An analytic model for the spatial clustering of dark matter haloes

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, b (M, R, δ) = δh(M, R, δ)/δ, where δh(M, R, where δ) is the mean overdensity of haloes of mass M within spheres which have radius R and mass overdensity δ; the results depend only very weakly on how haloes are identified in the simulations. 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 haloes. Our simulation results show, however, 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. Furthermore, the constant of proportionality is very close to that predicted on large scales by the analytic model. Since analytic formulae for the nonlinear autocorrelation function of the mass are already known, 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.