Standard perturbation theory density kernel ($F_n$)

The symmetrized kernel $F_n$ convolves $n$ linear density fields to produce the $n$th-order density perturbation. In an Einstein-de Sitter universe,

For plane-parallel pressureless matter in an Einstein-de Sitter universe, before shell crossing, the symmetrized growing-mode standard perturbation theory density kernel isIn particular $F_{2,s}=(q_1+q_2{)}^2/(2q_1{q}_2)$. An unsymmetrized representation of the one-dimensional beta mode-coupling kernel is $(q_1+q_2)/q_1$; its symmetric part is the expression above.

At fixed external momentum $\mathbf{k}$, mass and momentum conservation implyas $q\to \infty$. In particular, the constant term in the hard-momentum expansion must cancel.

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.

For Gaussian initial conditions, the tree-level connected matter five-point function has perturbative-order partitionsThey correspond respectively to a four-valent star, an $F_3$--$F_2$ tree, and a chain of three $F_2$ vertices. Every topology contains four linear power spectra.