Galaxy formation with a local bias

The relationship between galaxies (i.e., luminous objects) and the cosmological mass density field is discussed. A generalization of the idea of biased galaxy formation is introduced, in which galaxies form with a probability given by an arbitrary function of the local mass density. Constraints, in the form of inequalities, on various statistical measures of clustering are derived, which do not depend upon the form of this function or the underlying statistical distribution of the density fluctuations. Stronger constraints can be achieved if one specifies the underlying statistics. In particular, it is shown that for Gaussian fluctuations the ratio of the amplitudes of the galaxy correlation function and matter autocovariance function is a non-increasing function of scale. This means that change of the slope of the galaxy correlation function relative to that of the underlying density fluctuations in the correct sense to produce extra clustering on large scales can only be achieved by non-local effects, such as cooperative galaxy formation. The effect of a local bias is more subtle in Fourier space: the ratio of the mass and galaxy power spectra can be made to increase without bound in the small-wavenumber limit. Cell counts and discreteness effects in both local and non-local biasing models are also discussed.