Cosmological Perturbations: Entering the Nonlinear Regime

We consider one-loop corrections to the bispectrum and skewness of cosmological density fluctuations induced by gravitational evolution. As has been established by comparison with numerical simulations, tree-level perturbation theory (PT) describes these quantities at the largest scales. One-loop PT 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 less than nc, one-loop corrections increase the configuration dependence of the leading order contribution; for n greater than nc, one-loop 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 (ZA), 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 ZA. We also compute the skewness factor, including local averaging of the density field, for n=-2: S₃ (R) = 4. 02 + 3. 83 sigma² (R) for gaussian smoothing and S₃ (R) = 3. 86 + 3. 18 sigma² (R) for top-hat smoothing, where sigma² (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 PT can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.