72 An introduction to the physics of cosmology
The full version of Limber’s equation therefore gains two powers ofC(y),but
one of these is lost in converting betweenR 0 drand dx:
w(θ)=∫∞
0[C(y)]^2 y^4 φ^2 dy∫∞
−∞ξ(√
x^2 +y^2 θ^2)
dx
C(y)The net effect is therefore to replaceφ^2 (y)byC(y)φ^2 (y), so that the full power-
spectrum equation is
^2 θ=π
K∫
^2 (K/y)C(y)y^5 φ^2 (y)dy.It is also straightforward to allow for evolution. The power version of Limber’s
equation is really just telling us that the angular power from a number of different
radial shells adds incoherently, so we just need to use the actual evolved power at
that redshift. These integral equations can be inverted numerically to obtain the
real-space 3D clustering results from observations of 2D clustering; see Baugh
and Efstathiou (1993, 1994).
2.7.5 Nonlinear clustering: a problem for CDM?
Observations of galaxy clustering extend into the highly nonlinear regime,ξ.
104 , so it is essential to understand how this nonlinear clustering relates to the
linear-theory initial conditions. A useful trick for dealing with this problem is to
think of the density field under full nonlinear evolution as consisting of a set of
collapsed, virialized clusters. What is the density profile of one of these objects?
At least at separations smaller than the clump separation, the density profile of the
clusters is directly related to the correlation function, since this just measures the
number density of neighbours of a given galaxy. For a very steep cluster profile,
ρ∝r−, most galaxies will lie near the centres of clusters, and the correlation
function will be a power law,ξ(r)∝r−γ, withγ =. In general, because
the correlation function is the convolution of the density field with itself, the two
slopes differ. In the limit that clusters do not overlap, the relation isγ= 2 − 3
(for 3/ 2 <<3; see Peebles 1974 or McClelland and Silk 1977). In any case,
the critical point is that the correlation function may be be thought of as arising
directly from the density profiles of clumps in the density field.
In this picture, it is easy to see howξwill evolve with redshift, since clusters
are virialized objects that do not expand. The hypothesis ofstable clusteringstates
that, although the separation of clusters will alter as the universe expands, their
internal density structure will stay constant with time. This hypothesis clearly
breaks down in the outer regions of clusters, where the density contrast is small
and linear theory applies, but it should be applicable to small-scale clustering.
Regardingξas a density profile, its small-scale shape should therefore be fixed
inpropercoordinates, and its amplitude should scale as( 1 +z)−^3 owing to the
changing mean density of unclustered galaxies, which dilute the clustering at high