Modelling the matter bispectrum towards nonlinear scales - two and three
  loops in perturbation theories
  I compute the matter bispectrum of large-scale structure up to two loops in
the Standard Perturbation Theory and up to three loops in the MPTbreeze
renormalised perturbation theory, determining the contributing loop diagrams
and evaluating them numerically. In the process I remove the leading
divergences in the integrands, thus making them infrared-safe. By comparing the
results to numerical simulations, I show that in the case of the Standard
Perturbation Theory, the bispectrum at two loops is more accurate than at one
loop, up to $k_{\textrm{max}} \sim 0.09 \, h/\textrm{Mpc}$ at $z=0$ and
$k_{\textrm{max}} \sim 0.11 \, h/\textrm{Mpc}$ at $z=1$. The MPTbreeze can be
employed to accurately model the matter bispectrum up to $k_{\textrm{max}} \sim
0.17 \, h/\textrm{Mpc}$ at $z=0$ and $k_{\textrm{max}} \sim 0.24 \,
h/\textrm{Mpc}$ at $z=1$ using the results at three loops.
