A Universal Profile of the Dark Matter Halo and the Two-point Correlation Function

Abstract

We have investigated the relation between the two-point spatial correlation function and the density profile of the dark matter halo in the strongly non-linear regime. It is well known that when the density fluctuation grows into dark matter halo whose density profile is $\rho \propto r^{-\epsilon}$(${3/2}<\epsilon<3$) on almost all mass scales, the two-point spatial correlation function obeys a power law with the power index $\gamma = 2\epsilon -3$ in the strongly non-linear regime. We find the form of the two-point spatial correlation function, which does not obey the power law when the power index $\epsilon$ is smaller than 3/2, such as the density profile $\rho \propto r^{-1}$ around the center of the halo which is proposed by Navarro, Frenk & White (1996,1997). By using the BBGKY equation in the strongly non-linear regime, it is also found that velocity parameter $h \equiv - < v > / \dot{a}x$ is not a constant even in the strongly non-linear regime ($\tilde{x} \equiv x/x_{nl} \to 0$) although it is a constant when $\epsilon > 3/2$ and then the two-point spatial correlation function can be regarded as the power law. The velocity parameter $h$ becomes 0 at the non-linear limit of $\tilde{x} \to 0$, that is, the stable clustering hypothesis cannot be satisfied when $\epsilon < 3/2$. Comment: 15 pages with no figures, submitted to Astrophysical Journal

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arXiv:astro-ph/9906375v1 24 Jun 1999
A Universal Profile of the Dark Matter Halo and the Two-point
Correlation Function
Taihei Yano1and Naoteru Gouda
National Astronomical Observatory, Mitaka, Tokyo 181-8588, japan
E-mail: yano@pluto.mtk.nao.ac.jp, naoteru.gouda@nao.ac.jp
ABSTRACT
We have investigated the relation between the two-point spatial correlation
function and the density profile of the dark matter halo in the strongly non-linear
regime. It is well known that when the density fluctuation grows into dark
matter halo whose density profile is ρ ∝ r−ǫ(3
scales, the two-point spatial correlation function obeys a power law with the
power index γ = 2ǫ − 3 in the strongly non-linear regime. We find the form of
the two-point spatial correlation function, which does not obey the power law
when the power index ǫ is smaller than3
around the center of the halo which is proposed by Navarro, Frenk & White
(1996,1997). By using the BBGKY equation in the strongly non-linear regime,
it is also found that velocity parameter h ≡ −?v?/˙ ax is not a constant even in
the strongly non-linear regime (˜ x ≡ x/xnl→ 0) although it is a constant when
ǫ > 3/2 and then the two-point spatial correlation function can be regarded as
the power law. The velocity parameter h becomes 0 at the non-linear limit of
˜ x → 0, that is, the stable clustering hypothesis cannot be satisfied when ǫ < 3/2.
2< ǫ < 3) on almost all mass
2, such as the density profile ρ ∝ r−1
Subject headings: cosmology:theory-large scale structures-dark matter halo
1. INTRODUCTION
Formations of the large scale structures in the expanding universe is one of the
most important and interesting problems of cosmology. It is generally believed that
these structures have been formed owing to the gravitational instability. Hence it is very
important to clarify evolutions of density fluctuations by the gravitational instability. Here
1Research Fellow of the Japan Society for the Promotion of Science.

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we consider density fluctuations of collisionless cold particles such as cold dark matters, our
interest being mainly concentrated on effects of the self-gravity.
In the hierarchical clustering picture, small fluctuations grow and form small clusters
of dark matter (dark halo) at first. Then these small dark halos merge into a larger halo
or accrete surrounding matters and produce the larger dark matter halos. Evolutions of
density fluctuations in the non-linear regime are especially important to understand the
formation of the dark matter halos.
A central region of the dark halo profile was investigated by using N-body simulations
in 1990’s (Frenk et al. 1988, Navarro, Frenk & White 1996, 1997 : hereafter NFW ).
Hernquist (1990) investigated the elliptical galaxy analytically. NFW and Hernquist
claimed that dark halos are not well approximated by isothermal spheres. Instead, the
density profile is well approximated by the following form:
ρ ∝
1
(r
rs)ǫ(1 +
r
rs)µ−ǫ, (1)
where, rsis a characteristic scale of the dark halo. The parameter, ǫ, and µ are equal to 1
and 4, respectively, in the Hernquist’s case, and ǫ = 1, µ = 3 in the NFW’s case. In any
case, the density profile of the central region is shallower than the isothermal sphere.
On the other hand, the two-point spatial correlation function is well used in the analysis
of the density fluctuations of the dark matters. Of course, the two-point spatial correlation
function in the non-linear regime is related to the dark halo profile if the mass weight of the
dark halos dominates in the universe, that is, almost all regions of high density correspond
to dark halos. Davis & Peebles (1977) assumed the self-similar evolution and the stability
condition, that is, the mean relative physical velocity ?˙ r? is equal to 0 in the non-linear
regime. Then, they showed the index of the two-point spatial correlation function in the
non-linear regime by using the index of the initial power spectrum, n:
ξ ∝ r−γ,γ =3(3 + n)
5 + n
. (2)
When we do not assume the stability condition, the index of the two-point spatial
correlation function γ is expressed by using the relative velocity parameter h ≡ −?v?/˙ ax,
γ =
3h(3 + n)
2 + h(3 + n), (3)
where n > −3, and h is a value of order 1 (0 ≤ h ≤ 1) (Yano & Gouda (1997) ; hereafter
YG) in the non-linear regime. Here ?v? is the mean relative peculiar velocity ( see eq.(10))

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and a is the scale factor. Therefore γ have the value around the following:
0 < γ < 3. (4)
We consider the case that the halo dominates in the regions of high density and
then the density profile of the halo mainly contributes to the two-point spatial correlation
function in the non-linear regime. When the density profile of the dark halo is given by
ρ ∝ r−ǫ, then using the following equation,
ξ(r) ∼ ?ρ(s)ρ(|− →r +− →s |)?/?ρ?2, (5)
the two-point spatial correlation function in the strongly non-linear regime (r → 0) is
determined by the density profile of the dark halo and written in the following way
(McClelland & Silk 1977; Sheth & Jain 1997, Padmanabhan & Engineer 1998),
ξ ∝ r−γ′,γ′= 2ǫ − 3. (6)
The two-point spatial correlation function diverges at the center of the dark halo when
ǫ > 3 and it does so on ˜ r ≡ r/rnl→ ∞ when ǫ < 3/2, where rnlis the non-linear scale
of the density fluctuation. Therefore 3/2 < ǫ < 3 must be satisfied. When the relation
3/2 < ǫ < 3 is satisfied, γ′satisfies 0 < γ′< 3. This relation corresponds to the relation (4)
derived from eq. (3). In other words, when the two-point spatial correlation function obeys
the power law with 0 < γ < 3, the index ǫ satisfies 3/2 < ǫ < 3.
When the density profile is ρ ∝ r−ǫwith ǫ < 3/2 around the center of the dark halo,
the index of the density profile have at least following two values in order to converge the
integral.
ρ ∝ (r
rs)−κ(r)
: κ(r) =
?
ǫ
µ (r > rs
(r < rs
: ǫ <3
: µ >3
2)
2)
.(7)
Here rsis a characteristics scale. In this case, what is the form of the two-point
spatial correlation function ? The purpose of this paper is to obtain the two-point spatial
correlation function in the strongly non-linear regime when the dark halos have the above
density profile (7) and the self-similar evolution of the dark halos is satisfied. We also
investigate what is the form of the velocity parameter h, that is, the mean relative peculiar
velocity by using the BBGKY equation.
In §2, we will briefly review the power law solution of the two-point spatial correlation
function described in YG. We show in §3 the solution of the two-point spatial correlation

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function and the mean relative peculiar velocity when there exists the dark halo whose
density profile has the form of eq.(7). We will devote to the conclusions and discussions
in §4. In the Appendix, we will consider the relation between the power index of the
density profile of the dark halo and the initial power index n if the merging or accreting far
equilibrium dark halos could be regarded as a spherical halos.
2.POWER LAW SOLUTIONS
We will briefly review the power-law solutions of the two-point spatial correlation
function ξ, and the mean peculiar velocity ?v? described in YG. We use the second BBGKY
zeroth moment equation.
∂ξ
∂t+
1
ax2
∂
∂x[x2(1 + ξ)?v?] = 0, (8)
where, a is the scale factor, x is the comoving coordinate. The mean relative peculiar
velocity vector ?v? is defined as follows;
?v? ≡??(v2− v1)f(1)f(2)d3v1d3v2?
??f(1)f(2)d3v1d3v2?
Here ?...? shows the ensemble mean for any pair with the fixed distance x ≡ |x2− x1|
and f(i) ≡ f(xi,vi) (i = 1,2) is the phase space density at the point of the phase space
(xi,vi). We notice that ?f(1)? ≡ b(1) is the one-body probability distribution function,
?f(1)f(2)? ≡ b(1)b(2) + c(1,2) is the two-body probability distribution function, and c(1,2)
is the irreducible two-point correlation function. Here we assume that the background
universe is homogeneous and isotropic. Then, ?v? ∝ x2− x1≡ x and then the scalar
velocity ?v? ≡ ?v?x
mean relative peculiar velocity ?v? can be rewritten by
Therefore we can define the mean relative peculiar velocity ?v? as follows:
?v? =??x
??f(1)f(2)d3v1d3v2?
We assume that the two-point spatial correlation function ξ is given by the following
power-law form in the strongly non-linear regime:
. (9)
xis the only important quantity which determines ξ through eq.(8). The
x· (v2− v1)f(1)f(2)d3v1d3v2?
.(10)
ξ = ξ0aβx−γ.(11)

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Then we obtain from the dimensional analysis of the equation (8)
?v? = −h˙ ax,β = (3 − γ)h,(12)
where h is a constant. We call this parameter h, the velocity parameter. This parameter is
a value of order 1 (0 ≤ h ≤ 1)(YG) when we consider the collisionless cold dark matters in
the strongly non-linear regime. We notice that the stability condition proposed by Davis &
Peebles (1977) corresponds to h = 1 (stable clustering). In this case, the collapsed object
cannot be broken and clustered together to form the larger cluster. Therefore, the relative
physical velocity of two particles ?˙ r? = ?v? + ˙ ax is equal to 0 (?v? = −˙ ax) in the non-linear
regime. We can consider the other extreme case about the clustering picture. In this case,
the smaller objects have clustered and merged together and the completely virialized object
is newly formed. In this case, the separation of the particle is expanding with the Hubble
velocity and then h = 0(comoving clustering).
If the self-similarity solutions exist, the power index of the two-point spatial correlation
function in the non-linear regime can be represented by the following form (Padmanabhan
1996; YG):
γ =
3h(3 + n)
2 + h(3 + n), (13)
where n is the index of the initial power spectrum (n > −3). Therefore γ have the value
between 0 and 3. Here we consider the case that the halo mass dominates and the index
of the density profile ρ ∝ r−ǫis the same for all mass scales. The relation 3 − ǫ > 0 and
3−2ǫ < 0 must be satisfied so that the integral in eq.(5) should converge. Therefore, ǫ have
the following values:
3
2< ǫ < 3.
In this case, the relation between the two-point spatial correlation function and the density
profile is
(14)
ξ(r) ∝
?
d3sρ(s)ρ(|− →r +− →s |)
= c2(r
rnl)−2ǫ+3. (15)
[see McClelland & Silk 1977; Sheth & Jain 1997, Padmanabhan & Engineer 1998] where,
c2=
∞
?
m=0
2(
1
3 − ǫ + 2m−
1
3 − 2ǫ − 2m)(2 − ǫ)(1 − ǫ)···(2 − ǫ − 2m)
1
(2m + 1)!. (16)