Causality and quasi-normal modes in the GREFT

Eur. Phys. J. Plus (2024) 139:725 https://doi.org/10.1140/epjp/s13360-024-05520-5 Regular Article Causality and quasi-normal modes in the GREFT Scott Melvillea Queen Mary University of London, Mile End Road, London E1 4NS, UK Received: 4 January 2024 / Accepted: 29 July 2024 © The Author(s) 2024 Abstract 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. 1 Introduction A quantum theory of gravity has remained elusive for two reasons. The first is phenomenological: experimentally we either probe the small-scale small-curvature regime (e.g. in particle colliders) or the large-scale large-curvature regime (e.g. solar system, cosmology), but to see quantum and gravitational effects simultaneously would require both small scales and large curvatures. The second reason is theoretical: unlike the fundamental forces of the Standard Model, General Relativity (GR) is not a renormalisable field theory, and this typically leads to a loss of predictivity at high energies. Reconciling these difficulties and uncovering a complete quantum mechanical description of gravity has been a central aim of theoretical physics for the past century. Fortunately, these two difficulties also suggest a way forward. Since gravitational phenomena are typically observed on large scales (in the low-energy, or ‘IR’, regime), they are well described by an Effective Field Theory (EFT). An EFT description of gravity also resolves the theoretical issues surrounding renormalisation, since an EFT is renormalisable at any finite order in its derivative expansion. The goal of this work is to better understand how the physical principle of causality can be used as a guide when constructing and applying gravitational EFTs. The interpretation of General Relativity as an Effective Field Theory goes back several decades [1, 2] and is by now widely known. In this framework, the Einstein-Hilbert action of GR is extended by all possible higher-derivative interactions consistent with the symmetries of the problem: namely diffeomorphism invariance (plus any flavour/gauge symmetries of the matter sector). This produces an EFT extension of GR, also known as the “EFT of gravity” or the “General Relativity Effective Field Theory” (GREFT). This latter title best highlights the many parallels with the Standard Model Effective Field Theory (SMEFT), which is an analogous extension of the Standard Model by all possible interactions which are higher-order in derivatives and fields. Just as the SMEFT was developed for model-agnostic searches for BSM physics at colliders, the GREFT can be viewed as a parameterised framework in which to search for new physics beyond General Relativity. This is particularly important for gravitational wave (GW) astronomy. The number of black hole (BH) or neutron star binary mergers detected by gravitational wave observatories is now at least 90 [3], and is forecast to rise to thousands in the coming observing runs [4]. These gravitational waves were created by compact objects in very high-curvature environments, and therefore open an exciting new window into the gravitational Universe [5–11]. The GREFT framework has been used to study the inspiral [12–14], merger [15] and ringdown [16–21] phase of a binary merger, and compared with existing GW data in [22, 23]. With future observing runs and new GW observatories planned for the coming years, developing this framework both theoretically and phenomenologically will allow for the most precise tests of GR and its possible extensions. a e-mail: (corresponding author) 0123456789().: V,-vol 123 725 Page 2 of 30 Eur. Phys. J. Plus (2024) 139:725 The GREFT action is made up of two components: a gravitational sector and a coupling to matter. In four spacetime dimensions, all parity-preserving interactions in the gravitational sector with up to eight derivatives can be written as [12],    M P2 b1 (3) c1  (2) 2 c2  (2) 2 4 √ (1.1) Sgrav  d x −g R+ 4R + 6 R + 6 R̃ 2    where R (n) denotes the following contraction of n Riemann tensors, R (2)  R μν αβ R αβ μν , R̃ (2)  1 μν αβ R αβ ε ρσ R ρσμν , 2 R (3)  R μν αβ R αβ ρσ R ρσ μν , (1.2) where εμναβ is the antisymmetric Levi-Civita tensor. The constant coefficients {b1 , c1 , c2 } encode the underlying UV physics, i.e. different high energy theories (string theory,1 loop quantum gravity, etc.) will match onto different choices of these coefficients. The common energy scale  is used to track the EFT’s regime of validity, since (1.1) can be viewed as an expansion in powers of ∇μ / (which is therefore expected to break down when length/time scales become order 1/). The basis of interactions in (1.1) captures all physics involving only gravity2 : for instance graviton scattering amplitudes, as well as the physics of single black holes (e.g. their effective horizon, quasi-normal modes and BH-GW scattering). The grand ambition of the GREFT is to use gravitational observations to fix (or at least constrain) the coefficients {b1 /4 , c1 /6 , c2 /6 , ...}, and then use this information to infer properties of the underlying high-energy quantum theory. This work aims to answer two related questions: (i) Given the recent causality constraints that have been placed on the GREFT coefficients, what are the phenomenological consequences for GREFT black holes and in particular their quasi-normal mode (QNM) spectrum? (ii) Given a conjectured causality/stability property of black hole quasi-normal modes, what further constraints can be placed on the GREFT coefficients? The first is important for future analysis which fits GREFT coefficients to data, since it will inform more accurate GW templates for the GREFT and hence lead to stronger, more reliable constraints from experiment. The second is also important for phenomenology—since limiting the parameter space with theoretical priors before fitting to data can lead to qualitatively different results—but mainly it complements a growing theore (...truncated)