Self‐Similar Evolution of Gravitational Clustering. II.N‐Body Simulations of then= −2 Spectrum

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Abstract
The power spectrum P(k) ∝ kn with n = -2 is close to the shape of the measured galaxy spectrum on small scales. Unfortunately, this spectrum has proven rather difficult to simulate. Furthermore, two-dimensional simulations have suggested a breakdown of self-similar scaling for spectra with n < -1 due to divergent contributions from the coupling of long wave modes. This paper is the second (numerical) part of our investigation into nonlinear gravitational clustering of scale-free spectra, in particular, to test the scaling of the n = -2 spectrum. Using high-resolution N-body simulations, we find that the n = -2 power spectrum displays self-similar scaling. The phase shift of Fourier modes of the density show a dual scaling, self-similar scaling at early times and a scaling driven by the bulk velocity at late times. The second scaling was shown analytically to be a kinematical effect that does not affect the growth of clustering. Thus our analytical and N-body results verify that self-similarity in gravitational clustering holds for -3 < n < 1. The N-body spectrum is also compared with analytic fitting formulae, which are found to slightly underestimate the power in the nonlinear regime. The asymptotic shape of the spectrum at high k is a power law with the same slope as predicted by the stable clustering hypothesis.
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