4.4 Halo bias
4.4.2 Results from excursion set theory
Conditional mass function
A more thorough derivation of the halo bias is offered by Mo and White [51] and is reproduced very detailed in the review of Zentner [92]. In order to describe the clustering in an overdense regions, Mo and White needed a formula which relates halo abundances to the underlying density field on large scales. For their consideration, they were able to make use of the results by Bower [11] and Bond et al. [9]. These authors extended the Press-Schechter theory (see subsection 4.3.1) to find the de-sired conditional mass function, which gives the number density of halos assuming an overdense background density. In the following, we will briefly summarize the most important results.
Let us consider a region of comoving scale R1 at a redshift z1 characterized by a smoothed density contrast δR1(x) and a smoothed variance σ2(R1). This region is
enclosed in a larger region of comoving scaleR0defined byδR0(x) andσ2(R0). If we use a top-hat window as filter function, the average mass contained in a scaleRism(R) = (4π/3) ¯ρR3 = Vρ, which implies for the considered regions that¯ m1 < m0. Assuming that the density contrasts of these regions were initially Gaussian distributed, Bower and Bond et al. [9, 11] extended the results from the Press-Schechter formalism and found the conditional probability of attaining a density δ between δ and δ+ dδ given a background density δ0 to be
P(δ|δ0) dδ= 1
p2π(σ2 −σ02) exp
− (δ−δ0)2 2(σ2−σ02)
dδ , (4.77)
where we use here and in the following δi ≡ δRi(x) and σi ≡ σ(Ri) to shorten the notation. As before, we assume that an object is collapsed, if its density contrast exceeds a threshold δsc. The fraction of mass in bound halos contained in a region of mass m0 is then obtained by integrating the conditional probability density according to
F(m1|δ0, σ0) = Z ∞
δsc
P(δ|δ0)dδ= 1 2erfc
"
δsc−δ0 p2(σ21−σ02)
#
, (4.78)
where erfc(x) denotes the complementary error function. If the density of the sur-rounding region increases, the fraction F of masses in bound halos increases as well.
In the case that δ0 →δsc, one has F →1 and the entire region will be interpreted as one collapsed object. The fraction of mass in halos with mass in the range m1 and m1+ dm1 is then
dF(m1|δ0, σ0)
dm1 dm1 = 1
√2π
δsc−δ0 (σ21−σ02)3/2
dσ12 dm1 exp
−(δsc−δ0)2 2(σ21−σ02)
dm1. (4.79) In the limit δ0 → 0, when the surrounding region has the same smoothed density as the Universe, we obtain the unconditional solution from Eq. (4.58), remembering that σ1 > σ0. According to Mo et al. [52] the average comoving number density of halos in a region of mass m0 can then be defined as
n(m1|δ0, σ0)dm1 ≡ 1 V0
m0 m1
dF(m1|δ0, σ0) dm1 dm1,
≡ N(δ1|δ0, σ0)
V0 dm1. (4.80)
This expression is interpreted as the conditional mass function in a dense region with smoothed density contrast δ0 for masses in a rangem1 and m1+ dm1 at a redshift z1,
4.4 Halo bias 79
while N(δ1|δ0, σ0) denotes the average number of halos with masses between m1 and m1+ dm1.
Transformation from Lagrangian to Eulerian space
With the conditional mass function we define the halo density contrast as
δh(m1)≡ n(m1|δ0, σ0)−n(m1)
n(m1) = n(m1|δ0, σ0)
n(m1) −1, (4.81) where n(m) denotes the unconditional mass function as defined in Eq. (4.59) that corresponds to the average number density of halos. Up to now, we used only co-moving coordinates for our consideration and ignored the dynamical evolution of the regionR0. To calculate the Eulerian bias relation, we need an expression for the halo density contrast in terms of the gravitationally evolved region R. The corresponding conditional mass function is then defined as
nE(m1|δ0, σ0)dm1 ≡ N(δ1|δ0, σ0)
V dm1, (4.82)
where we distinguish between Eulerian and comoving Lagrangian conditional mass function by introducing the superscripts ‘E’ and ‘L’. In order to proceed, we need a relation between the initial region of scaleR0 and the gravitationally collapsed region R. This is provided by the spherical collapse model (see Sect. 4.2). Since the mass within a region about to collapse is assumed to be conserved in this model, the evolved volume of the region V is related to its initial volume V0 according to V0 ' V(1 +δ) for small initial overdensities. Additionally, the spherical collapse model provides a relation between the initial and evolved matter density of the form δ =δ(δ0). In the limit of small initial overdensities it yields δ ' δ0. With these relations at hand, we can transform the desired quantities from Lagrangian to Eulerian space. Starting with the Eulerian conditional mass function of Eq. (4.82), we find
nE(m1|δ0, σ0)dm1 = N(δ1|δ0, σ0)
V dm1
' N(δ1|δ0, σ0)
V0 (1 +δ) dm1
=nL(m1|δ0, m0) (1 +δ) dm1, (4.83) where nL(m1|δ0, m0) denotes the conditional mass function as defined in Eq. (4.79).
Making use of this, the Eulerian density contrast transforms into δEh = nE(m1|δ0, σ0)
n(m1) −1,
= nL(m1|δ0, m0)
n(m1) (1 +δ)−1
= (δhL+ 1)(1 +δ)−1
=δhL(1 +δ) +δ .
Remembering the bias relation δhL =bLδ0 and that δ 'δ0 on small scales, we obtain
δhL '(bL+ 1)δ0, (4.84)
where we only kept terms of first-order in δ and δ0. Thus, Lagrangian and Eulerian bias are related to each other according to bE=bL+ 1 for small density contrasts.
Press-Schechter bias
The introduced quantities allow us to calculate the bias relation for the Press-Schechter mass function in a perturbative approach. Following Mo, Jing, and White [52], we restrict the consideration to large scales in Eulerian space and assume that the halo density can be described by a smooth function F(δ) that depends only on the matter density. If the function is finite for δ around 0, we can expand F in a Taylor series around δ, such that
δh =F(δ) =
∞
X
k=0
bk
k!δk, (4.85)
where bk are the bias parameters and δ the non-linear matter density. The bias relation from Eq. (4.67) corresponds then to the first-order approximation of F and the linear bias we introduced there is now equivalent to b1. Since we remain during the subsequent considerations in Eulerian space, we omit the superscript ‘E’ here and in the following.
In order to find explicit expressions for the bias parameters bk, we expand in the definition of the halo density (Eq. 4.81), the conditional mass function around the matter density δ0 of the surrounding region and compare equal orders in δ0 to the Taylor expansion ofF(δ). Sinceδdenotes the non-linear density, we additionally have to use the expansion δ(δ0) as found from the spherical collapse model.
To determine the first order or linear bias b1, we have to calculate the first derivative
4.4 Halo bias 81
of the conditional mass function with respect to the density, since δh ' 1
n(m1)
dn(m1|δ0, σ0) dδ0
δ0=0δ0(1 +δ)−1 ' 1
n(m1)
dn(m1|δ0, σ0) dδ0
δ
0=0δ0 (4.86)
=b1δ0. (4.87)
Combining Eq. (4.80) and (4.79), we obtain for the conditional Press Schechter mass function on large scales
n(m1|δ0, σ0) = ρ¯ m1
(1 +δ)dF(m1|δ0, m0) dm1
dm1 (4.88)
' ρ¯
m1 (1 +δ) 1
√2π
δ1−δ0 σ13
dσ12 dm1 exp
−(δ1−δ0)2 2σ12
dm1, (4.89)
where we assumeR0 R1 and thus haveσ21 σ02 in hierarchical structure formation.
From this we can calculate the first derivative of the conditional mass function with respect toδ at the background density which gives two terms:
dn(m1|δ0, σ0) dδ0
= ρ¯ m1
√1 2π
dσ21 dm1
1 σ13
(δ1−δ0)2 σ21 −1
exp
−(δ1−δ0)2 2σ12
. (4.90)
Inserting this result into Eq. (4.86) and evaluating the found expression, we obtain finally the linear Eulerian bias
b(ν) = 1 +ν12−1
δsc (4.91)
in the extended Press-Schechter formalism, where ν1 ≡ δ1/σ1. This result is equiva-lent to equation (4.76) that we obtained in the peak-background split consideration.
Higher-orders are found in a similar way, but require tedious calculations, since the higher-order derivatives in the mass function have more terms and one additionally has to apply the expansion δ0(δ) = P
kakδk (see Eq. 4.54). For this work, we need the first four orders of the bias expansion in Eulerian space which are
b0 = 0, (4.92) b1 = 1 + ν12 −1
δ1 , (4.93)
b2 = 2(1 +a2)ν12−1 δ1 +
ν1
δ1 2
(ν12−3), (4.94)
b3 = 6(a2+a3)ν12−1
δ1 + 3(1 + 2a2) ν1
δ1 2
(ν12−3) +
ν1 δ1
2
ν14 −ν12+ 3
δ1 , (4.95)
whereν1 ≡δ1/σ1 anda2 =−17/21 anda3 = 341/567 are coefficients of the expansion δ0(δ) = P
kakδk.
Sheth-Tormen mass function
The same consideration can be made for the Sheth-Tormen mass function, where one has to use the corresponding conditional mass function to calculate the different orders of the Taylor expansion. From the Sheth-Tormen mass function [76, 77], one finds the following fraction of halos with masses between m and m+ dm in a background with density δ0:
dF(m1|δ0, σ0)
dm1 = q
2π 1/2
A(p) 1 +
q(δ1−δ0)2 σ12−σ20
−p!
δ1−δ0 (σ12−σ20)3/2
×exp
−q(δ1−δ0)2 2(σ12−σ02)
, (4.96)
where p, q, A(p) denote the constants that we already defined for the average Sheth-Tormen mass function in Sect. 4.3.3. In the large scale limit, where σ21 σ02, this becomes
dF(m1|δ0, σ0)
dm1 = q
2π 1/2
A(p) 1 +
q(δ1−δ0)2 σ21
−p!
δ1−δ0
σ13 . (4.97) Expanding the mass function in a Taylor series as in the case of the Press-Schechter formalism in the previous section, the Eulerian density contrast can be approximated.
The first four orders give according to [73] the following bias parameters
4.4 Halo bias 83
b0 = 0, (4.98)
b1 = 1 +1+E1, (4.99)
b2 = 2(1 +a2)(1+E1) +2+E2, (4.100) b3 = 6(a2+a3)(1+E1) + 3(1 + 2a2)(2+E2), (4.101) where
1 = qν12−1
δsc , 2 = qν12
δsc2 (qν12−3), (4.102) 3 = qν12
δsc3 (q2ν14−6qν12+ 3), (4.103) E1 = 2p/δsc
1 + (qν12)p , E1 E2 =
1 + 2p δsc + 21
, (4.104)
E3 E1 =
4(p2 −1) + 6pqν12 δsc2 + 321
, (4.105)
whereν1 ≡δ1/σ1 anda2 =−17/21 and a3 = 341/567 are coefficients of the expansion δ0(δ) = P
kakδk (see Eq. (4.54)). If one sets p = 0 and q = 1, the bias parameters reduce to the Press-Schechter bias as summarized in Sect. 4.4.2. In this case the En
parameters are 0. If not stated otherwise, we will use the best fit parameters as found by Sheth and Tormen [77], i.e. the values p= 0.3 and q= 0.707.