Biasing and Hierarchical Statistics in Large-scale Structure

TL;DR

This work establishes that local, nonlinear biasing preserves the hierarchical structure of large-scale matter statistics when fluctuations are small, although the hierarchical amplitudes can be altered by biasing. By using a Taylor expansion for the bias function and generating-function methods, Fry and Gaztañaga derive explicit expressions for the induced galaxy amplitudes S_g,j in terms of the matter amplitudes S_j and bias coefficients for j=3–7, and they prove general theorems showing the preservation of hierarchy for both one-point and multi-point statistics. The paper also introduces a bias transformation group and the concept of N-order biasing, providing tools to relate different galaxy samples (e.g., optical vs IRAS) and to invert observed galaxy statistics to recover matter statistics, provided bias is specified to the necessary order. Applying these ideas to optical and IRAS data reveals nonlinear bias between samples, challenging linear-bias assumptions and highlighting the need to account for bias when interpreting higher-order galaxy correlations. Overall, the results offer a rigorous framework for connecting galaxy clustering to underlying matter statistics and for testing bias models using higher-order statistics.

Abstract

In the current paradigm there is a non-trivial bias expected in the process of galaxy formation. Thus, the observed statistical properties of the galaxy distribution do not necessarily extend to the underlying matter distribution. Gravitational evolution of initially Gaussian seed fluctuations predicts that the connected moments of the matter fluctuations exhibit a hierarchical structure, at least in the limit of small dispersion. This same hierarchical structure has been found in the galaxy distribution, but it is not clear to what extent it reflects properties of the matter distribution or properties of a galaxy formation bias. In this paper we consider the consequences of an arbitrary, effectively local biasing transformation of a hierarchical underlying matter distribution. We show that a general form of such a transformation preserves the hierarchical properties and the shape of the dispersion in the limit of small fluctuations, i.e. on large scales, although the values of the