Bias expansion

A bias expansion relates the density of selected tracers to the matter distribution using allowed combinations of density contrast, tidal effects, spatial derivatives and stochastic contributions. For example, ${\delta }_g=b_1\delta +\frac{1}{2}{b}_2({\delta }^2-⟨{\delta }^2⟩)+\cdots$ illustrates a local part. The coefficients encode tracer-formation physics. Including further allowed operators makes the expansion systematically improvable on sufficiently large scales.^[ref]

Galaxy bias describes the statistical relation between fluctuations in galaxy counts and the underlying matter density contrast. In the simplest large-scale limit, ${\delta }_g=b_1{\delta }_m$, so the corresponding deterministic power spectrum is $b_1^2{P}_m$. Different selections can have different $b_1$ because they sample different formation environments. More generally, the bias expansion includes nonlinear, tidal and stochastic terms, so a single multiplicative constant is an approximation.^[ref]

Number conservation under the displacement from Lagrangian coordinates to Eulerian coordinates givesThe added one is the linear matter-density response of the coordinate Jacobian.

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]