Cosmological Perturbations: Entering the Nonlinear Regime

We consider one-loop corrections to the bispectrum and skewness ofcosmological density fluctuations induced by gravitational evolution. As hasbeen established by comparison with numerical simulations, tree-levelperturbation theory (PT) describes these quantities at the largest scales.One-loop PT provides a tool to probe the transition to the non-linear regime onsmaller 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-looppower spectrum, which shows a change in behavior at a critical index n_c =-1.4, where non-linear corrections vanish. For the bispectrum, for n less thann_c, one-loop corrections increase the configuration dependence of the leadingorder contribution; for n greater than n_c, one-loop corrections tend to cancelthe configuration dependence of the tree-level bispectrum, in agreement withknown results from n=-1 numerical simulations. A similar situation is shown tohold for the Zel'dovich approximation (ZA), where n_c = -1.75. Usingdimensional regularization, we obtain explicit analytic expressions for theone-loop bispectrum for n=-2 initial power spectra, for both the exact dynamicsof gravitational instability and the ZA. We also compute the skewness factor,including local averaging of the density field, for n=-2: S_3(R) = 4.02 + 3.83sigma^2(R) for gaussian smoothing and S_3(R) = 3.86 + 3.18 sigma^2(R) fortop-hat smoothing, where sigma^2(R) is the variance of the density fieldfluctuations smoothed over a window of radius R. Comparison with fullynon-linear numerical simulations implies that, for n < -1, one-loop PT canextend our understanding of nonlinear clustering down to scales where thetransition to the stable clustering regime begins.Comment: 38 pages, 10 figures. Submitted to ApJ, August 199