We present the results of a large library of cosmological N‐body simulations, using power‐law initial spectra. The non‐linear evolution of the matter power spectra is compared with the predictions of existing analytic scaling formulae based on the work of Hamilton et al. The scaling approach has assumed that highly non‐linear structures obey ‘stable clustering’ and are frozen in proper coordinates. Our results show that, when transformed under the self‐similarity scaling, the scale‐free spectra define a non‐linear locus that is clearly shallower than would be required under stable clustering. Furthermore, the small‐scale non‐linear power increases as both the power spectrum index n and the density parameter Ω decrease, and this evolution is not well accounted for by the previous scaling formulae. This breakdown of stable clustering can be understood as resulting from the modification of dark matter haloes by continuing mergers. These effects are naturally included in the analytic ‘halo model’ for non‐linear structure; we use this approach to fit both our scale‐free results and also our previous cold dark matter data. This method is more accurate than the commonly used Peacock–Dodds formula and should be applicable to more general power spectra. Code to evaluate non‐linear power spectra using this method is available from http://as1.chem.nottingham.ac.uk/~res/software.html. Following publication, we will make the power‐law simulation data publically available through the Virgo website http://www.mpa-garching.mpg.de/Virgo/.

We propose a heuristic model that displays the main features of realistic theories for galaxy bias. We show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes. A hypothetical galaxy catalogue depends on (i) the efficiency of galaxy formation, as manifested by the halo occupation number -- the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo. The first factor is constrained by the empirical luminosity function of groups. For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass. These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density. We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat $\Omega=0.3$ model ($\Lambda$CDM), but not for an $\Omega=1$ model with the same power spectrum ($\tau$CDM). This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high $\Omega$, and hence display enhanced small-scale clustering. Finally, the pairwise velocity dispersion of galaxies in the $\Lambda$CDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field. This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.Comment: 14 Pages; MNRAS in press. Minor changes to match accepted versio

In this, the first in a series of three papers concerning the SuperCOSMOS Sky Survey (SSS), we give an introduction and user guide to the survey programme. We briefly describe other wide‐field surveys and compare them with our own. We give examples of the data, and make a comparison of the accuracies of the various image parameters available with those from the other surveys providing similar data; we show that the SSS data base and interface offer advantages over these surveys. Some science applications of the data are also described and some limitations discussed. The series of three papers constitutes a comprehensive description and user guide for the SSS.