  The bispectrum, the three-point function of density fluctuations in Fourier
space, is the lowest order statistic that carries information about the spatial
coherence of large-scale structures. For Gaussian initial conditions, when the
density fluctuation amplitude is small, tree-level (leading order) perturbation
theory predicts a characteristic dependence of the bispectrum on the shape of
the triangle formed by the three wave vectors. This configuration dependence
provides a signature of gravitational instability, and departures from it in
galaxy catalogs can be interpreted as due to bias, that is, non-gravitational
effects. On the other hand, N-body simulations indicate that the reduced
three-point function becomes relatively shape-independent in the strongly
non-linear regime. In order to understand this non-linear transition and assess
the domain of reliability of shape-dependence as a probe of bias, we calculate
the one-loop (next-to-leading order) corrections to the bispectrum in
perturbation theory. We compare these results with measurements in numerical
simulations with scale-free and CDM initial power spectra. We find that the
one-loop corrections account very well for the departures from the tree-level
results measured in numerical simulations on weakly non-linear scales. In this
regime, the reduced bispectrum qualitatively retains its tree-level shape, but
the amplitude can change significantly. At smaller scales, the reduced
bispectrum in the simulations starts to flatten, an effect which can be
partially understood from the one-loop results. In the strong clustering
regime, where perturbation theory breaks down entirely, the simulation results
confirm that the reduced bispectrum has almost no dependence on triangle shape,
in rough agreement with the hierarchical ansatz.
