R. Scoccimarro et J. Frieman, LOOP CORRECTIONS IN NONLINEAR COSMOLOGICAL PERTURBATION-THEORY, The Astrophysical journal. Supplement series, 105(1), 1996, pp. 37-73
Using a diagrammatic approach to Eulerian perturbation theory, we calc
ulate analytically the variance and skewness of the density and veloci
ty divergence induced by gravitational evolution from Gaussian initial
conditions, including corrections beyond leading order. Except for th
e power spectrum, previous calculations in cosmological perturbation t
heory have been confined to leading order (tree level): we extend thes
e to include loop corrections. For scale-free initial power spectra, P
(k) similar to k(n) with -2 less than or equal to n less than or equal
to 2, the one-loop variance sigma(2) = [delta(2)] = sigma(l)(2) + 1.8
2 sigma(l)(4), and the skewness S-3 = [delta(3)]/sigma(4) = 34/7 + 9.8
sigma(l)(2), where sigma(l) is the rms fluctuation of the density fie
ld to linear order. (These results depend weakly on the spectral index
n, due to the nonlocality of the nonlinear solutions to the equations
of motion.) Thus, loop corrections for the (unsmoothed) density held
begin to dominate over tree-level contributions (and perturbation theo
ry presumably begins to break down) when sigma(l)(2) similar or equal
to 1/2. For the divergence of the velocity field, loop dominance does
not occur until sigma(l)(2) approximate to 1. We also compute loop cor
rections to the variance, skewness, and kurtosis for several nonlinear
approximation schemes, where the calculation can be easily generalize
d to one-point cumulants of higher order and arbitrary number of loops
. We find that the Zeldovich approximation gives the best approximatio
n to the loop corrections of exact perturbation theory, followed by th
e linear potential approximation (LPA) and the frozen flow approximati
on (FFA), in qualitative agreement with the relative behavior of tree-
level results. In LPA and FFA, loop corrections are infrared divergent
for spectral indices n less than or equal to -1; this is related to t
he breaking of Galilean invariance in these schemes.