Cosmology meets cluster algebra

In this paper we explore the mathematical properties of wavefunction coefficients in power-law FRW cosmologies, and establish their relation to cluster algebras. We focus on the particular contributions to the wavefunction coefficient coming from the path Feynman graphs, and show that the singularities of the wavefunction associated with a $n$-site path graph are related to the $\mathcal{X}$-coordinates of the cluster algebra $A_{2n-2}$. To establish this relation, we consider the symbol of the de Sitter wavefunction coefficients and show that the letters appearing there are the region variables associated to tubings on the path graph. These variables can be rewritten as simplicial coordinates of the moduli space $\mathcal{M}_{0,2n+1}$ and therefore identified with the $\mathcal{X}$-coordinates of type-$A_{2n-2}$ cluster algebras. We use this result to compute the wavefunction coefficients in terms of cluster functions.