Scott Melville

Using unitarity and causality, we derive an infinite tower of two-sided positivity bounds on the effective field theory coefficients which describe the propagation of heavy fields on de Sitter spacetime. We design this effective field theory (EFT) to describe propagators with the in-in boundary conditions that are relevant for cosmological correlators. Our positivity bounds therefore identify EFT correlators that can never emerge from a consistent underlying model of inflation. This implies nontrivial constraints on primordial non-Gaussianity; for instance the cosmological collider oscillations in the squeezed bispectrum from the exchange of a heavy scalar are tied to the shape of its EFT background.

We define an S matrix for massive scalar fields on a fixed de Sitter spacetime, in the expanding patch coordinates relevant for early Universe cosmology. It enjoys many of the same properties as its Minkowski counterpart, for instance: it is insensitive to total derivatives and field redefinitions in the action; it can be extracted as a particular “on shell” limit of time-ordered correlation functions; and for low-point scattering, kinematics strongly constrains its possible structures. We present explicit formulas relating inflationary observables—namely in-in equal-time correlators and wavefunction coefficients at the conformal boundary—to these S matrix elements. This new formalism will allow modern amplitude methods to be applied directly in cosmology and hence provide a wider range of more accurate theoretical predictions for upcoming sky surveys. Published by the American Physical Society 2024

A bstract: Extending scattering to states with unphysical mass values (particles “off their mass shell”) has been instrumental in developing modern amplitude technology for Minkowski spacetime. Here, we study the off-shell correlators which underpin the recently proposed S -matrix for scattering on de Sitter spacetime. By labelling each particle with both a spatial momentum and an independent “energy” variable (the de Sitter analogue of a 4-momentum), we find that the practical computation of these correlators is greatly simplified. This allows us to derive compact expressions for all 3- and 4-particle S -matrices at tree-level for scalar fields coupled through any derivative interactions. As on Minkowski, we find that the 3-particle and exchange part of the 4-particle S -matrices are unique (up to crossing). The remaining contact part of the 4-particle S -matrix is an analytic function of just two differential operators, which become the usual Mandelstam variables in the Minkowski limit. Finally, we introduce a spectral decomposition for the tree-level exchange of a heavy field responsible for a cosmological collider signal. Once projected onto physical mass eigenstates, these S -matrix elements encode the statistical properties of the early inflationary perturbations.

The General Relativity Effective Field Theory (GREFT) introduces higher-derivative interactions to parameterise the gravitational effects of massive degrees of freedom which are too heavy to be probed directly. The coefficients of these interactions have recently been constrained using causality: both from the analytic structure of 4-point graviton scattering and the time delay of gravitational waves on a black hole background. In this work, causality is used to constrain the quasi-normal mode spectrum of GREFT black holes. Demanding that quasi-normal mode perturbations decay faster in the GREFT than in General Relativity—a new kind of causality condition which stems from the analytic structure of 2-point functions on a black hole background—leads to further constraints on the GREFT coefficients. The causality constraints and compact expressions for the GREFT quasi-normal mode frequencies presented here will inform future parameterised gravitational waveforms, and the observational prospects for gravitational wave observatories are briefly discussed.

A number of diagrammatic “cutting rules” have recently been developed for the wavefunction of the Universe which determines cosmological correlation functions. These leverage perturbative unitarity to relate particular “discontinuities” in Feynman-Witten diagrams (with cosmological boundary conditions) to simpler diagrams, in much the same way that the Cutkosky rules relate different scattering amplitudes. In this work, we make use of a further causality condition to derive new cutting rules for Feynman-Witten diagrams on any time-dependent spacetime background. These lead to the cosmological analogue of Feynman’s tree theorem for amplitudes, which can be used to systematically expand any loop diagram in terms of (momentum integrals of) tree-level diagrams. As an application of these new rules, we show that certain singularities in the wavefunction cannot appear in equal-time correlators due to a cancellation between “real” and “virtual” contributions that closely parallels the KLN theorem. Finally, when combined with the Bunch-Davies condition that certain unphysical singularities are absent, these cutting rules completely determine any tree-level exchange diagram in terms of simpler contact diagrams. Altogether, these results remove the need to ever perform nested time integrals when computing cosmological correlators.

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic structure of wavefunction coefficients in Minkowski as a function of their kinematics. We introduce an off-shell wavefunction in terms of amputated time-ordered correlation functions and show that it is analytic in the complex energy plane except for possible singularities on the negative real axis. These singularities are determined to all loop orders by a simple energy-conservation condition. We confirm this picture by developing a Landau analysis of wavefunction loop integrals and corroborate our findings with several explicit calculations in scalar field theories. This analytic structure allows us to derive new UV/IR sum rules for the wavefunction that fix the coefficients in its low-energy expansion in terms of integrals of discontinuities in the corresponding UV-completion. In contrast to the analogous sum rules for scattering amplitudes, the wavefunction sum rules can also constrain total-derivative interactions. We explicitly verify these new relations at one-loop order in simple UV models of a light and a heavy scalar. Our results, which apply to both Lorentz invariant and boost-breaking theories, pave the way towards deriving wavefunction positivity bounds in flat and cosmological spacetimes.

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic structure of wavefunction coefficients in Minkowski as a function of their kinematics. We introduce an off-shell wavefunction in terms of amputated time-ordered correlation functions and show that it is analytic in the complex energy plane except for possible singularities on the negative real axis. These singularities are determined to all loop orders by a simple energy-conservation condition. We confirm this picture by developing a Landau analysis of wavefunction loop integrals and corroborate our findings with several explicit calculations in scalar field theories. This analytic structure allows us to derive new UV/IR sum rules for the wavefunction that fix the coefficients in its low-energy expansion in terms of integrals of discontinuities in the corresponding UV-completion. In contrast to the analogous sum rules for scattering amplitudes, the wavefunction sum rules can also constrain total-derivative interactions. We explicitly verify these new relations at one-loop order in simple UV models of a light and a heavy scalar. Our results, which apply to both Lorentz invariant and boost-breaking theories, pave the way towards deriving wavefunction positivity bounds in flat and cosmological spacetimes.

We derive new effective field theory (EFT) positivity bounds on the elastic 2 → 2 scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the t derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the s- and u-channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the t-derivative to the forward-limit EFT amplitude with no t derivatives, our bounds establish that the t-derivative alone must be strictly positive for sufficiently large helicities. Consequently, they can provide stronger constraints beyond the forward limit which can be used to constrain dimension-6 interactions with a milder assumption about the high-energy growth of the UV amplitude.