Gravitational clustering from scale-free initial conditions

We present results of a series of N-body experiments designed to study gravitational clustering from scale-free initial conditions in an Einstein–de Sitter universe. The initial density fluctuations are assumed to be Gaussian with power spectra of the form ||_ k|² kⁿ| where n = 1, 0, −1 and −2. Evolution from such initial conditions is expected to be self-similar in time, and we are able to verify the expected similarity scalings for a wide range of properties. As expected, we obtain steeper autocorrelation functions for more positive values of n, but in no case do we find power law behaviour over the range of clustering strengths for which it is exhibited by the real galaxy distribution. Deviations from the asymptotic scaling predicted for a stable clustering hierarchy are related to the behaviour of the mean relative velocity of particle pairs. For all n the reduced three-point correlation is well modelled by the standard quadratic form but the coefficient Q is anticorrelated with n. The abundance of non-linear clumps as a function of mass (the multiplicity function) is a strong function of n and is in good agreement with the theory of Press this effect could be important in biasing the galaxy distribution relative to that of the mass. We show how such biases can be understood using a simple modification of the Press–Schechter theory. The density profiles of clumps are shallower for more negative n, but the corresponding circular velocity curves are poorly represented by power laws. Their slopes depend on the overdensity at which they are measured as well as on n. At high overdensities the circular velocity curves for n=−2 are approximately flat; at overdensities of a few hundred they are falling for all values of n. We investigate the three-dimensional shapes, the angular momenta and the internal velocity distribution of clumps. Our results show that these quantities depend only slightly on n. In general, although non-linear clustering destroys the hierarchical arrangement of the initial conditions, we find that their scaling properties still determine the scaling properties of the mass distribution at high density contrast. Thus it should be possible to infer the initial conditions for galaxy formation from the structure of galaxy halos at radii r≳50 kpc.