The Speed of Gravity

TL;DR

The paper shows that in the standard effective field theory of General Relativity, gravitational waves propagate with a speed that generally deviates from the speed of light on backgrounds that spontaneously break Lorentz invariance, due to loop effects from heavy fields and, in some cases, tree-level exchanges of higher-spin states. It develops a systematic, EFT-based method to identify the GW speed on curved backgrounds, derives explicit corrections from curvature-squared and curvature-cubed operators on FLRW and static geometries, and connects the sign and magnitude of these corrections to positivity bounds and the field content of the UV completion. The authors show that, under reasonable positivity assumptions, the leading curvature-squared term with Weyl-squared coefficient $C_{W^2}>0$ yields a superluminal GW speed relative to minimally coupled light in NEC-preserving backgrounds, while the front (high-energy) velocity remains luminal; if these leading terms vanish, higher-dimension operators provide epoch- and species-dependent modifications. The results have broad implications for cosmological EFT model building and for using GW speed as a probe of high-energy content, including light scalar dark matter scenarios, by linking causal propagation to the spectrum and spins of heavy fields in the UV completion.

Abstract

Within the standard effective field theory of General Relativity, we show that the speed of gravitational waves deviates, ever so slightly, from luminality on cosmological and other spontaneously Lorentz-breaking backgrounds. This effect results from loop contributions from massive fields of any spin, including Standard Model fields, or from tree level effects from massive higher spins $s \ge 2$. We show that for the choice of interaction signs implied by S-matrix and spectral density positivity bounds suggested by analyticity and causality, the speed of gravitational waves is in general superluminal at low-energies on NEC preserving backgrounds, meaning gravitational waves travel faster than allowed by the metric to which photons and Standard Model fields are minimally coupled. We show that departure of the speed from unity increases in the IR and argue that the speed inevitably returns to luminal at high energies as required by Lorentz invariance. Performing a special tuning of the EFT so that renormalization sensitive curvature-squared terms are set to zero, we find that finite loop corrections from Standard Model fields still lead to an epoch dependent modification of the speed of gravitational waves which is determined by the precise field content of the lightest particles with masses larger than the Hubble parameter today. Depending on interpretation, such considerations could potentially have far-reaching implications on light scalar models, such as axionic or fuzzy cold dark matter.

Figures (4)

• Figure 1: TT amplitude: Graviton mediated loop contributions to matter interactions. $\chi$ symbolizes a matter field present in the stress--energy tensor $T_{\mu \nu}$, a wiggly line is a graviton propagator and solid purple lines are the loops of heavy fields.
• Figure 2: TT amplitude: Gravitational strength matter interactions arising from exchange of mass spin 0 and spin 2 states.
• Figure 3: Regions of infinitesimal sub-- and super--luminal GW speed on FLRW with a constant equation of state parameter $\varpi$, from integrating out scalars, vectors or spinors.
• Figure 4: Example diagrams indicating how loops of heavy fields correct the propagation of gravitational waves in a background geometry. For weak backgrounds, the effect can be entirely accounted for by perturbative QFT in Minkowski spacetime (as we show below), giving identical results to covariant approaches.