Substructure and the halo model of large-scale structure

We develop the formalism to include substructure in the halo model of clustering. Real haloes are not likely to be perfectly smooth, but have substructure which has, to date, been neglected in the halo model - our formalism allows one to estimate the effects of this substructure on measures of clustering. We derive expressions for the two-point correlation function, the power-spectrum, the cross-correlation between galaxies and mass, as well as higher order clustering measures. Simple forms of the formulae are obtained for the limit in which the size of the substructure and the mass fraction in it is small. Inclusion of substructure allows for a more accurate analysis of the statistical effects of gravitational lensing. It may also bring the halo model predictions into better agreement with the small-scale structure seen in recent high-resolution simulations of hierarchical clustering.