We investigate an analytic model to compute the non-linear power spectrum of dark matter, galaxies and their cross-correlation. The model is based on Press–Schechter haloes, which cluster and have realistic dark matter profiles. The total power spectrum is a sum of two contributions, one from correlations between the haloes and one from correlations within the same halo. We show that such a model can give dark matter power spectra which match well with the results of N-body simulations, provided that the concentration parameter decreases with the halo mass.
The galaxy power spectrum differs from the dark matter power spectrum because the pair-weighted number of galaxies does not scale with the halo mass and because most haloes harbour a central galaxy. If the pair-weighted number of galaxies increases less rapidly than the halo mass, as predicted by theoretical models and observed in clusters, then the resulting power spectrum becomes a power law with a slope close to the observed over several orders of magnitude in scale. Such a model also predicts a later onset of non-linear clustering in comparison with dark matter, which is needed to reconcile the cold dark matter (CDM) models with the data. A generic prediction of this model is that bias is scale-dependent and non-monotonic. This is particularly important for red or elliptical galaxies, which are preferentially found in larger mass haloes and for which the bias in the power spectrum may be scale-dependent even on large scales.
Our predictions for galaxy–dark matter correlations, which can be observed through galaxy–galaxy lensing, show that these cannot be interpreted simply as an average halo profile of a typical galaxy, because different halo masses dominate at different scales and because larger haloes host more than one galaxy. We compute predictions for the cross-correlation coefficient as a function of scale and discuss the prospects of using cross-correlations in combination with galaxy clustering to determine the dark matter power spectrum.