286 Gravitational instability in Newtonian theory
substituting (6.113) into (6.86), we find
ε(q,t)=ε 0[
1 −
(
(αβ+αγ+βγ)δi^2 − 2 αβγ δ^3 i)]
( 1 −αδi)( 1 −βδi)( 1 −γδi). (6.114)
On the other hand, it follows from (6.88) thatε should be given by the ex-
pression in (6.112). Hence the expected error of the Zel’dovich approxima-
tionis of order the disagreement between the results in (6.112) and (6.114),
that is, ∼O
(
(αβ+αγ+βγ)δi^2 ,αβγδi^3)
. When the perturbations are small,
αδi,βδi,γδi1, the Zel’dovich approximation reproduces the results of the lin-
ear perturbation theory. However, in the nonlinear regime, it is not very reliable. If,
for example,αβ,γ, then we can trust only the leading term, given by (6.110),
which describes one-dimensional contraction alongα-axis. The linear corrections
βδi,γδi1 become unreliable whenαδireaches a value of order unity. Ifα∼β,
then the formula in (6.112) fails to reproduce even the basic feature of the nonlinear
collapse.
6.4.3 Cosmic web
The strain tensor is very useful when we try to understand the nonlinear large-
scale structure of the universe. The initial inhomogeneities can be characterized
completely by the strain tensor, or equivalently by three functionsα
(
qi)
,β(
qi)
,
γ
(
qi)
. How an inhomogeneity will grow in a particular region depends on the
relation between the values ofα, β, γ. Based on the results above, we see that
the collapse is one-dimensional and produces two-dimensional pancakes (walls)
in those regions whereαβ,γ. In the places whereα∼βγ, we expect
two-dimensional collapse, leading to formation of one-dimensional filaments. For
α∼β∼γthe collapse is nearly spherical.
For initial Gaussian perturbations the probability distribution of the strain tensor
eigenvalues can be calculated exactly. A helpfullower dimensionalvisualization
of the initial density field is a mountainous landscape in which the mountain peaks
represent local maxima in the density and valleys correspond to local minima. In
the concordance model (cold dark matter plus inflationary perturbation spectrum),
inhomoheneities with significant amplitudes are present in nearly all scales and
hence mountains with nearly all base sizes are superimposed on each other. If we
are interested in the structure on scales exceeding some particular size, we have to
smear the inhomogeneities on smaller scales, in other words, remove the mountains
with small base sizes.
The first nonlinear structures are obviously formed near the tallest peaks where
the energy density takes its maximal value. Typically, the two curvature scales are
comparable near the mountain peak. Therefore, we expect that at a peak in the