High-order correlations of peaks and haloes: a step towards understanding galaxy biasing

We develop an analytic model for the hierarchical correlation amplitudes is the jth order connected moment of counts in spheres of radius R) of density peaks and dark matter haloes in the quasi-linear regime. The statistical distributions of density peaks and dark matter haloes within the initial density field (assumed Gaussian) are determined by the peak formalism of Bardeen et al. and by an extension of the Press-Schechter formalism, respectively. Modifications of these distributions caused by gravitationally induced motions are treated using a spherical collapse model. We test our model against results for S3,g(R) and S4,g(R) from a variety of N-body simulations. The model works well for peaks even on scales where the second moment of mass is significantly greater than unity. The model also works successfully for haloes that are identified earlier than the time when the moments are calculated. Because haloes are spatially exclusive at the time of their identification, our model is only qualitatively correct for haloes identified at the same time as the moments are calculated. For currently popular initial density spectra, the values of Sj,g at R ∼ 10 h−1 Mpc are significantly smaller for both haloes and peaks than those for the mass, unless the linear bias parameterb defined by for large R is comparable to or less than unity. The Sj,g depend only weakly on b for large b but increase rapidly with decreasing b at b ∼1. Thus if galaxies are associated with peaks in the initial density field, or with dark haloes formed at high redshifts, a measurement of Sj,g in the quasi-linear regime should determine whether galaxies are significantly biased relative to the mass. We use our model to interpret the observed high-order correlation functions of galaxies and clusters. We find that if the values of Sj,gfor galaxies are as high as those given by the APM survey, then APM galaxies should not be significantly biased.