Shapes of gravity: Tensor non-Gaussianity and massive spin-2 fields

TL;DR

This work analyzes how additional heavy spin-2 fields present during inflation can modify tensor non-Gaussianities. Using a model-independent framework based on the de Sitter isometries, the authors classify all on-shell cubic vertices among massless, partially massless, and massive spin-2 fields and compute the corresponding wavefunction coefficients. They derive explicit bulk-to-boundary propagators, establish analytic results for the partially massless case, and quantify how PM–graviton mixing can enhance the graviton bispectrum without heavily perturbing the power spectrum. The study reveals new, detectable tensor 3-point shapes beyond the standard Einstein gravity predictions, offering a potential observational window into extra high-spin degrees of freedom during inflation and hints at broader implications for cosmological collider phenomenology. The formalism connects cosmological correlators to conformal structures in de Sitter space and provides tools for AdS/CFT-type analyses with spinning operators.

Abstract

If the graviton is the only high spin particle present during inflation, then the form of the observable tensor three-point function is fixed by de Sitter symmetry at leading order in slow-roll, regardless of the theory, to be a linear combination of two possible shapes. This is because there are only a fixed number of possible on-shell cubic structures through which the graviton can self-interact. If additional massive spin-2 degrees of freedom are present, more cubic interaction structures are possible, including those containing interactions between the new fields and the graviton, and self-interactions of the new fields. We study, in a model-independent way, how these interactions can lead to new shapes for the tensor bispectrum. In general, these shapes cannot be computed analytically, but for the case where the only new field is a partially massless spin-2 field we give simple expressions. It is possible for the contribution from additional spin-2 fields to be larger than the intrinsic Einstein gravity bispectrum and provides a mechanism for enhancing the size of the graviton bispectrum relative to the graviton power spectrum.

Figures (10)

• Figure 1: Diagrams giving rise to the $\langle T^{2}\rangle$ and $\langle T^{3}\rangle$ wavefunction coefficients, which determine $\langle \gamma^{2}\rangle$ and $\langle \gamma^{3}\rangle$ to leading order. Wavy lines correspond to graviton bulk-to-boundary propagators. It should be noted that other references (such as Arkani-Hamed:2015bzaLee:2016vti) use similar diagrams to denote the entire in-in correlator, while for us the above diagrams only correspond to wavefunction coefficients. The three-point interaction vertex in the right diagram can arise from the Einstein--Hilbert term or a $W_{\mu\nu\rho\sigma}^{3}$ higher-derivative interaction (other diffeomorphism invariant interactions are redundant with these Maldacena:2011nz).
• Figure 2: Diagrammatic expressions for the correlation functions $\langle \gamma^{2}\rangle$, $\langle \gamma^{3}\rangle$, and $\langle\gamma^{4}\rangle$. Wavy graviton lines correspond to factors of $\, {\rm Re} \, \langle T^{2}\rangle^{-1}$, while three- and four-point vertices correspond to factors of $\, {\rm Re} \, \langle T^{3}\rangle$ and $\, {\rm Re} \, \langle T^{4}\rangle$, respectively. Note that these diagrams have a fundamentally different meaning than the wavefunction diagrams of Fig. \ref{['fig:Graviton2And3PointCoefficients']}, despite the similar notation. These instead represent equal-time correlation functions on the time slice $\tau=\tau_{\star}$.
• Figure 3: Wavefunction diagrams arising from interactions between the graviton and a partially massless spin-2 field. When combined with a mixing term $\langle T\Sigma\rangle$, these terms allow the partially massless field to imprint itself on $\langle \gamma^{3}\rangle$.
• Figure 4: Partially massless spin-2 contributions to $\langle \gamma^{3}\rangle$. Here, a mixing vertex corresponds to a factor of $\, {\rm Re} \, \langle T\Sigma\rangle$ while a PM line, , is a factor of $\, {\rm Re} \, \langle \Sigma^{2}\rangle^{-1}$.
• Figure 5: Corrections to the graviton power spectrum due to insertions of $\langle T\Sigma\rangle$ (middle) and $\langle T\Sigma^{2}\rangle$ (right). The first diagram is represents the familiar result $\langle \gamma^{2}\rangle \sim \frac{H^2}{M_{\rm Pl}^2k^{3}}$.