58 An introduction to the physics of cosmology
This all agrees with what we knew already: at early times the sphere expands with
thea∝t^2 /^3 Hubble flow and density perturbations grow proportional toa.
We can now see how linear theory breaks down as the perturbation evolves.
There are three interesting epochs in the final stages of its development, which
we can read directly from the above solutions. Here, to keep things simple, we
compare only with linear theory for an=1 background.
(1) Turnround. The sphere breaks away from the general expansion and reaches
a maximum radius atθ = π,t = πB. At this point, the true density
enhancement with respect to the background is just[A( 6 t/B)^2 /^3 / 2 ]^3 /r^3 =
9 π^2 / 16 5 .55.
(2) Collapse. If only gravity operates, then the sphere will collapse to a
singularity atθ= 2 π. This occurs whenδlin=( 3 / 20 )( 12 π)^2 /^3 1 .69.
(3) Virialization. Consider the time at which the sphere has collapsed by a
factor 2 from maximum expansion. At this point, it has kinetic energy
Krelated to potential energyVbyV =− 2 K. This is the condition for
equilibrium, according to thevirial theorem. For this reason, many workers
take this epoch as indicating the sort of density contrast to be expected as
the endpoint of gravitational collapse. This occurs atθ = 3 π/2, and the
corresponding density enhancement is( 9 π+ 6 )^2 / 8 147, withδlin 1 .58.
Some authors prefer to assume that this virialized size is eventually achieved
only at collapse, in which case the contrast becomes( 6 π)^2 / 2 178.These calculations are the basis for a common ‘rule of thumb’, whereby one
assumes that linear theory applies untilδlinis equal to someδca little greater than
unity, at which point virialization is deemed to have occurred. Although this only
applies for=1, analogous results can be worked out from the fullδlin(z,)
andt(z,)relations;δlin1 is a good criterion for collapse for any value of
likely to be of practical relevance. The full density contrast at virialization may
be approximated by
1 +δvir 178 −^0.^7
(although flat-dominated models show less dependence on;Ekeet al1996).
2.7 Quantifying large-scale structure
The next step is to see how these theoretical ideas can be confronted with
statistical measures of the observed matter distribution, and to summarize what
is known about the dimensionless density perturbation field
δ(x)≡ρ(x)−〈ρ〉
〈ρ〉.
A critical feature of theδfield is that it inhabits a universe that is isotropic
and homogeneous in its large-scale properties. This suggests that the statistical