Spectral Representation of Cosmological Correlators

Abstract: Cosmological correlation functions are significantly more complex than their flat-space analogues, such as tree-level scattering amplitudes. While these amplitudes have simple analytic structure and clear factorisation properties, cosmological correlators often feature branch cuts and lack neat expressions. In this paper, we develop off-shell perturbative methods to study and compute cosmological correlators. We show that such approach not only makes the origin of the correlator singularity structure and factorisation manifest, but also renders practical analytical computations more tractable. Using a spectral representation of massive cosmological propagators that encodes particle production through a suitable $i\epsilon$ prescription, we remove the need to ever perform nested time integrals as they only appear in a factorised form. This approach explicitly shows that complex correlators are constructed by gluing lower-point off-shell correlators, while performing the spectral integral sets the exchanged particles on shell. Notably, in the complex mass plane instead of energy, computing spectral integrals amounts to collecting towers of poles as the simple building blocks are meromorphic functions. We demonstrate this by deriving a new, simple, and partially resummed representation for the four-point function of conformally coupled scalars mediated by tree-level massive scalar exchange in de Sitter. Additionally, we establish cosmological largest-time equations that relate different channels on in-in branches via analytic continuation, analogous to crossing symmetry in flat space. These universal relations provide simple consistency checks and suggest that dispersive methods hold promise for developing cosmological recursion relations, further connecting techniques from modern scattering amplitudes to cosmology.