ABSTRACT. 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 = b δ between the density fluctuation fields of galaxies g and mass δ is replaced by the conditional distribution P(|gδ) 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_b²(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 slope of the regression of g on δ (replacing b), a non-linearity parameter , and a scatter parameter σ_b. The ratio of variances b_var² and the correlation coefficient r mix these parameters. The non-linearity and scatter lead to underestimates of order ² / ² and σ_b² / 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 for (not b_var) 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].

Table of Contents

• INTRODUCTION
• LOCAL MOMENTS: VARIANCES AND LINEAR REGRESSION
• TWO-POINT CORRELATIONS: REDSHIFT DISTORTIONS
• BIASING IN SIMULATIONS AND TOY MODELS
• OBSERVATIONAL CONSTRAINTS ON BIASING
• CONCLUSIONS
• REFERENCES