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

4.5.4 The Non-Linear Evolution of Two-Point Statistics
Self-similarity gives a powerful constraint on the space and time evolution of
correlation functions, by requiring that these depend only of the self-similarity
variables. However, different initial spectra can lead to very different functions
of the self-similarity variables. Hamilton et al. [289] suggested a useful way of
thinking about the non-linear evolution of the two-point correlation function,
by which the evolution from different initial spectra can all be described by
the same (approximately) universal formula, obtained empirically by fitting
to numerical simulations.
The starting point is conservation of pairs, Eq. (197), which implies
∂[x 3 (1 + ξav)]
∂τ
∂[x
+ u12
3 (1 + ξav)]
= 0. (209)
∂x
Thus, a sphere of radius x such that x3 (1 + ξav) ≡ x3 L is independent of time
will contain the same number of neighbors throughout non-linear evolution. At
early times, when fluctuations are small, xL ≈ x; as clustering develops and
becomes non-linear, x becomes smaller than xL. This motivated the ansatz
that the non-linear average two-point correlation function at scale x should
be a function of the linear one at scale xL [289]
ξav(x, τ) = Fmap[ξav L(xL, τ)], (210)
where the mapping Fmap was assumed to be universal, i.e. independent of
initial conditions. Using more recent numerical simulations [335] showed that
there is a dependence of Fmap on spectral index (particularly as n < −1);
in addition [493] extended the mapping above to the power spectrum and
arbitrary Ωm and ΩΛ. In this case, the non-linear power spectrum at scale k is
assumed to be a function of the linear power spectrum at scale kL, such that
k = [1 + ∆(k)] 1/3 kL, where ∆(k) ≡ 4πk 3 P(k),
∆(k, τ) = Fn,Ωm,ΩΛ [∆(kL, τ)], (211)
where it is emphasized that the mapping depends on spectral index and cosmological
parameters. Several groups have reported improved fitting formulae
that take into account these extra dependences [335,30,494]. In the most often
used version, the fitting function Fmap contains 5 free functions of the spectral
index n which interpolate between Fmap(x) ≈ x in the linear regime and
Fmap ≈ x 3/2 in the non-linear regime where stable clustering is assumed to
hold [494]
Fmap(x) = x 1 + Bβx + [Ax] αβ
1 + [(Ax) αg3 (Ω)/(V x1/2 )] β
1/β , (212)
where A = 0.482(1 + n/3) −0.947 , B = 0.226(1 + n/3) −1.778 , α = 3.310(1 +
n/3) −0.244 , β = 0.862(1 + n/3) −0.287 , V = 11.55(1 + n/3) −0.423 , and the linear
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