Bispectrum Islands

TL;DR

This work develops a model-independent positivity/bootstrap approach to constrain the inflationary bispectrum sourced by a hidden sector during inflation, anchored in unitarity, locality, analyticity, and de Sitter symmetry. By representing the bispectrum as a positive spectral integral over operator exchanges, the authors show the equilateral configuration must yield a negative signal and an island of allowed shapes (the Bispectrum Island) is carved out by unitarity bounds, with the maximum shape attained by a heavy-state exchange at the critical mass $m=3H/2$. They validate the bounds through explicit scenarios—a finite set of resonances and a bulk CFT operator—demonstrating shapes within the island and clarifying UV-convergence requirements. They also discuss how signals outside the island would indicate new physics, such as strong boost breaking, non-unitarity, or UV-divergent spectral densities, and outline future work on templates, higher-spin extensions, and nonperturbative bootstrap approaches.

Abstract

Inspired by the amplitude bootstrap program, the spirit of this work is to constrain the space of consistent inflationary correlation functions - specifically, the bispectrum of curvature perturbations - using fundamental principles such as unitarity, locality, analyticity, and symmetries. To this end, we assume a setup for inflation in which de Sitter isometries are only mildly broken by the slow roll of the inflaton field, and study the bispectrum imprinted by a generic hidden sector during inflation. Assuming that the hidden sector's contributions to primordial non-Gaussianity are dominated by the exchange of a scalar operator (which does not preclude high-spin UV completions), we derive nontrivial positivity constraints on the resulting bispectrum $B(k_1,k_2,k_3)$. In particular, we show that $B$ must be negative in a certain region around the equilateral configuration. For instance, for isosceles triangles (with $k_2=k_3$) this region is given by $0.027\lesssim k_3/k_1\leq 2$. Furthermore, we demonstrate that unitarity imposes upper and lower bounds on the bispectrum shape, thereby carving out a Bispectrum Island where consistent shapes in our setup can reside. We complement our analysis by contemplating alternative setups where the coupling to the hidden sector is allowed to strongly break de Sitter boosts. We also identify situations that would push the bispectrum off the island and the profound physical features they would reveal.

Figures (11)

• Figure 1: The exchange diagrams contributing to the bispectrum of $\zeta$, induced by the two-point function $\langle {\cal O}{\cal O}\rangle$ (left) and the three-point function $\langle {\cal O}{\cal O}{\cal O}\rangle$ (right).
• Figure 2: This figure shows that the equilateral bispectrum resulting from the exchange of a heavy field $f$, characterized by mass index $\mu$ and coupled through the perturbed Lagrangian \ref{['perturbedLag']} (with ${\cal O}$ replaced by $f$), is negative for all values of $\mu$. Note that the overall coefficient $\kappa$, defined in Eq. \ref{['kappaeq']}, is positive.
• Figure 3: The ansatz for the bispectrum shape ${\cal S}(k_i;\mu)$ is shown as a function of $\mu$, defined by Eq. \ref{['optimized']}. As the plots illustrate across various configurations, the upper bound on the bispectrum shape is always set by the ansatz evaluated at $\mu=0$, while the lower bound ${\cal S}_{\text{min}}$ corresponds to the exchange of fields with masses varying in the range $0<\mu_{\text{min}}<3$.
• Figure 4: The upper and lower bounds on the shape of the bispectrum ${\cal S}$ plotted as a function of the squeezing parameter $k_3/k_1\leq 2$, for isosceles triangles (with $k_2=k_3$).
• Figure 5: The bispectrum induced by the exchange of the hidden sector must be negative near the equilateral configuration, but it can switch sign for sufficiently squeezed triangles. The table shows the threshold values of the ratio $k_3/k_1$ at which the lower bound on the bispectrum vanishes, as a function of the angle $\theta$ between the short and the long mode.