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

Fig. 28. Non-linear evolution of the variance (left panels) and of the skewness parameter
S3 (right panels) from 10 realizations of flat CDM N-body simulations. Two
models are considered, ΛCDM with Ωm + ΩΛ = 1 and Γ = 0.2, and SCDM with
Ωm = 1 and Γ = 0.5, where Γ is the shape parameter of the power-spectrum [201].
In the left panels, symbols show the ratio of the non-linear to the linear variance as
a function of smoothing radius. The value of Γ is indicated on the panels, while σ 2 8
stands for the linear variance in a sphere of radius 8h −1 Mpc. The SC model predictions
are shown as a short-dashed line while one-loop PT predictions are shown
as a solid line. The arrows indicate where σl = 0.5. In the right panels, the output
times correspond to σ8 = 0.5 (top) and σ8 = 0.7 (bottom). Squares and triangles
correspond to measurements in Γ = 0.2 and Γ = 0.5 simulations, respectively. Each
case is compared to the corresponding PT tree-level predictions (solid lines) and SC
model (long-dashed). From [222].
a number of studies considered the evolution of higher-order moments from
non-Gaussian initial conditions given by cosmic strings [146,9] and texture
models [252] using numerical simulations. Recently, measurements of higher-
order moments in numerical simulations with χ 2 N
initial conditions with N
degrees of freedom were given in [689].
General properties of one-point moments evolved from non-Gaussian initial
conditions were considered using PT in [238,333,124,255,195]. To illustrate the
main ideas, let us write the PT expression for the first one-point moments:
〈δ 2 〉 = 〈δ 2 1
〉 +2
〈δ1δ2 〉
+ 〈δ 2 2 〉 +2 〈δ1δ3 〉 +O(σ 5 ), (289)
〈δ 3 〉 =
〈δ 3 1 〉
+ 3 〈δ 2 1δ2 〉 +
3 〈δ 2 2δ1 〉 +3 〈δ 2 1δ3 〉
+ O(σ 6 ), (290)
〈δ 4 〉 = 〈δ 4 1
〉 +4
〈δ 3 1δ2 〉
+ 6 〈δ 2 1δ2 2 〉 +4 〈δ3 1δ3 〉 +O(σ 7 ), (291)
where we simply use the PT expansion δ = δ1+δ2+. . .. Square brackets denote
terms which scale as odd-powers of δ1, and thus vanish for Gaussian initial
conditions. A first general remark one can make is that these additional terms
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