Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 312 / 4 / a / ii / Solution

Use the Fourier transform convention $\zeta (\mathbf{x})=\int d^{3}k{e}^{i\mathbf{k}\cdot \mathbf{x}}\zeta (\mathbf{k})/(2\pi {)}^3$. Statistical homogeneity and isotropy define the primordial bispectrum byThe Dirac delta function enforces momentum conservation; the three magnitudes describe a closed triangle. Set $\alpha =3f_{\mathrm{NL}}/5$. For the centered quadratic Gaussian transformation, its quadratic Fourier term isAt first order in $\alpha$, choose the quadratic field at one of the three external positions. For the first position, Wick contractions pair the two internal fields with the two remaining external fields in two ways. The self-pairing is precisely removed by the subtracted mean. The surviving contribution is $2\alpha P(k_2)P(k_3)$ times the momentum delta. Adding the other positions gives the tree-level local-type primordial non-Gaussianity resultThis expression is at leading order in $f_{\mathrm{NL}}$. The exact quadratic map also has a loop correction to the local primordial bispectrum, from three quadratic vertices:There is no quadratic-in-$\alpha$ bispectrum term, because it would involve five centered Gaussian fields. A regulator may be needed for this higher-order integral. The displayed leading formula uses the Gaussian power $P$, consistently neglecting its order-$f_{\mathrm{NL}}^2$ correction.