Impact of Dark Matter Substructure on the Matter and Weak Lensing Power Spectra

a r X i v :a s t r o -p h /0504557v 2 25 J u l 2005

D RAFT VERSION F EBRUARY 2,2008

Preprint typeset using L A T E X style emulateapj v.6/22/04

IMPACT OF DARK MATTER SUBSTRUCTURE ON THE MATTER AND WEAK LENSING POWER SPECTRA

B RADLEY H AGAN 1,

C HUNG -P EI M A 2,A NDREY V.K RAVTSOV 3

Draft version February 2,2008

ABSTRACT

We explore the effect of substructure in dark matter halos on the power spectrum and bispectrum of matter ?uctuations and weak lensing shear.By experimenting with substructure in a cosmological N =5123simu-lation,we ?nd that when a larger fraction of the host halo mass is in subhalos,the resulting power spectrum has less power at 1 k 100h Mpc ?1and more power at k 100h Mpc ?1.We explain this effect using an analytic halo model including subhalos,which shows that the 1 k 100h Mpc ?1regime depends sensitively on the radial distribution of subhalo centers while the interior structure of subhalos is important at k 100h Mpc ?1.The corresponding effect due to substructures on the weak lensing power spectrum is up to ～11%at angular scale l 104.Predicting the nonlinear power spectrum to a few percent accuracy for future surveys would therefore require large cosmological simulations that also have exquisite numerical resolution to model accurately the survivals of dark matter subhalos in the tidal ?elds of their hosts.

Subject headings:cosmology:theory —dark matter —large-scale structure of the universe

1.INTRODUCTION

One phenomenon to emerge from N -body simulations of increasingly higher resolution is the existence of substructure (or subhalos)in dark matter halos (e.g.,Tormen et al.1998;Klypin et al.1999b;Moore et al.1999;Ghigna et al.2000).These small subhalos,relics of hierarchical structure forma-tion,have accreted onto larger host halos and survived tidal forces.Depending on their mass,density structure,orbit,and accretion time,the subhalos with high central densities can avoid complete tidal destruction although many lose a large fraction of their initial mass.These small and dense dark matter substructures,however,are prone to numerical arti-facts and can be disrupted due to insuf?cient force and mass resolution.Disentangling these numerical effects from the actual subhalo dynamics is an essential step towards under-standing the composition and formation of structure.Quan-tifying the effects due to dark matter substructure is also im-portant for interpreting weak lensing surveys,which are sen-sitive to the clustering statistics of the overall density ?eld.The level of precision for which surveys such as SNAP are striving (Massey et al.2004)suggests that theoretical predic-tions for the weak lensing convergence power spectrum need to be accurate to within a few percent over a wide of range of scales (e.g.Huterer &Takada 2005).At this level,subhalos may contribute signi?cantly to the nonlinear power spectrum because they typically constitute about 10%of the host mass.In the sections to follow,we examine the effects of sub-structure on the matter and weak lensing power spectra with two methods.In §2we use the result of a high resolution N -body simulation and quantify the changes in the power spec-tra when we smooth out increasing amounts of substructures.Our other approach,detailed in §3,is to incorporate substruc-ture into the analytic halo model.The results are dependent on the parameters used in the model,but they provide useful physical insight into the results from N -body simulations.We summarize and discuss the results in §4.

1

Department of Physics,University of California,Berkeley,CA 947202Department of Astronomy,University of California,Berkeley,CA 947203Dept.of Astronomy and Astrophysics,Enrico Fermi Institute,Kavli Institute for Cosmological Physics,The University of Chicago,Chicago,IL 60637

2.SUBSTRUCTURE IN SIMULATIONS

We use the outputs of a cosmological dark-matter-only simulation that contains a signi?cant amount of substruc-ture.This simulation is a concordance,?at ΛCDM model:?m =1??Λ=0.3,h =0.7and σ8=0.9.The box size is 120h ?1Mpc,the number of particles is 5123,and the particle mass is 1.07×109h ?1M ⊙.The simulation uses the Adaptive Re?nement Tree N -body code (ART;Kravtsov et al.1997;Kravtsov 1999)to achieve high force resolution in dense re-gions.In this particular run the volume is initially resolved with a 10243grid,and the smallest grid cell found at the end of the simulation is 1.8h ?1kpc.The actual resolution is about twice this value (Kravtsov et al.1997).More details about the simulation can be found in Tasitsiomi et al.(2004).

To quantify the effects of subhalos on the matter and weak lensing power spectra,we ?rst identify the simulation parti-cles that comprise subhalos within each halo.This is achieved using a version of the Bound Density Maxima algorithm (Klypin et al.1999a),which identi?es all local density peaks and therefore ?nds both halos and subhalos.It identi?es the particles that make up each of the peaks and removes those not bound to the corresponding halo.As a controlled exper-iment,we then smooth out the subhalos within the virial ra-dius of each host halo by redistributing these subhalo parti-cles back in the smooth component of the host halo according to a spherically-symmetric NFW pro?le (Navarro,Frenk,&White 1996).For the concentration parameter c of the pro?le,we do not use the ?tting formulae (e.g.,Bullock et al.2001;Dolag et al.2004)but instead ?t each host halo individually to take into account the signi?cant halo-to-halo scatter in c .We therefore smooth over the subhalos and increase the normal-ization but not the shape of the spherically averaged pro?le of the smooth component to accommodate the mass from the subhalo component.

This smoothing procedure also serves as a simple model for the effects of resolution on the abundance of subhalos in sim-ulations,in which the lack of suf?cient resolution will cause an incoming small halo to be disrupted quickly and lose most of its particles over its short-lived orbit.We quantify this ef-fect by experimenting with different cut-offs on the subhalo mass:subhalos with masses below the cut-off are removed

2

F I

G . 1.—Effects of dark matter substructure on the matter ?uctuation

power spectrum (top)and the equilateral bispectrum (bottom)of a N =5123cosmological simulation.Plotted is the ratio of the original spectrum to that with a subset of the substructures smoothed out.The curves (from top down)correspond to increasing subhalo mass cut-offs,below which the mass in the subhalos is redistributed smoothly back into the host halo.The bottom panel is plotted to a lower k because B (k )becomes too noisy.The effects due to substructures are up to ～12%in ?2and 24%in B .

and have their particles spread over the host halo;subhalos with masses above the cut-off are left alone.Increasing mass cut-offs should roughly mimic increasingly lower resolution simulations because only higher mass subhalos will stay in-tact in the halo environment.

We then calculate the matter ?uctuation power spectrum P (k ),the Fourier transform of the 2-point correlation function,and the matter bispectrum B (k 1,k 2,k 3),the Fourier transform of the 3-point correlation function.In order to compute P and B at large k without using an enormous amount of mem-ory,we subdivide the simulation cube into smaller cubes and stack these on top of each other (called "chaining the power"in Smith et al.2003).A typical stacking level used is 8,mean-ing that we subdivide the box into 83cubes and stack these.We use stacked spectra for the high-k regime and unstacked spectra for low-k .Finally,we subtract shot noise (∝1/N )from the outputted spectra to eliminate discreteness effects.Fig.1shows the effects of substructure on the dimension-less power spectrum ?2(k )≡k 3P (k )/(2π2)and the equilat-eral bispectrum B (k 1)(k 1=k 2=k 3).Plotted is the ratio of the spectrum from the raw ART output divided by the spec-trum from the altered data.The altered data have no subha-los with masses below the labeled mass cutoff.The mass in the removed subhalos has been redistributed smoothly into the

host halo as described above.The deviation from the original spectrum becomes larger as the cutoff is increased because more subhalos have been smoothed out.For a given cutoff,the ?gure shows that a simulation with dark matter substruc-tures (such as the raw ART output)has more power at k >100h Mpc ?1and less power at 1 k 100h Mpc ?1than a simu-lation with smoother halos.We believe these opposite behav-iors re?ect the two competing factors present in our numeri-cal experiments:removal of mass within subhalos,which af-fects scales comparable to or below subhalo radii (and hence k 100h Mpc ?1),and addition of this mass back into the smooth component of the halo,which affects the larger scales of 1 k 100h Mpc ?1.The ratio approaches unity for k 1h Mpc ?1simply because the mass distribution on scales above individual host halos is unaltered.We will examine these ef-fects further in the context of the halo model in §3.

For a given curve in Fig.1,we have also calculated the contributions from subhalos in host halos of varying masses to quantify the relative importance of cluster versus galactic host halos.For the 1012.5h ?1M ⊙curve,e.g.,we ?nd that smooth-ing over the subhalos in host halos above 1014and 1013M ⊙account for 5%and 10%in the total 12%dip seen in Fig.1,respectively.For the 1011.5h ?1M ⊙cutoff,the numbers are 2%and 5%of the total 6%dip.

The halos found in N -body simulations are generally triax-ial.When we redistribute the subhalo particles,however,we assume for simplicity a spherical distribution.This assump-tion makes the altered halos slightly rounder.One can esti-mate how this effect changes the power spectrum by using the halo model without substructure.Smith &Watts (2005)in-corporated a distribution of halo shapes found by Jing &Suto (2002)from cosmological simulations into the halo model (ig-noring the substructure contribution).Compared with the case where all the halos are spherical,they observed a peak decre-ment in the power spectrum of about 4%for k ≈1Mpc ?1.The corresponding effect in our calculations would be much smaller since we redistribute only the subset of particles that belong to subhalos into the rounder shape (e.g.,about 10%of all particles in the case where the cutoff was 1012.5h ?1M ⊙).Thus,by extending the results of Smith &Watts,we expect the spurious rounder halos in our study to account for less than 0.5%of the total 12%drop.

Fig.2shows the weak lensing convergence power spectrum ?2

κ(l )corresponding to the matter power spectrum ?2(k )in Fig.1.It is calculated from ?2(k )using Limber’s approxima-tion and assumption of a ?at universe:

?2

κ(?)=

9πc 4 χmax

χ3d χW 2(χ)

3

F IG.2.—Effects of substructure on the weak lensing convergence power

spectrum for the same subhalo mass cutoffs as in Fig.1.The sources are

assumed to be at redshift1.The gray band is the1-σstatistical error assuming

Gaussian?elds.We take f sky=0.25,γrms=0.2,ˉn=100/arcmin2,and a band

of width?/10.

surements assuming Gaussian density?elds(Kaiser1998):

σ(?2κ)2

2πˉn?2κ ,(2)

where f sky is the fraction of the sky surveyed,γrms is the rms

ellipticity of galaxies,andˉn is the number density of galaxies

on the sky.The error is dominated by the sample variance

on large scales(?rst term in eq.[2])and the"shape noise"on

small scales.Our assumption of Gaussianity is not applicable

for the angular scales shown in the plot because the scales

plotted are near or below the size of individual halos,but the

errors shown should be a useful reference and have been used

in previous studies.A reliable estimate of the error would

presumably require a ray tracing calculation which is beyond

the scope of this paper.

3.SUBSTRUCTURE IN THE HALO MODEL

To gain a deeper understanding of the simulation results in

Figs.1and2,we use the semi-analytic halo model to build

up the nonlinear power spectrum from different kinds of pairs

of mass elements that may occur in halos(e.g.Ma&Fry

2000;Peacock&Smith2000;Seljak2000;Scoccimarro et

al2001).The original halo model assumes that all mass re-

sides in virialized,spherical halos without substructures.One

can then build the matter power spectrum from the different

kinds of pairs of particles that contribute to the2-point clus-

tering statistics by writing P(k)=P1h(k)+P2h(k),where the1-

halo term P1h contains contributions from particle pairs where

both particles reside in the same halo,and the2-halo term P2h

is from pairs where the two particles reside in different halos.

The1-halo term is a mass-weighted average of single halo

pro?les and dominates on the scales of interest(k 1h Mpc?1)

in Fig.1because close pairs of particles are more likely to be

found in the same halo.The2-halo term is closely related to

the linear power spectrum and is important only at large sepa-

ration(i.e.small k)where a pair of particles is more likely to

be found in two distinct halos.Similarly,the bispectrum can

be constructed from the different classes of triplets of particles

(see,e.g.,Ma&Fry2000).

The original halo model can be readily extended to take into

account a clumpy subhalo component in an otherwise smooth

host halo.Sheth&Jain(2003),e.g.,decompose the original

1-halo term into P1h=P ss+P sc+P1c+P2c,where"s"denotes

smooth and"c"denotes clump.The smooth-smooth term,P ss,

arises from pairs of particles that both belong to the smooth

component of the same host halo.This term is identical to the

original1-halo term except for an overall decrease in ampli-

tude by the factor(1?f)2,where f is the fraction of the total

halo mass that resides in subhalos.The smooth-clump term,

P sc,is due to having one particle in a subhalo(clump)and the

other in the host halo(smooth).The1-and2-clump terms,P1c

and P2c,come from having both particles in the same subhalo

and in two different subhalos,respectively.Explicitly,

P ss(k)=

(1?f)2

ˉρ2 dM N(M)MU(k,M)U c(k,M)

× dm n(m,M)mu(k,m)(4)

P1c(k)=

1

ˉρ2 dMN(M)U2c(k,M)

× dmn(m,M)mu(k,m) 2(6)

where U(k,M),u(k,m),and U c(k,M)are the Fourier trans-

forms of the host halo radial density pro?le,the subhalo radial

density pro?le,and the radial distribution of subhalo centers,

respectively.N(M)dM gives the number density of host ha-

los with mass M,and n(m,M)dm gives the number density of

subhalos of mass m inside a host halo of mass M.A similar

expression can be written down for the2-halo term P2h,which

is also included in our calculations.

Fig.3illustrates the contributions from the individual terms

in the halo model.We use the NFW pro?le(truncated at

the virial radius)for the input host halo U(k,M)and subhalo

u(k,m),and the concentration c(M)=c0(M/1014M⊙)?0.1with

c0=11that we?nd to approximate the ART host halos and

is identical to Dolag et al(2004)except for a15%increase

in amplitude.We use c sub

=3for the subhalos to take into

account tidal stripping but also compare different values in

Fig.4below.For the distribution of subhalo centers,U c(k),

we compare the pro?le of NFW with that of Gao et al.(2004),

who?nd the number of subhalos within a host halo’s virial ra-

dius r v to be

N(

1+acxα

,x=r/r v(7)

where a=0.244,α=2,β=2.75,c=r v/r s,and N tot is the to-

tal number of subhalos in the host.Since this distribution at

small r is shallower(∝r?0.25)than the inner part of the NFW

pro?le(∝r?1),its Fourier transform U c(k)at high k is about

a factor of10lower than that of the NFW pro?le.This decre-

ment results in a much lower?sc and?2c as shown in Fig.3

(dashed vs dotted curves).We?nd the subhalo centers in the

ART simulation to follow approximately the distribution of

Gao et al.although there is a large scatter.

4

F I

G . 3.—Comparison of individual subhalo terms in the halo model.

The smooth-clump ?2sc and 2-clump ?22c terms depend on the distribution

of subhalo centers U c within a host halo,having a much lower amplitude at k >1h Mpc ?1when U c has the cored isothermal pro?le of Gao et al.(2004)compared with the cuspy NFW pro?le.The smooth-smooth ?2ss and 1-clump ?21c terms are independent of U c .See text for parameters used in the model.

We use the mass function of Sheth &Tormen (1999)for the host halos N (M )and a power law n (m ,M )∝m ?1.9that well approximates the subhalo mass function in the ART sim-ulation.The latter is normalized so that the total mass of subhalos in a host halo adds up to f times the host mass M (f =0.14in Fig.3).To compare the halo model with simula-tions,we set the lower limit on the P ss integral to 1010h ?1M ⊙,which is the smallest halo present in the simulation (about ten times the simulation particle mass).The lower limit on the outer integrals of P sc ,P 1c ,and P 2c corresponds to the smallest halo that contains substructure,which we set to the small-est halo that we considered for erasing substructure in §2:2×1012h ?1M ⊙.Similarly,the lower limit on the inner inte-grals of these terms is set to the smallest subhalo that can be resolved (i.e.1010h ?1M ⊙).

Fig.4compares the sum of all the terms in the

halo model with the simulation result from Fig.1.As in Fig.1,we il-lustrate the effects due to substructures by dividing out the power spectrum from the original (smooth)halo model,i.e.,

?2smooth =k 3P ss /[(1?f )2(2π2

)],where P ss is given before in eq.(4).Fig.4shows that the halo model is able to reproduce qualitatively the simulation results when the subhalo centers in the halo model are assigned the shallower distribution of

Gao et al.The feature of ?2sub /?2

smooth <1at 1 k 100h Mpc ?1is mainly caused by the drop in the smooth-clump term relative to the smooth-smooth term at k 1h Mpc ?1shown in

Fig.3.The ratio ?2sub /?2

smooth becomes >1only at k 100h Mpc ?1when the 1-clump term ?nally takes over.Fitting the halo model to actual simulation results is clearly not exact in part due to the large scatters in the properties of simulated ha-los,e.g.,the concentration (for both hosts and subhalos),the subhalo mass fraction f ,and the maximum subhalo mass in each host halo.The halo model allows us to study the depen-dence of clustering statistics on these parameters (see Fig.4).In addition,a number of effects are neglected in the current halo model,e.g.,tidal effects are likely to reduce the number of subhalos (modeled by U c of Gao et al.here)as well as their

F I

G . 4.—Same ratio of ?2as in Fig.1but comparing simulation

(symbols;same 1012.5h ?1M ⊙curve in Fig.1)with halo model predictions (plain curves).The two agree qualitatively when the shallower distribu-tion of Gao et al.for subhalo centers U c (k )is used (bottom 3curves)in the halo model (but not for an NFW U c (k );dotted).The detailed model prediction depends on halo parameters:the solid curve uses the same pa-rameters as in Fig.3;the dashed shows how a larger subhalo concentration

(c sub =c sub 0(M /1014M ⊙)

?0.1;c sub 0=11vs 3)steepens the curve at high k ;the dash-dotted shows how a smaller subhalo mass fraction f (0.1vs.0.14)raises the dip.

outer radii (not modeled here)towards host halo centers;a larger amount of stripped subhalo mass may also be deposited to the inner parts of the hosts,resulting in a radius-dependent subhalo mass fraction f within the host.

Fig.4also shows that the halo model predicts the oppo-site effect due to substructure (i.e.?2sub /?2

smooth >1at all k )if the subhalo centers U c (k )are assumed to follow the NFW distribution like the underlying dark matter.The sign of this effect is consistent with the previous subhalo model study of Dolney et al.(2004),which assumed the same NFW pro?le for the subhalo centers and the hosts and obtained a matter power spectrum that had a higher amplitude for all k when the substructure terms were included.Their results differ slightly from ours because of different integration limits.Subhalos in recent simulations like that of Gao et al.,however,show a much shallower radial distribution in the central regions of the host halos,and inclusion of gas dynamics appears to have little effect on the survivability of subhalos (Nagai &Kravtsov 2005).The shallow distribution is apparently due to tidal disruptions,even though the precise shape of the distri-bution is still a matter of debate (e.g.,Zentner et al.2005).We have also experimented with a third distribution U c (k )that has the NFW form but is less concentrated.We are able to

bring ?2sub /?2

smooth below unity only when the concentration is reduced by a factor of more than 2.5,and only when this reduction factor is increased to ～100would we get a com-parable dip as the curves for ART simulation and Gao et al.in Fig.4.It is interesting to see if we can mimic the behav-ior of the ART simulation without using subhalos in the halo model.We try replacing the one-halo term,P 1h ,by one that is a simple superposition of a Gao et al.pro?le and the usual NFW pro?le.This accounts for the fact that ～90%of the mass is in a smooth NFW pro?le and that ～10%is in sub-halos,which follow a ?atter pro?le.One would not expect the high-k regime to agree as it is dominated by the subha-

5

los(the1-clump term,speci?cally).The intermediate range, 1 k 100h Mpc?1,is dominated by the host halo itself,but we?nd no similarity in this range either.Subhalos are there-fore needed if the halo model is to recreate the ART results.

4.DISCUSSION

The purpose of this work is to provide a physical under-standing of the effects of substructures on clustering statistics. By experimenting with dark matter substructures in a cosmo-logical simulation with5123particles,we have shown that the power spectra of matter?uctuations and weak lensing shear can change by up to～12%(and up to～24%in the bispec-trum)if a signi?cant amount of substructures is not resolved in a simulation.When a larger mass fraction of the host halos is in the form of lumpy subhalos,we?nd the effect is to lower the amplitude of the matter and weak lensing power spectra at the observationally relevant ranges of k～1to100h Mpc?1 and l 105,and to raise the amplitude on smaller scales.A similar drop in power is also seen in our analytic halo-subhalo model when the subhalo centers U c within a host halo are dis-tributed with a shallower radial pro?le than the underlying dark matter(as expected due to tidal effects).A way to un-derstand the drop involves looking at where the dense regions are.When U c has an NFW form the subhalos basically trace the smooth background.Thus,there is never a decrease in power when the smooth-smooth and smooth-clump terms are added because dense regions are in nearly the same relative positions.When we use a shallower pro?le for U c,the sub-halos are not as numerous in the denser inner regions of the background halo.This decrease in the overlap between dense clump regions and the dense inner regions causes the drop in power.

We have quanti?ed the effects of substructures on clustering statistics by erasing substructures in an N=5123simulation. An important related question is whether N=5123,single-mass resolution simulations such as the one used in our study has suf?cient resolution to measure the power spectrum to the few-percent accuracy required by future surveys.Note that at least hundreds of particles and force resolution of～kilo-parsec are required to ensure subhalo survival against tidal forces,placing stringent requirements on the dynamic range of simulations.Multi-mass resolution simulations designed for subhalo studies,on the other hand,do not give reliable predictions for P(k)on quasi-linear scales due to compro-mised resolution outside highly clustered regions.The fact that the curves in Figs.1and2continue to change at the few-percent level each time the mass threshold is lowered by 0.5dex from1012.5to1010.5h?1M⊙suggests that subhalos of M 1010.5h?1M⊙may still be affecting the power spectra at a comparable level and that N>5123would be required.We also?nd>3%changes in P(k)at k～10h Mpc?1in the halo model as the minimum subhalo mass in the integration limit of eq.(3)is lowered to107h?1M⊙(although the exact predic-tions are sensitive to the slope of the subhalo mass function, which is assumed to be?1.9here.)Careful convergence stud-ies with higher resolution aided by insight from this study and detailed semi-analytic models for halo substructure will likely be needed to determine N.

There are other challenges to predicting accurately the weak lensing signal on single halo scales.The effect of neutrino clustering could cause a rise in weak lensing convergence of ～1%at?～2000(Abazajian et al.2005).Two recent groups have investigated different aspects of baryon effects.White (2004)found that baryonic contraction and its subsequent im-pact on the dark matter distribution is capable of causing an increase in the weak lensing convergence power of a few per-cent at? 3000.Zhan&Knox(2004),on the other hand,use the fact that the hot intracluster medium does not follow the dark matter precisely and predict an opposite effect:a sup-pression of weak lensing power of a few percent at? 1000. Unlike the effects of substructure and neutrino clustering,the baryon effects cause departures from the pure dark matter weak lensing signal that only get larger with increasing?. We thank M.Boylan-Kolchin,W.Hu,D.Huterer,C.Vale, and P.Schneider for useful discussions.BH is supported by an NSF Graduate Student Fellowship.CPM is supported in part by NSF grant AST0407351and NASA grant NAG5-12173.A VK is supported by NSF grants AST-0206216and 0239759,NASA grant NAG5-13274,and the Kavli Institute for Cosmological Physics at the University of Chicago.

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