Self-Similar Evolution of Cosmological Density Fluctuations

TL;DR

This paper investigates whether self-similar evolution of cosmological density fluctuations persists for certain spectral indices, finding that despite divergences in long wave mode contributions, the core self-similar scaling remains intact.

Contribution

The study demonstrates that divergences from long wave modes do not break the self-similar scaling of density fluctuations for spectral indices between -3 and 4.

Findings

Divergent contributions cancel at all perturbative orders for n<-1.

Non-perturbative analysis shows phase shifts diverge but amplitude remains unaffected.

Self-similar scaling is preserved despite long wave mode divergences.

Abstract

The gravitational evolution of scale free initial spectra $P(k)\propto k^n$ 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 due to contributions from wavenumber $k\to 0$ (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 non-perturbative 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 {\it not} affected by the divergent terms. 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.