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.

The $P_{22}$ contribution correlates two second-order density fields. For symmetrized kernels,

The $P_{13}$ contribution correlates a linear density field with a third-order field:Both orderings contribute $2P_{13}$ to the power spectrum.

At equal time, leading loop contributions from a very long-wavelength displacement cancel between perturbative diagrams. This is required by the equivalence principle, because a coherent long displacement merely translates short-scale structure.

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.

For a matter power spectrum $P(k)=A{k}^n$ at a fixed epoch, the Poisson equation gives potential Fourier amplitudes proportional to ${\delta }_k/k^2$, and the growing-mode peculiar velocity is proportional to ${\delta }_k\mathbf{k}/k^2$. With a spherical top-hat window function, their variances are proportional to $\int k^{n-2}W(kR{)}^{2}dk$ and $\int k^{n}W(kR{)}^{2}dk$, respectively. Rescaling $kR$ gives the displayed powers when the integrals converge: $1<n<5$ for the averaged potential and $-1<n<3$ for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for $-3<n<1$, so these three absolute averages cannot all converge for one unbroken power law at every scale.

For an unsmoothed density spectrum $P(k)=A{k}^n$ and a scale-dependent smoothing window $\tilde{W}(kR)$, substituting $q=kR$ in the variance integral gives ${\sigma }_R^2\propto R^{-(n+3)}$, provided ${\int }_0^{\infty }{q}^{n+2}|\tilde{W}(q){|}^{2}dq$ converges. For a window tending to one at zero, infrared convergence requires $n>-3$; the ultraviolet condition depends on the window. A real-space spherical top-hat also requires $n<1$.