= 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 $a^nF_n$. 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.
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