Galaxy Biasing: Nonlinear, Stochastic and Measurable

Original · EN

I describe a general formalism for galaxy biasing (Dekel & Lahav 1998) and its application to measurements of beta (=Omega⁰.6/b), e.g. via direct comparisons of light and mass and via redshift distortions. The linear and deterministic relation g=b*d between the density fluctuation fields of galaxies g and mass d is replaced by the conditional distribution P(g|d) of these as random fields, smoothed at a given scale and at a given time. The mean biasing and its nonlinearity are characterized by the conditional mean <g|d>=b(d)*d and the local scatter by the conditional variance sb²(d). This scatter arises from hidden effects on galaxy formation and from shot noise. For applications involving second-order local moments, the biasing is defined by three natural parameters: the slope bₕ of the regression of g on d (replacing b), a nonlinearity parameter bₜ, and a scatter parameter sb. The ratio of variances bᵥ² and the correlation coefficient r mix these parameters. The nonlinearity and scatter lead to underestimates of order bₜ²/bₕ² and sb²/bₕ² in the different estimators of beta, which may partly explain the range of estimates. Local stochasticity affects the redshift-distortion analysis only by limiting the useful range of scales. In this range, for linear stochastic biasing, the analysis reduces to Kaiser's formula for bₕ (not bᵥ) independent of the scatter. The distortion analysis is affected by nonlinearity but in a weak way. Estimates of the nontrivial features of the biasing scheme are made based on simulations and toy models, and a new method for measuring them via distribution functions is proposed.