Top quark mass effects in Higgs boson pair production up to NNLO

We consider the production of pairs of Standard Model Higgs bosons via gluon fusion. Until recently the full dependence on the top quark mass $M_t$ was not known at next-to-leading order. For this reason we apply an approximation based on the expansion for large top quark masses up to $\mathcal{O}(1/M_t^{12})$. At next-to-next-to-leading order we avoid the calculation of real corrections via the soft-virtual approximation and obtain top quark mass corrections up to $\mathcal{O}(1/M_t^4)$. We use our results to estimate the residual uncertainty of the total cross section due to a finite top quark mass to be $\mathcal{O}(10\%)$ at next-to-leading order and $\mathcal{O}(5\%)$ at next-to-next-to-leading order.