SELF-SIMILAR EVOLUTION OF GRAVITATIONAL CLUSTERING - IS N=-1 SPECIAL

The gravitational evolution of scale-free initial spectra P(k) proport ional 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 t erms in the Fourier space fluid equations contain terms that diverge b ecause 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 differ ent analytical approaches. Perturbative contributions to the power spe ctrum are examined. It is shown that for n < -1 there are divergent co ntributions at all orders. However, at every order the leading order d ivergent terms cancel out exactly. This does not rule out the existenc e of a weaker but nevertheless divergent net contribution. The second approach consists of a nonperturbative approximation, developed to stu dy the nonlinear effects of long-wave mode coupling. A solution for th e phase shift of the Fourier space density is obtained which is diverg ent 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 amplitu de of the density is not affected by the divergent terms and should th erefore display the standard self-similar scaling. Thus, both analytic al approaches lead to the conclusion that the self-similar scaling of physically relevant measures of the growth of density perturbations is preserved.