We develop a simple analytic model for the gravitational clustering of dark
matter haloes to understand how their spatial distribution is biased relative
to that of the mass. The statistical distribution of dark haloes within the
initial density field (assumed Gaussian) is determined by an extension of the
Press-Schechter formalism. Modifications of this distribution caused by
gravitationally induced motions are treated using a spherical collapse
approximation. We test this model against results from a variety of N-body
simulations, and find that it gives an accurate description of a bias function.
This bias function is sufficient to calculate the cross-correlation between
dark haloes and mass, and again we find excellent agreement between simulation
results and analytic predictions. Because haloes are spatially exclusive, the
variance in the count of objects within spheres of fixed radius and overdensity
is significantly smaller than the Poisson value. This seriously complicates any
analytic calculation of the autocorrelation function of dark halos. Our
simulation results show that this autocorrelation function is proportional to
that of the mass over a wide range in R, even including scales where both
functions are significantly greater than unity. The constant of proportionality
is very close to that predicted on large scales by the analytic model. This
result permits an entirely analytic estimate of the autocorrelation function of
dark haloes. We use our model to study how the distribution of galaxies may be
biased with respect to that of the mass. In conjunction with other data these
techniques should make it possible to measure the amplitude of cosmic mass
fluctuations and the density of the Universe.Comment: 34 pages including 7 figs, gziped ps file, submitted to MNRA
