Se p 19 98 Galaxy Biasing : Nonlinear , Stochastic and Measurable 1

I describe a general formalism for galaxy biasing [20] and its application to measurements of β (≡ Ω 0.6 /b), e.g., via direct comparisons of light and mass and via redshift distortions. The linear and deterministic relation g = bδ between the density fluctuation fields of galaxies g and mass δ is replaced by the conditional distribution P (g|δ) of these as random fields, smoothed at a given scale and at a given time. The mean biasing and its non-linearity are characterized by the conditional mean g|δ ≡ b(δ) δ, and the local scatter by the conditional variance σ 2 b (δ). 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 slopê b of the regression of g on δ (replacing b), a non-linearity parameter˜b, and a scatter parameter σ b. The ratio of variances b 2 var and the correlation coefficient r mix these parameters. The non-linearity and scatter lead to underestimates of order˜b 2 / ˆ b 2 and σ 2 b / ˆ b 2 in the different estimators of β, 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 forˆb (not bvar) independent of the scatter. The distortion analysis is affected by non-linearity but in a weak way. Estimates of the nontrivial features of the biasing scheme are made based on simulations [54] and toy models, and a new method for measuring them via distribution functions is proposed [53].