Local quadratic galaxy bias

The simplest nonlinear local bias model isFor Gaussian matter fluctuations, its leading galaxy three-point function is proportional to $b_1^2{b}_2$.

The galaxy bispectrum measures the connected three-point correlation of the Fourier modes of the galaxy density contrast. For a mean-zero field, statistical homogeneity givesThe delta function enforces a closed wavevector triangle. With the additional assumption of statistical isotropy, one can write $B_g(k_1,k_2,k_3)$ as a function of triangle size and shape. Statistical homogeneity alone leaves possible orientation dependence.

The galaxy bispectrum adds information beyond the power spectrum. For example, take a zero-mean Gaussian random field $\delta$ and a local tracer model ${\delta }_g=b_1\delta +(b_2/2)({\delta }^2-⟨{\delta }^2⟩)$. Two Wick contractions at each quadratic insertion cancel its factor of two, yielding at leading orderThus nonlinear galaxy bias can generate a three-point signal even from Gaussian matter fluctuations. Nonlinear gravitational evolution supplies a further matter-bispectrum contribution. Both effects must be modeled before interpreting a measured galaxy three-point signal as primordial non-Gaussianity.^[ref]