BIASING AND HIERARCHICAL STATISTICS IN LARGE-SCALE STRUCTURE

In the current paradigm there is a nontrivial bias expected in the pro cess of galaxy formation. Thus, the observed statistical properties of the galaxy distribution do not necessarily extend to the underlying m atter distribution. Gravitational evolution of initially Gaussian seed fluctuations predicts that the connected moments of the matter fluctu ations exhibit a hierarchical structure, at least in the limit of smal l dispersion. This same hierarchical structure has been found in the g alaxy distribution, but it is not clear to what extent it reflects pro perties of the matter distribution or properties of a galaxy formation bias. In this paper we consider the consequences of an arbitrary, eff ectively local biasing transformation of a hierarchical underlying mat ter 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 t he values of the hierarchical amplitudes may change arbitrarily. We pr esent expressions for the induced hierarchical amplitudes S(g,j) of th e galaxy distribution in terms of the matter amplitudes S(j) and biasi ng parameters for j = 3-7. For higher order correlations, j > 2, restr icting to a linear bias is not a consistent approximation even at very large scales. To draw any conclusions from the galaxy distribution ab out matter correlations of order j, properties of biasing must be spec ified completely to order j-1.