Asymptotics of the $D^8 R^4$ genus-two string invariant

We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genus-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t \to \infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8 {\cal R}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $\ln (|v|)$ up to contributions which are power-behaved in $v$ as $v \to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.