Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 312 / 2

Use the one-loop matter power spectrum convention in which $P_{13}$ is one ordering of $⟨{\delta }^{(1)}{\delta }^{(3)}⟩$. Thenwith the P22 contribution to the one-loop matter power spectrumand the P13 contribution to the one-loop matter power spectrumThe factors two and three count Wick contractions. The mixed $P_{12}$ contains an odd number of zero-mean Gaussian fields, so $P_{12}=0$.

For $q\gg k$, the ultraviolet softness of the second-order density kernel gives $F_{2,s}=O(k^2/q^2)$, soThe third-order kernel has $F_{3,s}(\mathbf{k},\mathbf{q},-\mathbf{q})=O(k^2/q^2)$, givingThese integrals expose the sensitivity to unresolved short-scale dynamics. The effective sound-speed counterterm in large-scale structure has exactly the required deterministic shape:Here $c_{\mathrm{eff}}^2$ 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 $P_{22}$.

The linear density contrast is Gaussian, so its cubic moment vanishes. The leading contribution isTwo connected Wick contractions giveThe third pairing vanishes because $F_{2,s}(\mathbf{q},-\mathbf{q})=0$. Isotropy makes $⟨\mu ⟩=0$ and $⟨{\mu }^2⟩=1/3$, so the angular average of the kernel isSince ${\sigma }^2=D^2\int d^{3}q{P}_{\mathrm{lin}}(q)/(2\pi {)}^3$ to leading order, the unsmoothed matter-density reduced skewness isThis assumes finite unsmoothed variance; inserting a smoothing window generally changes $S_3$.

Use the imaginary-axis contour of the inverse transform of the moment-generating function, $J=iy$, so the Gaussian part gives the normal distributionBecause $J^3{e}^{-J\delta }=-{\partial }_{\delta }^3{e}^{-J\delta }$, the first non-Gaussian term isDifferentiating the Gaussian three times givesHence the Edgeworth expansion isEquivalently its correction is $(S_3\sigma /6){\mathrm{He}}_3(\delta /\sigma )$, with ${\mathrm{He}}_3(z)=z^3-3z$, a Probabilists' Hermite polynomial. It integrates to zero and reproduces the third moment $S_3{\sigma }^4$. The expansion is intended for weak non-Gaussianity in the bulk, rather than the far tails.