The Cosmological Optical Theorem

TL;DR

This work derives a Cosmological Optical Theorem (COT) from unitary time evolution, establishing infinite relations among the wavefunction coefficients $\psi_n$ that constrain in-in cosmological correlators beyond de Sitter symmetry. It provides explicit COT formulas for contact diagrams and for four-point exchange diagrams (including gravitons), and it extends these constraints to analytic correlators, linking trispectra to bispectra. The authors perform nontrivial checks in IR-finite and IR-divergent regimes, including inflationary trispectra from graviton exchange and $P(X)$-type models, and develop a detailed mapping between the total-energy pole residues of curved-space correlators and flat-space amplitudes. This framework offers a powerful, bootstrap-ready toolkit to bootstrap cosmological correlators from lower-point data and unitarity, with broad applicability beyond strictly de Sitter-invariant theories.

Abstract

The unitarity of time evolution, or colloquially the conservation of probability, sits at the heart of our descriptions of fundamental interactions via quantum field theory. The implications of unitarity for scattering amplitudes are well understood, for example through the optical theorem and cutting rules. In contrast, the implications for in-in correlators in curved spacetime and the associated wavefunction of the universe, which are measured by cosmological surveys, are much less transparent. For fields of any mass in de Sitter spacetime with general local interactions, which need not be invariant under de Sitter isometries, we show that unitarity implies an infinite set of relations among the coefficients $ ψ_{n} $ of the wavefunction of the universe with $ n $ fields, which we name Cosmological Optical Theorem. For contact diagrams, our result dictates the analytic structure of $ ψ_{n} $ and strongly constrains its form. For example, any correlator with an odd number of conformally-coupled scalar fields and any number of massless scalar fields must vanish. For four-point exchange diagrams, the Cosmological Optical Theorem yields a simple and powerful relation between $ ψ_{3} $ and $ ψ_{4} $, or equivalently between the bispectrum and trispectrum. As explicit checks of this relation, we discuss the trispectrum in single-field inflation from graviton exchange and self-interactions. Moreover, we provide a detailed derivation of the relation between the total-energy pole of cosmological correlators and flat-space amplitudes. We provide analogous formulae for sub-diagram singularities. Our results constitute a new, powerful tool to bootstrap cosmological correlators.

Figures (4)

• Figure 1: Diagrammatic form of the bulk perspective on the optical theorem for contact terms. The direction of the arrow indicates the sign of the frequency inserted in the corresponding bulk-to-boundary propagator. It is positive when the arrow is towards the boundary and negative otherwise.
• Figure 2: A diagrammatic representation of the optical theorem for exchange diagrams from a bulk perspective. The straight lines represent $\phi$ propagators while the wiggly lines are $\sigma$ propagators.
• Figure 3: A boundary depiction of the Cosmological Optical Theorem. The minus signs next to the $k_a$'s or $s$ indicate negative energies, which are accessible upon analytical continuation.
• Figure 4: The exchange diagram contribution to the trispectrum generated by inserting two copies of the cubic vertex $g\dot{\phi}^3$