Cosmological Perturbations: Entering the Non-Linear Regime

We consider one-loop corrections (non-linear corrections beyond leading order) to the bispectrum and skewness of cosmological density fluctuations induced by gravitational evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P (k) ∝ k. As has been established by comparison with numerical simulations, tree-level (leading order) perturbation theory describes these quantities at the largest scales. One-loop perturbation theory provides a tool to probe the transition to the non-linear regime on smaller scales. In this work, we find that, as a function of spectral index n, the one-loop bispectrum follows a pattern analogous to that of the one-loop power spectrum, which shows a change in behavior at a “critical index” nc ≈ −1.4, where non-linear corrections vanish. For the bispectrum, for n < ∼ nc, one-loop corrections increase the configuration dependence of the leading order contribution; for n > ∼ nc, oneloop corrections tend to cancel the configuration dependence of the tree-level bispectrum, in agreement with known results from n = −1 numerical simulations. A similar situation is shown to hold for the Zel’dovich approximation, where nc ≈ −1.75. Using dimensional regularization, we obtain explicit analytic expressions for the one-loop bispectrum for n = −2 initial power spectra, for both the exact dynamics of gravitational instability and the Zel’dovich approximation. We also compute the skewness factor, including local averaging of the density field, for n = −2: S3(R) = 4.02+3.83 σ 2 G(R) for gaussian smoothing and S3(R) = 3.86 + 3.18 σ 2 TH(R) for top-hat smoothing, where σ (R) is the variance of the density field fluctuations smoothed over a window of radius R. Comparison with fully non-linear numerical simulations implies that, for n < −1, one-loop perturbation theory can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.