 Solution

The displayed kernel is the growing-mode result of standard perturbation theory in cosmology. Its assumptions are a Newtonian, weak-field, subhorizon treatment of pressureless cold matter; a single-stream, irrotational velocity field; and negligible velocity-dispersion tensor of collisionless matter, so ${\sigma }_{ij}=0$. The background is the Einstein-de Sitter universe, which makes the growing mode proportional to $a$ and permits the separated powers $a^n{F}_n$. The density contrast is assumed perturbatively small and the decaying solutions are discarded. Gaussian initial conditions are not needed to derive $F_2$, but they are needed for the loop contractions in the later parts.