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

Fig. 16. The left panel shows the one-loop bispectrum predictions for CDM model
at scales approaching the non-linear regime, for k1/k2 = 2 and ∆ ≈ 1 (left) against
numerical simulations [560]. The right panel shows the saturation of ˜ Q at small
scales in the highly non-linear regime, for two different ratios for k1/k2 = 2,3 and
∆ > ∼ 100 [563]. Dashed lines in both panels correspond to tree-level PT results.
configuration dependence of the bispectrum present at the largest scales (and
given by tree-level perturbation theory), giving rise to the so-called hierarchical
form Q ≈ const in the strongly non-linear regime (see Sect. 4.5.5). One-loop
perturbation theory must predict this feature in order to be a good description
of the transition to the nonlinear regime. In fact, numerical integration [559]
of the one loop bispectrum for different spectral indices from n = −2 to
n = −1 shows that there is a change in behavior of the nonlinear evolution:
for n < ∼ −1.4 the one-loop corrections enhance the configuration dependence
of the bispectrum, whereas for n > ∼ −1.4, they tend to cancel it, in qualitative
agreement with numerical simulations. Note that this “critical index” nc ≈
−1.4 is the same spectral index at which one-loop corrections to the power
spectrum vanish, marking the transition between faster and slower than linear
growth of the variance of density fluctuations.
4.3 The Power Spectrum in the Zel’dovich Approximation
The Zel’dovich approximation (ZA, [705]) is one of the rare cases in which
exact (non-perturbative) results can be obtained. However, given the drastic
approximation to the dynamics, these exact results for the evolution of clustering
statistics are of limited interest due to their restricted regime of validity.
The reason behind this is that in the ZA when different streams cross they
pass each other without interacting, because the evolution of fluid elements is
local. As a result, high-density regions become washed out. Nonetheless, the
ZA often provides useful insights into non-linear behavior.
For Gaussian initial conditions, the full non-linear power spectrum in the
ZA can be obtained as follows [77,430,556,220,642]. Changing from Eule-
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