= Solution
Use the <one-loop matter power spectrum> convention in which $P_{13}$ is one ordering of $\langle\delta^{(1)}\delta^{(3)}\rangle$. Then
$$
P(k,t)=D^2P_{\rm lin}(k)+P_{22}(k,t)+2P_{13}(k,t)+\cdots,
$$
with the <P22 contribution to the one-loop matter power spectrum>
$$
\boxed{P_{22}=2D^4\int\frac{d^3q}{(2\pi)^3}
F_{2,s}(\mathbf q,\mathbf k-\mathbf q)^2
P_{\rm lin}(q)P_{\rm lin}(|\mathbf k-\mathbf q|)}
$$
and the <P13 contribution to the one-loop matter power spectrum>
$$
\boxed{P_{13}=3D^4P_{\rm lin}(k)
\int\frac{d^3q}{(2\pi)^3}
F_{3,s}(\mathbf k,\mathbf q,-\mathbf q)P_{\rm lin}(q)}.
$$
The factors two and three count <Wick contractions>. The mixed $P_{12}$ contains an odd number of zero-mean Gaussian fields, so \b[$P_{12}=0$].
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-312-one-loop-power.png]
{title=One-loop contributions to the matter power spectrum}
{description=The two one-loop power-spectrum topologies. Solid lines attach external density fields to kernels; dashed lines pair linear fields and carry a linear power spectrum. The P13 topology has two external orderings.}
For $q\gg k$, the <ultraviolet softness of the second-order density kernel> gives $F_{2,s}=O(k^2/q^2)$, so
$$
P_{22}^{\rm UV}=O\left(D^4k^4\int\frac{d^3q}{(2\pi)^3}
\frac{P_{\rm lin}(q)^2}{q^4}\right).
$$
The third-order kernel has $F_{3,s}(\mathbf k,\mathbf q,-\mathbf q)=O(k^2/q^2)$, giving
$$
2P_{13}^{\rm UV}=O\left(D^4k^2P_{\rm lin}(k)
\int\frac{d^3q}{(2\pi)^3}\frac{P_{\rm lin}(q)}{q^2}\right).
$$
These integrals expose the sensitivity to unresolved short-scale dynamics. The <effective sound-speed counterterm in large-scale structure> has exactly the required deterministic shape:
$$
\boxed{P_{\rm ct}(k,t)=-2c_{\rm eff}^2(t)\,k^2P_L(k,t)},
\qquad P_L=D^2P_{\rm lin}.
$$
Here $c_{\rm eff}^2$ absorbs the time and normalization conventions and has dimensions of length squared. A stochastic correction begins at $k^4$ and can absorb the corresponding short-scale sensitivity of $P_{22}$.
Back to article page
