On the Outskirts of Dark Matter Haloes Alice Y. Chenand Niayesh Afshordiy Department of Physics and Astronomy University of Waterloo

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On the Outskirts of Dark Matter Haloes

Alice Y. Chen∗and Niayesh Afshordi†

Department of Physics and Astronomy, University of Waterloo,

200 University Ave W, N2L 3G1, Waterloo, Canada

Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada and

Perimeter Institute For Theoretical Physics, 31 Caroline St N, Waterloo, Canada

Halo models of large scale structure provide powerful and indispensable tools for phenomenological

understanding of the clustering of matter in the Universe. While the halo model builds structures

out of the superposition of haloes, deﬁning halo proﬁles in their outskirts - beyond their virial radii -

becomes increasingly ambiguous, as one cannot assign matter to individual haloes in a clear way. In

this paper, we address this issue by ﬁnding a systematic deﬁnition of mean halo proﬁle that can be

extended to large distances - beyond the virial radius of the halo - and matched to simulation results.

These halo proﬁles are compensated and are the key ingredients for the computation of cosmological

correlation functions in an Amended Halo Model. The latter, introduced in our earlier work [1],

provides a more physically accurate phenomenological description of nonlinear structure formation,

which respects conservation laws on large scales. Here, we show that this model can be extended

from the matter auto-power spectrum to the halo-matter cross-power spectra by using data from

N-body simulations. Furthermore, we ﬁnd that this (dimensionless) deﬁnition of the compensated

halo proﬁle, r3×ρ(r)/M200c, has a near-universal maximum in the small range of 0.03−0.04 around

the virial radius, r'r200c , nearly independent of the halo mass. The proﬁles cross zero into negative

values in the halo outskirts - beyond 2-3×r200c - consistent with our previous results. We provide a

preliminary ﬁtting function for the compensated halo proﬁles (extensions of Navarro-Frenk-White

proﬁles), which can be used to compute more physical observables in large scale structure.

I. INTRODUCTION

Probing properties of the large scale structure

of the universe is an active area of research in

cosmology. While the interior structure of dark

matter haloes can be well modelled with the

Navarro-Frenk-White (NFW) or Einasto pro-

ﬁles [2, 3], the large scale distribution of dark

matter in the outskirts of individual haloes is

poorly understood in semi-analytic frameworks.

Previous studies have attempted to produce a

model using eﬀective ﬁeld theory (EFT) and

perturbation theory [4–7], but non-linearities in

structure formation make it diﬃcult to extrap-

olate the models beyond k≥1.0Mpc−1. For

smaller scales, the Standard Halo Model (SHM)

is often used as a phenomenological framework.

However, SHM suﬀers from pathologies that

stem from not enforcing the conservation laws

[8]. One proposal to address these pathologies,

namely the large-scale shot noise, has been to

impose an exclusion radius for haloes [9, 10],

but it is hard to see how this would distinguish

between the conserved and non-conserved quan-

tities, as the latter are expected to display shot

noise on large scales.

In order to address this issue in a system-

atic manner, we introduced an amended halo

model (AHM) [1] with compensated halo pro-

ﬁles and ﬁtted this to cold dark matter simu-

lations by Takahashi et al. [11] to model non-

linear dark matter density power spectrum on

scales of 10−2Mpc−1.k.100 Mpc−1. How-

ever, this analysis did not consider the pro-

ﬁle’s potential dependence on halo mass, nor

did it include non-linear biasing in halo-halo

correlations [12]. Consequently, the present

study aims to address these deﬁciencies. To do

this, we adopt simulation data from the Dark-

Emu cosmological emulator suite [13] - which

uses Planck 2015 cosmology [14] - to develop a

novel and systematic method to directly mea-

sure mean compensated proﬁles from simulated

(or emulated) halo-matter and halo-halo corre-

lations. We ﬁnd that the dimensionless com-

pensated halo proﬁles all peak around the virial

arXiv:2210.11499v3 [astro-ph.CO] 25 Apr 2023

radius at a near-universal maximum even across

diﬀerent mass bins, which is quite a striking re-

sult. We also ﬁnd that it is possible to “extrap-

olate” the NFW proﬁle beyond the virial radius

by using two extra parameters to ﬁt our com-

pensated halo proﬁle, and provide an approx-

imate functional form of the proﬁle (although

more parameters are likely necessary if we want

a highly accurate numerical ﬁt). Having such

a ﬁt for a semi-analytic framework allows us to

make predictions for halo power on a larger scale

(larger r and k range), beyond the resolution of

current N-body simulations.

While the physical meaning of the ﬁtting pa-

rameters used here is yet to be determined, our

main goal here is to show that a simple ﬁt

for the compensated halo proﬁle does exist and

can be used to predict the matter-halo cross-

correlation, while avoiding the pathologies of

the standard halo model. Our ﬁt matches NFW

in the halo regions r < rvirial, but is compen-

sated in the outer regions r > rvirial 1. This

paper is then structured as follows: Section II

outlines the amended halo model [1] for halo-

matter cross correlations, Section III discusses

our ﬁndings from the results of the simulation

data from DarkEmu [13], and Section IV sum-

marizes the results and potential avenues for

future research. The exact form of the ﬁtting

function we used, and its derivation and impli-

cations are outlined in the Appendix.

II. STANDARD VS AMENDED HALO

MODELS

In the standard halo model (SHM), the mat-

ter overdensity in the universe, δm(x), is de-

scribed as a superposition of individual halo

1In this paper, the virial radius rvirial is taken to be the

radius where density is 200 times the critical density

of the universe, or equivalent to r200c. Unless noted

otherwise, the mass of a halo is also deﬁned as the

total mass contained within this radius.

proﬁles, that we refer to here as uj:

δm(x) = ¯ρ−1X

Mjuj

SHM(x−xj),(1)

in real space, and

δm,k= ¯ρ−1X

Mjuj

SHM,kexp(ik·xj),(2)

in Fourier space. Mjand xjare the mass and

position of the j-th halo, respectively, while ¯ρ

denotes the mean density of the universe.

To get a complete understanding of matter

distribution, we need to know both how matter

is distributed within the individual halos, uj’s,

and how these halos are distributed throughout

space. Furthermore, conservation laws, such as

those of mass and linear momentum, require

a ﬁne balance between these two distributions,

which are often hard to enforce in the SHM for-

mulation (but see [4]).

In order to address this, we introduced the

Amended Halo Model (AHM), where we split

the nonlinear overdensities between the linear

δL(x) and halo contributions, revising the SHM

equations (1) and (2) to be:

δm(x) = δL(x) + ¯ρ−1X

Mjuj

AHM(x−xj) (3)

and

δm,k=δL,k+ ¯ρ−1X

Mjuj

AHM,kexp(ik·xj)

(4)

in real and Fourier spaces, respectively. The

nonlinear contribution to the halo proﬁle, which

is what we use to construct uj

AHM , is what we

will be calculating from simulations, since the

contribution from the linear term is already in-

cluded in δL(x).

Now, let us consider the overdensity of haloes

within a mass-bin b:

δb

halo,k=1

¯nb

halo X

bexp(ik·xj),(5)

where Nj

b= 1 if the j-th halo is within the

mass-bin b, but vanishes otherwise. Further-

more, ¯nb

halo is the number density of haloes

within the mass bin.

The auto matter, auto halo, and halo-matter

cross spectra can now be deﬁned as:

Pmm(k)≡hδm,kδ∗

m,ki

V(6)

Pbc

hh(k)≡hδb

halo,kδc∗

halo,ki

V(7)

hm(k)≡hδb

halo,kδ∗

m,ki

V,(8)

respectively, where Vis the volume of the simu-

lation, and band cstand for diﬀerent halo mass

bins. Now, by multiplying equations (5) and

(4), we ﬁnd the cross-power spectra:

hm(k)= b(¯

Mb)PL(k) + 1

¯nb

halo ¯ρV X

MjNj

buj

AHM,k

¯nb

halo ¯ρV X

j6=l

MjNl

buj

AHM,kexp[ik·(xj−xl)],

(9)

which we can write in the matrix form:







hm(k)

...

hm(k)





=PL(k)





b(¯

M1)

...

b(¯

Mq)





+





M1/¯ρ0... 0

0¯

M2/¯ρ ... 0

... ... ... ...

0... ... ¯

Mq/¯ρ











AHM(k)

...

AHM(k)





+







Phh(k|¯

M1,¯

M1)Phh(k|¯

M1,¯

M2)... Phh(k|¯

M1,¯

Mq)

Phh(k|¯

M2,¯

M1)Phh(k|¯

M2,¯

M2)... Phh(k|¯

M2,¯

Mq)

... ... ... ...

Phh(k|¯

Mq,¯

M1)Phh(k|¯

Mq,¯

M2)... Phh(k|¯

Mq,¯

Mq)





×





µ10... 0

0µ2... 0

... ... ... ...

0... ... µq











AHM(k)

...

AHM(k)





.(10)

Here, q≥1 is the number of mass bins used,

and we have used the following deﬁnitions:

Mb≡PjNj

bMj

¯nb

haloV,(11)

µb≡¯nb

halo ¯

¯ρ,(12)

AHM(k)≡PjNj

bMjuj

AHM,k

¯nb

halo ¯

MbV.(13)

Note that, at this level, AHM does not make

a prediction for the halo-halo auto-power spec-

trum, which can be impacted by nonlinear

structure formation, and here we rely on N-

body simulations to model it. Furthermore,

Equation (9) or (10) can be considered as pre-

cise deﬁnitions of (mean) halo proﬁles ub

AHM

and linear bias b(¯

Mb) (by setting ub

AHM(k=

0) →0), and thus make no assumptions about

the distribution of matter.

One may then gain an intuition about the na-

ture of the AHM vs SHM on large scales, using

the linear bias approximation for halo distribu-

tion, and the mass function n(M):

δm,k=δL,k+X

¯ρuj

AHM,kexp(ik·xj)

'1 + 1

¯ρZdMMn(M)b(M)uAHM(k, M )δL(k),

(14)

where the fact that ub

AHM(k= 0) →0 guaran-

tees the agreement with linear density predic-

tions on large scales. In contrast, for SHM, we

have:

δm,k'1

¯ρZdMMn(M)b(M)uSHM(k, M )δL(k),

(15)

where ub

SHM(k= 0) →1, and thus an additional

condition of RdMMn(M)b(M) = ¯ρis neces-

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OntheOutskirtsofDarkMatterHaloesAliceY.ChenandNiayeshAfshordiyDepartmentofPhysicsandAstronomy,UniversityofWaterloo,200UniversityAveW,N2L3G1,Waterloo,CanadaWaterlooCentreforAstrophysics,UniversityofWaterloo,Waterloo,ON,N2L3G1,CanadaandPerimeterInstituteForTheoreticalPhysics,31CarolineStN,Waterloo,Ca...

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