Good Triangulations of Cosmological Polytopes

Abstract Cosmological polytopes of graphs are a geometric tool in physics to study wavefunctions for cosmological models whose Feynman diagram is given by the graph. After their recent introduction by Arkani-Hamed, Benincasa, and Postnikov, the focus of interest shifted towards their mathematical properties, for example, their face structure and triangulations. Juhnke, Solus, and Venturello used toric geometry to show that these polytopes have a so-called good triangulation that is unimodular. Based on these results, Bruckamp et al. studied the Ehrhart theory of those polytopes and in particular the $h^{*}$-polynomials of cosmological polytopes of multitrees and multicycles. In this article, we complete this part of the story. We enumerate all maximal simplices in good triangulations of any cosmological polytope, hence finding a formula for their volume. Furthermore, we provide a method to turn such a triangulation into a half-open decomposition from which we deduce that the $h^{*}$-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph. This settles several open questions and conjectures of Juhnke, Solus, and Venturello as well as Bruckamp et al.