One-loop matter power spectrum

For Gaussian linear perturbations, the one-loop matter power spectrum iswhere $P_{22}$ correlates two second-order fields and $P_{13}$ correlates a linear field with a third-order field.

A scale-free linear matter spectrum has $P(k)=A{k}^n$. Dimensional analysis then makes its one-loop $P_{22}$ contribution proportional to $A^2{k}^{2n+3}$ whenever the loop integral converges.