Biasing and hierarchical statistics in large-scale structure

Abstract

In the current paradigm there is a nontrivial 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 hierarchical amplitudes may change arbitrarily. We present expressions for the induced hierarchical amplitudes S_g,j_ of the galaxy distribution in terms of the matter amplitudes S_j_ and biasing parameters for j = 3-7. For higher order correlations, j > 2, restricting to a linear bias is not a consistent approximation even at very large scales. To draw any conclusions from the galaxy distribution about matter correlations of order j, properties of biasing must be specified completely to order j - l.