The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization

TL;DR

The paper extends the cosmological bootstrap to massless spinning correlators in de Sitter space by combining conformal Ward identities with current conservation via Ward–Takahashi identities. It develops two complementary approaches: (i) a weight-shifting framework that generates spinning correlators from scalar seeds and (ii) a factorization-based method that fixes four-point functions from their singularity structure, including total-energy and partial-energy poles. The analysis yields explicit three- and four-point functions for massless spin-1 and spin-2 fields with conformally coupled scalars, uncovering boundary manifestations of charge conservation, the equivalence principle, and Yang–Mills structure, and provides phenomenological inflaton-relevant scalar-tensor correlators. Together, these results illuminate how bulk locality and symmetry constrain cosmological observables and demonstrate the potential to map bulk dynamics to boundary data in a highly constrained bootstrap program.

Abstract

We extend the cosmological bootstrap to correlators involving massless particles with spin. In de Sitter space, these correlators are constrained both by symmetries and by locality. In particular, the de Sitter isometries become conformal symmetries on the future boundary of the spacetime, which are reflected in a set of Ward identities that the boundary correlators must satisfy. We solve these Ward identities by acting with weight-shifting operators on scalar seed solutions. Using this weight-shifting approach, we derive three- and four-point correlators of massless spin-1 and spin-2 fields with conformally coupled scalars. Four-point functions arising from tree-level exchange are singular in particular kinematic configurations, and the coefficients of these singularities satisfy certain factorization properties. We show that in many cases these factorization limits fix the structure of the correlators uniquely, without having to solve the conformal Ward identities. The additional constraint of locality for massless spinning particles manifests itself as current conservation on the boundary. We find that the four-point functions only satisfy current conservation if the s, t, and u-channels are related to each other, leading to nontrivial constraints on the couplings between the conserved currents and other operators in the theory. For spin-1 currents this implies charge conservation, while for spin-2 currents we recover the equivalence principle from a purely boundary perspective. For multiple spin-1 fields, we recover the structure of Yang-Mills theory. Finally, we apply our methods to slow-roll inflation and derive a few phenomenologically relevant scalar-tensor three-point functions.

Figures (14)

• Figure 1: Correlations measured at the end of inflation are created during a period of quasi-de Sitter expansion in the early universe. These correlations capture information about the production and decay of massive particles during inflation ( left). For weakly interacting particles, the boundary correlators are constrained by the isometries of the spacetime, which act as conformal transformations on the boundary. These constraints take the form of "conformal Ward identities," which are differential equations that dictate how the strength of the boundary correlations changes when the external momenta are varied ( right). This momentum dependence encodes features of the inflationary time evolution.
• Figure 2: Illustration of the different singularities of four-point correlators in the $s$-channel. All correlators have a total energy singularity, while correlators arising from the exchange of a particle also have partial energy singularities when the energy at an interaction vertex is conserved. Requiring these singularities to have the correct residues is a powerful constraint on the structure of the correlators and in many cases fixes them completely.
• Figure 3: Schematic illustration of the different weight-shifting operators used in this paper.
• Figure 4: Feynman diagrams of the $s$- and $t$-channel contributions to photon-induced pion production.
• Figure 5: Diagrammatic representation of the $s$, $t$, and $u$-channel contributions to $\langle J\varphi \varphi \varphi \rangle$.