Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 312 / 3 / a / Solution

For nonzero momentum, the Fourier transform of the local bias expansion isThe subtracted variance contributes only at $\mathbf{k}=0$. Because $\delta$ is a Gaussian random field, its three-point function vanishes and its four-point function factorizes by Wick theorem. At leading order one of the three galaxy fields supplies the quadratic term and the other two supply linear terms. The two cross-contractions cancel the factor $1/2$, givingThus even Gaussian matter fluctuations acquire a nonzero galaxy bispectrum through local quadratic galaxy bias.