Symbol Recursion for the dS Wave Function

TL;DR

The paper introduces a compact recursive rule for the symbol of perturbative wave-function contributions of a conformally coupled scalar in FRW cosmologies, exact for a class including $\lambda\phi^3$ in $dS_4$. It derives the rule from time-integral residues and interprets it via the cosmological polytope, then demonstrates its practicality through explicit calculations of the two-site chain, two-site loop, and three-site chain symbols. By integrating these symbols, the authors reproduce the four-point result and obtain new results for the five-point tree-level contribution, while also discussing the limitations of expressing higher-weight functions with classical polylogarithms and the extension to general couplings and backgrounds. The work bridges cosmological bootstrap ideas with symbol calculus, offering a systematic, geometry-driven approach to construct, factorize, and integrate perturbative wave-function contributions, with potential applications to Euclidean AdS Witten diagrams and inflationary correlators.

Abstract

We present a recursive rule for the symbol of perturbative contributions to the vacuum wave function of a conformally coupled scalar in FRW cosmologies. The rule applies exactly for a class of interactions and cosmologies, which contains λφ^3 in dS_4, a case of particular relevance as a source of building blocks for inflationary correlators. We use the rule to efficiently reproduce the tree-level four-point contribution and present novel computations of the bubble integrand and the tree-level five-point contribution. Our results apply equally well to the computation of Witten diagrams in Euclidean AdS.

Figures (1)

• Figure 1: The dumbbell is a Feynman diagram where we have truncated the particles propagating to the boundary (dashed lines). The vertex energies are $X_1 = k_1+k_2$ and $X_2 = k_3+k_4$ and the solid edge is the propagation of an intermediate fluctuation.