In this section we derive the explicit form of the galaxy power spectrum using the halo model formalism.
The galaxy power spectrum is defined by the two-point correlator
hδg(k)δg(k0)i= (2π)3δD(k+k0)Pgg(k). (4.25) The one-halo term of the galaxy power spectrum is weighted by the mean number of pairs (including repetition) and in order to correctly normalize the equation we divide by
the square of the mean number density of galaxies. Considering a halo which contains only satellite galaxies we find
P1-h (nc)gg (k) = Z
dm n(m)hN(N −1)(m)i
¯
n2g u˜2g(k, m). (4.26) and we label this contribution with the subscript “(nc)” in the following. When we furthermore assume that the first galaxy in each halo is placed at the center of the host halo, terms involving correlations with central galaxies are reduced by a factor of one density profile compared to terms involving correlations with only satellite galaxies (see discussion in Appendix B.7). Under this assumption the one-halo term of the galaxy power spectrum is given by
P1-hgg(k) = Z
dm n(m)
[ ¯Nsatu˜g(k, m)]2+ 2 ¯NcenN¯satu˜g(k, m)
¯ n2g
, (4.27) where we used Eq.(4.16)to relate the second-order moment of the HOD to the first-order moments of satellite and central galaxies. The first term in the integrand originates from satellite-satellite correlations and the second from central-satellite correlations.
Note that there is no central-central correlation because each halo contains only one central galaxy. Similarly, the two-halo term is given by
P2-h (nc)gg (k) = Z
dm n(m)hN(m)i
¯
ng u˜g(k, m)bh1(m) 2
Ppt(k). (4.28) Assuming that each halo contains a central galaxy, the two-halo term is given by
P2-hgg(k) =
"
Z
dm n(m)
N¯cen + ¯Nsatu˜g(k, m)
¯
ng bh1(m)
#2
Ppt(k), (4.29) where we have satellite-satellite, central-satellite and central-central correlations. The first term in square brackets is sometimes neglected because the contribution of central galaxies only dominates the two-halo term on small scales where the one-halo term is anyway larger than the two-halo term. This is due to the fact that on small scales the dominant contribution comes from small mass halos, which is discussed in detail at the end of this section. The total galaxy power spectrum is the sum of the one- and two-halo terms
Pgg(k) = P1-hgg(k) +P2-hgg(k). (4.30) Using the building blocks, introduced in Eq. (4.24), we can write both terms in the compact form
Pgg(k) =G02(k, k) +Ppt(k)[G11(k)]2. (4.31) where
G11(k) = Z
dm n(m)
N¯cen+ ¯Nsatu˜g(k, m)
¯
ng bh1(m) (4.32)
and
G02(k, k) = Z
dm n(m)
N¯satu˜g(k, m)2
+ 2 ¯NcenN¯satu˜g(k, m)
¯
n2g . (4.33)
Note that for large scales (k knl)8 the one-halo term (4.27) does not go to zero but converges to a constant scale-independent white noise spectrum sinceu(k, m)˜ →1 in this case. As the linear power spectrum is proportional tokat small k, the constant one-halo term eventually gives the dominant contribution making the halo model inconsistent with linear perturbation theory on the largest scales9. In the review paper by Cooray
& Sheth (2002), we find an ansatz to solve this problem by using compensated density profiles which are constructed such that u(k, m) =˜ u˜g(k, m)→0 at smallk. However, in this case there will be no power on large scales, since the two-halo term also depends on the density profile. Thus, this weakness of the halo model remains and it is not clear how to solve this problem. On the other hand, the two-halo term (4.29) is in accordance with the results from perturbation theory since on large scales we find P2-hgg(k) =b2Ppt(k), where b is the linear bias factor defined by the large-scale limit of G11 in Eq. (4.32).
In the presented formalism several processes have been neglected. First of all, we did not account for the effect of halo exclusion. Different halos in simulations are never separated by distances smaller than the sum of their virial radii since otherwise they would have been linked to one larger halo. This effect is not included in our description of the two-halo terms of the dark matter and galaxy power spectra. To correct for this effect, we need to exclude overlapping halos in the two-halo terms. Secondly, simulations indicate that the halo bias is scale-dependent, whereas the halo model calculation in Sect. 3.4 predicts a scale-independent halo bias (see for example the Press-Schechter halo bias in Eq. 3.76). One possibility is to determine the scale-dependent bias factor directly from simulations. However, then we would need to use the nonlinear dark matter power spectrum for the clustering of halo centers that is obtained from simulations for consistency instead ofPpt. The drawback of using simulations to obtain the halo bias is that there is no simple way to generalize the results to higher-order spectra, like the galaxy bispectrum. In addition, the cosmology dependence of the scale-dependent halo bias is not well tested. A detailed description of all these effects can be found in the appendix of Tinker et al. (2005). We refrain from using these halo model extensions because they are only valid and tested at the two-point level. As we mainly aim at modeling higher-order correlation functions there is no simple extension of this formalism.
Also, all these corrections are only important for the transition of the two-halo to the one-halo term at intermediate scales. In particular, the nonlinear part which is probed by the one-halo term is unaffected by these corrections.
The dimensionless galaxy power spectrum (see definition of dimensionless power
8Here,knlis the nonlinear wave-number defined as the scale where the dimensionless power spectrum is 1.
9Note that the one-halo term of the dark matter power spectrum (see Eq. 3.103) suffers from the same inconsistency.
Figure 4.3: The plot shows in the left panel the dimensionless galaxy power spectrum defined in Eq.(2.107) and in the right panel the dark matter power spectrum against wave-number k at redshiftz= 0. Both plots show the separate contributions from the one- and two-halo terms to the total power spectrum.
spectrum in Eq. 2.107) is plotted in the left panel of Fig. 4.3 and the right panel shows the dimensionless dark matter power spectrum for comparison. Both panels show the one- and the two-halo term contributions to the corresponding total power spectrum for z = 0. The first point to note is the approximate power-law behavior of the galaxy power spectrum over the whole range of scales. The change from the large-scale regime described by the two-halo term to the small-scale regime dominated by the one-halo term at around k ' 0.5hMpc−1 is not well modeled by our halo model method and results in a small dip in the total power spectrum. The dark matter power spectrum is clearly not described by a single power law. Furthermore, we note both power spectra have a similar amplitude on large scales. The ratio of the dark matter and galaxy power spectrum will be quantified in Sect. 4.3.3 where we introduce the scale-dependent bias between galaxy and dark matter clustering.
In summary, the halo model reproduces the observed power-law behavior of the galaxy power spectrum. The origin of this behavior is best seen by comparing the one- and two-halo terms of the galaxy power spectrum in Eqs. (4.27) and (4.29) to the dark matter power spectrum in Eqs. (3.103) and(3.106). Both power spectra are similar if the radial distribution of galaxies trace the dark matter density profile, the HOD is described by a Poisson distribution over the whole mass range (i.e., there is no minimal mass) and the first moment of satellites, N¯sat, depends linearly on the host halo mass.
The only difference left between the power spectra is the different normalization either by the mean galaxy density or by the mean dark matter density.
To illustrate the importance of specific assumptions made in the galaxy halo model, we show in the left panel of Fig. 4.4 the one- and the two-halo terms of the dimensionless
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.