COSMOLOGICAL PERTURBATIONS - ENTERING THE NONLINEAR REGIME

We consider next-to-leading-order (one-loop) nonlinear corrections to the bispectrum and skewness of cosmological density fluctuations induc ed by gravitation evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P(k) proportional to k ''. As has been established by comparison with numerical simulations , leading order (tree-level) perturbation theory describes these quant ities at the largest scales. The one-loop perturbation theory provides a tool to probe the transition to the nonlinear regime on smaller sca les. In this work, we find that, as a function of spectral index n, th e one-loop bispectrum follows a pattern analogous to that of the one-l oop power spectrum, which shows a change in behavior at a ''critical i ndex'' n(c) approximate to -1.4, where nonlinear corrections vanish. T he tree-level perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle defined by its argumen ts. For n less than or similar to n(c), one-loop corrections increase this configuration dependence of the leading order contribution; for n greater than or similar to n(c), one-loop corrections tend to cancel the configuration dependence of the tree-level bispectrum, in agreemen t with known results from n = -1 numerical simulations. A similar situ ation is shown to hold for the Zeldovich approximation, where n(c) app roximate to -1.75. 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 Zeldovich approximation. We also compute the skewness factor, including local averaging of the density field, for n = -2: S-3(R) = 4.02 + 3.83 sigma(G)(2)(R) for Ga ussian smoothing and S-3(R) = 3.86 + 3.18 sigma(TH)(2)(R) for top-hat smoothing, where sigma(2)(R) is the variance of the density field fluc tuations smoothed over a window of radius R. A comparison with fully n onlinear numerical simulations implies that, for n < -1, the one-loop perturbation theory can extend our understanding of nonlinear clusteri ng down to scales where the transition to the stable clustering regime begins.