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 induced by gravitational evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P(k) ∝ kⁿ. As has been established by comparison with numerical simulations, leading order (tree-level) perturbation theory describes these quantities at the largest scales. The one-loop perturbation theory provides a tool to probe the transition to the nonlinear 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" n_c ≈ 1.4, where nonlinear corrections vanish. The tree-level perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle defined by its arguments. For n ≲ n_c, one-loop corrections increase this configuration dependence of the leading order contribution; for n ≳ n_c, 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 Zeldovich approximation, where n_c ≈ -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: S₃(R) = 4.02 + 3.83σ_G²(R) for Gaussian smoothing and S₃(R) = 3.86 + 3.18σ_TH²(R) for top-hat smoothing, where σ²(R) is the variance of the density field fluctuations smoothed over a window of radius R. A comparison with fully nonlinear numerical simulations implies that, for n < -1, the one-loop perturbation theory can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.