Loop Corrections in Non-Linear Cosmological Perturbation Theory II. Two-point Statistics and Self-Similarity

Abstract

We calculate the lowest-order non-linear contributions to the power spectrum, two-point correlation function, and smoothed variance of the density field, for Gaussian initial conditions and scale-free initial power spectra, $P(k)\sim k^n$. These results extend and in some cases correct previous work in the literature on cosmological perturbation theory. Comparing with the scaling behavior observed in N-body simulations, we find that the validity of non-linear perturbation theory depends strongly on the spectral index $n$. For $n<-1$, we find excellent agreement over scales where the variance $σ^2(R) \la 10$; however, for $n\ge -1$, perturbation theory predicts deviations from self-similar scaling (which increase with $n$) not seen in numerical simulations. This anomalous scaling suggests that the principal assumption underlying cosmological perturbation theory, that large-scale fields can be described perturbatively even when fluctuations are highly non-linear on small scales, breaks down beyond leading order for spectral indices $n\ge -1$. For $n<-1$, the power spectrum, variance, and correlation function in the scaling regime can be calculated using dimensional regularization.

48 pages, 19 figures, uses axodraw.sty; also available at http://fnas08.fnal.gov/

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• Pattern in Nonlinear Gravatational Clustering: A Numerical Investigation

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