Ehrhart theory of cosmological polytopes

The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive geometries dictate that this wavefunction may be computed as a sum of rational functions associated to the facets in a triangulation of the cosmological polytope. The normalized volume of the polytope then provides a complexity estimate for these computations. In this paper, we examine the (Ehrhart) $h^\ast$-polynomial of cosmological polytopes. We derive recursive formulas for computing the $h^\ast$-polynomial of disjoint unions and $1$-sums of graphs. The degree of the $h^\ast$-polynomial for any $G$ is computed and a characterization of palindromicity is given. Using these observations, a tight lower bound on the $h^\ast$-polynomial for any $G$ is identified and explicit formulas for the $h^\ast$-polynomials of multitrees and multicycles are derived. The results generalize the existing results on normalized volumes of cosmological polytopes. A tight upper bound and a combinatorial formula for the $h^\ast$-polynomial of any cosmological polytope are conjectured.