Bias expansion

A bias expansion expresses a tracer density, such as a galaxy or halo density, as a renormalized sum of local and nonlocal operators built from matter density, tidal fields, derivatives, and stochastic fields. Its coefficients encode unresolved tracer formation physics.

Table of contents
- Lagrangian-to-Eulerian linear bias relation Bias expansion
- Local quadratic galaxy bias Bias expansion
  - Galaxy bispectrum Local quadratic galaxy bias

Lagrangian-to-Eulerian linear bias relation

Number conservation under the displacement from Lagrangian coordinates to Eulerian coordinates gives

b_{1}^{(E)} = 1 + b_{1}^{(L)} .

The added one is the linear matter-density response of the coordinate Jacobian.

Local quadratic galaxy bias

The simplest nonlinear local bias model is

δ_{g} = b_{1} δ + \frac{b _{2}}{2} (δ^{2} - ⟨ δ^{2} ⟩) .

For Gaussian matter fluctuations, its leading galaxy three-point function is proportional to

b_{1}^{2} b_{2}

.

Galaxy bispectrum ( $B_{g} (k_{1}, k_{2}, k_{3})$ )

The galaxy bispectrum is the connected momentum-space three-point function of the galaxy density contrast after removing its momentum-conserving delta function.

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