Self-similar evolution of gravitational clustering: Is n=-1 special?

The gravitational evolution of scale-free initial spectra P(k) proportional to k(n) in an Einstein-de Sitter universe is widely believed to be self-similar for -3 < n < 4. However, for -3 < n < -1 the existence of self-similar scaling has not been adequately demonstrated. Here we investigate the possible breaking of self-similar scaling due to the nonlinear contributions of long-wave modes. For n < -1 the nonlinear terms in the Fourier space fluid equations contain terms that diverge because of contributions from wavenumber k --> 0 (the long-wave limit). To assess the possible dynamical effects of this divergence the limit of long-wave contributions is investigated in detail using two different analytical approaches. Perturbative contributions to the power spectrum are examined. It is shown that for n < -1 there are divergent contributions at all orders. However, at every order the leading order divergent terms cancel out exactly. This does not rule out the existence of a weaker but nevertheless divergent net contribution. The second approach consists of a nonperturbative approximation, developed to study the nonlinear effects of long-wave mode coupling. A solution for the phase shift of the Fourier space density is obtained which is divergent for n < -1. A kinematical interpretation of the divergence of the phase shift, related to the translational motion induced by the large-scale bulk velocity, is given. Our analysis indicates that the amplitude of the density is not affected by the divergent terms and should therefore display the standard self-similar scaling. Thus, both analytical approaches lead to the conclusion that the self-similar scaling of physically relevant measures of the growth of density perturbations is preserved.