Loop corrections in nonlinear cosmological perturbation theory .2. Two-point statistics and self-similarity

We calculate the lowest order nonlinear 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) similar to 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 nonlinear perturbation theory depends strongly on the spectral index n. For n < -1, we find excellent agreement over scales where the variance sigma(2)(R)less than or similar to 10; however, for n greater than or equal to -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, namely, that large-scale fields can be described perturbatively even when fluctuations are highly nonlinear on small scales, breaks down beyond leading order for spectral indices n greater than or equal to -1. For n < -1, the power spectrum, variance, and correlation function in the scaling regime can be calculated using dimensional regularization.