Imprints of flat space analyticity in de Sitter S-matrix

TL;DR

The paper addresses how flat-space analyticity constraints on the S-matrix can be extended to de Sitter space, where time-translation symmetry is broken. It introduces the Hubble flat-space (HFS) limit, taking $E\to0$ and $H\to0$ with $\alpha=E/H$ fixed, to map de Sitter amplitudes to their flat-space counterparts for tree-level exchanges with a massive scalar and derivative interactions. A central result is an explicit integral transform relating the de Sitter amplitude $\mathcal{A}'_{2\to2}$ to the flat-space amplitude $\mathcal{M}_{2\to2}$, $\mathcal{A}'_{2\to2}(s;m,c_n) = (\sqrt{s})^{2-d}\frac{H}{2}\int_0^{\infty}d\tilde{s}\,\tilde{s}^{\frac{d-4}{2}}e^{-i\alpha\sqrt{\tilde{s}/s}}\mathcal{M}_{2\to2}(\tilde{s};m,c_n)$, and shows that in this limit the Mandelstam variable $s$ serves as the unique energy scale for EFT. The work delineates how mass information appears at next-to-leading order in energy-conservation or enters at leading order in the HFS limit, providing a concrete framework to carry flat-space EFT intuition into de Sitter cosmology and motivating future extensions to spinning particles, loops, and inflationary phenomenology.

Abstract

The analytic structure of the flat-space S-matrix provides non-perturbative constraints on low-energy effective field theories based on the properties of high-energy theory. While the analytic structure of the flat-space S-matrix is well understood, extending this framework to de Sitter space is challenging, as the expanding background complicates the definition of asymptotic states and breaks time-translation symmetry. This paper investigates how flat-space analyticity is imprinted on the de Sitter S-matrix. We derive a relation between flat-space amplitude and de Sitter S-matrix on a specific limit called the Hubble flat-space limit. Specifically, we show that the relation holds for tree-level amplitude exchanging a massive scalar field with any local derivative interactions. Finally, we argue that the Hubble flat-space limit is more compatible with the description of effective field theory, as the total energy dependence of de Sitter S-matrix becomes trivial, allowing the Mandelstam variable to be identified as the unique energy scale, just as in flat space.