Generalised Cluster Adjacency for Cosmology

In this paper we study the cluster algebraic properties of wavefunction coefficients for conformally coupled scalar theories in de Sitter cosmology. We show that the symbol of the wavefunction coefficient of the $n$-site path graph $P_n$ obeys a generalisation of cluster adjacency, where all letters in a given word belong to the same cluster of an $A_{2n-3}$ algebra, with certain additional constraints on the order of the letters. We call this property the ordered single cluster condition, and provide its physical interpretation. This condition is stronger than the usual cluster adjacency obeyed by neighbouring letters, and thereby constrains the symbol bootstrap far more tightly. We also show that for an arbitrary graph the alphabet carries a cluster-like structure, described by tubes and tubings on the graph, which allows for a similar bootstrap approach.