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 ) \propto k ^ { n } .
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 ” n _ { c } \approx - 1.4 , where non-linear corrections vanish .
For the bispectrum , for { { { { n \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfill < $ \cr$% \displaystyle \hfill \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfill < $% \cr$ \textstyle \hfill \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfill% < $ \cr$ \scriptstyle \hfill \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $% \scriptscriptstyle \hfill < $ \cr$ \scriptscriptstyle \hfill \sim$ } } } } n _ { c } , one-loop corrections increase the configuration dependence of the leading order contribution ; for { { { { n \mathrel { \mathchoice { \lower 3.6 pt \vbox { \halign { \cr } $ \displaystyle \hfill > $ \cr$% \displaystyle \hfill \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \textstyle \hfill > $% \cr$ \textstyle \hfill \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $ \scriptstyle \hfill% > $ \cr$ \scriptstyle \hfill \sim$ } } } { \lower 3.6 pt \vbox { \halign { \cr } $% \scriptscriptstyle \hfill > $ \cr$ \scriptscriptstyle \hfill \sim$ } } } } 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 Zel ’ dovich approximation , where n _ { c } \approx - 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 : S _ { 3 } ( R ) = 4.02 + 3.83 \sigma ^ { 2 } _ { G } ( R ) for gaussian smoothing and S _ { 3 } ( R ) = 3.86 + 3.18 \sigma ^ { 2 } _ { TH } ( R ) for top-hat smoothing , where \sigma ^ { 2 } ( 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 .