal prediction of L16 in Section 4.4. In Section 4.5 we usethis model to correct the rescaled profiles and show the re-sulting improvements. Attempts at further ameliorations forthe halo outskirts based on models for the position of thesplashback radius are discussed in Section 4.7. We will focuson representative cases using one of the cosmologies studiedwhere the others manifest similar trends and primarily re-port on the findings for (0 . , . in Appendix B as theseparameters strongly deviate from current observational con-straints. One of the most basic quantities predicted by simulationsis the halo mass function. The cumulative halo mass func-tion (HMF) N ( > M ) defines the number of haloes above acertain mass M per comoving volume. In AW10, the num-ber densities were properly matched with a bias of order (cid:46)
10 %. To avoid numerical artefacts, we only compare HMFsfor haloes with (rescaled) masses exceeding h − M (cid:12) (i.e.objects resolved with > particles).In Fig. 1, we show N ( > M ) for all haloes in the di-rect and rescaled cosmologies with the fractional differencein the bottom panel. In numbers, there are 100 154, 28 427,47 519, 33 123 and 8 325 haloes with M m > h − M (cid:12) in the direct simulations (listed according to increasing Ω m ),and 97 232, 28 145, 46 620, 32 888 and 8 999 haloes in therescaled snapshots. As seen in Fig. 1, the error in the num-ber counts is in the range ±
10 % for all simulations exceptfor (0 . , . and for masses < h − M (cid:12) . At highermasses, Poisson noise is significant. In addition, these clus-ters are the last structures to have collapsed and thus are − − − − − N ( > M m ) (cid:20) h M p c − (cid:21) (0 . , . . , . . , .
81) (0 . , . . , . M [ h − M (cid:12) ]0.000.50 − N r / N d Direct simulationRescaled simulation
Figure 1.
Cumulative halo mass function at z = 0 (in 0.05dex bins) for simulations with different values for ( Ω m , σ ) asindicated by the legend. For each cosmology, we display results fordirect and rescaled simulations. The fractional differences betweenthese two cases are shown in the bottom panel where solid linesmark ±
10 % . M [ h − M (cid:12) ] − . − . − . . . . . ( N p a ss / N a ll ) d − ( N p a ss / N a ll ) r (0 . , . . , .
60) (0 . , . . , . Figure 2.
Difference in the fraction of relaxed haloes betweenthe direct and rescaled simulations per 0.1 dex mass bin with the d off + f sub cuts enforced (the results are similar if only the d off cut is applied). most sensitive to changes in the growth rate governed bythe background cosmology. Since we opt for a minimisationscheme covering a large range of halo masses, the rescal-ing parameters are not necessarily the best ones for cluster-size haloes. This could then bias the predicted masses. Thebest matches are found for the (0 . , . and (0 . , . cosmologies, with fractional differences (cid:46) . Overall, thisperformance is similar to that stated in AW10.Trends for passing the relaxation cuts are similar in thedirect and rescaled simulations, with cuts more effective atthe high mass end, and peak passing rates between 54 and73 % for the . − . h − M (cid:12) mass bin. As Fig. 2 illus-trates, there are however some differences between the directand rescaled simulation in the fraction of haloes per massbin which satisfy the relaxation criteria. For (0 . , . MNRAS000
Difference in the fraction of relaxed haloes betweenthe direct and rescaled simulations per 0.1 dex mass bin with the d off + f sub cuts enforced (the results are similar if only the d off cut is applied). most sensitive to changes in the growth rate governed bythe background cosmology. Since we opt for a minimisationscheme covering a large range of halo masses, the rescal-ing parameters are not necessarily the best ones for cluster-size haloes. This could then bias the predicted masses. Thebest matches are found for the (0 . , . and (0 . , . cosmologies, with fractional differences (cid:46) . Overall, thisperformance is similar to that stated in AW10.Trends for passing the relaxation cuts are similar in thedirect and rescaled simulations, with cuts more effective atthe high mass end, and peak passing rates between 54 and73 % for the . − . h − M (cid:12) mass bin. As Fig. 2 illus-trates, there are however some differences between the directand rescaled simulation in the fraction of haloes per massbin which satisfy the relaxation criteria. For (0 . , . MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations -0.40-0.200.000.200.40 (0 . , . Mean relation per 0.1 dex binMedian relation per 0.1 dex bin (0 . , . -0.40-0.200.000.200.40 (0 . , . (0 . , . − M r m / M d m M [ h − M (cid:12) ] Figure 3.
Fractional difference in the mass of matched haloesidentified in direct and rescaled simulations. Each panel showsresults for a different combination of Ω m and σ indicated in thelegend. Contours enclose 68 % and 95 % of the distributions, andsymbols mark the mean and median values per mass bin. and (0 . , . , fewer haloes per mass bin survive the cuts,which may indicate a possible redshift dependence of the cutefficiency, as the rescaled signals come from fiducial snap-shots at higher redshifts. This implies that we do not onlyhave a slight scatter in the number of haloes but also in theproperties of the haloes which pass the relaxation cuts.Almost all haloes ( ∼
99 % ) with M m ≥ h − M (cid:12) in the direct simulations have matches in the rescaled simu-lation (and the few non-matches have no significant impacton the profile statistics considered here). However, proper-ties of matching haloes are usually not identical. The frac-tional difference in recorded M m between the matchedhaloes in the direct simulation and their matched rescaledcounterparts is shown in Fig. 3. Both a scatter and a system-atic trend with mass and cosmology are discernible. For ex-ample, haloes in the rescaled simulation tend to be less mas-sive than their counterparts for (0 . , . . These trendsare in part responsible for differences in the halo profiles be-tween the direct and rescaled simulations discussed in thefollowing sections. In Fig. 4 we plot the median density profiles for five massbins in the (0 . , . cosmology in 40 log -equidistant binsbetween . − h − Mpc. The halo profiles in the directand rescaled simulations display remarkable agreement, withdifferences of at most
20 % over two orders of magnitudein density and scale. The differences likely reflect differentmass accretion histories and formation times for the directand rescaled haloes. They are characterised by two features:(i) an underestimation (overestimation) of the density nearthe halo centre, and (ii) an overestimation (underestima-tion) of the density near the transition scale between the1-halo and 2-halo terms for the (0 . , . , (0 . , . ρ ( r ) / ¯ ρ m (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r [ h − Mpc]-0.30-0.150.000.15 − ρ r / ρ d Direct simulationRescaled simulation
Figure 4.
3D comoving matter density profiles ρ ( r ) in unitsof the cosmic mean density ¯ ρ m as function of radius r for allhaloes in direct and rescaled simulations of the Ω m = 0 . , σ =0 . cosmology for five different mass bins (see legend). Fractionaldifferences between the results from the two simulations are shownin the bottom panel. and (0 . , . cosmologies, with the opposite signs for (0 . , . and (0 . , . .Fig. 5 shows the fractional difference for four of our testsimulations for haloes in four to six mass bins, where morethan twenty haloes have been recorded in the direct andrescaled simulations. The magnitude (though not always thesign) of the differences is similar to that for the (0 . , . cosmology. From approx. . to r m , the rescaled pro-files have an outer bias with the opposite sign to the inner( r (cid:46) . r m ) profile bias, until they reach better agree-ment at larger scales ( r > r m ). This suggests that thesimulations have a similar halo bias. Fewer haloes in thehigher mass bins lead to a larger scatter, predominantly inthe outskirts where the active evolution takes place. Per-forming the same tests with just haloes passing the relax-ation cuts or matched haloes yield similar results as for thewhole population, indicating that the biases are universalfeatures. We show the corresponding fractional differencesfor matched haloes only in Appendix C. As shown in Fig. 6, the small differences in the 3D den-sity profiles propagate to small differences in the weak lens-ing profiles. The best agreement between the profiles of therescaled and direct simulations is reached for (0 . , . .The other cosmologies show larger differences, in partic-ular in the inner profiles. In contrast, the outer profilebias is barely discernible except for the low mass bins for (0 . , . , implying that it is washed out by taking themean and calculating the projection. If we increase the massbin width to 0.2 dex and recompute the profiles, the outerprofile bias almost completely vanishes in 2D but it is stilldiscernible in 3D for median profiles. The transition regime MNRAS , 1–26 (2018)
Renneby, Hilbert, & Angulo -0.30-0.150.000.15 (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) -0.45-0.30-0.150.000.15 (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r/r -0.09-0.06-0.030.000.030.06 (0 . , . − r/r -0.16-0.080.000.080.160.24 (0 . , . − ρ r / ρ d Figure 5.
Fractional differences in the 3D density profiles of haloes in the direct and rescaled simulation snapshots. Each panel displaysresults for a different background cosmology and for five to six disjoint halo mass bins, as indicated by the legend. The x -axis is in unitsof the halo r m radius, which highlights that the differences are almost independent of mass. − ∆ Σ ( r ) [ h M (cid:12) p c − ] (0 . , . -0.150.000.150.30 − ∆ Σ r / ∆ Σ d (0 . , . -0.100.000.100.2010 ∆ Σ ( r ) [ h M (cid:12) p c − ] (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r [ h − Mpc]-0.12-0.060.000.060.12 − ∆ Σ r / ∆ Σ d (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r [ h − Mpc]-0.16-0.080.000.080.16
Figure 6.
Differential excess surface mass density profiles ∆Σ( r ) for stacks of haloes in the direct and rescaled simulations. Differentcolours indicate the different halo mass bins displayed whereas different panels show results for different cosmologies, where the bottomsub-panels show fractional differences with the same mass bin line styles as in Fig. 5. In the upper sub-panels, the best fit NFW profilesare indicated by dotted lines. The results for the (0 . , . cosmology are presented in Appendix B. MNRAS000
Differential excess surface mass density profiles ∆Σ( r ) for stacks of haloes in the direct and rescaled simulations. Differentcolours indicate the different halo mass bins displayed whereas different panels show results for different cosmologies, where the bottomsub-panels show fractional differences with the same mass bin line styles as in Fig. 5. In the upper sub-panels, the best fit NFW profilesare indicated by dotted lines. The results for the (0 . , . cosmology are presented in Appendix B. MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations scatter does not necessarily dampen at larger scales . Forthe total maximum and median values of the residuals be-low r m , we refer to Table 2.As for the 3D density profiles, we find negligible differ-ences between all haloes and all matched haloes. However,the scatter in the 2-halo transition regime is dampened, andthe inner and outer profile biases are accentuated, especiallyfor (0 . , . . In addition, there are no conspicuous differ-ences between the profiles for all haloes, for those whichpass the d off relaxation cut and for those which pass both d off + f sub relaxation cuts.Fig. 6 also illustrates that the ∆Σ profiles for r (cid:46) r m are well described by NFW lens profiles. We fit the measuredmean profiles by minimising Eq. (22) with both c m and r m as free parameters. Fig. 7 shows the relative differencebetween the mean M m recorded by the halo finder andthe value fitted from the ∆Σ profiles. For the simulationswith rescaled fiducial snapshots close to z = 0 , the rescaledand direct simulation mass biases have similar amplitudesand show a similar evolution in mass with additional scatterat the high mass end. Introducing relaxation cuts shifts theamplitude consistently in the direct and rescaled simulationtowards zero and for some high mass bins the bias changessigns, presumably due to scatter. The results with only the d off cut enforced are similar to the ones where both cuts areimposed.The negative bias for low mass haloes, particularly for (0 . , . , is likely due to a lack of spatial resolution,which causes the measured lensing profiles to fall belowthe analytic profiles in the innermost regions. Moreover, for (0 . , . and (0 . , . , there is a visible systematicoffset between fit masses of the rescaled and direct simula-tions, which is preserved with the introduction of cuts. Smallbut significant cosmology-dependent deviations from the an-alytic NFW lens profiles even for relaxed haloes might causethis offset. This requires further investigation in future work. In Fig. 8, we compare the values of the concentration param-eter from the 3D and 2D NFW fits to the predictions of themodel described in Section 2.3. At the low mass end, the fi-nite force resolution of the simulations affects the inner haloprofiles and thus the concentrations estimates noticeably, inparticular for (0 . , . due to its larger softening scale.The vertical dotted lines in Fig. 8 and 9 mark the halo massabove which the scale radius exceeds r s > l s for the theorypredictions, and thus the concentrations estimates are lessaffected by the finite force resolution.In 3D, the model fails to predict the concentration-massrelation within the statistical errors for the general popula-tion. Additional cuts remove the tension, as Fig. 9 showsfor (0 . , . . For low mass haloes, the Einasto fits favourhigher c -values than the NFW fits (see Appendix D) andhave the best agreement with the L16 model with the cutsenforced (which is encouraging since the model is supposedto match such relations). We are able to reach a complete We calculated the large scale ∆Σ for (0 . , . for the samemass bins for − h − Mpc and there are small differences atthe level of the scatter over this range. agreement with the model with the cuts enforced with theEinasto parameterisation for all cosmologies where we usesnapshots close to z = 0 in the fiducial run.Yet, the model cannot describe the measured rescaled c ( M ) -relation for (0 . , . . This is caused by a failureto model the signal at z = 0 . in the fiducial cosmology.We have also computed the unscaled M c concentration-mass relations for median Einasto c ( M, z ) relations with thecorresponding cuts implemented , which yield the highestavailable concentrations per mass bin. Even in this case, themodel predicts higher than observed concentrations. Thiscould be due to the neglect of the redshift evolution of thecollapse threshold.In 2D, the model fits the measured values well athigh masses, particularly for the relaxed subpopulations.Due to limited resolution, we cannot discern the expectedmonotonous c ( M ) -relation in 2D below ≈ . h − M (cid:12) for (0 . , . . This effect is present in the low mass bins for (0 . , . as well. The relations in 2D and 3D mainly dif-fer due to different binning choices; in 3D we follow the ap-proach in L16 whereas we opt for an observation conformingchoice in 2D. Fewer bins in the inner projected regions of thestacked haloes combined with the down-weighting of thesebins result in less sensitivity to the concentration, which ex-plains the flat relations for low mass haloes. On the otherhand, the masses are still determined well which is reflectedin the small horizontal error bars.As Fig. 10 illustrates, the difference in concentration ∆ c between the direct and rescaled simulations is approximatelyconstant for haloes in the mass range − h − M (cid:12) ,and moreover roughly consistent with the model predictions.The deviation for (0 . , . results in a discrepancy be-tween the model and the measured difference relation, butfor (0 . , . , (0 . , . and partly for (0 . , . atthe high mass end, there is consistency both in 3D and forthe lensing profiles. For low mass haloes, resolution effectsand the relatively higher amplitude of the (not modelled)2-halo term obscure the results. At the high mass end, thelow number of haloes cause a larger scatter.The constant ∆ c relations hold for the relaxed pop-ulations as well, especially for ∆Σ , though the varianceincreases. The small changes for the 3D density profilesare quantified by comparing ∆ c relaxed / ∆ c all haloes for haloeswith − h − M (cid:12) masses and record the median dif-ferences in the mass bins where we have more than twentyhaloes for each imposed cut. This produces variations of theorder of but there are no consistent trends present forboth the NFW and Einasto parameterisations. This meansthat whereas the L16 model fails to accurately predict theconcentration-mass relations for halo samples containingboth relaxed and unrelaxed systems, it can predict the dif-ference in this relation between two simulations for such amixed population very well both for 3D density and ∆Σ profiles. Hence, it is suitable for modern surveys. Since M ∆ m > M ∆ c generally holds, the cuts are more conser-vative with a M ∆ c mass definition as neither the centre-of-mass,the position of the most bound particle nor the mass containedin substructure are altered for the same halo.MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo -0.100.000.100.20 (0 . , .
00) (0 . , . All haloes (d)All haloes (r) d off + f sub cut (d) d off + f sub cut (r)10 -0.100.000.100.20 (0 . , . (0 . , . − M l e n s m / M s i m . m M [ h − M (cid:12) ] Figure 7.
Fractional differences between the true mean mass of haloes in our simulations, M sim. m , and that inferred from their ∆Σ profiles, M lens m . Each panel focuses on a particular cosmology, and it shows results from the direct and rescaled simulations for all haloesand only for those relaxed according to two different criteria. The coloured regions mark the
68 % and
95 % percentiles, estimated fromthe bootstrap resample. . , . Model (direct)Model (rescaled) Direct sim.Rescaled sim. (0 . , . . , .
81) 10 (0 . , . M [ h − M (cid:12) ] c Density profiles . , . Model (direct)Model (rescaled) Direct sim.Rescaled sim. (0 . , . . , .
81) 10 (0 . , . M [ h − M (cid:12) ] c ∆Σ profiles Figure 8.
The concentration-mass relation of haloes in rescaled and direct simulations. Concentrations were estimated from NFW fitswith the halo mass as a free parameter. The left and right plots show results from employing 3D density profiles and ∆Σ , respectively,and each sub-panel focuses on a different cosmology. Dotted and dashed lines show the predictions of the model by L16. Symbols markthe mean relations, and shaded regions show the
68 % and
95 % of the distribution at a fixed mass. Horizontal error bars indicate thespread in the fitted M m masses. The vertical dotted lines denote the mass limit below which the finite force resolution affects theconcentration estimates. Note that the disagreement between the model and the measurements originates mostly from unrelaxed haloes(cf. Fig. 9). For an analogous plot using concentrations obtained with Einasto profiles, see Appendix D. MNRAS000
95 % of the distribution at a fixed mass. Horizontal error bars indicate thespread in the fitted M m masses. The vertical dotted lines denote the mass limit below which the finite force resolution affects theconcentration estimates. Note that the disagreement between the model and the measurements originates mostly from unrelaxed haloes(cf. Fig. 9). For an analogous plot using concentrations obtained with Einasto profiles, see Appendix D. MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations M [ h − M (cid:12) ]468101214 c (0 . , . Model (direct)Model (rescaled)All haloes (direct)All haloes (rescaled) d off cut (direct) d off cut (rescaled) d off + f sub cut (direct) d off + f sub cut (rescaled) Figure 9.
The impact of unrelaxed haloes in the concentration-mass relation. Different lines show the results for direct andrescaled halo catalogues after different cuts were applied to elim-inate unrelaxed systems. Note that applying relaxation cuts in-creases the amplitude of the relation and produces a better agree-ment with the theoretical models.
Motivated by the good agreement in Fig. 10, we correct therescaled profiles by multiplying the measured values withthe ratio between the fitted profile to the rescaled simulationdata and a modified profile with the concentration bias fromthe model, ∆ c ( r m ) : ρ ( r ) ρ NFW ( r, c + ∆ c ( r m ) , r m ) ρ NFW ( r, c, r m ) ρ ( r ) , (23) ∆Σ ( r ) ∆Σ NFW ( r, c + ∆ c ( r m ) , r m )∆Σ NFW ( r, c, r m ) ∆Σ ( r ) , (24)for all radii r (cid:46) r m . We will refer to these correctionfactors as γ ( r i ) . The Einasto correction is calculated in thesame manner (see Appendix D). Since ∆ c ( M ) only weaklydepends on M , there are no significant differences betweenusing the fitted M m or halo finder value.Correcting the profiles up to h − Mpc does not signif-icantly affect the lensing signal, but jeopardises the agree-ment for the 2-halo term in 3D (see Fig. 18). We find thatrestricting the correction to r < . r m reduces differencesin the 1-to-2-halo transition region without compromisingthe agreement on larger scales.The concentration correction could be additive insteadof multiplicative. This gives a slightly better performanceon scales r > r m , since the field differences are small,but this correction also induces a small bias and should thusbe applied below a cutoff radius. The multiplicative cor-rection preserves the shape of the residual throughout thetransition regime slightly better. Otherwise, we have checkedthat there are no significant differences between the two forall halo mass bins and cosmologies with NFW or Einastoparametrizations for matched haloes, in bootstrapped stacksor individually. Both largely preserve the width and shapeof the ∆ c distribution around the median or the mean con-centration, with no obvious advantages, and yield ∆ c = 0 if we correct the rescaled profiles with the measured directconcentrations.The residuals for the corrected 3D density profiles areshown in Fig. 11 and for the corrected ∆Σ profiles in Fig. 12.The maximum and median pre- and post-correction profiledifferences are listed in Table 2 for the 40 radial bins setup.Typically, the largest differences occur in the most or sec-ond most massive halo mass bin. In most cases, the correc-tion reduces the differences by factors of two to five. For (0 . , . , both the residual profiles and residual concen-tration differences indicate that a larger concentration cor-rection than predicted by the L16 model could improve theagreement between direct and rescaled profiles.However, when comparing the measured halo concentra-tions pre- and post-correction, we find significant improve-ment in the concentration mismatch between rescaled anddirect simulations for all considered cosmologies, as Fig. 13illustrates for all haloes (see Appendix C for the result formatched populations). We also examine how the correction in Eq. (23) af-fect the concentrations from 3D profile fits to individualhaloes. The joint distribution of concentrations for haloesabove . h − M (cid:12) in the (0 . , . -simulation and theirrescaled counterparts is shown in Fig. 14. Applying the con-centration correction translates the distribution towards thediagonal in a similar manner for high and low concentrationhaloes. This is a consequence of the modest mass evolutionof the concentration bias for the cosmologies in this study.However, the correction cannot account for a slight tilt be-tween the two simulations, with low- c (high- c ) haloes havinghigher (lower) concentrations in the direct simulation thanin the rescaled simulation. The tilt is stronger for cosmologies with ∆Ω m > away from the fiducial simulation with a clockwise tilt rela-tive to the diagonal (see Appendix C). For (0 . , . and (0 . , . , there is a slight counter-clockwise tilt. The re-sults are robust to changes in the fitting scheme. We havechecked that there are negligible differences for all cosmolo-gies between the c ( M ) relations computed from the medianprofiles and the median c ( M ) relations from fits to individ-ual haloes, and that the tilt in the distributions persist whenone corrects the individual halo concentrations with the me-dian measured relations.The tilt in the joint distribution is also present for halosamples selected in narrower mass ranges. The asymmetryis partly washed out in the results for the median profiles,as both high c and low c haloes contribute to the effectivedensity field per mass bin. However, this secondary rescaling This tilt persists when relaxation cuts are enforced, regardlessof whether r m is fixed or a free parameter, and is also presentwith Einasto parameterisations (Appendix D). For all profile fits we use the Levenberg-Marquardt algorithmwith ( c = 4 , r m = r m , sim. ) as a starting point. We havechecked that the results are insensitive to the starting point choicefor physically viable parameter values. In addition we have com-puted the parameters with the limited-memory BFGS algorithmwith bounds c ∈ [1 , and r ∈ [0 . r m , sim. , r m , sim. ] and obtain consistent results.MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Concentration bias 3D M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Concentration bias ∆Σ Figure 10.
The difference in concentrations measured in the direct and rescaled simulations, ∆ c (direct − rescaled) , as a function halo massat z = 0 . Concentrations were measured by fitting NFW profiles to 3D density profiles (left panel), and to ∆Σ profiles (right panel).Different colours indicate results for different combinations of ( Ω m , σ ). Dotted lines correspond to the predictions for this quantity basedon the L16 model. The shaded regions mark the
68 % and
95 % percentiles, and horizontal error bars the range of fitted halo masses. -0.30-0.150.000.15 (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) -0.45-0.30-0.150.000.15 (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r/r -0.09-0.06-0.030.000.030.06 (0 . , . − r/r -0.16-0.080.000.080.160.24 (0 . , . − ρ r / ρ d Figure 11.
Same as Fig. 5 but after correcting the inner profiles of rescaled haloes.
Table 2.
Total and median maximum deviation between the direct and rescaled simulation, − ρ r /ρ d and − ∆Σ r / ∆Σ d , for 3D medianand for 2D mean profiles per mass bin for radial bins in the given range before and after the concentration correction.Residuals: ρ ( r ) , h − kpc < r < r m ∆Σ( r ) , h − kpc < r < r m Pre-correction Post-correction Pre-correction Post-correctionSimulation Halo mass range Max Median Max Median Max Median Max Median (0 . , .
00) 10 . − . h − M (cid:12)
35 % 22 % −
17 % − . . . , .
60) 10 . − . h − M (cid:12)
25 % 15 % −
17 % − . . . , .
81) 10 . − . h − M (cid:12)
16 % 2 . −
16 % 1 . . . . − . . , .
70) 10 . − . h − M (cid:12)
25 % 7 . −
18 % 3 . −
26 % − . −
15 % − . . , .
40) 10 . − . h − M (cid:12)
43 % 22 % 11 % 6 . −
42 % −
29 % 13 % − . MNRAS000
29 % 13 % − . MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations -0.150.000.150.30 (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) -0.16-0.080.000.080.160.24 (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r [ h − Mpc]-0.15-0.10-0.050.000.050.10 (0 . , . − r [ h − Mpc]-0.16-0.080.000.080.16 (0 . , . − ∆ Σ r / ∆ Σ d Figure 12.
Same as Fig. 6 but after correcting the inner profiles of rescaled haloes. M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Concentration bias 3D post-correction M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Concentration bias ∆Σ post-correction Figure 13.
Same as Fig. 10 but after applying our corrections in Eqs. (23) and (24) to the rescaled profiles. The concentration bias forthe corrected profiles is reduced considerably. concentration bias could influence analyses where the halopopulation is split into different concentration samples atfixed mass, such as assembly bias studies. Further studieswith larger simulation volumes are required to accuratelyquantify this effect.
The concentration correction does not fully account for dif-ferences in the halo outskirts, as it focuses on rearrangingmaterial within the halo. Subsequent outer corrections couldredefine the halo boundary and potentially improve agree-ment in the halo mass function. Fig. 15 highlights that theprofile bias in the inner halo regions is mostly an amplitudeoffset, whereas the bias in the halo outskirts is rather a ra-dial offset. Hence, correcting the rescaled profiles by shiftingthem radially in the outskirts can mitigate the outer profilebias. In Fig. 16, we plot the measured differences in the lo-cation of the steepest slope of the density field for matchedhaloes. We adjust the position of the rescaled profile’s steep-est slope with r (d) m /r (r) m to account for the mismatch inhalo mass between the matched samples, which has a minorimpact on the result. We compare these differences to theexpected offset between the splashback radii r sp , the apoc-entre of the first orbit of accreted material (e.g. Diemer &Kravtsov 2014; Adhikari et al. 2014; More et al. 2015; Shi2016; Mansfield et al. 2017; Diemer et al. 2017), betweenthe direct and rescaled profiles ∆ r sp = r (d) sp − r (r) sp . We applythe recent fit provided in Diemer et al. (2017) to simula-tion results in Diemer (2017) to predict the median splash-back radius as a function of halo mass and cosmology. Thismodel has been fitted by tracing billions of particle orbitsin haloes spanning from typical cluster to dwarf galaxy hostmasses in different cosmological simulations up to z = 8 .Percentiles correspond to the fraction of the first apocenters MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo c rescaled c d i r ec t (0 . , . InitialCorrected
Figure 14.
Effect of the density field correction on the NFW es-timated concentration distribution for individual matched haloesin the direct and rescaled simulation with (0 . , . where thehaloes in the direct simulation have M m > . h − M (cid:12) . Forsmoother contours, the distributions have been convolved with aGaussian filter with σ = 1 . ρ ( r ) / ¯ ρ m · ( r / r m ) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r/r − − d l og ρ / d l og r Direct simulationRescaled simulation
Figure 15.
Comparison between direct and rescaled profilesand their radial derivatives for matched haloes for (0 . , . forthree mass bins. The concentration bias is visible as an amplitudeoffset close to the halo centres whereas the outer profile bias cor-responds to a radial shift of the profiles at the halo boundaries.This shift is visible in the radial derivatives of the field (computedwith a fourth-order Savitzky-Golay filter with a window length of15 bins) in the lower panel as well, where there are offsets in thepositions of the steepest slope between the direct and rescaledprofiles. M [ h − M (cid:12) ]-0.40-0.200.000.200.400.60 ∆ r s p / r m Model(0 . , . . , .
60) (0 . , . . , . . , . Figure 16.
Measured differences in the location of the steepestslope of the density field for matched haloes w.r.t. to the Diemeret al. (2017) model, for the 75th percentile. Error regions for
95 % and
68 % are computed from resampled medians from stacks ofmatched haloes in the direct and rescaled simulation snapshots. of the particle orbits contained inside a given radius. Parti-cles which were contained in a subhalo with mass exceeding1 % of the host halo mass at infall are excluded to minimisebias from dynamical friction. In accordance with previousstudies (e.g. Diemer & Kravtsov 2014; More et al. 2015),the relation between the halo accretion rate Γ and the ra-tios r sp /r m and M sp /M m is found to be well describedby a functional form X sp = A + Be − Γ /C where X sp is ei-ther ratio and A, B and C are free parameters where B and C depend on the matter fraction Ω m and halo peak height ν = δ c / ( D ( z ) σ ( M h )) with M h as the halo mass. In addition,the median accretion rate Γ( ν, z ) can be well captured bya parameterisation Γ = A ν + B ν / , where A and B arepolynomials in z . We use this expression for the median ac-cretion rate to compute the radii. The measurements tracethe model prediction, except for (0 . , . where the scat-ter is driven by poor statistics due to the small box size.We also compute the radial shifts that minimise thelargest relative difference between the direct and rescaledouter density profiles. Between . < r/r m < . , welocate the maximum of the − ρ r ( r ) /ρ d ( r ) residual defin-ing r = r max and then shift the interpolated rescaled pro-file radially to find the radius r min that minimises − ρ r ( r min ) /ρ d ( r max ) . The resulting shifts r max − r min are shownin Fig. 17 for matched haloes with the r (d) m /r (r) m cor-rection. This shift is almost constant for haloes, all andmatched, with M m between − h − M (cid:12) . Forhigher masses the result is obscured by scatter. The pre-dicted splashback bias do not exactly match the required As we are probing the median 3D density profiles, we opt forthe 75th percentile of the model which was found to best matchthe median profiles in More et al. (2015), especially at the highmass end. The splashback radius rescales as r sp αr sp and thepredicted position r (r) sp is hence given as the fitted solution inthe fiducial simulation at the fiducial redshift with β − m M m determining the peak height and r m .MNRAS000
68 % are computed from resampled medians from stacks ofmatched haloes in the direct and rescaled simulation snapshots. of the particle orbits contained inside a given radius. Parti-cles which were contained in a subhalo with mass exceeding1 % of the host halo mass at infall are excluded to minimisebias from dynamical friction. In accordance with previousstudies (e.g. Diemer & Kravtsov 2014; More et al. 2015),the relation between the halo accretion rate Γ and the ra-tios r sp /r m and M sp /M m is found to be well describedby a functional form X sp = A + Be − Γ /C where X sp is ei-ther ratio and A, B and C are free parameters where B and C depend on the matter fraction Ω m and halo peak height ν = δ c / ( D ( z ) σ ( M h )) with M h as the halo mass. In addition,the median accretion rate Γ( ν, z ) can be well captured bya parameterisation Γ = A ν + B ν / , where A and B arepolynomials in z . We use this expression for the median ac-cretion rate to compute the radii. The measurements tracethe model prediction, except for (0 . , . where the scat-ter is driven by poor statistics due to the small box size.We also compute the radial shifts that minimise thelargest relative difference between the direct and rescaledouter density profiles. Between . < r/r m < . , welocate the maximum of the − ρ r ( r ) /ρ d ( r ) residual defin-ing r = r max and then shift the interpolated rescaled pro-file radially to find the radius r min that minimises − ρ r ( r min ) /ρ d ( r max ) . The resulting shifts r max − r min are shownin Fig. 17 for matched haloes with the r (d) m /r (r) m cor-rection. This shift is almost constant for haloes, all andmatched, with M m between − h − M (cid:12) . Forhigher masses the result is obscured by scatter. The pre-dicted splashback bias do not exactly match the required As we are probing the median 3D density profiles, we opt forthe 75th percentile of the model which was found to best matchthe median profiles in More et al. (2015), especially at the highmass end. The splashback radius rescales as r sp αr sp and thepredicted position r (r) sp is hence given as the fitted solution inthe fiducial simulation at the fiducial redshift with β − m M m determining the peak height and r m .MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations M [ h − M (cid:12) ]-0.30-0.20-0.100.000.100.200.30 ( r m a x − r m i n ) / r m Model(0 . , . . , .
60) (0 . , . . , . . , . Figure 17.
Measured density field outer profile bias for matchedhaloes vs. the predicted ∆ r sp /r m bias using the model inDiemer et al. (2017). ρ ( r ) / ¯ ρ m · ( r / r m ) DirectRescaled × γ ( r i ) − ∆ r × r min /r max Joint (with sigmoids)10 − r/r -0.30-0.150.000.150.30 − ρ r / ρ d Figure 18.
Profiles for matched haloes for (0 . , . for M m ∈ [10 , . ) h − M (cid:12) in the direct simulation withdifferent corrections applied (see the text for more detailed de-scriptions). Although not perfect, the concentration correction’ × γ ( r i ) ’ mitigates the residual in the centre and the shifts removethe outer profile bias. These two corrections can be combined withsigmoids. shifts to remove the radial bias , but they show similar rela-tive amplitudes, signs and weak mass dependence. A splash-back radius model may thus provide a good starting pointfor further improvements of the rescaled profiles and halomasses (an initial attempt to correct the masses is presentedin Appendix E).As Fig. 18 illustrates the outer profile bias vanishes,if we shift the rescaled density field values radially by r r min /r max × r or r r − ∆ r with ∆ r = r max − r min .Whereas the multiplicative correction performs better in thehalo centre, the additive correction has a better large scale Moreover, typically r max ≈ . r m , which does not coincidewith the predicted position of the splashback radius for all massesand cosmologies. behaviour. To combine the radial shift correction with theconcentration correction, we modulate each by a sigmoidfunction to restrict their actions to their intended radialrange: ρ ρ = ρ ( r − ζ ( r )) + ξ ( r ) , ζ ( r ) = 11 + e − k ( r − r ) ∆ r,ξ ( r ) = 11 + e − k ( r − r ) · (cid:0) ρ NFW − ρ NFW (cid:1) , (25)where r marks the transition scale, k and k control thesharpness of the onsets of the corrections, and the concen-tration correction is evaluated at the unshifted radius. Fit-ting these parameters, r in the vicinity of r m seems pre-ferred, but all parameters vary with mass and cosmologywhen fitting the rescaled simulation to the direct simulation.In Fig. 18 we plot one possible solution with ( r , k , k ) as ( r m , . , . , where ∆ c is obtained from the L16 modeland ∆ r is measured. Future investigations are required tofind the best set of parameters. The rescaling predictions for the halo matter and lensingprofiles are reasonably accurate even before applying theconcentration correction. Partly, this is due to the matchedinitial conditions. This ensures similar peak heights, proto-halo regions, environments, and tidal fields, which leads tosimilar growth histories, as the growing density perturba-tions subsequently cross the collapse threshold.After our additional correction, the predictions becomeaccurate at the level. In this section, we discuss theexpected cosmology dependence of the corrections (Sec-tion 5.2), the method’s accuracy in light of the expectedimpact of baryons (Section 5.3) and large-scale corrections(Section 5.4), as well its application for lensing mass estima-tions (Section 5.5).
Our approach differs from the setup in Mead & Peacock(2014a) since it is a nonlocal operation on the density pro-files built from the full 3D and 2D rescaled particle distribu-tions whereas their method involve shifting the halo particlepositions. They work with a subset of particles randomlysampled from the fiducial distribution to fill up the pre-dicted density profile where information from the tidal ten-sor helps to account for the asphericity (this produces betteragreement in halo morphology but does not take substruc-ture into account which is problematic for satellite galax-ies). It is not evident how much this sampling scheme differsfrom a refined method working on the actual 3D distribu-tion of particles within the halo. A possible way to imple-ment our algorithm as a localised, discrete mapping is toperform a local measurement of the spherically binned den-sity field around each halo, use the correction to find theclosest NFW/Einasto profile and shift the particles betweenthe shells accordingly till some convergence criteria has beenmet. Preferably, this should prioritise displacements betweenadjacent shells. One could also account for the shape of the
MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . z -4.00-2.000.002.004.006.00 ∆ c ( M , z ) (0 . , . . , . . , .
81) (0 . , . . , . Figure 19.
Expected bias in the concentration of rescaled haloesbased on the L16 model, evaluated as the median bias for haloeswith < M m / ( M (cid:12) h − ) < , as a function of redshift. tidal tensor, compute Penna-Dines surfaces for accretion re-sponses (cf. Mansfield et al. 2017) and extract additionalphase-space information to preserve the halo shape, compo-sition, stream structure and extension. Due to the few simulations in our study, we cannot putstrong constraints on a model-independent fitting functionfor the concentration bias. All cosmologies, with the excep-tion of (0 . , . , trace the Ω m − σ degeneracy favouredby weak lensing, which means that we have few constraintsperpendicular to this line. We thus use the L16 model topredict the rescaled concentration bias for cosmologies andredshifts where we do not have access to a correspondingdirect simulation.Firstly, we investigate the redshift evolution in the cos-mologies already covered. We use the linear growth factorrelation in Eq. (6) to calculate the redshifts in the fiducialsimulation which correspond to the higher redshifts in thedirect simulation. We plot the median concentration biasfor haloes with M m in − h − M (cid:12) as a functionof redshift from z = 0 to z = 2 in the direct simulationin Fig. 19. Overall the difference in concentration decreaseswith redshift and there is a turnover point for all cosmolo-gies expect (0 . , . where the bias changes sign. This isa consequence of the rescaling parameters being determinedby the locally matched growth history. Yet, caution musttaken as we have already seen that the model predictionworks less well at higher redshifts in Fig. 10. To bring abouta better agreement with the measurements, the model couldbe modified to feature a slight redshift dependence whicheither decreases C and/or raises f since these changes lowerthe amplitude of the c ( M ) − relation.In Fig. 20, we plot the expected median ∆ c bias forhaloes with masses M m in − h − M (cid:12) whenrescaling the Millennium simulation (Springel et al. 2005)to match target cosmologies with different Ω m and σ at z = 0 , with the target matter power spectra generated by m σ -4.00-2.000.002.004.006.00 ∆ c Figure 20.
Expected bias in the concentration of rescaled haloesat z = 0 as a function of the value of Ω m and σ . Our assumedfiducial cosmology is Ω m = 0 . and σ = 0 . (marked by thewhite cross). The white diamonds mark the test simulations cos-mologies employed in this paper. CAMB (Lewis et al. 2000) combined with linear growth fac-tors (e.g. Hamilton 2001) assuming a constant baryon frac-tion Ω baryons / Ω m . The corresponding contours for the rescal-ing parameters ( α, z ∗ ) are shown in Appendix F. Rescalingto a lower σ at fixed Ω m or a lower Ω m with a higher σ induces a positive ∆ c , whereas raising Ω m and lowering σ will produce negative ∆ c .If one relaxes the growth history constraint to permitmatches in the future, negative redshifts represent the pre-ferred solutions for the ∆Ω m > , ∆ σ > quadrant. Suchsolutions yield ∆ c < . If we instead restrict our redshiftrange to z ∗ (cid:38) − . , the concentration bias becomes positiveagain as we move further away from the degeneracy plane.The contours for the predicted ∆ r sp -bias (see Appendix F)partly trace the ∆ c contours with the opposite sign overmost of the plane except in the ∆Ω m > , ∆ σ > quad-rant.The concentration bias is a smooth function of cosmol-ogy, i.e. small changes in the cosmological parameters pro-duce small concentration offsets. A set of well-placed simula-tions could thus be used together with rescaling to efficientlycover a large region of parameter space accurately.Lastly, we discuss rescaling to emulate a WMAP7 cos-mology (Komatsu et al. 2011) and Planck (2014) cosmol-ogy (Planck Collaboration 2014) at z = 0 using the Millen-nium simulation with SAMs in Guo et al. (2013) with theAW10 weighting scheme and in Henriques et al. (2015) withthe AH15 scheme, respectively. The corresponding ( z ∗ , α ) are (0 . , . and (0 . , . , respectively, which pro-duce ∆ c ( M ) relations with shallow slopes with median bi-ases ∆ c = 0 .
88 (∆ c min = 0 . , ∆ c max = 0 . and ∆ c =0 .
06 (∆ c min = 0 . , ∆ c max = 0 . for M m between − h − M (cid:12) . This means that the concentration biasfor haloes in Henriques et al. (2015) is almost negligible.We plot these relations in Appendix G with the predicted An existing N -body simulation can cheaply be evolved intothe future (see e.g. Angulo & Hilbert 2015).MNRAS000
06 (∆ c min = 0 . , ∆ c max = 0 . for M m between − h − M (cid:12) . This means that the concentration biasfor haloes in Henriques et al. (2015) is almost negligible.We plot these relations in Appendix G with the predicted An existing N -body simulation can cheaply be evolved intothe future (see e.g. Angulo & Hilbert 2015).MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations redshift evolutions, where the biases also are reduced at ear-lier times. Hence, we can predict the bias of the measuredlensing signal around central SAM galaxies in rescaled sim-ulation snapshots. Our method currently does not account for effects baryonicprocesses have on halo profiles. The impact of baryonic pro-cesses on the matter distribution has been investigated insimulations (e.g. by van Daalen et al. 2014; Velliscig et al.2014; Schaller et al. 2015; Leauthaud et al. 2017; Mummeryet al. 2017). Baryon physics affects the matter clustering by ∼
10 % on scales (cid:46) Mpc. The impact on ∆Σ is similar.By matching the haloes in Illustris with their counterpartsin a dark matter-only run, the baryonic physics has beenfound to suppress ∆Σ by ∼
20 % from r (cid:38) . h − Mpc to r (cid:54) h − Mpc (Leauthaud et al. 2017).Even for cosmologies far from the fiducial cosmology,the rescaling predictions without the concentration correc-tions are at most off by 40 % in the innermost radial bins,and the disagreement decreases to ∼
10 % at r ≈ h − Mpc.The concentration correction substantially improves agree-ment in the inner region. Moreover, the discrepancies aremuch smaller for cosmologies closer to the fiducial cosmol-ogy. This means that the bias induced by rescaling is sub-dominant to the baryonic feedback effects below h − Mpc,except for extreme cosmologies.
Here, we do not attempt any corrections at very large scales.We have computed the difference between the matter powerspectrum in the weakly nonlinear to the nonlinear regime for (0 . , . with and without the large-scale displacementfield correction from AW10 and it was found to be negli-gible. The large-scale halo-matter correlations do not differsignificantly between the rescaled and direct simulations forthe halo masses we are investigating in 3D. There appears atmost a small offset with surrounding scatter. The connectionand coupling between this offset and the detected mass bias,as well as the proper response of the large-scale correlationsto the rescaling transform are topics for future studies. Inhalo models of GGL (e.g. Oguri & Takada 2011), the large-scale lensing signal (2-halo term) is directly related to theprojected linear power spectrum. It should thus be straight-forward to compute its response to rescaling. Moreover, theproposed recipe in AW10 to correct the displacement fieldusing the Zel’dovich approximation (Zel’dovich 1970) shouldimprove the agreement.For the linear regime, there already exist fast, accuratelarge-scale structure solvers, e.g. COLA (Tassev et al. 2013,2015) and
FastPM (Feng et al. 2016). Thus, corrections forexclusive large-scale analyses using the rescaling approachare of limited practical importance. However, the benefitsof rescaling the small scales become manifest when success-fully coupled to such a large-scale solver, as a wide range ofcosmologies can be explored on multiple refinement levels. × × M [ h − M (cid:12) ]020406080100120140 χ Before correctionAfter correction
Figure 21. χ -parabolae for rescaled ∆Σ profiles fitted to adirect ∆Σ profile for a stack of galaxy group-size haloes withmean M m marked by the vertical dashed line according toEq. (26). The minimum determines the best fit rescaled profile,and the corresponding simulation mass the best fit mass. Theconcentration correction shifts the parabola to be more symmetricaround the direct simulation’s mean mass, reducing the differenceto the best-fit rescaled mass. One application for galaxy-galaxy lensing is halo mass esti-mation for a selected foreground galaxy sample. We thus ex-amine how the residual statistical and systematic differencesin the profiles translate to errors in the measured masses.For simplicity, we focus on the (0 . , . cosmology, andwe choose a series of mass-selected samples in the direct sim-ulation: haloes in mass bins of 0.05 dex or 0.1 dex centred onslightly different masses with bin borders shifted with 0.005dex w.r.t. one another around . h − M (cid:12) (i.e. massivegalaxy haloes) or . h − M (cid:12) (galaxy group haloes). Themean ∆Σ profiles for these bins constitute our mock weaklensing observations.If we fit NFW profiles to these mock lensing observa-tions, we obtain mass estimates that are approx. to
10 % below the true mean halo masses as recorded by the halofinder (see Fig 7). We should be able to bypass this bias ifwe employ the rescaled simulation’s stacked profiles (whichshould be ‘biased’ in the same way) as model predictions(instead of analytic NFW profiles) to estimate the meanmass of our mock halo sample. This however requires thatthe rescaled halo profiles are close enough to the true haloprofiles (i.e. the direct simulation’s profiles in this exercise),since a mismatch, e.g., in concentration of ∆ c = 1 causesan error ∼ in the inferred masses (e.g. Applegate et al.2016; Schrabback et al. 2018). For the considered example,the concentration mismatches are already small before thecorrection ( ∆ c ∼ . and ∆ c ∼ . ), and vanish afterthe correction. Thus, mass errors due to concentration mis-matches are well below here (this is not necessarily thecase for rescaling to the other, more extreme cosmologies).To fit the rescaled mean profiles (our predictions) to thedirect profiles (our mock data), we minimise the figure-of- MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo merit χ = N r X i r i [∆Σ direct ( r i ) − ∆Σ rescaled ( r i )] , (26)for radial bins . < r i /r m < . . Fig. 21 illustrateshow the figure of merit changes when the mean profile ofhaloes in the direct simulation in a bin with width 0.1 dexcentred on . h − M (cid:12) is fit with rescaled mean profiles ofmass bins with the same width but varying mean mass. Theconcentration correction shifts the χ -parabola to be moresymmetric around the direct simulation’s mean mass.The results from the different sweeps are listed in Ta-ble 3. For smaller halo samples, the χ -parabolae featureconsiderable scatter which cause larger errors for the best-fit mass. As the number of haloes grow, the χ -parabolaebecome smoother and the errors on the best-fit masses de-crease. This behaviour is in line with previous work (Becker& Kravtsov 2011; Hoekstra et al. 2011) where the relativeerror on the mass was found to be ∼
30 % per system forgroup haloes (and around 20 % for more massive systems).For example, this yields a relative mass error of ∼ . forstacks of ∼ ∼ . for ∼
10 000 haloes.For future dark energy task force stage IV surveys, suchas Euclid, statistical errors on mass estimations from ∆Σ profiles are expected to shrink substantially compared tocurrent surveys. We can acquire a rough estimation by scal-ing corresponding values from CFHTLenS (Velander et al.2014), which has a similar depth but a smaller survey areaof deg , to an area of
15 000 deg for Euclid (Laureijset al. 2011; Amendola et al. 2013). A hundred times largersurvey area roughly translates to a reduction of the statis-tical errors by a factor of ten. As example, we consider thesample L7 of 344 lenses in Velander et al. (2014) with abso-lute r -band magnitudes in the range [ − . , − . , averageredshift ¯ z = 0 . , fraction of blue galaxies f blue = 0 . . Themean halo mass of these lenses estimated from CFHTLenSis . h − M (cid:12) with a quoted
20 % error. The statisticalerror for Euclid would shrink to . This suggests that ourproposed method is accurate enough for current halo weaklensing data, and moreover may be viable for much largerfuture surveys, once baryonic effects on halo profiles havebeen properly accounted for.
We have demonstrated the prowess of a refined rescaling al-gorithm with growth history constraints in predicting halo3D and GGL profiles. Residual differences in the inner pro-files have been parametrised as concentration biases thatcan be predicted using linear theory combined with excur-sion sets. Differences in the profile outskirts can be expressedin terms of a shift in the splashback radius. This enables usto correct the profiles and improve the method’s accuracy.This represents an important step towards the reusability of N -body simulations for cosmic structure analyses.The algorithm’s accuracy is satisfactory for currentGGL data. However, small remaining discrepancies in thehalo profile outskirts and for the lens mass estimates mayrequire further treatment depending on the application. Fur-ther studies could clarify, which of these discrepancies aredue to systematic biases, and which are due to scatter in, e.g., halo shapes and line-of-sight structure. With possiblyimproved corrections capturing biases not addressed so farand large N -body simulations to minimise statistical errors,the method may be made suitable for analysing future large(dark energy task force stage IV) surveys. ACKNOWLEDGEMENTS
We would like to thank the anonymous referee for a compre-hensive report which has improved the structure and pre-sentation of our results in this paper. M.R. and S.H. ac-knowledge support by the DFG cluster of excellence ‘Originand Structure of the Universe’ ( ).M.R. and S.H. thank the Max Planck Institute for Astro-physics and the Max Planck Computing and Data Facilityfor computational resources. R.E.A. acknowledges supportfrom AYA2015-66211-C2-2 and support from the EuropeanResearch Council through grant number ERC-StG/716151.
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Residuals from three different mass bins’ ∆Σ pro-files for (0 . , . in the rescaled simulation w.r.t. the directsimulation. − r [ h − Mpc]-0.40-0.200.000.20 − ∆ Σ r / ∆ Σ d (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) Figure A2.
Residuals from concentration corrected ∆Σ profilesfor (0 . , . .Zu Y., Mandelbaum R., 2015, MNRAS, 454, 1161van Daalen M. P., Schaye J., McCarthy I. G., Booth C., VecchiaC. D., 2014, MNRAS, 440, 2997van den Bosch F. C., Tormen G., Giocoli C., 2005, MNRAS, 359,1029 APPENDIX A: IMPACT OF RADIAL BINNINGAND FIELD RESIDUAL VARIANCES FOR ∆Σ PROFILES
The measured differences between direct and rescaled haloprofiles presented in Section 4 could depend on the radialbinning. To investigate the impact of the bin width, wecompute ∆Σ profiles with twice as many bins. For ∆Σ ,the new values for (0 . , . are
41 % and
32 % (pre-correction) and
15 % and . (post-correction), which rep-resent the largest differences owing to the lower resolution ofthis simulation. For (0 . , . , the differences increase to . and . (pre-correction) and . and − . (post-correction) which implies an increase with for the total M [ h − M (cid:12) ]-0.40-0.200.000.200.40 − M r m / M d m (0 . , . Mean relation per 0.1 dex binMedian relation per 0.1 dex bin
Figure B1.
Mass bias for matched haloes in the (0 . , . sim-ulation. The rescaled haloes are consistently more massive thantheir counterparts in the direct simulation across the whole massrange, with some outliers among galaxy class haloes. maxima and less than for the median maximum values.For (0 . , . and (0 . , . , the resulting changes arebelow or maximally 1 %. The same is true for (0 . , . ,though the median maximum deviation changes signs to − . post-correction.Concerning the cosmic variances of these residuals,we plot the residuals from the bootstrapped profiles for (0 . , . using all haloes in three mass bins in Figs. A1and A2, before and after applying concentration correction(the results are qualitatively the same for the other simula-tions). For galaxy and galaxy group class haloes, the spreadin the differences in the inner regions are quite narrow andthey widen as one approaches the 1-halo to 2-halo transitionregime. For cluster size haloes, there is a larger variance inthe inner regions which is both driven by poor statistics andthe impact of unrelaxed systems. This is reflected in thespread in concentrations. Overall, the correction preservesthe variance with slightly larger error bars for cluster masshaloes as the haloes are not necessarily matched in eachbootstrapped stack w.r.t. one another. APPENDIX B: RESULTS FOR (0 . , . The almost Einstein-de Sitter cosmology represents our mostextreme sample, and its cosmological parameters deviatestrongly from what is favoured by observations. The massesdiffer substantially between the matched haloes in the di-rect and rescaled simulation, see Fig. B1, with haloes in therescaled simulation on average more massive. In Fig. B2,we show the measured ∆Σ profiles together with the fittedNFW lens profiles and in Fig. B3 the profiles post-correction.Due to the small volume of the simulation as listed in Ta-ble 1, we do not have any mass bins beyond h − M (cid:12) with more than twenty haloes in both the direct and rescaledsnapshot. Since the amplitude of the 2-halo term is directlyproportional to the matter fraction of the Universe, its influ-ence kicks in at smaller scales than for the other simulations.The inner profile bias is negative and can be quantified as ∆ c ≈ − as seen in Figs. 10 and B4 where we plot the MNRAS000
Mass bias for matched haloes in the (0 . , . sim-ulation. The rescaled haloes are consistently more massive thantheir counterparts in the direct simulation across the whole massrange, with some outliers among galaxy class haloes. maxima and less than for the median maximum values.For (0 . , . and (0 . , . , the resulting changes arebelow or maximally 1 %. The same is true for (0 . , . ,though the median maximum deviation changes signs to − . post-correction.Concerning the cosmic variances of these residuals,we plot the residuals from the bootstrapped profiles for (0 . , . using all haloes in three mass bins in Figs. A1and A2, before and after applying concentration correction(the results are qualitatively the same for the other simula-tions). For galaxy and galaxy group class haloes, the spreadin the differences in the inner regions are quite narrow andthey widen as one approaches the 1-halo to 2-halo transitionregime. For cluster size haloes, there is a larger variance inthe inner regions which is both driven by poor statistics andthe impact of unrelaxed systems. This is reflected in thespread in concentrations. Overall, the correction preservesthe variance with slightly larger error bars for cluster masshaloes as the haloes are not necessarily matched in eachbootstrapped stack w.r.t. one another. APPENDIX B: RESULTS FOR (0 . , . The almost Einstein-de Sitter cosmology represents our mostextreme sample, and its cosmological parameters deviatestrongly from what is favoured by observations. The massesdiffer substantially between the matched haloes in the di-rect and rescaled simulation, see Fig. B1, with haloes in therescaled simulation on average more massive. In Fig. B2,we show the measured ∆Σ profiles together with the fittedNFW lens profiles and in Fig. B3 the profiles post-correction.Due to the small volume of the simulation as listed in Ta-ble 1, we do not have any mass bins beyond h − M (cid:12) with more than twenty haloes in both the direct and rescaledsnapshot. Since the amplitude of the 2-halo term is directlyproportional to the matter fraction of the Universe, its influ-ence kicks in at smaller scales than for the other simulations.The inner profile bias is negative and can be quantified as ∆ c ≈ − as seen in Figs. 10 and B4 where we plot the MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations ∆ Σ ( r ) [ h M (cid:12) p c − ] (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r [ h − Mpc]-0.30-0.150.000.15 − ∆ Σ r / ∆ Σ d Figure B2. ∆Σ profiles for (0 . , . . ∆ Σ ( r ) [ h M (cid:12) p c − ] (0 . , . . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) − r [ h − Mpc]-0.30-0.150.000.15 − ∆ Σ r / ∆ Σ d Figure B3.
Concentration corrected ∆Σ profiles for (0 . , . . M [ h − M (cid:12) ]468101214 c (0 . , . Model (direct)Model (rescaled)All haloes (direct)All haloes (rescaled) d off cut (direct) d off cut (rescaled) d off + f sub cut (direct) d off + f sub cut (rescaled) Figure B4.
NFW c ( M ) -relations for (0 . , . for all haloesand with different relaxation cuts enforced.
3D density profile NFW c ( M ) -relations. The Einasto c ( M ) -relations, see Fig. D2, perform slightly better at the low massend w.r.t. the L16 predictions. APPENDIX C: MATCHED HALO RESULTS
In Fig. C1 we show the fractional differences in the mediandensity profiles between matched haloes in the direct andrescaled simulations binned according to the mass in thedirect run for all test cosmologies. The error regions are cal-culated from comparing the median differences between thesame bootstrapped matched haloes in the direct and rescaledsimulations. With respect to the differences shown in Fig. 5,the two biases are slightly more discernible, especially theouter profile bias and the (small) concentration bias for (0 . , . . Re-sampling the matched population for eachmass bin yields similar results. For all cosmologies and massbins the profile bias changes signs at ≈ . − . r/r m which was also observed previously for all haloes. The me-dian ∆ c -biases for these matched haloes are illustrated inFig. C2 where the error regions are computed from boot-strap resamples of the same matched haloes in the directand rescaled simulations.In Fig. C3 we plot the individual concentration relationsin the direct and rescaled simulation for all cosmologies ex-cept for (0 . , . which was already shown in Fig. 14with the same setup. We only correct the profiles if thefitted c + ∆ c > . This chiefly affects massive haloes inthe (0 . , . simulation and it has a negligible impact onthe shape of the contours. The concentration correction in-duces a translation towards the diagonal but rotations are re-quired for (0 . , . and (0 . , . to bring about agree-ment. Slight rotational adjustments might improve the con-cordance for (0 . , . and (0 . , . . For (0 . , . ,a larger translation correction is required. Imposing relax-ation cuts and demanding that haloes pass them in bothsimulations does not affect the tilt of the distributions, butremoves low concentration haloes as expected. APPENDIX D: EINASTO CONCENTRATIONS
In Figs. D1 and D2 we plot the measured c ( M ) -relationsand relaxation cut impacts for an Einasto parametrisationof the density field, and in Fig. D3 the corresponding ∆ c biases. To compute the rescaling mappings we rephrase thedensity profile in Eq. (9) in terms of the average density h ρ Einasto i ( r ) for the enclosed mass M ( < r ) : h ρ Einasto i ( r ) = M ( < r )4 π/ r = ∆ y γ (3 /α ; 2 /α ( yc ∆ ) α ) γ (3 /α ; 2 /αc α ∆ ) ρ crit ( z ) , (D1)where y = r/r ∆ and γ ( a ; b ) is the lower incompletegamma function, readily replace the NFW density profilein Eq. (23) and calculate the correction accordingly. Evalu-ating Eq. (D1) at the scale radius, the concentration w.r.t.the mean density c ∆ m is then the solution to c m γ (3 /α ; 2 /αc α ∆ m ) = c c γ (3 /α ; 2 /αc α ∆ c ) E ( z ) Ω m (1 + z ) . (D2) MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo -0.60-0.40-0.200.000.20 (0 . , . -0.60-0.40-0.200.000.20 (0 . , . -0.20-0.100.000.10 (0 . , . − r/r -0.40-0.200.000.20 (0 . , . − r/r -0.40-0.200.000.200.40 (0 . , . − ρ r / ρ d . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) . − . h − M (cid:12) Figure C1.
Matched halo density field residuals from 64 log -equidistant radial bins with error regions from bootstrapped stacks ofmatched haloes in each simulation. These error regions trace the median results well. M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Matched haloes: before correction M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Matched haloes: after correction
Figure C2.
Difference in concentration estimated from density profiles (as in the left panel of Fig. 10) for matched haloes before andafter applying our correction to rescaled haloes. MNRAS000
Difference in concentration estimated from density profiles (as in the left panel of Fig. 10) for matched haloes before andafter applying our correction to rescaled haloes. MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations c rescaled c d i r ec t (0 . , . InitialCorrected 2 4 6 8 10 12 14 16 c rescaled c d i r ec t (0 . , . InitialCorrected2 4 6 8 10 12 14 16 c rescaled c d i r ec t (0 . , . InitialCorrected 2 4 6 8 10 12 14 16 c rescaled c d i r ec t (0 . , . InitialCorrected
Figure C3.
Concentration difference for matched haloes quantified with 3D NFW profiles, pre- and post-correction. Note that the lowermass threshold in the direct simulation for (0 . , . is M m > . h − M (cid:12) instead of > . h − M (cid:12) for the others to mitigateresolution effects. The masses are rescaled in the same manner as in Section 2.3and the resulting c ( M ) -relations with α = 0 . and ∆ = 200 differ negligibly from the NFW curves.With relaxation cuts enforced, the measured Einasto c ( M ) -relations are close to the L16 model predictions, asFig. D2 shows. Overall the the model predictions bettermatch measured relations for Einasto profiles (see Fig. D1)than for NFW profiles (see Fig. 8). While the concentrationbiases are similar to those measured for the NFW relationsin Fig. 10, we have a slightly larger bias for (0 . , . and (0 . , . , and for the low mass bins for (0 . , . in Fig. D3. Since the masses are fixed, the small horizontalscatter stems from the different median M m masses of thebootstrap samples. These values do not deviate significantly from one another until the sparsely populated high mass endfor some cosmologies.Fig. D4 shows the Einasto estimated concentrationdistribution for individual haloes pre- and post-correctionfor (0 . , . . Compared to the NFW distributions, theEinasto fits favour higher concentrations for low mass haloeswhich is seen for the median c ( M ) − relations in Fig. D1 andalso in the shift of the distributions between Fig. D4 andthe (0 . , . panel in Fig. C3. In addition, the slightlylarger mismatch between the L16 model prediction and themeasured median concentration relations for (0 . , . isvisible as an offset between the diagonal and the centre ofthe densest contour in Fig. D4 (cf. Figs. C3, C2 and 10). Alarger spread of concentrations is also possible, which can be MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo . , . Model (direct)Model (rescaled) Direct sim.Rescaled sim. (0 . , . . , .
81) 10 (0 . , . M [ h − M (cid:12) ] c Figure D1.
Concentration-mass relations for Einasto fits with α = 0 . for direct and rescaled simulations w.r.t. the L16 modelpredictions. M [ h − M (cid:12) ]468101214 c (0 . , . Model (direct)Model (rescaled)All haloes (direct)All haloes (rescaled) d off cut (direct) d off cut (rescaled) d off + f sub cut (direct) d off + f sub cut (rescaled) Figure D2.
Einasto c ( M ) -relations for (0 . , . for all haloesand with different relaxation cuts enforced. The correspondingNFW relations are similar though the Einasto measurements cor-respond better to the theory values at the low mass end. noted by comparing the contours for (0 . , . in Fig. 14(NFW) to those in Fig. D5 (Einasto). Still, the tilt is pre-served by the two parameterisations for all cosmologies. Theresults in general are qualitatively quite similar. APPENDIX E: SPLASHBACK MASSCORRECTION
The outer profile correction can be used to build a na¨ıvemass correction, if we redefine the M m masses in therescaled simulation as masses within the perturbed r m ,which is set such that r (d)sp /r (d) m = r (r)sp /r m . Assumingthat the density field just beyond r m is dominated by the1-halo term which is well captured by an NFW profile, one M [ h − M (cid:12) ] − − ∆ c ( d i r ec t − r e s c a l e d ) Model(0 . , . . , .
60) (0 . , . . , . . , . Figure D3.
The measured differences for Einasto concentrationswith α = 0 . and r s and ρ s free. c rescaled c d i r ec t (0 . , . InitialCorrected
Figure D4.
Einasto estimated concentrations for matchedhaloes in the direct and rescaled simulation with M m > . h − M (cid:12) for haloes in the direct simulation. could extend the integration to r m = (1 + δ ) r m where δ = 1 / (1 + ∆ r sp ) . This simplifies to the following expres-sion for the mass correction: M eff. m M m = 1 − / (1 + c/ (1 + ∆ r sp )) − ln (1 + c/ (1 + ∆ r sp ))1 − / (1 + c ) − ln (1 + c ) , (E1)where c = c ( M ) , and ∆ r sp could be predicted with the L16and Diemer et al. (2017) model fits, respectively. The weakmass evolution of this correction factor for the different cos-mologies is plotted in Fig. E1 for the uncorrected and cor-rected rescaled density field. The concentration correctionaffects the relation marginally. Due to the mismatch betweenthe detected outer profile bias for (0 . , . , (0 . , . MNRAS000
Einasto estimated concentrations for matchedhaloes in the direct and rescaled simulation with M m > . h − M (cid:12) for haloes in the direct simulation. could extend the integration to r m = (1 + δ ) r m where δ = 1 / (1 + ∆ r sp ) . This simplifies to the following expres-sion for the mass correction: M eff. m M m = 1 − / (1 + c/ (1 + ∆ r sp )) − ln (1 + c/ (1 + ∆ r sp ))1 − / (1 + c ) − ln (1 + c ) , (E1)where c = c ( M ) , and ∆ r sp could be predicted with the L16and Diemer et al. (2017) model fits, respectively. The weakmass evolution of this correction factor for the different cos-mologies is plotted in Fig. E1 for the uncorrected and cor-rected rescaled density field. The concentration correctionaffects the relation marginally. Due to the mismatch betweenthe detected outer profile bias for (0 . , . , (0 . , . MNRAS000 , 1–26 (2018) alo profiles in rescaled simulations c rescaled c d i r ec t (0 . , . InitialCorrected
Figure D5.
Einasto estimated concentrations for matchedhaloes in the direct and rescaled simulation with M m > . h − M (cid:12) for haloes in the direct simulation. M [ h − M (cid:12) ]0.800.901.001.101.20 M e ff m / M m (0 . , . . , . . , . . , . . , . c ( M ) direct c ( M ) rescaled Figure E1.
Effective mass correction with the NFW density fieldcorrection before and after the concentrations are corrected. and (0 . , . and the model prediction in Fig. 17, as wellas the mismatch between the L16 model and the measured c ( M ) -relations, the correction is too large. This is reflectedin the cumulative halo mass function in Fig. E2 for matchedhaloes pre- and post-mass correction, where the agreementis worse. For (0 . , . and (0 . , . , however, the biaschanges signs at the low mass end, and for (0 . , . , thesituation improves somewhat at the low mass end.We can interpret these results in light of the discrepan-cies in mass between the matched direct and rescaled haloesin Fig. 3, where the median relations for these two simula-tions are off (see appendix B for (0 . , . ) and the masscorrection shifts these median levels in the right direction.Still, there is a mass evolution of the discrepancy between − − − N ( > M m ) (cid:20) h M p c − (cid:21) (0 . , . . , . . , .
81) (0 . , . . , . M [ h − M (cid:12) ]-0.500.000.50 − N r / N d Direct sim.Rescaled simRescaled sim, corrected
Figure E2.
Halo mass function before and after the mass cor-rection. m σ -0.50-0.40-0.30-0.20-0.100.000.100.200.300.40 ∆ r s p / r m Figure F1.
Predicted offset in splashback radius for matchedhaloes in a direct and rescaled fiducial simulation with WMAP1parameters from the Diemer et al. (2017) model (75th percentile). the direct and rescaled haloes which must be modelled bya more elaborate correction. For the other simulations, thistilt dominates over the wrong offset level, and for (0 . , . there is a very small predicted shift. APPENDIX F: COSMOLOGICAL CONTOURPLOTS FOR THE RESCALING PARAMETERS
In Fig. F1 the predicted offsets computed with the Diemeret al. (2017) model in the position of the splashback radiusw.r.t. r m for matched halo samples in different target cos-mologies is shown. In large sections of the parameter plane, ∆ r sp /r m has the opposite sign as ∆ c although this is notnecessarily true for small changes from the fiducial run norfor the ∆Ω m > , ∆ σ > quadrant. We plot the ( α, z ∗ ) pairs to emulate these different cosmologies in Figs. F2 andF3. They are smooth functions depending on ∆Ω m and ∆ σ to the fiducial cosmology. Shrinking the simulation box ispreferable to emulate a cosmology with a higher matter frac-tion, and expanding the box for lower matter fractions. Sim- MNRAS , 1–26 (2018) Renneby, Hilbert, & Angulo m σ α Figure F2.
The length scale parameter α as a function of ∆Ω m and ∆ σ w.r.t. a fiducial simulation with WMAP1 parameters. m σ -0.800.000.801.602.403.204.004.805.60 z ∗ Figure F3.
The time scale parameter z ∗ as a function of ∆Ω m and ∆ σ . ilarly intuitively, going to a higher redshift in the fiducialsimulation could be used to match a cosmology with a lower σ , i.e. with a lower amplitude of the fluctuations of the mat-ter field. This puts the Millennium simulation in a suitableposition for rescaling as the WMAP1 σ = 0 . is compa-rably high to the current best fit matter power spectrumamplitudes. APPENDIX G: BIASES FOR A RESCALEDMILLENNIUM SIMULATION TO WMAP ANDPLANCK COSMOLOGIES
In Fig. G1 and Fig. G2, we illustrate the predicted con-centration biases for the Millennium simulation (Springelet al. 2005) with its WMAP1 parameters (Spergel et al.2003) rescaled to a range of cosmologies (WMAP3, WMAP5,WMAP7, WMAP9, Planck 2014) (Spergel et al. 2007; Ko-matsu et al. 2009, 2011; Hinshaw et al. 2013; Planck Col-laboration 2014) at z = 0 according to the parameters inHenriques et al. (2015) and Guo et al. (2013). For the cos-mologies where there is a mass evolution of the concentration M [ h − M (cid:12) ]-1.50-1.00-0.500.000.501.001.502.002.50 ∆ c ( d i r ec t − r e s c a l e d ) WMAP3, Henriques+15WMAP5, Henriques+15WMAP7, Henriques+15 WMAP7, Guo+13WMAP9, Henriques+15Planck14, Henriques+15
Figure G1.
Predicted concentration biases for haloes rescaledusing the parameters in Henriques et al. (2015) and Guo et al.(2013). .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 .
75 2 . z -1.50-1.00-0.500.000.501.001.502.002.50 ∆ c ( M , z ) WMAP3, Henriques+15WMAP5, Henriques+15WMAP7, Henriques+15 WMAP7, Guo+13WMAP9, Henriques+15Planck14, Henriques+15
Figure G2.
Redshift evolution of the concentration biases inFig. G1. bias in Fig. G1, the slope decreases at higher redshift. Thepredicted concentration bias for haloes with a Millenniumsimulation rescaled to Planck 2014 with the parameters inHenriques et al. (2015) is very small and decreases for higherredshifts, which is fortuitous for future lensing analyses.
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