Analyticity and Unitarity for Cosmological Correlators

TL;DR

The paper develops a perturbative framework that translates quantum field theory on rigid de Sitter space to computations in Euclidean AdS, enabling efficient evaluation of late-time cosmological correlators and revealing a boundary Euclidean CFT structure. It analyzes analyticity through conformal partial waves and OPE in the EAdS setting, and shows that de Sitter unitarity manifests as positivity of the spectral density rather than operator unitarity, a property proven non-perturbatively and checked perturbatively with explicit tree- and one-loop diagrams. The work highlights a resonance-centric picture for de Sitter, where heavy particles appear as narrow resonances in the spectral density and potentially offer phenomenological signatures, while also drawing connections to quasi-normal modes and Liouville-like structures. Overall, the results provide a non-perturbative positivity framework, a perturbative confirmatory calculation program, and a practical bridge between cosmological correlators and AdS/CFT-inspired techniques with potential holographic and observational implications.

Abstract

We study the fundamentals of quantum field theory on a rigid de Sitter space. We show that the perturbative expansion of late-time correlation functions to all orders can be equivalently generated by a non-unitary Lagrangian on a Euclidean AdS geometry. This finding simplifies dramatically perturbative computations, as well as allows us to establish basic properties of these correlators, which comprise a Euclidean CFT. We use this to infer the analytic structure of the spectral density that captures the conformal partial wave expansion of a late-time four-point function, to derive an OPE expansion, and to constrain the operator spectrum. Generically, dimensions and OPE coefficients do not obey the usual CFT notion of unitarity. Instead, unitarity of the de Sitter theory manifests itself as the positivity of the spectral density. This statement does not rely on the use of Euclidean AdS Lagrangians and holds non-perturbatively. We illustrate and check these properties by explicit calculations in a scalar theory by computing first tree-level, and then full one-loop-resummed exchange diagrams. An exchanged particle appears as a resonant feature in the spectral density which can be potentially useful in experimental searches.

Figures (22)

• Figure 1: Left: Penrose diagram of dS, the expanding Poincaré patch is indicated in green. Constant global (red) and Poincaré (purple) slices are also indicated. Middle: global dS. Right: No-boundary state geometry in which contracting part of the global dS is analytically continued to become a half-sphere.
• Figure 2: Left: Penrose diagram of dS, cutoff at the reheating surface (red) and glued to the non-expanding spacetime (yellow). The lightcone of an observer that sees the entire reheating surface is indicated in purple. Right: static patch indicated in blue. It coincides with the causal past of a point (the black dot) on the late-time surface, in the expanding part of dS.
• Figure 3: Examples of dS Feynman diagrams for the $\phi^3$ theory. Vertices are labeled by $r/l$ to denote if they are coming from the expansion of the time-ordered "right" time-evolution operator, or the anti time-ordered "left" one, in eq. \ref{['ininpert']}. The propagators connecting the vertices to the insertions at the future boundary, or extending between two vertices, depend on the $r/l$ labels through the different $i\epsilon$ prescriptions.
• Figure 4: "Wick rotation" from dS to EAdS, in Poincaré coordinates. The branch-cut for real negative values of $\eta$ denotes the light-cone singularity and the associated discontinuity for time-like interval. The $r/l$$i\epsilon$ prescriptions are illustrated with a displacement of the $\eta$ variable in the positive/negative imaginary direction. To connect to EAdS, we rotate the variable $\eta$ all the way to the imaginary axis avoiding the singularity on the real negative axis, i.e. clockwise for $\eta^r$ and anti-clockwise for $\eta^l$. This implies the identifications \ref{['eq:lrint']}-\ref{['eq:llint']} between the dS and EAdS two-point invariants.
• Figure 5: A contour deformation which produces the conformal block expansion \ref{['eq:confblock']} from the spectral decomposition (or equivalently the conformal partial wave expansion) \ref{['eq:EAdSCP']}. After rewriting the integrand, we can move the contour of integration to the right half plane. As a result, the integral picks up contributions from poles, which generate the conformal block expansion.