power spectrum by dropping specific assumptions as indicated in the figure caption.
We find that the one- and two-halo term drop below the “real” galaxy power spectrum on small scales if the halos contain only satellite galaxies. This is due to the fact that on small scales the normalized halo density profile begins to decrease from unity as we showed in Fig. 3.3. In particular, central galaxy correlations are dominant on small scales because they are weighted by only one density profile. Therefore, the suppression of the halo terms due to the density profile on small scales is reduced for central galaxy correlations. If we further assume that the HOD distribution is Poissonian for all halo masses and in addition neglect the threshold mass for the central galaxy, we obtain a one-halo term which is proportional to the one-halo term of the power spectrum in the right panel of Fig. 4.3.
The right panel in Fig. 4.4 depicts the mass contributions to the one-halo term from halos divided into certain mass ranges as indicated in the figure. On large scales the dominant contributions come from high-mass halos 1014h−1M < m < 1016h−1M. Going to smaller scales, also smaller mass halos begin to dominate. On the smallest scales the largest contributions come from low-mass halos. This is due to the fact that low-mass halos contain only a very small number of galaxies, which enhances the probability to have central-satellite correlations in the one-halo term. Note that for the right panel of Fig. 4.4 we changed the minimal mass to mmin = 1011h−1M. Reducing mmin leads to a higher number density of galaxies n¯g resulting in a suppression of the one-halo term which is normalized by the inverse square of the number density.
10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105
101 100
10-1 10-2
∆gg (k,z=0)
k [h Mpc-1] mmin=1012 h-1 M⊙ 1-halo
2-halo
10-2 10-1 100 101 102 k [h Mpc-1]
mmin=1011 h-1 M⊙
Figure 4.4: Dimensionless galaxy power spectrum at redshift z= 0. The left panel depicts the influences on the one- (red line) and two-halo terms (blue line) by neglecting particular assumptions of the galaxy halo model. First the green and the magenta curve show the results for the one-halo and two-halo terms if we consider halos that contain only satellite galaxies.
The light blue curve shows the effect of additionally neglecting the sub-Poissonian nature of the HOD at low halo masses. Finally, the black curve shows the effect of neglecting the threshold mass for building galaxies. The right panel shows the contributions to the one-halo term (red line) as thin black lines from halos in a particular mass range: for large scales (k.1hMpc−1) the dominant contribution comes from the most massive halos in the range 1014h−1M<
m <1016h−1M. The contribution for the mass ranges 1013h−1M < m <1014h−1M, 1012h−1M< m <1013h−1M and 1011h−1M< m <1012h−1M becomes successively smaller. However, on the smallest scales (k'102hMpc−1) this behavior is reversed and the dominant contribution comes from small mass halos. For both panels we used the fiducial model (see Model Ain Table A.1) but for the right panel we changed the minimal mass from mmin= 1012h−1M tommin = 1011h−1M which reduces the power spectrum.
cosmological model. Here we employ the halo model formalism for dark matter and galaxy clustering as developed in the previous sections. Encouraged by the agreement of the results from the halo model and simulations on the two-point level (e.g., Zehavi et al. 2004) we want to extend it to the three- and four-point levels which are needed for galaxy-galaxy-galaxy lensing (see Chapter 6) and to quantify the errors for galaxy-galaxy lensing (see Chapter 7), respectively.
The halo model for dark matter and galaxy clustering can be combined to calculate the cross-spectra without introducing new concepts. More precisely, the cross-spectra in any order can be constructed using the building blocks of the dark matter spectra in Eq. (3.125) and the building blocks of galaxy clustering in Eq.(4.24) discussed in the previous sections.
4.3.1 Galaxy-Dark Matter Cross-Power Spectrum
The cross-power spectrum Pδg(k) is defined as the two-point correlator of the dark matter and the galaxy density contrast, i.e.,
hδ(k)δg(k0)i= (2π)3δD(k+k0)Pδg(k). (4.34) We present the explicit form of the cross-power spectrum making use of the clustering formalism of galaxies and dark matter developed above. The one-halo term is given as
P1-hδg (nc)(k) = Z
dm n(m) m
¯ ρ
hN(m)i
¯
ng u˜dm(k, m)˜ug(k, m). (4.35) When we place one galaxy in the center of the halo the result is
P1-hδg(k) = Z
dm n(m) m
¯ ρ
˜
udm(k, m)
N¯cen + ¯Nsatu˜g(k, m)
¯ ng
. (4.36) Similarly, we find for the two-halo term including central galaxies
P2-hδg(k) =Ppt(k) Z
dm n(m) m
¯ ρ
˜
udm(k, m)bh1(m)
×
"
Z
dm n(m)
N¯cen+ ¯Nsatu˜g(k, m)
¯
ng bh1(m)
#
. (4.37)
Using the building blocks for the dark matter and galaxy spectra we can write the two-halo term in the compact notation
P2-hδg(k) =I11(k)G11(k)Ppt(k), (4.38) where we used Eq. (4.32) and
I11(k) = Z
dm n(m) m
¯ ρ
˜
udm(k, m)bh1(m). (4.39)
10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105
101 100
10-1 10-2
∆gδ (k,z=0)
k [h Mpc-1] 1-halo
2-halo Total
10-2 10-1 100 101 102
∆δδ (k,z=0)
k [h Mpc-1]
Figure 4.5: The plot shows the dimensionless galaxy-dark matter cross-power spectrum (left panel) and the dark matter power spectrum (right panel) against wave-number kat redshift z = 0. Both plots show the separate contribution from the one- and two-halo term to the total power spectrum.
The total cross-power spectrum is then the sum of the one- and two-halo terms Pδg(k) = P1-hδg(k) +P2-hδg(k). (4.40) The dimensionless cross-power spectrum is shown in the left panel of Fig. 4.5. The right panel depicts the dimensionless dark matter power spectrum for comparison. The cross-power spectrum is well described by a single power law and is very similar to the galaxy power spectrum. This is due to the fact that both power spectra have the same dependence on the normalized density profile if galaxies follow the dark matter profile.
In addition, they are both influenced by the low-mass cutoff mmin. We will analyze the ratio of both spectra in detail in Sect. 4.3.3 where we discuss scale-dependent bias.
4.3.2 Large-Scale Galaxy Bias Parameters
The theoretical prediction of the bias between galaxy and dark matter clustering from first principles is not feasible up to now because the bias is in general nonlinear and stochastical. But on sufficiently large scales one can assume that galaxy formation is a local process and thus depends only on the local matter density field. Hence, in this case there is a local mapping of the dark matter density contrast to the galaxy density contrast
δg(x)≡f[δ(x)], (4.41)
where the functionf describes the mapping. As on large scales |δ| 1, one can expand the galaxy overdensityδg at comoving position xin a Taylor series (see Fry & Gaztanaga
1993) around δ= 0 of the underlying matter overdensity, i.e., δg(x) = bL1δ(x) + bL2
2δ2(x) + bL3
6δ3(x) +. . . , (4.42) where we introduced the constant first-, second- and third-order bias parameters bL1, bL2 and bL3, respectively. The bias parameters are defined by the derivatives of the mapping function:
bLi ≡ di dδif(δ)
δ=0. (4.43)
Transforming Eq. (4.42) into Fourier space yields δg(k) = bL1δ(k) + bL2
2
Z d3q1
(2π)3 δ(q1)δ(k−q1) +bL3
6
Z d3q1 (2π)3
Z d3q2
(2π)3 δ(q1)δ(q2)δ(k−q1−q2) +. . . (4.44) Using this expression, we can easily calculate the galaxy two-point correlator to lowest order in δ and we find for the galaxy power spectrum (see definition in Eq. 4.25)
Pgg(k) = bL12
Ppt(k). (4.45)
The cross-power spectrum defined in Eq. (4.34) is accordingly
Pδg(k) = bL1Ppt(k). (4.46) Similarly, we find for the galaxy bispectrum (Matarrese et al. 1997)
Bggg(k1, k2, k3) = (bL1)3Bpt(k1, k2, k3) + (bL1)2bL2 [Ppt(k1)Ppt(k2) +Ppt(k1)Ppt(k3)
+Ppt(k2)Ppt(k3)] . (4.47)
Note that we define the galaxy bispectrum in Eq. (4.54) below.
The framework of the galaxy halo model allows us to calculate these large-scale bias parameters. On large scales the n-point spectra are dominated by the n-point halo correlation function between the n halo centers. As the normalized density profile goes asymptotically to u˜g(k = 0, m)→1, we find for thei-th order large-scale galaxy bias factor (Scoccimarro et al. 2001)
bLi(z) = 1
¯ ng
Z
dm n(m, z)hN(m)ibhi(m), (4.48) wherebhi is the bias between halos and the underlying dark matter field and we explicitly wrote the redshift dependence. This equation allows the determination of the large-scale galaxy bias parameters from the well-known halo bias parameters that are extensively tested in dissipationless simulations. This method circumvents the modeling of galaxy
clustering to compute the bias parameters. For example setting u˜g(k, m) → 1 in Eq. (4.29) yields
P2-hgg(k, z) =
bL1(z)2
Ppt(k, z) (4.49)
with the first-order bias parameter bL1(z) = 1
¯ ng
Z
dm n(m, z)[ ¯Nsat+ ¯Ncen]bh1(m). (4.50) Similar we can derive the results for the higher-order bias parameters which fulfill Eq. (4.48).
4.3.3 Scale-Dependent Power Spectrum Bias
In the previous section we explored the large-scale bias between galaxies and the dark matter field. Here, we extend this concept with the help of the halo model now to small scales. On large scales the bias goes to a constant value as described by Eq. (4.48).
However, on small scales the bias factor is non-monotonic and scale-dependent, as will be shown below.
The relation between the galaxy and the dark matter power spectra can be parame-trized by
Pgg(k, w) =b2(k, w)Pδδ(k, w), (4.51) whereb(k, w) is the in general scale- and redshift-dependent bias parameter where the redshift dependence is encoded in the comoving distance, i.e., w≡w(z). Similarly, the relation between the cross- and the dark matter power spectra is given by
Pδg(k, w) =b(k, w)r(k, w)Pδδ(k, w), (4.52) where we additionally introduced the scale- and redshift-dependent galaxy-mass cor-relation coefficient r(k, w). For large scales the bias factor is described by the linear deterministic bias, whereb(k, w)→bL1(w), and the correlation coefficient goes to 1. This follows from the large-scale solutions in Eqs. (4.45)and (4.46). We build the following ratios
b(k, w) = s
Pgg(k, w)
Pδδ(k, w), b(k, w)
r(k, w) = Pgg(k, w)
Pδg(k, w), (4.53) and present them in dependence of the wave-number k in Fig. 4.6. The figure shows both quantities for three different minimal masses. The galaxy bias and the ratio b/r resemble the linear deterministic bias on large scales. On small scales the galaxy bias becomes highly scale-dependent, whereas the ratio of the galaxy and cross-power spectra is nearly constant over the scales considered. We note that a higher threshold mass results in a smaller number density n¯g which leads to an enhancement of the galaxy power spectrum.
Figure 4.6: The left panel shows the scale-dependent bias between galaxies and dark matter.
The right panel shows the ratio of the galaxy power spectrum to the cross-power spectrum (see definitions in Eq. 4.53). Both functions are evaluated at z= 0 and shown for three different threshold masses mmin as indicated in the figure.