Large-Scale Structure of the Universe and Cosmological ...

proximation. In fact, Eq. (616) is the Fourier analog of the so-called “streaming
model” [508], as modified in [219] to take into account the density-velocity
coupling.
The physical interpretation of this result is as follows. The factors in square
brackets denote the amplification of the power spectrum in redshift space due
to infall (and they constitute the only contribution in linear theory, giving
Kaiser’s [362] result). This gives a positive contribution to the quadrupole
(l = 2) and hexacadupole (l = 4) anisotropies. On the other hand, at small
scales, as k increases the exponential factor starts to play a role, decreasing
the power due to oscillations coming from the pairwise velocity along the line
of sight. This leads to a decrease in monopole and quadrupole power with
respect to the linear contribution; in particular, the quadrupole changes sign.
In order to describe the non-linear behavior of the redshift-space power spectrum,
it has become popular to resort to a phenomenological model to take
into account the velocity dispersion effects [493]. In this case, the non-linear
distortions of the power spectrum in redshift-space are written in terms of
the linear squashing factor and a suitable damping factor due to the pairwisevelocity
distribution function
Ps(k) = Pg(k)
(1 + βµ 2 ) 2
[1 + (kµσv) 2 . (617)
/2] 2
Here σv is a free parameter that characterizes the velocity dispersion along
the line-of-sight. This Lorentzian form of the damping factor is motivated by
empirical results showing an exponential one-particle 104 velocity distribution
function [489]; comparison with N-body simulations have shown it to be a
good approximation [132]; however, these type of phenomenological models
tend to approach the linear PT result faster than numerical simulations [301].
In addition, although σv can be chosen to fit, say, the quadrupole-to-monopole
ratio at some range of scales, the predictions for the monopole or quadrupole
by themselves do not work as well as for their ratio.
Accuracy in describing the shape of the quadrupole to monopole ratio as a
function of scale is important, since this statistic gives a direct determination
of β from clustering in redshift surveys [290,131,132,302]. An alternative to
phenomenological models, is to obtain the redshift-space power spectrum using
approximations to the dynamics, as we now discuss.
In the case of the Zel’dovich approximation (ZA), it is possible to obtain the
redshift-space power spectrum as follows [220,642]. In the ZA, the density field
104 Alternatively, if one assumes the two-particle velocity distribution is exponential,
the suppression factor is the square root of that in Eq. (617), with σv the
pairwise velocity dispersion along the line of sight, see e.g. [18]. The observational
results regarding velocity distributions and their interpretation is briefly discussed
in Sect. 8.3.2.
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