Loop corrections in nonlinear cosmological perturbation theory. II. 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) ∼ kⁿ. 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 σ²(R) ≲ 10; however, for n ≥ -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≥ -1. For n < -1, the power spectrum, variance, and correlation function in the scaling regime can be calculated using dimensional regularization.