Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 57 / 3 / d

Let $G={\Phi }_G$ and $f=f_{NL}^{\mathrm{local}}$. The Gaussian random field covariance in Fourier space isThe Fourier transform of the centered quadratic term isThe Gaussian three-point function vanishes. At first order in $f$, the only terms are $f⟨Q_1{G}_2{G}_3⟩$ and its two cyclic placements. For the first, Wick theorem pairs four Gaussian factors in three ways. Pairing the two factors inside $Q_1$ together is canceled by the mean subtraction. Pairing one of those factors with $G_2$ and the other with $G_3$ gives $P(k_2)P(k_3)$ and the overall momentum delta; exchanging the two factors gives the same contribution. ThusAdding the three placements givesThere is no second-order term because it would contain five centered Gaussian factors. The two-point function receives its first correction at $f^2$, whereas the displayed local-type primordial non-Gaussianity gives a three-point correlation already at $f$. For $P\propto k^{-3}$ the squeezed limit is $B_{\Phi }(q,k,k)\simeq 4fP(q)P(k)$; this explains the strong correlation between a long potential and short-scale power. A finite variance or regulator is needed to define the subtracted $⟨G^2⟩$, as noted in part (b).