  We study the matter bispectrum of the large-scale structure by comparing
different perturbative and phenomenological models with measurements from
$N$-body simulations obtained with a modal bispectrum estimator. Using shape
and amplitude correlators, we directly compare simulated data with theoretical
models over the full three-dimensional domain of the bispectrum, for different
redshifts and scales. We review and investigate the main perturbative methods
in the literature that predict the one-loop bispectrum: standard perturbation
theory, effective field theory, resummed Lagrangian and renormalised
perturbation theory, calculating the latter also at two loops for some triangle
configurations. We find that effective field theory (EFT) succeeds in extending
the range of validity furthest into the mildly nonlinear regime, albeit at the
price of free extra parameters requiring calibration on simulations. For the
more phenomenological halo model, we confirm that despite its validity in the
deeply nonlinear regime it has a deficit of power on intermediate scales, which
worsens at higher redshifts; this issue is ameliorated, but not solved, by
combined halo-perturbative models. We show from simulations that in this
transition region there is a strong squeezed bispectrum component that is
significantly underestimated in the halo model at earlier redshifts. We thus
propose a phenomenological method for alleviating this deficit, which we
develop into a simple phenomenological "three-shape" benchmark model based on
the three fundamental shapes we have obtained from studying the halo model.
When calibrated on the simulations, this three-shape benchmark model accurately
describes the bispectrum on all scales and redshifts considered, providing a
prototype bispectrum HALOFIT-like methodology that could be used to describe
and test parameter dependencies.
