Does gravitational clustering stabilize on small scales?

The stable clustering hypothesis is a key analytical anchor for the non-linear dynamics of gravitational clustering in cosmology. It states that on sufficiently small scales, the mean pair velocity approaches zero, or equivalently that the mean number of neighbours of a particle remains constant in time at a given physical separation. N-body simulations have only recently achieved sufficient resolution to probe the regime of correlation function amplitudes xi similar to 100-10(4) in which stable clustering might be valid. In this paper we use N-body simulations of scale-free spectra P (k) proportional to k(n) with -2 less than or equal to n less than or equal to 0 and of the CDM spectrum to apply two tests for stable clustering: the time evolution and shape of xi (x, t), and the mean pair velocity on small scales. We, solve the pair conservation equation to measure the mean pair velocity, as it provides a more accurate estimate from the simulation data. For all spectra the results are consistent with the stable clustering predictions on the smallest scales probed, x < 0.07 x(nl)(t), where x(nl)(t) is the correlation length. The measured stable clustering regime corresponds to a typical range of 200 less than or similar to xi less than or similar to 2000, although spectra with more small-scale power (n similar or equal to 0) approach the stable clustering asymptote at larger values of xi. We test the amplitude of xi predicted by the analytical model of Sheth & Jain, and find agreement to within 20 per cent in the stable clustering regime for nearly all spectra. For the CDM spectrum the non-linear xi is accurately approximated by this model with n similar or equal to -2 on physical scales less than or similar to 100-300 h(-1) kpc for sigma(8) = 0.5-1, and on smaller scales at earlier times. The growth of xi for CDM-like models is discussed in the context of a power-law parametrization often used to describe galaxy clustering at high redshifts. The growth parameter epsilon is computed as a function of time and length-scale, and is found to be larger than 1 in the moderately non-linear regime thus the growth of xi is much faster on scales of interest than is commonly assumed.