The Massive Flat Space Limit of Cosmological Correlators

TL;DR

This paper introduces the Massive Flat-Space (MFS) limit, a double-scaling approach in which external energies are taken to zero while internal masses grow, preserving finite-mass effects in cosmological correlators. It derives a universal reduction formula expressing in-in diagrams in the MFS limit as amputated flat-space graphs, and proves this via in-in integrals and an effective-action perspective in curved spacetime. The authors apply the formalism to inflationary correlators with heavy field exchanges in a small sound-speed regime, uncovering novel bispectrum shapes that a local EFT cannot reproduce, instead requiring spatially non-local operators. By connecting massive Feynman integrals to cosmological correlators, the work offers a new tool for extracting heavy-field signals (the cosmological phonon collider) and motivates further exploration of on-shell MFS limits and AdS/CFT flat-space connections with potential observational implications.

Abstract

Identifying useful flat-space limits for cosmological correlators, where they can be expressed in terms of observables in Minkowski space is nontrivial due to their scale-invariant nature. In recent years, it has been shown that momentum-space correlators encode flat-space amplitudes at specific singularities that emerge in the complex plane of their kinematics after analytical continuation. This flat-space limit is massless in the sense that the amplitude corresponds to the ultraviolet regime of the associated flat-space process, where the masses of the internal propagators are effectively zero. In this paper, we introduce a novel massive flat-space (MFS) limit, in which the internal masses in the corresponding flat-space Feynman graph remain finite. Our proposal applies to arbitrary graphs with light external legs and heavy internal lines, using a double-scaling limit. In this limit, the external energies, treated as independent variables, approach zero in inverse proportion to the propagator masses, which are sent to infinity. We present a general reduction formula that expresses diagrams in this limit in terms of amputated Feynman graphs in flat space. Our findings underscore the deep connections between the rich structure of massive Feynman integrals and the properties of cosmological correlators involving the exchange of heavy fields. Using this reduction formula, we compute sample one-loop contributions from heavy particles to inflationary correlators in the small sound-speed regime, revealing novel bispectrum shapes. The non-Gaussian signals we uncover, which are especially pronounced around the equilateral configuration, cannot be reproduced by adding local terms to the effective field theory of single-field inflation. Instead, they are captured by incorporating prescribed spatially non-local operators into the EFT.

Figures (14)

• Figure 1: The amplitude limit of correlators, Eq. \ref{['amplitudelimit']}.
• Figure 2: The massive flat-space limit of off-shell diagrams.
• Figure 3: The scalar-exchange (SE) and scalar one-loop, bubble diagrm (SB) diagrams contributing to the four-point function of the conformally coupled field.
• Figure 4: An illustration of the elements involved in the MFS limit reduction formula \ref{['flatspace']} for a one-loop graph of the four-point function with three vertices (left). $p_i^\mu = (0, \bm{k}_i / a)$ ($i = 1, \dots, 4$) represent a set of fictitious four-momenta that characterize the amputated graph in flat-space (right). $\bar{q}_{bc}$ ($b, c = 1, 2, 3$) denote the internal line four-momenta of this graph, which can be expressed in terms of the loop momentum $\bar{q}$ and the $p_i$'s, i.e., $\bar{q}_{12} = q$, $\bar{q}_{13} = p_2 + p_3 - q$, and $\bar{q}_{23} = q + p_1$. In the MFS limit, the correlator reduces to a contact diagram (middle), with its vertex corresponding to the amputated Feynman diagram.
• Figure 5: Comparison between the exact single-exchange diagram $\hat{F}_{\text{SE}}$ with varying masses (the colored curves) and their massive flat-space limits $\hat{F}^{\text{MFS}}_{\text{SE}}$ (the black curve), all plotted as functions of $r\times (m/H)$ with a fixed value of $r'\times (m/H) = 0.2$.