Mastering Cosmological Amplitudes Using Generalized Ramanujan's Theorem

TL;DR

This paper introduces a novel systematic approach combining the Method of Brackets and diagrammatic rules to compute cosmological amplitudes in FRW spacetime, enabling efficient, scalable, and analytic evaluation of massive field effects.

Contribution

It develops a new method that uses a multivariate extension of Ramanujan's theorem and diagrammatic rules for calculating cosmological correlators involving massive scalars.

Findings

Infinite series representations valid for all momenta and energies

Efficient diagrammatic rules for scalable computations

Expressions that interpolate smoothly to the conformal limit

Abstract

We present a systematic method for computing cosmological amplitudes, including in-in correlators and wavefunction coefficients, in FRW spacetime. Specializing to cases with conformally-coupled external scalars and massive scalar exchanges, we introduce a decomposition into massive family trees, which capture the nested time structure common to these observables. We then evaluate these building blocks using the Method of Brackets (MoB), a multivariate extension of Ramanujan's master theorem that operates directly on the integrand, translating integrals into discrete summations via a compact set of algebraic rules. This yields infinite series representations valid across the full space of external momenta and internal energies. We also develop Feynman-like diagrammatic rules that map interaction graphs to summand structures, enabling efficient and scalable computation. The resulting expressions make time evolution manifest, smoothly interpolate to the conformal limit, and are well suited for both numerical evaluation and analytic analysis of massive field effects in cosmology.