Biasing and hierarchical statistics in large-scale structure

In this paper we consider the consequences for galaxy formation 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 greater than 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 - 1.