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arXiv:astro-ph/0301341v1 16 Jan 2003

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 February 2008(MN LATEX style file v2.2)

Halo Concentration and the Dark Matter Power Spectrum

Kevin M. Huffenberger & Uroˇ s Seljak

Department of Physics, Jadwin Hall

Princeton University, Princeton, NJ 08544

January 2003

ABSTRACT

We explore the connection between halo concentration and the dark matter power

spectrum using the halo model. We fit halo model parameters to non-linear power

spectra over a large range of cosmological models. We find that the non-linear evolu-

tion of the power spectrum generically prefers the concentration at non-linear mass

scale to decrease with the effective slope of the linear power spectrum, in agreement

with the direct analysis of the halo structure in different cosmological models. Using

these analyses, we compute the predictions for non-linear power spectrum beyond the

current resolution of N-body simulations. We find that the halo model predictions

are generically below the analytical non-linear models, suggesting that the latter may

overestimate the amount of power on small scales.

1 INTRODUCTION

In recent years, computational N-body simulations have

provided important insights into the non-linear formation of

structure by the dark matter. Navarro et al. (1997) (NFW)

observed a ‘universal’ density profile for dark matter haloes,

suggesting that the profiles depend only on the mass of the

halo, with relatively little scatter between them. They also

noticed that the halo concentration parameter, describing

the degree of profile steepness (defined more precisely be-

low) decreases with increasing halo mass, though the exact

relation depends on the cosmology at hand. Several formulae

have been suggested for this relation (Navarro et al. 1997;

Bullock et al. 2001; Eke et al. 2001) which rely on fitting

some mechanism to haloes culled from N-body simulations.

These studies found that the concentration dependence on

the power spectrum is caused primarily by the slope of the

linear power spectrum. Both this and the mass dependence

of concentration can be explained by the epoch of the halo

formation, since less massive haloes form at higher redshift

when the universe was denser and this leads to a more con-

centrated profile.

Another relation which appears to be universal is that of

the mass function (Jenkins et al. 2001). The distribution of

halo masses in N-body simulations appears to be universal

in the sense that when mass is related to the fluctuation

amplitude of the linear power spectrum, the mass function

has a universal form. This relation has also been explored in

detail in recent work (Sheth & Tormen 1999; Jenkins et al.

2001; White 2001, 2002), defining precisely what is meant

by the mass and confirming the universality of the mass

function for a large set of cosmological models.

While the above approaches have focused on individ-

ual haloes in the simulations by counting them and explor-

ing their structure, for many cosmological applications all

that we need is the dark matter power spectrum. This can

be computed from N-body simulations as well, but com-

putational limitations prevent one from exploring a large

range of cosmological models or from achieving a high reso-

lution on small scales. For this reason many fitting formulae

have been developed with increasing accuracy over the years

(Hamilton et al. 1991; Jain et al. 1995; Peacock & Dodds

1996; Smith et al. 2002), although these are still significantly

limited on small scales by the dynamical range of simula-

tions.

The close connection between the two descriptions

has been highlighted recently with the revival of the halo

model (Seljak 2000; Peacock & Smith 2000; Ma & Fry 2000;

Scoccimarro et al. 2001). For a few cosmological models, it

was shown that with an appropriate choice of concentra-

tion mass dependence and mass function one can use the

halo model to accurately predict the non-linear dark mat-

ter power spectrum. One of the questions we explore in this

paper is whether this agreement can be extended to a wider

range of models and whether the trends seen in the analy-

sis of individual haloes are confirmed also with the power

spectrum analysis. There are several reasons why this is of

interest: since the selection of individual haloes is somewhat

subjective it is possible that those left out may be system-

atically different. For example, they could be less relaxed,

more recently formed and so less concentrated. An opposite

effect is that because the power spectrum is a pair-weighted

statistic, if there is a scatter in the mass-concentration re-

lation then pair weighting increases the mean concentration

relative to the simple particle weighting.

On the other hand, if the analysis of individual haloes

is in agreement with the power spectrum analysis over the

range where both are reliable, then the same analysis can be

extended to smaller scales which are not resolved by cosmo-

logical N-body simulations that compute the power spec-

trum. The reason for this is that the resolution scale for the

halo structure can be extended significantly by resimulating

the representative regions of the haloes with a higher resolu-

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2 K. Huffenberger & U. Seljak

tion simulation zoomed on that halo. On the other hand, a

power spectrum calculation requires a large simulation vol-

ume, so that the largest scales are in the linear regime, which

then implies that the mass and force resolution cannot be

as high as in the simulations which focus on single haloes.

This paper is structured in the following way. We sum-

marize the halo model and give relevant background infor-

mation in §2. In §3, we give our method and results. In §4, we

compare our results to other models and in §5 we summarize

our conclusions.

2 BACKGROUND

2.1 The Halo Model

We begin by reviewing the formalism of the halo model to

compute the non-linear dark matter power spectra, following

the notation in Seljak (2000). The haloes are characterized

by their mass M, density profile ρ(r,M), number density

n(M) and bias b(M). N-body simulations suggest a family

of density profiles:

ρs

(r/rs)−α(1 + r/rs)3+α,

ρ(r) =

(1)

where ρsis a characteristic density, rsis the radius where the

profile has an effective power law index of −2, and −1.5 <

α < −1 (Navarro et al. 1997; Moore et al. 1998). Here we

use α = −1, since power spectra are not sensitive to the

inner parts of the halo.

As is conventional, we re-parametrize ρsand rsin equa-

tion (1) in terms of a halo mass and concentration. We

define the virial radius rvir of a halo to be the radius of

a sphere with some characteristic mean density, discussed

below. Insisting that the mass contained inside rvir is M

fixes ρs for a given rs. We introduce the concentration pa-

rameter, the ratio c ≡ rvir/rs. There are several defini-

tions of virial radius used in the literature. In this paper,

the virial radius is defined as the radius of a halo-centreed

sphere which has a mean density 180 times the mean den-

sity of the universe. This may be denoted r180Ω. Other au-

thors set rvir to the radius inside which the mean density is

200 times the critical density (r200), a measure independent

of Ω (Navarro et al. 1997, for example). Still others use a

spherical collapse model to estimate the radius of a virial-

ized halo (r∆). These lead to different concentrations and

masses. For Ω = 0.3 and Λ = 0.7, r200 < r∆ < r180Ω, so for

the same halo with c180Ω = 10 one has c200 ∼ 6, c∆ ∼ 8,

M200/M180Ω ∼ 0.72 and M∆/M180Ω ∼ 0.87. Assuming the

NFW halo profile it is straightforward to translate values

for concentrations and masses between these conventions.

We choose c180Ω as it permits the use of the universal mass

function (White 2002). To avoid cumbersome notation, in

this paper we refer to c180Ω as c and M180Ω as M.

For the mass dependence of concentration we choose a

power law parametrization

c = c0(M/M∗)β, (2)

where c0and β are free parameters, and M∗is the non-linear

mass scale, which we will define shortly. This is of course not

the most general parametrization, but as we will show below

it suffices for the current dynamical range.

The halo number density is written in terms of the mul-

tiplicity function via

n(M)dM =

¯ ρ

Mf(ν)dν, (3)

where ¯ ρ is the mean matter density of the universe and mass

is written in terms of peak height

ν ≡

?

δc(z)

σ(M)

?2

, (4)

where δc(z) is the spherical over-density which collapses at

z (δc ≈ 1.68) and σ(M) is the rms fluctuation in the matter

density smoothed on a scale R = (3M/4π¯ ρ)1/3. ν(M∗) = 1

defines the non-linear mass scale. A cosmology with Ω0 =

0.3, Λ = 0.7, Γ = 0.2, and σ8 = 0.9 has M∗ ≈ 1013h−1M⊙.

The power spectrum has two terms. The first corre-

sponds to correlations in density between pairs of points

where each member of the pair lies in a different halo, and

so is named the “halo-halo” (hh) term. The second corre-

sponds to correlations between pairs in the same halo, and is

known as the one-halo or Poisson (P) term. For convenience

in calculating convolutions, we work in Fourier space, intro-

ducing the Fourier transform of the halo profile, normalized

by the virial mass,

y(M,k) =

1

M

?

4πr2ρ(r)sin(kr)

kr

dr.(5)

The mass of the NFW profile is logarithmically diver-

gent, so in order to evaluate this integral, we must impose

a cutoff. This does not have to be at the virial radius, since

we know that haloes are not completely truncated there. In-

stead, NFW profile typically continues to 2 − 3rvir. In this

regime there is already some overlap between the haloes, so

for the purpose of correlations one can count the same mass

element in more than one halo. Thus the mass function inte-

grated over all the haloes may even exceed the mean density

of the universe (but it can also be below it, since it is not

required that all the matter should be inside a halo).

The halo-halo contribution is

Phh(k) = Plin(k)

??

dνf(ν)dν b(ν)y(M(ν),k)

?2

,(6)

where b(ν) is the (linear) bias of a halo of mass M(ν), for

which we use the N-body fit of Sheth & Tormen (1999).

Since we want this term to reproduce the linear power spec-

trum on large scales we impose this constraint onto the form

for b(ν) (Seljak 2000). The Poisson contribution is

PP(k) =

1

(2π)3

?

dνf(ν)M(ν)

¯ ρ

|y(M(ν),k)|2. (7)

If a cutoff beyond the virial radius is used, the mass weight-

ing should reflect the increase. The treatment of the Poisson

term on large scales is only approximate. Mass and momen-

tum conservation require that on very large scales the non-

linear term should scale as k4, rather than as a constant

implied by equation 7, so the contribution from this term

on large scales is overestimated for k ≪ r−1

compensates the increase in power from the matter outside

the virial radius and for this reason we chose to use rvir as

the radial cutoff for the halo. The halo-halo term is only ap-

proximate, since we do not include the exclusion of haloes,

vir. This partially

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Halo Concentration and the Dark Matter Power Spectrum3

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Figure 1. Using the halo model, several power spectra were gen-

erated from a cosmology with Ω0 = 0.3, Λ = 0.7, Γ = 0.2, and

σ8= 0.9 at z = 0. In the upper panel, c0varies and β = −0.2. In

the lower panel, β varies and c0= 10.

which would suppress the term. Because of these approxi-

mations one would not expect the halo model to be perfect,

especially in the transition between the linear and non-linear

regime.

The total power spectrum is the sum of the two contri-

butions, P(k) = Phh(k)+PP(k). On small scales, the Pois-

son term dominates. On larger scales, the halo-halo term

dominates and reduces to the linear power spectrum. From

here forward, we express the power spectrum in dimension-

less form, ∆2(k) = 4πk3P(k).

The effect on the power spectrum of varying c0 and β in

a typical ΛCDM cosmology is shown in Figure 1. The con-

centration controls how tightly matter is correlated within a

single halo. Therefore, higher concentration means a larger

one-halo term. Increasing c0 increases the concentration at

the non-linear mass, affecting the amplitude of the power

spectrum where it is dominated by one-halo term.

Varying β keeping c0 constant produces a tilt in the

non-linear power spectrum around the fiducial value, which

is set by the scale where the non-linear mass dominates the

power spectrum contribution (k ∼ 30–40 h Mpc−1). Haloes

less massive than the non-linear mass dominate at higher

k. Steeper (more negative) β means that haloes less mas-

sive than the non-linear mass will have their concentrations

enhanced, leading to an enhanced one-halo term, and more

power at high k. Haloes more massive than the non-linear

mass will have their concentration reduced. These dominate

at intermediate k in the non-linear regime, so power there is

reduced. The changes in β shown in the figure may not have

much effect on the power spectrum at k < 100 h Mpc−1,

which is the range where N-body simulations are reliable.

On the other hand, they have a substantial effect on the

concentration of typical-sized haloes. For the same concen-

tration at the non-linear mass (∼ 1013h−1M⊙), a halo at

1012h−1M⊙ has a concentration 50% lower with β = −0.1

than with β = −0.4. These masses do not dominate the

power spectrum until k > 100 h Mpc−1, so this change does

not make as much of a difference on the power spectrum at

scales larger than this. This discussion suggests that while

power spectrum analysis cannot provide strong constraints

on the mass dependence of concentration, one can use con-

centration mass relation to predict the power spectrum at

small scales which are not resolved by simulations.

3 METHOD AND RESULTS

We consider two sets of cases which have been extensively

simulated, self-similar initial conditions and more realistic

cold dark matter initial conditions. In both cases we use the

fitting formulae for non-linear ∆2(k) given by Smith et al.

(2002).

3.1Self-Similar Case

The simplest case to consider is Ω = 1 Einstein-de Sitter

universe with a power law linear power spectrum. In this

case, β has an analytic form, provided we assume that once

a halo collapses, the scale radius rs is fixed in proper coordi-

nates. This is suggested by the simple model which assumes

the haloes remain unchanged once formed, which seems to

hold in numerical simulations (Bullock et al. 2001). We fol-

low the evolution of a single halo, as it traces out a portion

of the c(M) relation.

In an Ω = 1 universe, density perturbations grow as the

scale factor a = 1/(1+z). For a power law linear spectrum,

this means Plin(k) ∝ a2kn. Top-hat smoothing at the scale

corresponding to M yields σ(M) ∝ aM−n+3

universe, δc ≈ 1.68 is constant in time, so σ(M∗) is also

n+3

6

∗

c ∝ rvir ∝ a follows from the definitions of c and rvir and

from ¯ ρ ∝ a−3. Since c ∝ M−β

∗

M is constant. In reality M also increases with a, which

decreases β somewhat, although not by more than 20%.

We calculated halo model power spectra with n = −2.0

and n = −1.5, for which β = −0.16 and −0.25 as calculated

above, with several values of c0. These spectra are shown

in Figure 2. The agreement is quite good, given all the lim-

itations of the halo model. At higher k in the n = −2.0

case, it appears that the slope of the power spectrum dis-

agrees independent of c0. For k/k∗ > 2 this model is better

fit by a power spectrum with β = −.037, c0 = 2.6. How-

ever, according to figure 12 of Smith et al. (2002), simula-

tion data do not exist for k/k∗ greater than a few tens in

the n = −2 case, so the discrepancy is not really significant.

6 . For an Ωm = 1

constant. This means M

∝ a. If rs is fixed in time,

then β ≈ −n+3

6

if we assume

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4 K. Huffenberger & U. Seljak

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Figure 2. Non-linear power spectra in scale free models, divided

by the linear power. The power law linear spectra are ∆2

4πk3(k/k∗)n, where n = −2 and n = −1.5. In each case the halo

model spectra for several different c(M) are shown in solid lines,

and the fitting formula of Smith et al. (2002) is in dashed lines.

For the n = −2 case we show three models with β = . − 1667,

which is the analytically predicted value from the text. These

have c0= 3,4,5 in increasing amplitude, and fit poorly. We also

show the best-fitting β = −.037, c0= 2.6 with a thicker line. For

n = −1.5, we show the predicted β = −.25 with c0 = 6,8, and

the best-fitting β = −.245, c0= 7.1.

lin=

In the n = −1.5 case, which is tested against simulations to

k/k∗ = 100, the predicted β = −0.25 is close to the best fit-

ted value. This agreement therefore confirms the assumption

that haloes have a fixed scale radius once they formed.

3.2 Cold Dark Matter Models

Next we consider several flat CDM models (we do not con-

sider open or closed models here, since they are observation-

ally disfavored). To compare the halo model power spectrum

to the Smith et al. (2002) power spectrum, we use a simple

χ2statistic:

χ2=

N

?

i=1

?∆2

Smith(ki) − ∆2(ki)

σi

?2

. (8)

We distributed the N = 40 sample ki evenly in logk, consid-

ering 0.01 < k < 40 h Mpc−1. This range takes us from the

linear regime to some of the highest k that the fitting formula

has been tested with, according to figure 16 of Smith et al.

(2002). For the errors σi, we take 30% of the Smith et al.

(2002) power. This is somewhat arbitrary, but roughly the

size of the combined error in the mildly non-linear regime,

and is probably conservatively large in the fully non-linear

regime where the Poisson term dominates. We are ignoring

the correlations between the points, so the actual value of

χ2is just a qualitative measure of the goodness of fit.

For a variety of cosmologies, we minimized χ2over the

two-dimensional parameter space of c0 and β from equa-

tion (2). We employed a Powell minimization as described

in Press et al. (1992) to give a best-fitting c0 and β. All runs

were given the same initial point: c0 = 10 and β = −0.2, and

terminated when the the minimum changed by less than

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Figure 3. The values of c0 and β at several values of the mass

density parameter Ω0 for the fitted halo model power spectrum

and the models compared in §4. The vertical line denotes the

fiducial model with Ω0= 0.3. The other cosmological parameters

are given in §3.2.

0.1%. We place an error bar on each minimization by calcu-

lating χ2on a grid about the minimum point. We compute

an error contour where exp(−χ2/2) falls to half-maximum.

The error bars in c0 and β show the maximum extent of

this contour in each direction. With this definition, these er-

ror bars cannot properly be used to set confidence limits or

rule out models, but are included simply to gauge in which

cosmologies parameters are better or worse constrained and

to explore parameter degeneracies. Indeed, because of the

range of k we consider, c0 and β are somewhat degenerate:

making β more negative has a similar effect to lowering c0

(compare Figure 1).

We varied one cosmological parameter at a time, choos-

ing the other parameters from a fiducial model with Ω0 =

0.3, Ω0 + Λ = 1, Γ = 0.2, σ8 = 0.9, and z = 0. We

examined the ranges 0.1 < Ω0 < 1.0, 0.1 < Γ < 0.4,

0.6 < σ8 < 1.5,and 0 < z < 1. We plot the variation of

c0 and β with Ω0 with solid lines in Figure 3. The dashed

lines are discussed in §4. The relation for c0 is well fit by:

c0(Ω0) = 11(Ω0/0.3)−0.35, (9)

while β is consistent with being constant around zero or

slightly negative. Note that the Ω dependence depends on

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Halo Concentration and the Dark Matter Power Spectrum5

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Figure 4. The values of c0 and β as a function of the effec-

tive spectral index at the non-linear scale neff for the fitted

halo model power spectrum and for the models compared in

§4. The line type indicates which model. We vary one param-

eter at a time. The fiducial model, given in §3.2, is marked

the vertical line at neff

≈ −1.7. Each parameter increases

as neff increases except for z, which decreases. In their re-

spective panels, the points (left to right) correspond to: Γ =

0.1,0.2,0.3,0.4, σ8 = 0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4, and

z = 1.0,0.8,0.6,0.4,0.2,0.0.

the definition of the virial radius and with a different defi-

nition there would be a different Ω dependence.

Although c0 and β vary with all of the other cosmologi-

cal parameters, it is more instructive to present its variation

with the effective power law index of the linear power spec-

trum at the non-linear scale,

neff =d(logPlin(k))

d(logk)

????

k∗

, (10)

where ∆2

is that the slope of the linear power spectrum determines the

epoch of formation of small haloes that merge into the larger

halo. For lower effective slope neff these haloes formed later,

when the density of the universe was smaller, and the final

halo concentration of the larger halo is also lower. This is

depicted with solid lines in Figure 4. Our fiducial cosmology

has neff ≈ −1.7. Varying Ω0 has no effect on neff. When

varying z, a portion of the variation of c0 and β should be

due to the change in Ω(z). The variation of c0 with neff is

consistent with

lin(k∗) = 1 defines the non-linear scale. The reason

c0 = 11(Ω(z)/0.3)−0.35?neff

while β has large errors, but is typically slightly negative.

−1.7

?−1.6

, (11)

4 COMPARISON TO HALO ANALYSIS

We now compare our results to the models of Eke et al.

(2001) and Bullock et al. (2001). These are designed to re-

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Figure 5. A comparison of the variation of the concentration pa-

rameter with mass for our fiducial ΛCDM cosmology (§3.2). The

masses and concentrations here have been translated to a common

virial radius convention, discussed in the text. The shaded region

covers power law models within our error contour, discussed in

the text, and should not be considered a limit on c(M) relations

of arbitrary shape.

produce the concentrations of haloes found in N-body sim-

ulations. Throughout we translate to our virial convention.

We begin by considering concentration as a function of

halo mass for the fiducial ΛCDM cosmology defined in §3.2.

This is plotted in Figure 5. It is clear that our results are

in agreement with those from direct halo structure analy-

sis. However, most of the information in our model comes

from the haloes with masses between 1013and 1014h−1M⊙,

because haloes smaller than this dominate at scales smaller

than the range we considered for our fit. This means we have

a relatively short lever arm with which to determine the

mass concentration relation and consequently we are unable

to strongly constrain the slope β. In direct analysis of haloes,

concentration is a decreasing function of mass, so β < 0. Our

best fitted values for β are also typically negative, although

the errors are large and positive values also cannot be ex-

cluded. The strongest confirmation of this prediction comes

from the analysis of scale-free models in previous section,

where β is negative for n = −1.5. It is clear that measuring

the mass-concentration directly is a better way to determine

the mass dependence of concentration rather than using the

power spectrum, particularly at lower masses.

Since determining mass dependence of concentration

from the power spectrum does not appear promising let

us ask the opposite question: how well can we predict the

power spectrum using the mass concentration relations mea-

sured directly from the analysis of individual haloes in

the simulations? Figure 6 shows the power spectra calcu-

lated with power law approximations to the c(M) func-

tions shown in Figure 5, as well as the Smith et al. (2002)

power spectrum. While there is good agrement between

them for k < 40hMpc−1, the halo models always predicts

less power at k > 40hMpc−1compared to the fits to the

power spectrum. However, the direct fits are not reliable

in this regime, since the N-body simulations become unre-

liable for k > 40hMpc−1, which is why the fits presented

above only use information from k < 40hMpc−1.

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