Speed of Gravity

PHYSICAL REVIEW D 101, 063518 (2020)

† Claudia de Rham * and Andrew J. Tolley Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom and CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA

(Received 19 September 2019; accepted 3 March 2020; published 17 March 2020)

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 ≥ 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 null energy condition 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 effective field theory 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.

DOI: 10.1103/PhysRevD.101.063518

I. INTRODUCTION plus curvature corrections that grow away from x. Since the Einstein-Hilbert action is second order, modifications from In this new era of gravitational wave astronomy, it is the background curvature terms to the propagation of especially important to understand how gravitational waves gravitational waves on this background can only arise as propagate. The recent simultaneous observation of gravi- an effective mass term (simply from power counting tational waves from the coalescence of two neutron stars, derivatives), and never as corrections to the kinetic or GW170817, together with its gamma-ray counterpart, GRB gradient terms. For example, in Friedmann-Lemaître- 170817A, has put the cleanest constraint on the propaga- Robertson-Walker (FLRW) spacetime, gravitational waves tion speed of gravitational waves relative to photons [1–3]. have an “effective mass” from the background expansion of In classical general relativity (GR) minimally coupled to 2 ˙ matter, gravitational waves always travel luminally, as order H ; H, in terms of the Hubble parameter H, but their sound speed defined by the ratio of kinetic to gradient terms defined by the light cones of the metric gμν with respect 1 ’ to which matter is coupled, by virtue of the equivalence is luminal. Hence it is the two derivative nature of Einstein s principle. For instance, when considering the propaga- theory, together with diffeomorphism invariance, that guar- tion of linearized gravitational waves across some general antees luminality in general relativity. curved background geometry, the background metric may 1 always be put in a Riemann normal coordinate system where Canonically normalizedR tensor fluctuations in GR have ¼ η 3 1 ð 02 − ð∇⃗ Þ2 þ a00 2Þ it is locally Minkowski in the vicinity of a spacetime point x, quadratic action SGR d d x 2 h h a h . Despite the effective mass −a00=a the actual mass is zero. In the well- known case of the propagation of gravitational waves during * inflation, this effective mass is negative and drives an instability [email protected] † which generates long wavelength scale invariant tensor fluctua- [email protected] tions, but the retarded propagator vanishes outside the light cone c ¼ 1 Published by the American Physical Society under the terms of defined by s . In what follows, in FLRW we define the speed via the light cone of the effective metricR on which the Creative Commons Attribution 4.0 International license. ¼ η 3 1 ð 02− Further distribution of this work must maintain attribution to modes propagate, i.e., via cs in the action Shh d d x 2 h ’ 2ð∇⃗ Þ2 − 2 2Þ the author(s) and the published article s title, journal citation, cs h meff h , so the effective mass does not play a role in and DOI. Funded by SCOAP3. the definition of the speed.

2470-0010=2020=101(6)=063518(37) 063518-1 Published by the American Physical Society CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020)

Classical general relativity is not, however, the real family of them depending on the choice of scale M above world. At a minimum, gravitational effects generated from which physics has been integrated out. quantum loops of known particles, e.g., the electron, will Even in the absence of matter, the speed of gravitational already generate modifications to Einstein gravity which waves is modified by the addition of higher curvature alter this process. Many of these effects are finite (meaning terms, precisely because the earlier argument based on free from renormalization ambiguities) and calculable. power counting of derivatives is no longer valid. Higher These effects are of course highly suppressed, being derivative curvature terms can, and do, modify the second induced by loops, but any potential departure of the speed order derivative terms in the equation for propagation when of propagation of gravitational waves from luminality is expanding around a background. Since gravitational waves itself significant, not least because it impacts our under- are luminal in pure GR, the sign of higher curvature terms standing of the causal structure of a given theory. Various will typically lead the resulting corrections to make the proposed extensions to four-dimensional general relativity, waves either superluminal or subluminal. Typically cau- such as extra dimension models, or string theory, will also sality is imposed by demanding that the modified gravi- induce modifications to the Einstein-Hilbert Lagrangian tational waves are subluminal (with respect to the light cone that can potentially change the above picture. defined by the metric gμν) based on similar arguments for The general framework to account for such corrections is scalar fields [9,10], hence fixing the signs for the higher well understood and goes under the umbrella of “effective curvature coefficients. For instance, for Ricci flat back- field theory for gravity” [4–7] (for a recent example of this grounds this is done in [11] for quartic curvature correc- methodology, see [8]). Historical issues with nonrenorma- tions of the type that arise in the low energy EFT from lizability and the artificial separation of quantum fields on string theory. Within the context of the EFTs of inflation/ curved spacetimes are replaced with the general effective dark energy, where matter sources the background, a field theory (EFT) framework that allows us, if desired, to potential modification to the speed of gravitational waves simultaneously quantize matter and gravity despite the has been noted in [12–16].Forpp waves such an effect was nonrenormalizability of the Lagrangian. The price to pay is also noted in [17]. the need to introduce an infinite number of counterterms, More generally this procedure of demanding sublumin- but in practice at low energies only finite numbers are ever ality of all fluctuations is problematic because in a relevant. The low energy effective theory is defined by an gravitational EFT the metric itself is ambiguous (see effective Lagrangian valid below some scale E ≪ M which [18] for related discussions). It is always possible to accounts for all tree and loop level corrections from perform field redefinitions, schematically of the form particles of masses greater than or equal to M, and loop X α 2 n;p 2p n processes of light fields at energies greater than M. The μν → μν þ ð∇ Þ ð Þ g g 2ðnþpÞ Riem μν; 1:1 starting point is then the Wilsonian action which includes p;n MPl all possible covariant operators built out of the Riemann 2p n tensor, its derivatives, and combinations of light fields and where ð∇ Riem Þμν is a tensor constructed out of n their derivatives. For instance, assuming no other light contractions of Riemann and 2p covariant derivatives, fields than gravity, the leading corrections are powers of which leave invariant the leading Einstein-Hilbert term. curvature and derivatives thereof, i.e., very schematically Those are consistent with the gravitational EFT, but modify Z the light cone of the metric. In this way, in some cases, pffiffiffiffiffiffi M2 some spacetime with superluminal fluctuations may be ¼ 4 − −Λ þ Pl þ 2 þ 2 SEFT d x g 2 R C1R C2Rμν rendered subluminal and vice versa.3 A related effect known to occur at one loop is that the paths of massless X∞ C Xn þ 2 þ n;p ∇2p 2þn−p particles of different spins do not receive the same amount C3Rμναβ 2n Riem ; n¼1 MPl p¼0 of bending as they pass a massive object (e.g., the Sun) [19–21], which means that despite being massless they where by ∇2pRiem2þn−p we mean all possible scalar local effectively do not see the same metric which further operators constructed out of contractions of this number of confuses the question of how to describe the causal powers of Riemann tensor and covariant derivatives. The structure. 2n precise energy scale of suppression Cn;p=MPl will depend In response to this, more recent discussions of causality on the origin of a given term; e.g., it will in general be in the EFT context have focused on causality constraints different for interactions coming from tree level processes implied by S-matrix analyticity. These have the virtue of or from loops. In general there is not one such EFT, but a being invariant under field redefinitions and are in some sense true avatars of causality. One such idea is to demand 2Precisely how this is achieved depends on the renormalization prescription, and we work with the most convenient which is 3Although in general there is no universal procedure to render dimensional regularization. all fields (sub)luminal.

063518-2 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) causality a` la Wigner by imposing positivity of the Our principal focus will be to ask what is the speed, by scattering time delay [22,23]. These arguments may which we mean the speed defined by the effective light plausibly apply for weakly coupled UV completions cone of the low energy equations of motion (specified where the irrelevant operators in the EFT arise from tree precisely in Sec. III A), of gravitational waves on a level effects of high mass modes, but are already known spacetime with a long range spontaneous breaking of to fail for QED where the corrections come from loops Lorentz invariance. Thus we will not be interested in the [24]. Another proposal is to use S-matrix positivity rather special shock wave or asymptotically flat geom- bounds [10,25–28] which constrain the 2 − 2 scattering etries considered for example in [17,22,24], but in cases amplitude of gravitons and other particles. For example for which the departure of the speed of sound from unity S-matrix positivity arguments have been applied to can significantly build up over time to lead to clearly quartic curvature interactions in [29]. Recent works have noticeable differences. The clearest example is FLRW applied these ideas more specifically to the weak gravity spacetimes since they spontaneously break time diffeo- conjecture, which focuses on the EFT for gravity and a morphisms, have a clearly identifiable sound speed, have Uð1Þ gauge field [30–32]. In what follows we shall see sufficient symmetry to be simple such that gravitational that positivity bounds will provide very useful guidance modes decouple from matter at linear order, and have on corrections to propagation speeds. obvious phenomenological relevance. Nevertheless much Phenomenologically it is still pertinent to ask what is of what we will discuss will be relevant to other more the speed of gravitational waves relative to the metric to generic backgrounds. Crucially this means we do not which photons and the Standard Model fields couple consider vacuum spacetimes, but require some light fields minimally, and this is independent of field redefinitions. to source the breaking. The inclusion of light fields in the It is well known that the photon speed can be modified EFT is useful since they themselves provide a clock, in a curved background due to loop corrections from and their interactions with other matter can be used as we charged particles, e.g., electrons, even leading to super- will see, to impose S-matrix analyticity and unitarity luminal group velocities at low energies for certain requirements. backgrounds [33,34]. The fact that this low energy Our main conclusions are the following: superluminal group velocity is not in conflict with (i) In the frame in which matter is minimally coupled, causality has been discussed extensively in a series of the leading EFT corrections with signs imposed by papers [24,35–42], which essentially identify the require- S-matrix locality, unitarity, and analyticity (if the ment that the front velocity is luminal as the key contribution from the massless graviton t-channel requirement for causality. Apparent low energy violations pole can be ignored) enforce that gravitational waves of causality in, for instance, scattering time delays are are superluminal for any matter satisfying the null absent in the UV theory [24]. In the EFT description energy condition (NEC). these effects come from nonminimal Riemann curvature (ii) On performing a field redefinition, this is equivalent coupling to the Maxwellpffiffiffiffiffiffi field strength squared, specifi- at leading order (alone) to the generation of universal ΔL ∝ −2 − ab cd cally me gRabcdF F , which would arise gravitationally induced matter interactions (TT de- from electron loops. These effects arise in the EFT formation) which ensure that standard matter fluc- ∇ ≪ defined below the scale of the electron, me,or tuations propagate slower than gravity. whatever charged particle has been integrated out. While (iii) The precise coefficient that determines the departure those operators are present in the EFT we will consider, of the propagation speed from unity is connected they do not affect the speed of gravitational waves in the with the elastic scattering amplitudes for matter same way and are not the focus of our discussion. fields, both those that drive the expansion and Moreover they typically enter at a scale much larger spectator fields. than what we have in mind in the cosmological context. (iv) If the leading order EFT corrections are set to zero, For the rest of this discussion, we shall therefore consider the next to leading order corrections that arise from that matter (including photons) minimally couples to the loops give rise to an epoch and species dependent metric, and equivalence principle violating curvature modification of the speed of gravitational waves. If terms are not present. We shall rather focus on pure the lightest particle with mass above the Hubble curvature interactions that can arise equally from inte- parameter has spin-0, the speed of gravity is super- grating out charged and chargeless particles and are luminal throughout the whole standard cosmological applicable for any matter (dark matter, Standard history of the Universe. Model, inflaton, etc.). In other words, photons will (v) Our results remain valid when considering purely always be luminal, and the relevant question is what quartic curvature corrections, such as those known to is the speed of gravitational waves as compared with a arise in the low energy string theory effective action, luminal photon, or at least the metric to which the photon for which gravitational waves also travel super- is minimally coupled. luminally on NEC preserving backgrounds.

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Stated differently, given the assumed sign of the leading II. EFFECTS OF HEAVY MODES ON EFT coefficients (either as inferred from explicit integration GRAVITY AT LOW ENERGY of fields or from positivity bounds or as implied from string Throughout this work, we consider gravity as a low– theory), the light cone inferred from the low energy sound energy EFT and look at the effects that heavy fields speed of (minimally coupled) matter always lies inside the minimally coupled to gravity have on the EFT. In other light cone of gravitational waves and is never exactly at the words we shall focus on the Wilsonian effective action for same speed. The superluminality of the propagation of the light fields which shall include gravity as well as some gravitational waves in the calculation in the original frame light field that sources the background expansion (e.g., is (far from being in conflict with) consistent with the radiation, quintessence, inflaton) and look at the influence requirements of causality implied by S-matrix analyticity. of those corrections that arise from integrating out massive Arguably we must accept it as a price for the associated fields for which the masses satisfy Mi ≫ H. Our focus will field redefinition ambiguity in metric (1.1); however, as is be on identifying the speed of tensor gravitational waves on already clear from the QED case [33], there are no field non–maximally symmetric backgrounds, and for most of redefinitions which render all modes luminal or sublumi- Section III onwards we shall focus on cosmological back- nal. We will further show that when additional nonminimal grounds. Working on FLRW has the advantage that at the EFT interactions between gravity and the light fields are linear level tensor fluctuations cleanly decouple from scalar included, the same results hold, namely that positivity and vector perturbations, the former of which is usually bounds (if valid) enforce the overall superluminality. coupled to whatever matter drives the cosmological expan- The rest of this paper is organized as follows: In Sec. II sion. The FLRW symmetry also allows us to cleanly we start by reviewing what we need of the standard EFT of distinguish between the speed of gravitational waves and general relativity and how curvature corrections are gen- the speed of matter perturbations. However, we emphasize erated in the low energy EFT. We emphasize the role played that our results hold more generically beyond FLRW as is, by field redefinitions and how to take care of them. We then for instance, illustrated in Sec. III C dealing with static explore the leading curvature-squared contributions to the warped geometries. low energy EFT for gravity in Sec. III and identify their effect on the speed of gravitational waves on FLRW and on A. Effective field theory for gravity at low energy static warped backgrounds. We then discuss the implications of our findings within the context of standard causal We shall have in mind two different scenarios: and local UV completions in Sec. IV and argue that such (i) Tree level corrections to the EFT, whereby tree level completions favor superluminal gravitational waves for effects of massive particles potentially generate NEC preserving backgrounds. In Sec. V we explore the higher curvature interactions. possibility of tuning the EFT so that the leading quadratic (ii) Loop level corrections to the EFT coming from curvature corrections cancel, such that the dominant effect integrating out standard matter (e.g., Standard ≤ 1 comes from higher (cubic and quartic) curvature correc- Model fields) of any spin, including s . tions. In particular we show that the low energy speed of The latter case is the most interesting since it does not the gravitational wave depends on the field content of require any assumptions about an unknown UV comple- the high-energy completion, and particularly on the spin tion, but rather relies on the calculable gravitational effects of the lightest massive particle that is integrated out to of known particles, in particular from Standard Model derive the low energy Wilsonian action. We provide an particles. outlook of our results in Sec. VI. Appendix A derives the exact curvature-cubed operators in the one-loop effective 1. Tree level interactions action obtained from integrating out a massive scalar. At tree level, a massive field which is minimally coupled Appendix B provides the details for the derivation of to gravity (i.e., without explicit curvature couplings) has no the speed of gravitational waves on FLRW in the pre- effect on the low energy gravitational propagator if the spin sence of curvature-cubed (dimension-6) operators. Finally of that field is less than two s<2. This can be seen Appendix C highlights subtleties in defining the retarded straightforwardly as a consequence of the scalar-vector- propagator perturbatively and justifies the approach we tensor (SVT) decomposition. At tree level, we can work follow in identifying the speed. with the (partial) UV completion that includes the addi- Throughout this paper we work in natural units where tional massive mode as part of the Lagrangian. If this mode the speed of light in vacuum, without any quantum has spin s<2, then it has no tensor component and so will correction effects, is c ¼ 1. We shall also, as is standard, at the level of quadratic fluctuations completely decouple slightly abuse the EFT operator counting terminology and from the gravitational tensor modes. Thus integrating out refer to Riemannn operators as dimension-2n operators the massive mode at tree level will provide no contributions even though they includeP an infinite number of operators of to the gravitational fluctuations. Stated differently, it is not kð∂2 Þn kþ3n−4 þ 2 various dimensions k h h =MPl . possible to integrate out massive states with s< at tree

063518-4 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) level and obtain higher curvature interactions that change which light loops may be computed afterwards. The the speed of gravitational waves. former effects are captured by the Wilsonian effective The situation changes if we integrate out at tree level action, the latter by the 1PI effective action. When massive spins with s ≥ 2. Any spin of s ≥ 2 will by virtue focusing on the Wilsonian effective action, gravity, being of the helicity or SVT decomposition effectively contain a light field, is treated “classically,” and so our results are spin 2 states, or more precisely tensor modes. Even at largely independent of the precise details of quantum quadratic order these states could mix with the usual gravity and the UV completion of gravity at the Planck massless graviton, generating modifications to its speed scale (or string scale). of propagation. We shall see explicit examples of this To be more concrete, we consider gravity (standard GR) below. Obvious examples are string theory, where an minimally coupled to light fields one or more of which will infinite tower of massive spin states arise, or extra dimen- be used to generate the cosmological backgrounds, and sional models where the massless graviton in higher heavy fields of mass Mi which define the UV completion. dimensions can be viewed as a massless graviton in four, In the case of loop calculations we shall consider heavy together with an infinite tower of massive spin 2 states. See fields of spin-s ¼ 0; 1=2, or 1, but for tree level UV Refs. [43,44] for examples from superstring theory where completions we have in mind any spin. We denote the integrating out massive higher spins at tree level leads to massive spin fields generically by Φ, and so schematically specific types of quadratic and cubic curvature operators to we have the action the low energy EFT for gravity. 2 Related arguments tell us that as soon as we allow pffiffiffiffiffiffi M ð Þ L ¼ − Pl þ L l:e: ð ψÞþLðh:e:Þð ΦÞþL for massive spin 2 states, in order to construct a weakly UV g 2 R ψ g; g; c:t ; coupled UV completion of gravity we must necessarily ð Þ include an infinite number of spin particles. Recent versions 2:1 of these arguments have been given in [22,45,46] but they ðl:e:Þ follow straightforwardly from the observation that the where Lψ is the low energy Lagrangian for the low scattering amplitude of a massive spin 2 particle violates energy fields (denoted generically as ψ) with masses mψ ∼ the fixed t Froissart bound by virtue of the s2 growth of its OðHÞ ≪ M (hence including massless modes), whose role t-channel pole, and in a weakly coupled UV completion this will be to generate the cosmological background, while can only be turned around by an infinite number of powers Lðh:e:Þðg; ΦÞ represents the dynamics of the heavy fields, of s resumming into a softer behavior, which necessitates an with masses M ≳ M. We focus in what follows on minimal 4 i infinite number of spin states (e.g., see [47]). couplings between both the light and heavy fields and gravity, meaning that the fields Φ and ψ do not directly 2. Loop level interactions couple to the curvature below the Planck scale.5 At loop level the situation is quite different. An internal At loop level, it is well known that integrating out any Φ loop, even of a particle of spin s ≤ 1, effectively contains massive field would lead to divergent contributions to states of total angular momenta of arbitrary spin, as is the cosmological constant, as well as to R and curvature- 2 2 2 implied by the partial wave expansion. As such loop squared terms R , Rμν, and Rμνρσ. In the EFT context,pffiffiffiffiffiffi in corrections from the standard matter can, and do, correct order to deal with these divergences we must add −g, pffiffiffiffiffiffi pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 2 the propagation of gravitational waves. This effect is of −gR and −gR , −gRμν, −gRμνρσ counterterms. course tiny, being loop suppressed; nevertheless it is finite Hence we must include in the UV action (up to local counterterms), calculable, and controlled from 1 the EFT point of view. It is thus not necessary to know 2 UV UV 2 UV 2 L ¼ −Λ þ M a R þ C R þ C Rμν what the appropriate theory of quantum gravity is in order c:t UV 2 Pl 1 2 to determine the magnitude of this effect. In what follows UV 2 þ C3 Rμνρσ þ; ð2:2Þ we will be integrating out heavy modes of mass M, with ≪ ≪ H M MPl,whereH is the typical scale at which we in addition to any other matter counterterms. The first two are interested in probing our low energy EFT for gravity terms are just a redefinition of the cosmological constant (for instance, H is the typical scale of the curvature, and and Planck mass and may be ignored in what follows as ≪ ≪ we will consider modes with frequency H k M). We their consequences are straightforward. The latter terms are “ ” will only be integrating out loops of matter fields ; i.e., as we will see nontrivial and directly affect the speed of there will be no gravitons in the loops. This is consistent propagation of gravitational waves. with standard Wilsonian EFT, whereby we first integrate out massive states to construct the low energy EFT, from 5We would of course expect the EFT for gravity to include operators that mix the Riemann curvature and the other fields 4The pole itself cannot be canceled as its residue is positive by through Planck scale suppressed terms. Such types of interactions unitarity. are considered in Sec. IV D.

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1 3. Wilsonian effective action IR IR IR C 2 ¼ C2 þ 2C3 > 0: ð2:8Þ To construct the low energy Wilsonian effective action, W 2 we integrate out (both at tree and loop level) the heavy It is of course not possible in the EFT context to fix the modes in the schematic manner6 precise values of CUV or CIR in the absence of an explicit Z 1;2;3 1;2;3 ð ψÞ ð ψ ΦÞ matching calculation onto a UV completion. Hence we are iSIR g; ¼ Φ iSUV g; ; ð Þ e D e : 2:3 instructed to compare them with observations. Precisely one such observation which is at least in principle possible The resulting low energy effective theory will take the form to measure is the speed of gravitational waves relative to that of light. We shall begin in Sec. III B by focusing on the pffiffiffiffiffiffi 2 MPl ðl:e:Þ 2 case where these terms are present. In Sec. VAwe shall set L ¼ −g −ΛIR þ R þ Lψ ðg; ψÞþCIRR IR 2 1 them to zero and focus on the finite R3 terms that arise from integrating out matter loops. IR 2 IR 2 þ C2 Rμν þ C3 Rμνρσ þ : ð2:4Þ 4. Inclusion of light loops Lðl:e:Þð ψÞ L More generally ψ g; will also receive corrections if As we have discussed the Wilsonian effective action IR the light field couples to the heavy fields integrated out. An includes loops from heavy fields but not from light fields. obvious and well-known example is if the light field is the As such it is local and is the typical starting point for photon, and integrating out charged particles will result at cosmological and phenomenological analyses. It is inter- leading order in addition to the Euler-Heisenberg terms, the esting to ask what would happen if we integrated out the RFF interactions considered in [33] (see also [34,49]). In light fields, in particular the massless graviton and photon. the interests of simplicity we will neglect these corrections In this case we should be working with the 1PI effective for now, but consider examples of them in Sec. IV D. action which is nonlocal and difficult to deal with. There is The IR coefficients that enter the low energy EFT will considerable work on this in the literature [56–64] and differ from their UV values by virtue of both loop and tree results are often presented in terms of a curvature expan- effects, sion. At the level of curvature-squared terms, the contributions from loops of massless (or light) fields may be IR ¼ UV þ Δ ð Þ C1;2;3 C1;2;3 C1;2;3: 2:5 modeled by the following proxy effective action Δ The natural scale of the corrections C1;2;3 is of order the −□ − iϵ Δ ∼ Δ ¼ ˆ IR number of fields integrated out N, C1;2;3 N. In what Llight−loops C1 R ln μ2 R follows we will see that positivity bounds generically imply 1 −□ − ϵ that two specific combinations of these coefficients satisfy ˆ IR i μν þ C Rμν R 2 ln μ2 1 2 Δ 2 ¼ Δ þ 2Δ 0 ð Þ −□ − iϵ CW C2 C3 > ; 2:6 þ ˆ IR μνρσ ð Þ 2 C3 Rμνρσ ln 2 R : 2:9 μ3 1 1 Δ 2 ¼ Δ þ Δ þ Δ 0 ð Þ CR C1 3 C2 3 C3 > : 2:7 This has been used in the cosmological context in [65].Itis clear that due to the logarithm, the massless loops can Indeed, we shall further argue, provided we may apply dominate over the heavy loop contributions, in particular in positivity bounds even in the presence of massless graviton the IR. However, if as we will assume, the number of heavy t-channel poles, as for example recently argued in [32], fields is much greater than the number of massless or light 7 ˆ that (see also Ref. [55] for related discussions on whether fields, then we expect Ci ≫ Ci and so it will be sufficient to this argument is justified) focus on the heavy contributions.

6At loop level the process of “integrating out” may lead to B. A word of caution on field redefinitions incorrect conclusions about the scales of coefficients, and so it is 2 2 better to phrase this in terms of a “matching” calculation [48]. As is well known, the R and Rμν interactions are This subtlety will, however, not be important for us since we will redundant operators and are therefore removable with field be able to rephrase our result in terms of dispersion relations redefinitions. Since the S-matrix is invariant under field which are universal. 7 redefinitions, it seems appropriate to ignore these contri- It is worth noting that this is the opposite sign from what is butions. This would be true for pure gravity, but when required for the quadratic gravity scenario [50–53]. However, it is expected that these models will have different causality and gravity is coupled to matter, all the field redefinition does is analyticity structure [51,54] and hence the usual positivity shift the same effect into another operator that arises at the bounds are unlikely to apply. same scale, specifically into a pure matter contribution that

063518-6 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) produces the same effect. In general field redefinitions of could be removed via a field redefinition and are hence the metric change the “speed” of propagation by virtue of irrelevant for the low energy EFT relevant for gravitational modifying the background metric with respect to which the waves (GWs). While it is true that such operators could be speed is identified. However, field redefinitions do not removed via field redefinitions, this would then affect ðl:e:Þ change the relative speed. For instance, if gravitational Lψ ðg; ψÞ and lead to nonminimal couplings with low “‘ ” waves travel faster than photons in one field frame, they energy matter fields (and in particular photons), hence do so in all field frames. That is in the cosmological context leading to a nonstandard light cone for light and other light particles. This effect is less important for the analysis there, 2ð Þ cs tensor ð Þ but is crucially important for cosmological analyses. Here 2 is invariant under field redefinitions: 2:10 cs ðmatterÞ we largely insist on keeping a minimal coupling for the low energy matter fields and hence avoid performing such field The relative causal structure is kept intact [18]. Thus the redefinitions except where it is useful to give an alternative question of whether gravitational waves are superluminal explanation of the same phenomena and in deriving or subluminal with respect to light is a frame independent positivity bounds. question. For backgrounds with FLRW symmetry, it is always C. Relevance of the dimension-six curvature operators technically possible to perform a field redefinition that renders the gravitational waves luminal. This is largely a As for the dimension-six curvature operators, a subset of triviality, due to the symmetry, the difference between the them are not removable by field redefinitions (namely the metric which matter couples to and the metric on which Weyl cubed terms). We shall consider those that arise in the ≤ 1 gravitational waves propagate is just a rescaling of the time specific computations of loops from particles of spin s component of the metric combined with an overall con- in Sec. V. It was argued in [67] in the application to formal factor (see e.g., [66]). Given whatever field is used gravitational waves from mergers that those terms should to spontaneously break time diffeomorphism, ψ, we can be suppressed, and one should focus instead on dimension- always perform a field redefinition in the manner (as an eight operators. This argument was on the grounds that for example) weakly coupled UV completions, these terms would arise 2 3 48 at a scale MPlRiemann =M where M is the scale of gμν → AðψÞgμν þ BðψÞ∇μψ∇μψ; ð2:11Þ particles that have been integrated out. In order for the Riemann3 term to have an interesting effect for gravita- and engineer the functions A and B so that for a given tional wave astronomy, the scale M would have to be taken background the metric travels luminally. However, there is so low that we would have observed the effect of the in general no single local and covariant background field associated additional gravitationally coupled states that redefinition that would render gravitational modes luminal arise at the scale M. However, if the dimension-six around all backgrounds and so this procedure while operators are suppressed, the dimension-eight operators comforting is also misleading. At one loop level and higher must be further suppressed since it equally holds for order, we find Riemann cubed terms in the effective action dimension-eight operators that if they arise in a weakly generated from loops (5.1), part of which are Weyl cubed coupled UV completion, they do so in the manner 2 4 6 terms. These terms cannot be removed with a local field MPlRiemann =M , and then the UV completion would redefinition since they are not proportional to the leading need an infinite number of states of spin s>2 arising at the equations of motion. Although these terms do not contrib- same low scale M. This is transparent by their influence on ute on FLRW backgrounds, for backgrounds with less the speed of propagation of gravitational waves, effects symmetry they do change the speed of gravitational waves which could not arise at tree level in a theory of massless and yet there is clearly no local field redefinition that spin 2 and massless/massive spin s<2. Thus it does not removes them. make sense to argue on phenomenological grounds that the In this work we shall mainly focus on dimension-four R2 dimension-eight operators are larger than the dimension-six and dimension-six (R3 and R∇2R) curvature operators, as ones. Lower dimension operators are always more signifi- well as specific dimension-eight R4 curvature operators in cant, unless they are suppressed by a symmetry, which is Sec. VB. Dimension-four curvature operators are naturally not the case here. the leading contributions, but if those vanish (as will be considered in Sec. V) the dimension-six curvature oper- 8In a weakly coupled UV completion, we assume that there ators are then the leading contributions. is a small dimensionless coupling constant g which controls the −2 loop expansion of the UV theory SUV ¼ g S0. The implicit 2 ¼ 2 2 assumption is that MPl M =g which explains why tree level 1. Taking care of the dimension-four curvature operators 2 effects will come with MPl in front but one loop effects will have In discussing EFT descriptions of gravitational waves no power of MPl. In the case of string theory, M is the string scale from mergers, in [67] it was argued that the dimension-four and g the string coupling constant (dilaton).

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In this work we shall not assume any such preconditions excludes massive gravity [68,69]) and couple minimally to and will consider the (albeit small) effect of all dimension- gravity.9 four and -six operators from loop corrections where they 2 3 2 rather arise at the respective scales Riemann ,Riemann =M A. Identifying speeds in an EFT context from integrating out particles of any spin, including Standard Model particles of mass M. By relinquishing ourselves 1. Reorganizing EFT expansion from the constraints of purely tree level effects, we may consider lower mass scales M without being ruled out by Before proceeding to explicit calculations, it is worth other gravitational constraints. We refer to Appendix A discussing how we identify the speed of propagation in a for details on how integrating out a massive scalar field time or space dependent setting in which we are working, leads to specific Riemann2 and Riemann3=M2 operators. with a truncated EFT with higher derivative operators. The contributions from integrating out generic spin s<2 Throughout the following discussion we mainly have fields can be found in [59]. We will also consider those FLRW in mind, although the reorganization of the EFT dimension-eight operators known to arise in the string and the way we identify the speed is fully generalizable to effective action in from α0 corrections in Sec. VB. any other type of background. In an EFT higher derivative operators should be dealt III. TENSOR MODES ON A BACKGROUND, with perturbatively, and we may only draw conclusions LEADING EFT CORRECTIONS from them in the regime in which perturbation theory in these higher derivative operators is valid. For instance, In what follows we shall remain agnostic on the precise working with the curvature-squared interactions introduces low energy field content that leads to the cosmological fourth order derivatives in the equations of motion. Directly solution and only assume that it is as “standard” as perturbing will give an effective equation of motion for possible; in particular we assume that the effective degrees gravitational waves written in momentum space of the of freedom relevant for the low energy dynamics (and the schematic form cosmological background) are of spin s<2 (in particular it

˜ ðη Þ ˜ ðη Þ ˜ ðη Þ 1 þ b2 ;k ∂2 ðηÞþ ˜ ðηÞþb1 ;k ∂ ðηÞþ ˜ ðη Þþb0 ;k ðηÞ 2 ηhk a1 2 ηhk a0 ;k 2 hk MPl MPl MPl ˜ ðηÞ ˜ ðηÞ þ b4 ∂4 ðηÞþb3 ∂3 ðηÞþOð −4 −2 −2 −4Þ ≈ 0 ð Þ 2 ηhk 2 ηhk M ;MPl M ;MPl : 3:1 MPl MPl

To simplify the procedure we may first perform a rescaling of field variables hkðηÞ → ΩðηÞhkðηÞ so that for the resulting equation the leading friction term a˜ 1ðηÞ vanishes and the resulting momentum space equation is schematically of the form ðη Þ ðη Þ ðη Þ 1 þ b2 ;k ∂2 ðηÞþb1 ;k ∂ ðηÞþ ðη Þþb0 ;k ðηÞ 2 ηhk 2 ηhk a0 ;k 2 hk MPl MPl MPl ðηÞ ðηÞ þ b4 ∂4 ðηÞþb3 ∂3 ðηÞþOð −4 −2 −2 −4Þ ≈ 0 ð Þ 2 ηhk 2 ηhk M ;MPl M ;MPl : 3:2 MPl MPl In the limit H ≪ k=a ≪ M we may use the Wentzel-Kramers-Brillouin (WKB) approximation to determine a dispersion relation which has four powers of frequency and hence twice as many solutions as that of a second order differential equation.10 The additional solutions of this dispersion relation are the ghostly states that arise from the truncation and whose solutions should be ignored in the EFT context. What we are interested in are only the solutions that are continuously → 0 connected with the solutions that arise in the limit MPl for which the equation of motion is second order,

∂2 ðηÞ¼− ðη Þ ðηÞþOð −2Þ ð Þ ηhk a0 ;k hk MPl : 3:3

9Conformal couplings to gravity can be dealt with by first diagonalizing and then working with the appropriate minimally coupled fields. 10The higher powers in the dispersion relation are always unphysical and simply signal the breakdown of perturbation theory when k ∼ M. At high energies the dynamics of the modes that have been integrated out should be accounted for. Note that the additional modes one would obtain in any truncated theory are not and should not be directly identified with the degrees of freedom present in the high energy theory [70].

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To identify these, we can use the lower order equations and substitute them either into the higher order ones or more consistently at the level of the Lagrangian and perform field redefinitions to remove higher order time derivatives. Then to the desired order, the equation of motion can be reexpressed as a second order in time derivatives equation of motion ðη Þ ðη Þ ðη Þ 1 þ b2 ;k ∂2 ðηÞþb1 ;k ∂ ðηÞþ ðη Þþb0 ;k ðηÞ 2 ηhk 2 ηhk a0 ;k 2 hk MPl MPl MPl ðηÞ ðηÞ þ b4 ∂2ð− ðη Þ ðηÞÞ þ b3 ∂ ð− ðη Þ ðηÞÞ þ Oð −4 −2 −2 −4Þ ≈ 0 ð Þ 2 η a0 ;k hk 2 η a0 ;k hk M ;MPl M ;MPl : 3:4 MPl MPl

2. Identifying the speed velocities which as we see are poorly defined in this time We could at this stage use the WKB approximation to dependent setting, but the causal properties of the hyper- ’ define an effective dispersion relation. Indeed we will in bolic equations defining the retarded Green s functions. general only be interested in the effective speed of This is entirely determined by the light cones of the propagation in the region H ≪ k=a ≪ M in which the hyperbolic metric defining the equation. EFT is under control and the modes are sufficiently With this in mind we reorganize the EFT expansion in a subhorizon that the WKB approximation is valid and it manner suitable to determine the retarded propagator is meaningful to talk about waves. However, already for perturbatively, as a second order in the time hyper- gravitational waves in GR, such an analysis would imply a bolic system plus (perturbative) higher spatial derivative superluminal group velocity when the effective mass is corrections. In doing so it is worth emphasizing that in negative. For instance for GR, to implement the WKB general the effective friction term, in the above example ð − ∂ − Þ∂ approximation we begin with the ansatz b1 b4 ηa0 a0b3 ηhk, is typically k-dependent and an additional rescaling h ðηÞ → ð1 þ Ωðk; ηÞ=M2 Þh ðηÞ is R k Pl k η 0 0 k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiA ikz−i dη ωðη ;kÞ helpful in order to remove any -dependence in the friction hkðηÞ¼ e ; ð3:5Þ ωðη;kÞ term before determining the propagation speed. Once this is done, the equation of motion for tensors can be put in the which leads to the exact equation form ∞ 00 00 02 X a 1 ω 3 ω 2 2n ω2ðη Þ¼ 2 − − − ð Þ ∂ηh ðηÞ¼− β ðηÞk h ðηÞ; ð3:7Þ ;k k 2 : 3:6 k n k a 2 ω 2 ω n¼0

The WKB approximation amounts to solving this equation which is naturally reorganized as (to any desired order in iteratively to any desired order. The leading iteration the EFT expansion) ω2ðη;kÞ¼k2 − a00=a would for example during inflation, 00 2 2 where −a =a is negative, give superluminal dω=dk. This is ∂ηhkðηÞþβ1ðηÞk hkðηÞþβ0ðηÞhkðηÞ clearly meaningless since an exact construction of the X∞ 2n retarded propagator on FLRW shows that it only has ¼ − βnðηÞk hkðηÞ: ð3:8Þ support on and inside the light cone [71]. The WKB n¼2 approximation assumes k2 ≫−a00=a and hence is not accurate enough to account for the effective mass term The left-hand side (LHS) defines a hyperbolic system 2 2 2 ¼ β ðηÞ in the exponent; all we can infer from it is ω ≈ k and with propagation speed cs 1 and effective mass 2 ¼ β ðηÞ that if this were the exact equation the front velocity meff 0 . Temporarily ignoring the right-hand side 11 (RHS), just as in the GR case, the presence or not of limk→∞ ω=k would be luminal. What is relevant from the perspective causality is neither the phase or group the effective mass is irrelevant to the causal structure of the retarded propagator. The latter is determined by the 2 11 effective light cone of the two derivative=k terms. The full The WKB approximation is still under control, and it is ’ just its interpretation that is failing. For example in the explicit Green s function can be determined perturbatively by case of gravitational waves in de Sitter a ¼ −1=ðHηÞ, the iterating the relation exact solutions are Hankel functionsP whose WKB form is ∼ −1=2 ikz−ikηð1 þ α ð ηÞþ ∞ α ð ηÞ2Þ 2 k 0 2 k 0 k 0 h k e 1= k n¼2 n= k . This follows ∂ηG ðη; η Þþβ1ðηÞk G ðη; η Þþβ0ðηÞG ðη; η Þ ω ¼ ret ret ret from (3.6) by taking the leading iteration as k and treating ∞ all high order terms perturbatively outside the exponent rather X ¼ − β ðηÞ 2n k ðη η0Þþδðη − η0Þ ð Þ than inside it. In fact in this special case the series terminates n k Gret ; : 3:9 ¼2 αn ¼ 0 for n ≥ 2, and so this is the exact solution. n

063518-9 CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020) 2 At any finite order in this expansion the causal structure of pffiffiffiffiffiffi M L ¼ −g Pl R þ Ll:e:ðg; ψÞ Green’s function will be determined by zeroth order 2 k 0 Green’s function G0 ðη; η Þ ret 2 2 2 þ C1R þ C2Rμν þ C3Rμνρσ þ ; ð3:13Þ ∂2 k ðη η0Þþβ ðηÞ 2 k ðη η0Þþβ ðηÞ k ðη η0Þ ηG0 ret ; 1 k G0 ret ; 0 G0 ret ; ¼ δðη − η0Þ; ð3:10Þ where the ellipses represent higher-order operators in the EFT expansion, Ll:e:ðg; ψÞ is the Lagrangian for the low ψ which has support on and inside the light cones defined by energy fields which we assume here are all minimally ψ the effective metric [71] coupled to gravity and do not include fields with spin 2 or more. Here and in what follows Ci denotes the IR value IR 2 2 2 2 2 C . In four dimensions Rμνρσ can be written in terms of the ds˜ ¼ −cs ðηÞdη þ dx⃗: ð3:11Þ i Gauss-Bonnet term, which does not contribute to local Hence this defines what we mean by the low energy speed dynamics, plus the remaining curvature-squared terms, and 2 so these coefficients are better written in terms of the cs ¼ β1ðηÞ. The above procedure may be easily generalized to any order in time derivatives and hence any order in the coefficients of Gauss-Bonnet, Weyl squared, and Ricci EFT expansion.12 scalar squared In a fundamentally Lorentz invariant theory, all coef- 2 2 2 C1R þ C2Rμν þ C3Rμναβ ficients βn → 0 for n ≥ 2 when the spontaneous breaking is 2 2 removed, and similarly β1 → 1. Thus while the corrections ¼ 2 þ 2 þ ð Þ CR R CW Wμναβ CGBGB; 3:14 to the sound speed from unity will be small, suppressed by at least one power of the maximal symmetry breaking, e.g., where the Gauss-Bonnet term is ˙ 2 by one power of H=MPl in FLRW, the same will be true of ¼ 2 − 4 2 þ 2 ð Þ all the coefficients βn for n ≥ 2 which must similarly be GB Rμναβ Rμν R : 3:15 suppressed by one power of H=M˙ 2 .13 Hence, as long as Pl 1 ¼ 2 þ þ 2 ¼ k=a is small in comparison to the momentum EFT cutoff, The precise relations are C1 CR CGB 3 CW , C2 −2 2 − 4 ¼ 2 þ the dominant low energy modification to the propagation CW CGB, C3 CW CGB. Since the Gauss- will be captured by β1 − 1, that is, Bonnet term is topological in four dimensions effectively for the rest of this section, we shall be working with the X∞ leading EFT corrections to GR as follows: jðβ ðηÞ − 1Þ 2j ≫ jβ ðηÞj 2n ð Þ 1 k n k : 3:12 n¼2 2 pffiffiffiffiffiffi M 2 2 L ¼ − Pl þ Ll:e:ð ψÞþ 2 þ 2 g 2 R g; CR R CW Wμναβ : The way in which we have defined the low energy speed is easily generalizable to perturbations around any spacetime ð3:16Þ and will always predict cs ¼ 1 for pure GR minimally coupled to classical matter. The merit in its definition will As an example, we show in Appendix A how loops of a be seen in that it is naturally connected with precise terms massive scalar field of mass M leads to a contribution to in scattering amplitudes that are potentially constrained by such curvature-squared contributions with [see Eq. (A28)] means of S-matrix positivity bounds as we discuss 2 in Sec. IV. 5 1 Λ C 2 ¼ log ; R 8 16 × 240π2 M2 2 B. Curvature-squared corrections on FLRW 1 1 Λ C 2 ¼ log : ð3:17Þ W 4 16 × 240π2 M2 1. Dimension-four curvature operators We begin in this section with the leading curvature These expressions need to be supplemented by UV 2 Λ corrections to our EFT, the R corrections. To reiterate, counterterms which remove the dependence. They 2 ð Þ − these may arise either from tree level or loop level effects of confirm, however, the general expectation that CW M1 2 ð Þ 0 heavy fields, and are to this order CW M2 > for M1

063518-10 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) P ¼ εσ and traceless tensor fluctuations hij σ hσ ij, where the may independently arise from tree level corrections from σ ¼þ εþ;× fields of spin s ≥ 2. The contributions from the R2- sum is over the two polarizations ; × and ij represents the two polarization tensors. In what follows, operators are of the form we shall omit any mention of those two polarizations and LðhhÞ ¼ 2 ˆ ð Þ simply denote the tensor modes as h, while it is of course dim 4 a hOdim 4h; 3:22 understood that a sum over both polarizations is implicit. ˆ The tensor modes are normalized so that the full metric is where Odim 4 is a fourth order operator given by given by 1 1 1 1 ˆ ¼ □2 þ □ ∂ þ □ þ ∂ Odim 4 4 g1 η 3 g3 η η 2 g4 η g5 η gij ¼ γij þ ahij: ð3:18Þ a a a a 1 0 2 ⃗2 We use the standard notation, H ¼ a=a˙ ¼ a =a is the þ g7∇ þ g8; ð3:23Þ a2 Hubble parameter, with dots referring to derivatives with respect to physical time t, and primes with respect to with the coefficients expressed as conformal time η.

¼ 2 ð Þ g1 CW ; 3:24 3. Einstein-Hilbert

¼ 2 2 ð Þ The Einstein-Hilbert term then leads to the standard g3 CW H; 3:25 canonical kinetic term for the tensor modes, 2 ¼ 6 2 ð2 þ ˙Þþ6 2 ˙ ð Þ g4 CR H H CW H; 3:26 2 ðhhÞ MPl 2 −2 2 ˙ L ¼ a hða □η − 4H − 3HÞh; ð : Þ EH 3 19 ¼ −6 2 ð4 ˙ þ ̈Þ − 4 2 ð ˙ þ ̈Þ ð Þ 4 g5 CR HH H CW HH H ; 3:27 □ ¼ −∂2 þ ∇⃗2 ’ ˙ where η η is the d Alembertian on Minkowski g7 ¼ −4C 2 H; ð3:28Þ ⃗2 W spacetime, with ∇ being the standard three-dimensional 4 2 2 ¼ 6 2 ð4 − 10 ˙ − 8 ˙ − 11 ̈− 2 ⃛Þ Cartesian Laplacian (that will be replaced by the momenta g8 CR H H H H HH H 2 −k below). 2 ˙ ˙ 2 ̈ ⃛ − C 2 ð2H H þ H þ 3HH þ HÞ: ð3:29Þ Assuming that all the couplings to gravity involved in W Lðl:e:Þð ψÞ ψ ψ g; are minimal and there are no fields with spin Working perturbatively in the dimension-four curvature 2 or more, then the matter field Lagrangian leads to the operators, following the approach discussed in Sec. III A, following “effective mass” term for the tensor fluctuations □ 14 we may substitute the relation for ηh in terms of h as on FLRW, derived from (3.21), 2 pffiffiffiffiffiffi ð Þ M 2 2 ˙ − L l:e: ð ψÞðhhÞ ¼ Pl 2ð3 2 þ 2 ˙Þ 2 ð Þ □ηh ¼ a ð−2H − HÞh: ð3:30Þ g ψ g; 2 a H H h ; 3:20 This perturbative substitution can be performed on the first leading to the standard low energy contribution ˆ three terms of the operator O 4 defined in (3.23) so that dim 2 1 only the last three terms remain with slightly altered ðhhÞ MPl 2 2 ˙ L ¼ a h □η þ 2H þ H h: ð3:21Þ coefficients, EH;ψ 4 a2 1 1 ˆ ¼ ˜ ∂ þ ∇⃗2 þ ˜ ð Þ Odim 4 g5 η 2 g7 g8: 3:31 4. Curvature-squared contribution a a

We shall now derive the equation of motion for tensor The expressions of g˜5;8 is irrelevant for the rest of the modes on including the R2-operators which either arise as discussion but we include them for completeness, logarithmically running terms coming from matter loops or 3 ˙ ̈ 2 ˙ g˜5 ¼ g5 þ 2g1ð4H þ 6HH þ HÞþg3ð−2H − HÞ; 14 This result follows for quite general Lagrangians; for ð3:32Þ instance, for a single scalar ψ it follows for all interactions of ð ψÞ¼ ðð∂ψÞ2 ψÞ the form P X; P ; as well as a generalized cubic ˜ ¼ þ ð16 4 þ 34 2 ˙ þ 9 ̈þ ⃛ þ 7ð ˙ Þ2Þ Galileon Gðð∂ψÞ2; ψÞ□ψ. However, we do not consider other g8 g8 g1 H H H HH H H Horndeski operators as well as beyond Horndeski, as these 3 ˙ ̈ 2 ˙ þ g3ð−4H − 6HH − HÞþg4ð−2H − HÞ: ð3:33Þ involve nonminimal couplings to gravity and already induce a sound speed that differs from luminal at low energy on cosmological backgrounds without even accounting for the effect of At this stage we see directly that to this order, the modified heavier modes. equation of motion for the tensor modes on FLRW is

063518-11 CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020) 2 2 2 2 2 2 1 16 2 ˙ 4 ˜ ¼ ð Þ ð − þ þ Þþ ð Þ a b −∂2 þ 1 − CW H ∇2 þ ag5 ∂ ¼ 2 ds a y dy dt dx dz a y habdx dx ; 2 η 2 2 η h m0h: a MPl MPl ð3:36Þ ð3:34Þ where hab are transverse and traceless with respect to the The friction term can easily be taken care of by performing ðt; x; zÞ subspace. Due to the fact that these solutions a rescaling of the field which will keep the second space have the same amount of symmetry as the FLRW solutions, and time derivatives unaffected and simply modify the the equivalently defined tensor modes decouple from the 2 2 matter degrees of freedom which source the background y effective mass term by order H =MPl corrections. As a result we can directly read off the effective low energy dependence. Repeating the previous analysis, we find sound speed which as we see gets affected by the Weyl term similarly a fourth order differential equation which may (and solely the Weyl term) on this spontaneously Lorentz be reorganized into a second order differential equation breaking background, with associated propagation speed along the y direction (the speed in the x-z plane is luminal by symmetry) ˙ 4 16C 2 ð−HÞ H 0 4 2 W 16 2 ð− ð ÞÞ c ¼ 1 þ þ O : ð3:35Þ 2 CW Hy y H s 2 4 c ðyÞ¼1 þ þ O ; ð3:37Þ MPl MPl s 2 4 MPl MPl Interestingly, we see that the effective sound speed is dH ðyÞ where H ðyÞ¼dlnaðyÞ=dy and H0 ðyÞ¼ y . As in the superluminal on a null-energy condition satisfying back- y y aðyÞdy ˙ ground H<0 as soon as C 2 > 0. At this stage we may be cosmological case, for matter satisfying the null energy W condition we have inclined to conclude that CW2 ought to be negative for any consistent (causal) UV completion; however, this conclu- H0 ðyÞ < 0; ð3:38Þ sion would be wrong, or at least highly premature, as we y

2 0 will argue in what follows (see Sec. IV). and so we again conclude that if CW > , then the tensor In a weakly coupled UV completion, the natural scale modes propagate superluminally. This is consistent with the 2 Λ2 Λ for CW is MPl= where is the scale of new tree level arguments given below in Sec. IV which apply for any physics. Hence the correction to the sound speed may be as geometry. large as ∼jH˙ j=Λ2. This is particularly interesting in the case of inflationary models where the hierarchy between jH˙j1=2 D. Sound speed frequency dependence Λ and the scale of new physics is not necessarily large. We have seen that the speed of gravitational waves is 2 0 superluminal in the low energy region for CW > . Since C. Static warped geometries this calculation is performed in an EFT context, we can 2 1 2 Although our main focus is on cosmological spacetimes, only trust the calculation of cs up to and including =MPl it is worth noting that the above analysis trivially general- corrections without including higher order operators in the izes to static warped geometries with ISOð1; 2Þ symmetry. EFT Lagrangian. Nevertheless it is instructive to see what By analytic continuation we can equally well consider happens if we temporarily assume that the R2 and W2 terms metrics with nontrivial dependence on only one space define the exact Lagrangian and compute the speed to dimension, e.g., y and associated matter profiles higher order. The next order correction takes the form

2 2 2 4 4 16 2 ð− ˙Þ 3 ˙ − 6 ˙ þ 5 ̈− ⃛ ω2 ≈ 1 þ CW H − 1024 3 H H H HH H k þ O H k 2 ð Þ 2 CW2 4 2 2 8 k : 3:39 MPl MPl a MPl MPl

We see that at higher momenta, the departure of the speed which spontaneously breaks Lorentz invariance, for mo- from unity is reduced regardless of the sign of CW2 . Indeed, menta much larger than the scales of the background it will determining the exact form of the dispersion relation shows always be the leading Lorentz invariant operators which that the speed of sound always asymptotes to unity as dominate and guarantee a luminal front velocity k → ∞. This is simply because, at high energies in the 2 2ð Þ¼1 ð Þ truncated Lagrangian, the W terms dominate the dispersion lim cs k : 3:40 k→∞ relation, the leading term is a Lorentz invariant □2 operator. We stress again that we cannot trust this calculation in the Indeed taking seriously the fourth order equation inferred EFT context since other operators will kick on; however, it is from (3.22) and (3.21), it is helpful to note that to this order indicative of a general expectation that even on a background the effective action can be rewritten as

063518-12 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) Z 2 ðhhÞ 4 g1 2 MPl accurate determination of the low energy speed. In par- S ¼ d x ð□ηhÞ þ þ g4 a2 4 ticular, we see from performing a Taylor expansion of this approximation, the order k4 term is incorrect. It is, however, ˙2 ⃗ 2 ⃗ 2 2 2 × ðh − ð∇hÞ Þ − g7ð∇hÞ þ a DðηÞh : ð3:41Þ correct in the higher momentum limit if the equation were taken as exact. This can be rewritten as a second order system by intro- IV. SUPERLUMINALITY AND CAUSAL ducing an auxiliary variable Ψ, UV COMPLETIONS Z 1 2 ðhhÞ 4 2 −1 2 MPl On NEC preserving backgrounds, we have seen that S ¼ d x Ψ□ηh − a g Ψ þ þ g4 4 1 4 gravitational waves have superluminal low energy speeds if the coefficient CW2 of the Weyl squared operator in the ˙2 2 ⃗ 2 2 2 × ðh − ð∇hÞ Þ − g7ð∇hÞ þ a DðηÞh : ð3:42Þ EFT of gravity is positive. If one were to jump to conclusions at this stage, one may be inclined to argue that consistency of the EFT requires setting CW2 to be Performing a standard WKB approximation with ansatz negative (this is indeed the logic followed in much of the standard literature). However, as we shall argue in this ¼ ðη Þ −iWðη;kÞ Ψ ¼ Ψ ðη Þ −iWðη;kÞ ð Þ h h0 ;k e ; 0 ;k e ; 3:43 section, this conclusion is premature and likely erroneous. Indeed as already argued earlier, contributions to the −iWðη;kÞ with e varying rapidly in time and h0ðη;kÞ; Ψ0ðη;kÞ 2 – ≥ 2 coefficient CW from tree level massive spin s fields slowly in time, we obtain at leading order in k ≫ aH the or from loops of massive spin–s<2 fields lead to a approximate equations 2 positive contribution to CW . In what follows we shall show that a positive sign for C 2 typically follows from 1 W 02 2 2 −1 standard positivity bound arguments (if applicable to ðW − k Þh0 ≈ a g1 Ψ0; ð3:44Þ 2 gravity) and ensures subluminality in the matter sector. 2 02 2 MPl 02 2 2 ðW − k ÞΨ0 þ 2 þ g4 ðW − k Þh0 − 2k g7h0 ≈ 0: A. Gravity versus matter light cones 4 and null-energy condition ð3:45Þ 1. Matter frame Combining together the dispersion relation can be deter- Both for NEC preserving FLRW backgrounds and for mined from NEC preserving static warped geometries, it was shown in (3.35) and (3.37) that the effective low energy sound speed M2 1 k2 1 k2 2 of gravitational waves is (ever so slightly) superluminal as Pl þ 02 − þ 02 − g4 W g1 W 2 0 4 a2 a2 a2 a2 soon as CW > . This result can actually be derived very 2 simply by recognizing that it is entirely a consequence − k ≈ 0 ð Þ of the field redefinition between the metric frames in 2 g7 ; 3:46 a which matter minimally couples and that in which gravity minimally couples, following from (4.6). To see this which has the exact solution (taking only that solution which explicitly, we can start with our EFT Lagrangian for GR is continuously connected with the usual GR solution) including the leading order curvature corrections as given qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in (3.16) 2 2 2 4g1g7k MPl 2 −4g4 − M þ 4 2 þðg4 þ Þ ω2 ¼ 02 ¼ 2 þ 2 Pl a 4 pffiffiffiffiffiffi 2 W k a : MPl 2 2 8 L ¼ − þ 2 þ 2 þ g1 g 2 R CR R CW Wμναβ CGBGB ð3:47Þ þ L ð ψÞ ð Þ matter g; ; 4:1 Interestingly this solution is always real, meaning no decay, regardless of the sign of CW2 (i.e., g1)aslongasthenull 2 ˙ pffiffiffiffiffiffi M 2 2 2 0 ¼ − Pl þ 2 − 2 þ 2 2 energy condition is satisfied H< , has the desired low g R CR CW R CW Rμν 2 02 2 2 3 energy behavior, and asymptotes to ω ¼ W → k at high energies. þðC 2 þ C Þ GB þ L ðg; ψÞ ; ð4:2Þ We have avoided performing a WKB analysis of the W GB matter fourth order equation in order to determine the speed of L ð ψÞ propagation in the previous sections since for these higher where matter g; is the Lagrangian for the (low energy) order derivative systems they generally do not give an matter fields that we assume (for now) are minimally

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FIG. 1. TT amplitude: Graviton mediated loop contributions to matter interactions. χ symbolizes a matter field present in the stress- energy tensor Tμν, a wiggly line is a graviton propagator, and solid purple lines are the loops of heavy fields. coupled to gravity. For definiteness, we refer to this frame price of nonminimal interactions in the matter sector. Such as the frame in which matter is minimally coupled and types of operators were considered within the context of matter denote the metric in this frame as gμν ¼ gμν . The leading EFTs for cosmic acceleration [73,74]. It is clear that to this matter ¼ −2 order, in this representation, gravitational waves will travel order equation of motion in this form is Gμν MPl Tμν, tensor where Tμν is the stress energy of the matter field at the speed defined by the light cones of the metric gμν , but light itself will no longer travel at this speed since 2 1 pffiffiffiffiffiffi Maxwell’s equations are modified by the inclusion of Tμν ¼ − pffiffiffiffiffiffi ð −gL ðg; ψÞÞ: ð4:3Þ −g δgμν matter higher order operators.

3. Gravitationally induced matter interactions 2. Tensor frame This leading order “TT deformation” of the matter Consider now the following field redefinition: Lagrangian in (4.7) can be understood diagrammatically 1 as arising from the process given in Figs. 1 and 2. The matter ¼ tensor þ δ ð Þ gμν gμν 2 gμν; 4:4 diagram in Fig. 2 represents the tree level process whereby MPl a massive heavy state of spin-2 or -0 is exchanged between the two stress energies. This corresponds to the explicit for which the Lagrangian picks up a new interaction example given in Sec. IV C 1. The diagram in Fig. 1 describes a loop process from a heavy field mediated via 1 μν −2 μν 1 μν ΔL ¼ − δgμνðG − M T Þþ OðRR Þ: ð : Þ tree level massless graviton exchange. This corresponds to 2 Pl 2 4 5 MPl the explicit example given in Sec. VA1, at least after field redefinition. We stress again that while the perspective If we make the choice that obtained by performing these field redefinitions is useful, at least at these low orders, once we consider higher order 1 3 δgμν ¼ 4C 2 Gμν þ Tμν interactions, gravitational couplings arise (e.g., Riemann ). W M2 Pl These cannot be removed via local field redefinitions alone, 2 1 and so it will not be possible to give such a simple effective − 2 C 2 − C 2 R − T gμν; ð4:6Þ R 3 W M2 description in terms of gravitationally induced matter Pl interactions. Of course S-matrix elements are invariant then the field redefined Lagrangian is15 under these field redefinitions, and the on-shell process described by Figs. 1 and 2 can be computed in any frame. 2 pffiffiffiffiffiffi M 2C 2 L ¼ − Pl þ þ L þ W μν g 2 R CGBGB matter 4 TμνT MPl 4. Connection with the NEC 2 ðC 2 − C 2 Þ After performing the field redefinition to remove the þ R 3 W 2 þ ð Þ 4 T ; 4:7 curvature-squared terms, we have the Einstein (or tensor) MPl where again ellipses represent higher order curvature 3 4 operators (e.g., of order R =M ) and gμν is now the tensor tensor (Einstein) frame metric gμν . At the order at which we are working, the dimension-four curvature-squared interactions have now disappeared, other than the Gauss- Bonnet term which does not affect local physics, at the

15To this order this is similar to a four-dimensional TT¯ FIG. 2. TT amplitude: Gravitational strength matter inter- deformation; however, the coincidence ends at this order [72]. actions arising from exchange of mass spin-0 and spin-2 states.

063518-14 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) frame metric in which the gravitational tensor fluctuations The previous argument works for any background given are minimally coupled the field redefinition (4.6).Ifnμ denotes a vector which is μ μ null with respect to the matter light cone gμνn n ¼ 0, then 4C 2 1 μ ν 0 tensor ¼ matter − W matter þ if matter satisfies the NEC n n Tμν > , we have gμν gμν 2 Gμν 2 Tμν MPl MPl 4 2 2 − 2 1 CR 3 CW matter matter μ ν tensor − −R þ T gμν þ: n n gμν < 0; if C 2 > 0; 2 2 W ð Þ MPl MPl μ ν 4:13 tensor 0 2 0 n n gμν > ; if CW < ; ð4:8Þ

Evaluating this on-shell, then to the same order we have meaning that the vector nν is timelike with respect to the 2 0 8 gravitational wave light cone if CW > and spacelike if 8C 2 4C 2 − C 2 ν tensor ¼ matter − W matter − R 3 W matter matter C 2 < 0. Since the vector n is timelike with respect to gμν gμν 2 Gμν 2 R gμν W MPl MPl the gravitational wave light cone, in the case where 16 2 0 þ ð4:9Þ CW > , the matter light cone always lies inside the gravity light cone regardless of the choice of background, 8 8 2 4 2 − 2 for all NEC respecting matter. Violating the NEC or matter CW matter CR 3 CW matter matter ¼ gμν − Tμν þ T gμν þ: having C 2 < 0 automatically reverses this (arguably M4 M4 W Pl Pl natural) order. ð4:10Þ B. Positivity bounds for light fields For concreteness, we now focus on the FLRW metric considered in Sec. III B, although the following argu- From an EFT point of view the coefficients CR2 and ment clearly applies for any background (as for the CW2 are a priori undefined unless we match them with a example of the static warped geometry in Sec. III C), UV completion as will be performed shortly in Sec. IV C. not just FLRW. In the case where we are dealing solely with curvature- When the matter metric has the FLRW form squared corrections in the gravitational EFT, a local field matter μ ν 2 2 2 gμν dx dx ¼ aðηÞ ð−dη þ dx⃗Þ, then to this order redefinition is always possible so as to move the the Einstein (tensor) frame metric is corrections into the matter sector as was performed in the previous subsection. Before considering a matching tensor μ ν 2 2 2 2 2 with a UV completion, we shall first consider how the gμν dx dx ¼ Ω a ð−csdη þ dx⃗Þþ; ð4:11Þ standard positivity bounds from S-matrix analyticity, 2 2 ¼ 1 þ 16 2 ð− ˙ Þ locality, and unitarity may be used to constrain the sign with cs CW H =MPl which is exactly the result obtained in (3.35). The conformal factor Ω2 is given by of CR2 and CW2 of the resulting matter interactions, provided we argue or assume that the contribution of the 8C 2 graviton exchange t-channel pole can be neglected. Ω2 ¼ 1 þ W ð2H˙ þ 3H2Þ M2 Pl 1. Scaling Limit ð24C 2 − 16C 2 Þ − R W ðH˙ þ 2H2Þ : ð4:12Þ M2 Assuming we are free to choose the coefficients in the Pl → 0 EFT, at this point we can take a decoupling limit MPl 2 2 One of the interesting features about the above results for keeping CR2 =MPl and CW2 =MPl fixed. For instance, in the the sound speed is that the correction is proportional to H˙ case in which the R2 terms arise from loop corrections from and so changes sign if we consider a field theory with NEC integrating out fields, we expect the C’s to scale with N, the violation. A violation of the NEC, required to achieve number of fields. At the same time the Planck mass is ϖ ¼ ρ −1 2 ¼ Λ2 p= < , typically leads to instabilities [75,76] or related to the species scale as MPl N species [79]. Hence even a breaking of the low energy effective field theory → ∞ → ∞ this decoupling limit is simply the limit N , MPl [77], unless it is accompanied with superluminal modes in Λ keeping species fixed. In this limit, gravity may be the sector responsible for the breaking of the NEC [78].Our described by a linearized massless spin-2 on Minkowski findings naturally complement these results in the case coupled to matter with Lagrangian 2 0 2 0 where CW > . Indeed, for CW > , gravitational waves ˙ 0 become subluminal as soon as H> , which, in the field 16 frame in which gravity is luminal, is equivalent to the As would, for instance, be the case if the curvature-squared corrections were solely arising from integrating out massive statement that the matter fluctuations become superluminal scalar fields, or as implied from the spectral representation as soon as the NEC is violated (as soon as H>˙ 0). discussion in Sec. IV C.

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1 1 region t<0 to t>0, precluding any statement of L ¼ habEh þ habT þ L0 þ; ð4:14Þ 17 8 ab 2M ab matter positivity even for small positive t. Pl However, a recent argument given in [32] has suggested where E ¼ □ þis the Lichnerowitz operator, Tab is the a potential solution. The idea is to regulate the IR L t ¼ 0 stress energy of matter, and the last term is a modified divergence at by compactification to three dimen- matter Lagrangian sions and apply positivity bounds there. Assuming the validity of this reasoning (see Ref. [55] for subtleties that 2 2 2 ð 2 − 2 Þ 0 CW μν CR 3 CW 2 may need to be accounted for), it makes it possible to L ¼ L þ TμνT þ T þ: matter matter 4 4 discard the contribution from the massless graviton t MPl MPl channel pole. If correct, then in the present context the ð4:15Þ forward limit positivity bounds [10,25,26] applied to the Since this is now a local field theory living on Minkowski pole subtracted amplitude impose spacetime, we may ask what are the constraints on the 4 2 2 C 2 þ C 2 > 0: ð4:19Þ coefficients CR and CW based on positivity bounds R 3 W applied to the scattering of these light matter states, including these N−1 ∼ M−2 corrections. In the next section we shall argue for this positivity in a Pl different manner which is consistent with this ability to ignore the t-channel pole. 2. Light scalar fields For instance, suppose we consider matter to be a single (nearly) massless scalar field whose stress energy 3. Electromagnetism 1 2 takes the form Tμν ¼ ∂μϕ∂νϕ − 2 ημνð∂ϕÞ . The effective Similarly, taking the example of the matter being α 1 αβ Lagrangian for the scalar is then electromagnetism for which Tμν ¼ FμαFν − 4 ημνFαβF 4 then the effective matter Lagrangian is 1 ðC 2 þ C 2 Þ L0 ¼ − ð∂ϕÞ2 þ R 3 W ð∂ϕÞ4 þ Oð −4Þ matter 4 M : 1 2 2 1 2 M Pl 0 αβ CW 4 2 2 Pl L ¼ − FαβF þ ðTrðF Þ − ðTr½F Þ Þ matter 4 M4 4 ð4:16Þ Pl þ OðM−4Þð4:20Þ −1 ∼ −2 2 − 2 Pl At order N MPl , the tree scattering amplitude 1 2 describing the process ϕϕ → ϕϕ will receive two types ¼ − αβ þ CW ð αβÞ2 4 FαβF 2 4 FαβF of contributions. Contact interactions which come from MPl ð∂ϕÞ4 the interactions and s, t, and u channel poles that CW2 ˜ αβ 2 −4 þ ðFαβF Þ þ OðM Þ: ð4:21Þ come from the exchange of massless spin-2 graviton, 8M4 Pl Pl 1 −tu −st −su A ðs; tÞ ∼ þ þ Familiar arguments on the absence of superluminalities for ϕϕ→ϕϕ 2 2 − − − MPl s u t photons in different backgrounds [10], or equivalently 4 positivity bounds applied assuming the graviton t-channel ðC 2 þ C 2 Þ þ 2 R 3 W ð 2 þ 2 þ 2Þ 4 s t u pole can be neglected [32] imply that the coefficients of MPl both of the above dimension-six operators are positive þ Oð −4Þ ð Þ MPl : 4:17 which is satisfied with the single condition

2 0 ð Þ The direct application of forward limit positivity bounds CW > : 4:22 [10,25,26] is famously problematic due to the massless t-channel pole. Defining as A0, the fixed t, s-channel pole We emphasize that what has been performed here is an subtracted amplitude, we have inverted logic as compared to what is typically considered in the literature when imposing bounds on curvature 2 0 1 1 4 4 −4 operators. Rather than applying positivity bounds directly ∂ A ðs; tÞ ∼− þ C 2 þ C 2 þ OðM Þ: s 2 4 R 3 W Pl on the gravitational sector, we have applied it on scattering MPl t MPl ð Þ amplitudes of the matter sector. Subtleties related to the 4:18 t-channel pole are of course equivalent in both cases, as it 2 0 The forward scattering limit of ∂s A ðs; tÞ is dominated by 17In the case of massive gravity, this problem is conveniently the contribution from the t-channel pole and potentially 2 4 avoided since the pole is at t ¼ m and so one can analytically bears no relevance for the sign of C 2 þ C 2 . More 2 R 3 W continue from t<0 to m >t>0. Extensions of positivity importantly, the pole at t ¼ 0 prevents analytic continu- bounds to t>0 have recently been considered in [27,28] with ation of the partial wave expansion from the physical particular application to massive gravity in [47,80].

063518-16 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) should be. The related arguments of [32] similarly deter- integrating out massive spin-2 and higher states which mine the positivity bounds on photon scattering. naturally coupled to the stress energy through the −1=3 polarization factor and massive spin-0 states which could C. TT amplitude couple to the trace of the stress energy. The previous arguments allow us to impose constraints on the sign of the curvature-squared operators using 1. Weakly coupled UV completion positivity bounds applied on a scattering amplitude of To make the previous argument explicit, imagine a the matter sector provided the t-channel pole that appears in weakly coupled UV completion, with a potentially infinite those amplitudes can be discarded. A stronger form of these tower of massive spin states. Let us imagine that matter arguments follows from considering the TT-contributions from the matter Lagrangian to arise from the integration of couples to an effective metric build out of the Einstein massive spin-2 fields. Indeed, the effective matter frame zero mode metric gμν and some combination of all Lagrangian may also be written as the other spin states. If the spin states are only weakly excited, as would be expected in the regime of validity of 2 2 1 the EFT, we may treat them as approximately linear, even L0 ¼ L þ CW μν − 2 matter matter 4 TμνT T while the zero mode metric gμν is nonlinear. Matter may M 3 Pl eff then be taken to effectively couple to gμν ¼ gμν þ C 2 P P þ R T2 þ OðM−4Þ; ð4:23Þ 1 α i þ 1 β ϕ i 4 Pl M i iHμν M j j jgμν where Hμν are any number MPl Pl Pl of massive spin-2 particles of mass Mi and ϕj any number which are the natural combinations following from the of scalar particles of mass Mj. Other spin states will not Käll´en-Lehman spectral representation for the TT two- couple at this order. point function. This emphasizes the fact that this interaction The UV Lagrangian describing this setup may then be could be viewed as the TT amplitude obtained from taken to be (ignoring Gauss-Bonnet terms)

pffiffiffiffiffiffi 2 2 X 1 1 MPl UV UV 2 UV 2 μν i 2 2 2 L ≈ −g R þ C 2 − C 2 R þ 2C 2 Rμν þ H EHμν − M ðH − H Þ UV 2 R 3 W W 2 i 2 i iμν i i X 1 1 1 X 1 X þ ϕ □ϕ − 2ϕ2 þ L þ α i μν þ β ϕ μν þ ð Þ 2 j j 2 Mj j matter 2 iHμνT 2 j jgμνT ; 4:24 j MPl i MPl j with E the covariant version of the Lichnerowitz operator and ellipses indicating higher order interactions for the additional spin fields. The obvious examples of this kind of effective Lagrangian are extra dimensional braneworld setups where matter is localized on a brane whose induced metric is not equivalent to the Einstein frame zero mode metric. The induced μν ϕ metric will indeed include Kaluza-Klein modes Hi as well as potentially other scalar moduli fields j. Now integrating out the massive fields to obtain the low energy effective theory, then due to the Fierz-Pauli structure of the mass term from the spin-2 fields we obtain the −1=3 factor, namely pffiffiffiffiffiffi 2 2 MPl UV UV 2 UV 2 L ≈ −g R þ C 2 − C 2 R þ 2C 2 Rμν þ L IR 2 R 3 W W matter 1 X α2 1 1 X β2 i μν 2 j 2 þ T Tμν − T þ T þ : ð : Þ 4 2 2 3 4 2 2 4 25 MPl i Mi MPl j Mj μ Here we have made use of the fact that at leading order ∇ Tμν ≈ 0, and so the action of the polarization tensors is simplified. 2 2 By rewriting the Tμν and T back in terms of curvature-squared interactions we may identify X 2 X 2 IR UV MPl 2 IR UV MPl 2 C 2 ¼ C 2 þ α ;C2 ¼ C 2 þ β ; ð4:26Þ W W 8 2 i R R 4 2 j i Mi j Mj that is, X 2 X 2 M 2 M 2 Δ 2 ¼ Pl α 0 Δ 2 ¼ Pl β 0 ð Þ CW 8 2 i > ; CR 4 2 j > ; 4:27 i Mi j Mj which gives the desired positivity properties. The equality could be saturated only if no fields coupled, i.e., if αi ¼ βj ¼ 0.

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2. Generic UV completion gravitational waves to be superluminal on NEC preserving 2 ≡ 0 Although the previous argument was made explicitly for backgrounds unless one insists on having CW in a weakly coupled UV completion, it follows equally well in which case gravitational waves would be luminal to this general from the spectral representation for the TT two- order (but not to higher orders as we see in Sec. V). point function [81] for a conserved source, which would Related spectral representation arguments are given in also apply when loops are included and hence for an [81,82]. In these approaches the t-channel pole is effec- arbitrary UV completion tively neglected by focusing on graviton pseudoamplitudes which are essentially on-shell stress energy correlators [83]. Z 1 ∞ 1 1 Similar arguments applied to the Gauss-Bonnet term or Δ ¼ μρ ðμÞ μν − LTT 4 d 2 T Tμν gμνT equivalent Weyl squared term in higher dimensions are M 0 μ − □ − iϵ 3 Pl Z given in [84] with the same implied choice of sign. S-matrix 1 ∞ 1 þ μρ ðμÞ ð Þ positivity arguments have been applied to quartic curvature 4 d 0 T μ − □ − ϵ T; 4:28 MPl 0 i interactions in [29,30] complementing previous superluminality arguments [11]. Entropic arguments that constrain where standard unitarity arguments imply ρ2ðμÞ ≥ 0 and the curvature-squared terms are given in [31,55] and are ρ0ðμÞ ≥ 0. Crucially though this expression assumes that consistent with these implied signs. no subtractions are necessary in writing this dispersion relation. The leading subtraction term can be directly 3. Neglecting the t-channel pole UV UV absorbed into C 2 and C 2 so that we may formally write W R The Käll´en-Lehman spectral representation for a Z 1 ∞ ρ ðμÞ 2-tensor can also include contributions from massless IR UV 2 C 2 ¼ C 2 þ dμ ; ð4:29Þ graviton exchange. In the above we did not include this W W 2 μ 0 as we have intentionally written the action (4.14) in a Z ∞ ρ ðμÞ representation in which the massless graviton has not been IR UV 0 C 2 ¼ C 2 þ dμ ; ð4:30Þ integrated out. Had we done so then we would have R R μ 0 obtained an effective matter Lagrangian

IR with the understanding that C 2 2 are finite quantities. We L00 ¼ L þ Δ 0 ð Þ W ;R matter matter L ; 4:35 α2 2 TT see that i =Mi is replaced by the spin-2 spectral density μ β2 2 divided by the spectral mass squared and j =Mj replaced where by the spin-0 spectral density divided by μ. Again we 1 1 1 see that 0 μν ΔL ¼ T Tμν − gμνT TT 2M2 −□ − iϵ 2 Pl Z Δ 2 0 Δ 2 0 ð Þ CW > ; CR > : 4:31 1 ∞ ρ ðμÞ 1 þ μ μν 2 − 4 d T Tμν gμνT M 0 μ − □ − iϵ 3 We may define these coefficients at an arbitrary scale, Pl Z ∞ which may be interpreted as the coefficients in the EFT 1 ρ0ðμÞ þ dμT T; ð4:36Þ M 4 μ − □ − ϵ defined with all states of energies greater than integrated MPl 0 i out, in the manner Z which is the full Käll´en-Lehman spectral representation ∞ 1 ρ2ðμÞ between two conserved sources. The first term in (4.36) is 2 ð Þ¼ UV þ μ ð Þ CW M CW2 d ; 4:32 2 M2 μ of course the term that gives the pole terms in (4.17),in Z particular the massless t-channel pole which is responsible ∞ ρ0ðμÞ for the problems with applying standard forward limit 2 ð Þ¼ UV þ μ ð Þ CR M CR2 d ; 4:33 M2 μ dispersion relations. In the above representation (4.36), it is arguably obvious so that the RG flow is finite (independent of subtraction/ why we can ignore the contribution of the t channel renormalization considerations) and positive in the sense pole. Unitarity imposes positivity of ρ2ðμÞ and ρ0ðμÞ through the requirement ImðΔL0 Þ ≥ 0 regardless of Z 2 TT M2 ρ2ðμÞ whether the massless pole part is present. This will allow 2 ð Þ − 2 ð Þ¼ μ ð Þ CW M1 CW M2 d : 4:34 2 μ C 2 C 2 M1 us to determine a positivity bound for W and R . This argument is slightly different from that given in [32] since it We thus see that standard spectral representation arguments focuses on only those interactions that are written in terms Δ 2 0 imply the expectation that CW > and related positivity of a single Tμν; nevertheless it explains at least in part why 2 0 bounds strengthen this to the expectation that CW > .Itis one would expect the result of [32] with regards to the precisely with this sign that we found in (3.35) and (3.37) neglect of the t-channel to be correct.

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The real issue, however, is not the t-channel pole per se, dispersion relation (4.28) in powers of □=μ we infer at next but the required number of subtractions of the remaining order terms. The dispersion relation (4.36) is valid only if the Z integrals 1 ∞ □ 1 Δ ¼þ μρ ðμÞ μν − LTT 4 d 2 T 2 Tμν gμνT Z Z M 0 μ 3 ∞ ∞ Pl ρ2ðμÞ ρ0ðμÞ Z dμ and dμ ð4:37Þ 1 ∞ □ 0 μ 0 μ þ μρ0ðμÞT T þ; ð : Þ 4 d μ2 4 39 MPl 0 converge. If not, then it is necessary to perform subtractions. For instance, performing one subtraction is equiv- which after a field redefinition is equivalent to a higher Δ 0 derivative curvature correction alent to rewriting LTT as Z 1 1 1 ∞ ρ ðμÞ Δ 0 ¼ μν − Δ ¼þ μ 2 □ μν LTT 2 T Tμν gμνT LTT d 2 Rμν R 2M −□ − iϵ 2 0 μ Pl Z Z 1 ∞ ρ ðμÞ 1 ∞ ρ ðμÞ μν 2 þ μ 0 − μ 2 □ þ þ a2T Tμν − gμνT þ a0T d 2 d 2 R R : 3 0 μ 3 0 μ Z 1 ∞ ρ ðμÞ□ 1 ð Þ þ μ μν 2 − 4:40 4 d T Tμν gμνT M 0 μðμ − □ − iϵÞ 3 Pl Z 1 ∞ ρ ðμÞ□ As a nontrivial check on this, comparing with the explicit þ μ 0 ð Þ 4 d T T; 4:38 one-loop effective action from massive particles of spin-0, 0 μðμ − □ − iϵÞ MPl −1=2, -1 given in (5.1) or Appendix A for spin-0, then we have where a2 and a0 are the subtraction constants. After a field redefinition, the addition of these constants is equivalent to ð Þ Z X si ∞ 2 2 1 d ρ2ðμÞ adding R and Rμν counterterms, which are known to be 2 ¼ μ ; ð : Þ 12ð2πÞ4 2 d μ2 4 41 needed already in one-loop calculations to remove diver- i Mi 0 UV IR gences, in other words, the inclusion of CR2 and CW2 .In R ρ ðμÞ ð Þ Z Z IR UV 1 ∞ 2 X si ∞ ∞ the relations C 2 ¼ C 2 þ dμ , etc., the RHS is 1 d ρ0ðμÞ 1 ρ2ðμÞ W W 2 0 μ 1 ¼ μ − μ ð Þ 4 2 d 2 d 2 : 4:42 12ð2πÞ 0 μ 3 0 μ the difference of two infinite quantities, and it is hence not i Mi possible to conclude positivity of the LHS. Hence the ability to apply positivity bounds with the t-channel pole The positivity of ρ2ðμÞ and ρ0ðμÞ at all scales implies that neglected comes down to whether the integrals (4.37) converge. This is not surprising since it is equivalent at ð Þ ð Þ 1 ð Þ si 0 si þ si 0 ð Þ the scattering amplitude level to requiring some Froissart- d2 > ;d1 3 d2 > : 4:43 like bound on the t-channel pole subtracted amplitude. Alternatively this comes down to the question of how many From the results of Table I, or from (A25) for spin-0, we subtractions are needed in specifying a dispersion relation see that all computed values of d2 are positive and similarly ð Þ ð Þ for the graviton two-point function. si 1 si 17 1 3 d1 þ 3 d2 ¼ð840 ; 840 ; 280Þ > 0 for si ¼ð0; 1=2; 1Þ. These ðsiÞ results are nontrivial since several of the d1 are negative. 4. Higher derivative corrections Again we emphasize that since the one-loop effective The positivity bounds implied by the TT amplitude action is finite at this order, Eq. (5.1) needs no renormal- argument also apply to higher derivative terms. In particular ization counterterms which corresponds to the statement if one subtraction is sufficient for convergence (as is known that the dispersion relation (4.28) does not need any to be the case at the one-loop level), then expanding the subtractions at this order.

TABLE I. Coefficients entering the dimension-six operators in the one-loop effective action, for scalars, spinors, and vectors. From [85].

Spin sd1 d2 d3 d4 d5 d6 d7 d8 d9 d10 0 1=56 1=140 1=63 −1=180 1=180 −8=945 2=315 1=1260 17=7560 −1=270 1=2 −3=280 1=28 1=864 −1=180 −7=1440 −25=756 47=1260 19=1260 29=7560 −1=108 1 −27=280 9=28 −5=72 31=60 −1=10 −52=63 −19=105 61=140 −67=2520 1=18

063518-19 CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020) 1 5. EFT matching Lðl:e:Þð ψÞ¼− ð∂ψÞ2 − ðψÞ ð Þ ψ g; 2 V : 4:48 In Sec. III D, we have shown how already within the truncated EFT (only including the quadratic-curvature It is of course consistent to imagine that these light fields corrections), the sound speed is frequency dependent. also have nonminimal couplings to gravity. These may Moreover, to the level we have expanded the sound speed arise if the light field is itself nontrivially coupled to the (while remaining in the regime k ≪ M), the departures heavy field, and then on integrating out the heavy fields we from luminality are always reduced at higher frequency generate new curvature interactions for the light fields. regardless of the sign of CW2 and regardless of whether the Following the discussion in Sec. IV B we already know that background satisfies the NEC (i.e., regardless of whether interactions of the form ð∇ψÞ4 could be viewed as having the speed is sub- or superluminal at low frequencies). arise from field redefinitions. Far less trivial are the Putting those arguments aside, the clearest argument that following dimension-six operators which will contribute the speed of sound returns to unity at high energy is at the same order as the curvature-squared corrections,18 obtained from the EFT matching. The speed we have calculated reflects the tree level speed identified from a ΔLðl:e:Þð ψÞ¼ C4 μν∇ ψ∇ ψ þ C5 ð∇ψÞ2 ψ g; 2 G μ ν 2 R Wilsonian effective action in which states heavier that some MPl MPl mass scale M have been integrated out. As we have argued, ðψÞ þ U ð Þ in general we anticipate C6R 2 ; 4:49 MPl ΔC 2 ¼ C 2 ðM1Þ − C 2 ðM2Þ where UðψÞ is a function of ψ with the same overall scale W Z W W 2 ðψÞ M2 ρ2ðμÞ as V . The addition of these operators modifies the ¼ dμ > 0;M1

2 ð Þ perturbations. Following almost verbatim the previous where CW M denotes the associated coefficient in an EFT with masses above M integrated out. It follows that the recipe we find that the speed of propagation for tensors associated speed of gravitational waves on FLRW defined with the addition of these three nonminimal interactions in a given EFT is becomes ˙ 4 4 4ðC4 þ 4C 2 Þð−HÞ H 16 2 ð Þð− ˙Þ 2 W 2 CW M H H c ¼ 1 þ þ O : ð4:50Þ c ðMÞ¼1 þ þ O ; ð4:45Þ s 2 4 s 2 4 MPl MPl MPl MPl ˙ 0 It would be tempting to suppose that the value of C4 should H<0 H ðyÞ 2 and so for any fixed [or in a static warped 2 ≤ 1 be chosen in terms of CW so that either cs or, indeed geometry] 2 ¼ −8 2 ¼ 1 by taking C4 CW , cs so as to enforce GWs (sub) 2 2 ˙ luminality. The problem with this is that it presupposes that c ðM1Þ >cðM2Þ;M1

2 ð Þ¼ UV ¼ 0 ð Þ dark matter or dark energy degrees of freedom that may lie lim CW M C 2 ; 4:47 M→∞ W in some dark sector arbitrarily weakly coupled to Standard where by M → ∞ we mean at energy scales well above the Model fields. Furthermore the precise light fields determin- cutoff of the low energy EFT and the scale at which Lorentz ing the cosmological expansion are epoch dependent, invariance is spontaneously broken. inflaton, radiation, dark matter, dark energy, and there is no reason to suppose each of these distinct light fields should be nonminimally coupled in the precise manner D. Curvature couplings to light fields 2 needed to enforce cs ¼ 1. More importantly, however, just In the previous discussion we assumed that the light as in Sec. IV B, positivity bounds (to the extent where the fields that generate the cosmological background are t-channel pole may be ignored) lead to the inevitable þ 4 2 0 minimally coupled to gravity. For instance, the light field conclusion that C4 CW > and hence that gravita- Lagrangian may be that describing a minimally coupled tional waves remain superluminal. scalar field ψ with a potential VðψÞ. Alternatively, the light field Lagrangian may model a perfect fluid with a 18 2 A special case of these interactions has been considered in Pðð∂ψÞ ; ψÞ Lagrangian. To be concrete let us consider the [87] from the perspective of time delays/advances. Our discus- example of the former so that we may take sion here complements the one in [87].

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To see how positivity bounds constrain the sign of at the price of introducing additional matter interactions. þ 4 2 χ C4 CW , we note that the operators in (4.49) can Let us consider a spectator field which we take to be a free be removed with a field redefinition similarly as in massless scalar living on the original metric. The above Sec. IVA, field redefinition will induce interactions between the scalar χ and the field ψ. Further removing the R2 and Weyl 2 → − ð ∇ ψ∇ ψ − ð∇ψÞ2 − ðψÞ Þ squared terms via the field redefinition (4.6), then to this gμν gμν 4 C4 μ ν C5 gμν C6U gμν ; MPl order we obtain Einstein gravity minimally coupled to an ð4:51Þ effective matter Lagrangian

0 1 2 1 2 1 2 2 L ¼ − ð∇ψÞ − VðψÞ − ð∇χÞ þ 4C6UðψÞVðψÞþ4 4C 2 − C 2 VðψÞ matter 2 2 M4 R 3 W Pl 1 2 4 þ ð∇ψÞ 2C6UðψÞþ2 8C 2 − C 2 − C4 þ 4C5 VðψÞ 2M4 R 3 W Pl 8 2 8 2 þ 2 2 þ 2 þ þ 2 ð∇ψÞ þ 4 2 − 2 − þ 2 ð∇χÞ CR 3 CW C4 C5 CR 3 CW C4 C5 1 2 4 4 2 þ ð∇χÞ ðψÞþ 8 2 − 2 ðψÞþ 2 þ 2 ð∇χÞ 4 C6U CR 3 CW V CR 3 CW MPl

ð4 2 þ Þ þ CW C4 ð∇μχ∇ ψÞ2 ð Þ 4 μ : 4:52 MPl 1. Subluminal spectator → ∞ 2 2 L0 We may now take the double scaling limit MPl , keeping C=MPl and C4=MPl finite, so that matter may be taken as a nongravitational field theory living on Minkowski spacetime. The effective equation for linearized fluctuations of the spectator field is

μν ∂μðZ ∂νχÞ¼0; ð4:53Þ with the effective metric 2 μν μν 8 ð∂ψÞ 2 4 ¼ η 1 − 4 2 − 2 − þ 2 − ðψÞþ 8 2 − 2 ðψÞ Z CR 3 CW C4 C5 2 4 C6U CR 3 CW V MPl MPl 2 μ ν − ð4 2 þ Þ∂ ψ∂ ψ ð Þ 4 CW C4 : 4:54 MPl The effective speed of propagation on a background in which ψðtÞ is time dependent is seen to be to this order

2 2ð þ 4 2 Þψ˙ 4ð þ 4 2 Þð− ˙Þ 2ðχÞ¼1 − C4 CW þ Oð −4Þ¼1 − C4 CW H þ Oð −4Þ ð Þ cs 4 MPl 2 MPl : 4:55 MPl MPl

Thus in performing the field redefinition to the frame is field frame independent. Hence demanding that, in this in which gravity is minimally coupled, and hence gravi- decoupling limit, the spectator field fluctuations are (sub) tational waves are luminal, we have made the spectator luminal requires field propagate subluminally by exactly the same amount.

þ 4 2 ≥ 0 ð Þ This is consistent with the general expectation that C4 CW : 4:57 the ratio 2. Positivity bounds c2ðtensorsÞ s ð : Þ We may provide a better argument by focusing on 2ð Þ 4 56 cs spectator S-matrix positivity bounds applied to ϕψ → ϕψ scattering,

063518-21 CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020) assuming as per our previous discussion that we can neglect curvature-squared terms cannot be determined since they the massless t-channel pole. Regardless of the choice of arise logarithmically divergent and must be renormalized ðψÞ ðϕÞ UV potentials V and U we can see by power counting by introducing an appropriate counterterm CR2;W2 . The IR derivatives that the only terms in the Lagrangian (4.52) that resulting renormalized coefficient CR2;W2 can be consis- will potentially contribute to the twice subtracted scattering tently chosen to take any value without contradicting amplitude at tree level are (neglecting again the contribu- the requirements of consistency of the EFT. We have tion from exchange of massless gravitons) argued from positivity bounds that it is reasonable to IR suppose C 2 > 0, which would then lead to superluminal ð4 2 þ Þ W L0 ⊃ CW C4 ð∇μχ∇ ψÞ2 matter 4 μ gravitational waves on NEC preserving backgrounds. MPl Nevertheless it remains technically possible that we 8 IR ð4C 2 − C 2 − C4 þ 2C5Þ have C 2 ¼ 0. Although a seemingly technically unnatural þ R 3 W ð∇ψÞ2ð∇χÞ2; W 2 4 tuning, it would ensure that the nonluminal propagation we MPl have uncovered so far is removed. This forces us to look to ð Þ 4:58 next order in the EFT expansion where things become more interesting. which gives a contribution of the form In this section, we shall assume that the IR curvature- IR IR squared terms are set to zero, C 2 ¼ C 2 ¼ 0, and that in ð4C 2 þ C4Þ 1 1 W R A ð Þ ⊃ W ð − 2 Þ2 þ ð − 2 Þ2 such a basis, SM fields are minimally coupled to the metric. ψχ→ψχ s; t 4 2 s mψ 2 u mψ MPl In this case any effect arising on the speed of propagation of 8 ð4 2 − 2 − þ 2 Þ gravitational waves will arise at next order in the EFT CR 3 CW C4 C5 2 þ tðt − 2mψ Þ; 2M4 expansion, which in the present context means dimension- Pl six operators, i.e., curvature-cubed and higher derivative ð4:59Þ curvature-squared terms, which we consider in Sec. VA before considering dimension-eight operators (fourth 2 where u ¼ 2mψ − t − s and mψ is that mass of ψ. power of curvature) in Sec. VB. Unlike the curvature- Consequently squared terms, the curvature-cubed terms that arise from matter loops are finite, are calculable, and have explicit 2 1 ∂ ð4C 2 þ C4Þ dependence on the number of species. There is no need to A0ðs ¼ 0;t¼ 0Þ¼ W ; ð4:60Þ 2 ∂ 2 4 assume the existence of a UV contribution (at least in the s MPl absence of graviton loops), and so we can meaningfully and so a standard application of the forward limit positivity consider unambiguous finite contributions to the sound bounds implies [10,25,26] speed. Moreover, unlike the case of the curvature-squared operators considered so far, there is no covariant and local

þ 4 2 0 ð Þ C4 CW > : 4:61 field redefinition that can remove all the higher-order curvature operators that we consider in this section. The equality cannot be saturated since the right-hand side of the dispersion relation is determined from the total A. Dimension-six curvature operators scattering cross section which is necessarily nonzero. Such considerations would then ensure that in this field frame the 1. One-loop effective action spectator field fluctuations are necessarily subluminal, or The general form of the dimension-six curvature oper- that in the original field frame the gravitational waves are ators that are expected to arise in a gravitational EFT are necessarily superluminal. Although we have run this argu- well-known and can be parametrized by [43] ment introducing a light spectator field, we could equally 1 pffiffiffiffiffiffiX 1 h apply it to any Standard Model field. For instance, the role ð1-loopÞ ðsiÞ ðsiÞ μν Γ ¼ −g d R□R þ d Rμν□R χ dim-6 12ð2πÞ4 2 1 2 of could have been played by the Higgs field, indeed Mi χ i introducing a mass for that field does not affect the ð Þ ð Þ ð Þ ð Þ þ si 3 þ si 2 þ si 2 þ si 3 relevant conclusions. d3 R d4 RRμν d5 RRμναβ d6 Rμν

ðsiÞ μν αβ ðsiÞ μν αβγ þ d R R Rμανβ þ d R RμαβγRν 7 8 i V. EFFECTS FROM HIGHER-DIMENSION ð Þ ð Þ þ si μναβ γσ þ si μ ν α β γ σ CURVATURE OPERATORS d9 R RμνγσRαβ d10 R α βR γ σR μ ν ; In the previous section, we discussed the effect of the ð5:1Þ leading dimension-four operators in the EFT expansion, ðsiÞ the curvature-squared corrections. In the context of loop where dI denotes the contribution from integrating corrections from matter the precise coefficients of the out a particle of mass Mi and spin si. The sign of the

063518-22 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) dimension-six curvature operators is not constrained by 2. Dimension-six curvature operators on FLRW known positivity bounds and fields of various spins can In what follows it will be convenient to define the lead to different sign coefficients of these operators, as is effective number N of scalars s ¼ 0, vectors s ¼ 1, and ¼ 0 1 2 1 s illustrated here for fields of spin s ; = ; . In Ref. [88], spinors s ¼ 1=2 as it was shown how the spin of intermediary states may affect the sign of six derivative operators in a scalar field theory X 2 M and higher spins may have interesting effects. N ¼ ; ð5:2Þ s M2 As usual, the above form can be simplified by using field field i of spin s i redefinitions to remove for example all the R and Rμν terms (see [8] for a recent discussion), but doing so will only where we only include fields with masses above the scale of ≫ introduce interactions in the matter sector which capture the the EFT we are interested in, i.e., Mi H on FLRW. same basic one-loop process, such as those described in Unless there are large number particles N ⋙ 1 at the same Fig. 1. Performing this field redefinition would take us out mass, we would typically expect this effective number to be of the frame in which we have chosen to minimally couple dominated by the lightest massive particle beyond the low SM fields and so we prefer to remain in this frame, being energy EFT. the natural one from the perspective of a path integral Given our assumption that the IR curvature-squared calculation. terms have been set to zero, at energy scales below the Unlike the case for the curvature-squared corrections, mass M, then the leading terms in our gravitational EFT are ð Þ ð Þ si si other than the coefficients d1 and d2 there are no known pffiffiffiffiffiffi 2 MPl ðl:e:Þ ð1-loopÞ positivity bounds that fix the signs of the remaining L ¼ −g R þ Lψ ðg; ψÞ þ Γ IR 2 dim -6 coefficients. That is because, even if we were to rewrite 1 these interactions as pure matter ones, they would corre- þ Oðð 2 ∇2 Þ2Þ ð Þ 4 R ; R ; 5:3 spond to TabTcdTef interactions and would only contribute M (at tree level) to 3 − 3 scattering and higher order ampli- Γð1-loopÞ tudes, or to three point Källén-Lehman dispersion relations, where dim -6 is given in (5.1), where the terms omitted are for which clean statements of positivity are not known operators of dimension-eight and higher that are further (although see [46] for statements in the holographic/CFT considered in Sec. VB. Expanding to quadratic order context). around an FLRW background, using the same conventions We can bypass this problem by, however, focusing on the as in Sec. III B with tensor modes normalized as in (3.18), explicit example of loop corrections from particles of spin the contributions from the dimension-six operators are of si ≤ 1 for which the coefficients are known and are finite. the form This finiteness is crucial since it tells us there is no need to 1 2 add any counterterms at this order, and so there is no ðhhÞ a ˆ L ¼ hO 6h; ð5:4Þ ambiguity about the signs of the resulting coefficients. For dim -6 12ð2πÞ4 420M2 dim - these dimension-six operators, the explicit one-loop effective action was computed exactly in [59,85] for massive particles where the differential operator Oˆ includes up to fourth of spin 0; 1=2, and 1, where the dimensionless coefficients order in derivatives. More specifically, the operator can be ðsiÞ dn are given in Table I of Appendix A and depend on the put in the form spin si of the particle integrated. We reproduce these results X 1 1 for spin-0 explicitly in Appendix A using dimensional ˆ ðsÞ 2 ðsÞ ⃗2 O 6 ¼ N f □η þ f □η∇ regularization for convenience. dim - s 4 1 4 2 s a a As before, we shall see that the very existence of these 1 1 þ ðsÞ□ ∂ þ ðsÞ□ dimension-six operators leads to a sound speed for gravi- 3 f3 η η 2 f4 η tational waves at low energy which can differ from a a 1 1 1 luminality. On a cosmological background they lead to þ ðsÞ∂ þ ðsÞ∇⃗2∂ þ ðsÞ∇⃗2 þ ðsÞ ˙ 2 2 2 f5 η 3 f6 η 2 f7 f8 ; corrections to the sound speed of order NHH =MPlM , a a a where N is the number of fields integrated out and M their ð5:5Þ mass. What is crucially different at this order is, depending on the field content of heavy modes, the speed of 2 η □ ¼ −∂2 þ ∇⃗ gravitational waves can effectively turn both superluminal where again is the conformal time and η η is ’ and subluminal. This is true even for matter forced to the d Alembertian on Minkowski spacetime. The functions ðsÞ respect the null energy condition, for signs we know to be fn depend on the background (and on the spin s of consistent with positivity since they are derived from an the particles integrated). Their precise expressions are explicitly unitary calculation from a local and well-behaved rather nonilluminating and are provided in Appendix B, field content. Eqs. (B4)–(B27).

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As before, on pure de Sitter, the functions f2;5;6;7 vanish coefficients that differ away from de Sitter. We may thus and the one-loop contribution is a simple combination of infer an effective sound speed symbolically of the form 2 B−A flat d’Alembertian and effective mass terms acting on the ¼ 1 þ 2 2 cs M M , derived explicitly in Appendix B to be tensor fluctuations, leading to the standard dispersion Pl relation of the form ω2 ¼ k2 þ m2. On FLRW, however, 1 0 2 ¼ 1 − the breaking of the maximal symmetry implies the exist- cs 12 105ð2πÞ4 2 2 × M MPl ence of additional time-derivative operators that break the 2 2 2 ð2ð163 − 39 − 659 Þ ˙ trivial relation between ω and k in the dispersion relation. × N0 N1=2 N1 H H − ð46 þ 62 þ 530 Þ ˙ 2 N0 N1=2 N1 H 3. Modification of the dispersion relation þð−93 þ 46 þ 617 Þ ̈ N0 N1=2 N1 HH Working perturbatively in the corrections from (5.4),we þð−37N − 10N þ 57NÞH⃛Þ; ð5:10Þ may use the equations of motion inferred from the standard 0 1=2 1 Einstein and matter term L ψ into (5.4) so as to trade the EH; and therefore departs from unity as soon as the back- higher derivatives for lower ones as explained in Sec. III A. ≠ This is performed explicitly in Appendix B, and we are then ground departs from pure de Sitter H const. At this level, we can also directly see that there is also no field left with a perturbative second order action of the form content (no tuned values of N0;N1=2;N1) that would lead 2 1 ðhhÞ MPl 2 2 ˙ to an exactly luminal speed for different cosmological L ¼ a h □η þ 2H þ H h 4 a2 epochs. 2 1 a ˆ þ hO 6h ð5:6Þ 12ð2πÞ4 420M2 dim - 4. Field content dependence Assuming a constant equation of state ϖ on FLRW,20 the with now effective speed of GWs is then X 1 1 2 ˆ ˜ ðsÞ ˜ ðsÞ ⃗ 4 O 6 ¼ N f ∂η þ f ∇ ∂η 1 dim - s 5 3 6 2 ¼ 1 − H ð1 þ ϖÞ ¼0 1 2 1 a a cs 4 2 4 s ; = ; 12ð2πÞ 140 M M 1 Pl ˜ ðsÞ ⃗2 ˜ ðsÞ 2 þ f ∇ þ f ; ð5:7Þ × ½N0ð−349 þ 1302ϖ þ 999ϖ Þ a2 7 8 þ 6N ð86 þ 105ϖ þ 45ϖ2Þ ˜ 1=2 and the coefficients f5–8 are given in Eqs. (B30)–(B34). þ ð3209 − 966ϖ − 1539ϖ2Þ ð Þ The friction term can then be removed as usual with a field N1 ; 5:11 redefinition, where the field redefinition is now momentum dependent (unless we are on de Sitter), of the symbolic where Ns is the effective number of spin-s particles form19 integrated out [as defined in (5.2)]. The regions where the effective speed cs is sub- vs superluminal for scalars, 1 2 → 1 þ 4 þ ˙ k ð Þ vectors, and spinors is depicted in Fig. 3. h 2 2 H H 2 h; 5:8 MPlM a Scalar vs other fields.—Crucially, if the effective number of where the exact expression is provided in (B34) and (B35). scalars integrated out dominates over that of vectors and The apparent nonlocality of this field redefinition does not spinors, then as we have defined it, cs would be super- cause a problem in identifying the speed of propagation, luminal for most of the late-time cosmological history of ’ since it is perturbatively local and the associated Green s our Universe (since radiation-matter equality). If, on the function should be determined perturbatively in it. The other hand, the effect is dominated by vectors or spinors, resulting equation of motion is then (symbolically) of the then c would be subluminal for the whole (standard) form s cosmological history of the Universe so long as one never Aðk; ηÞ Bðk; ηÞ crosses down to ϖ < −1 corresponding to the breaking of 1 þ h00 þ 1 þ k2h þ m2h ¼ 0; ð5:9Þ M2 M2 M2 M2 0 the null energy condition. This is an entirely novel effect Pl Pl which did not arise at the previous order. where the two functions Aðk; ηÞ and Bðk; ηÞ are symboli- ∼ 4 þ k2 2 — cally of the form A; B H a2 H , but with precise Subluminal radiation era. Interestingly, independently of the precise field content, the finite contributions from 19In this symbolic notation, H should be understood to also 4 include all derivatives of H, so, for instance, H is really a 20In this section ϖ represents the background equation of state placeholder for H4;H2H;˙ ðH˙ Þ2;HḦ, and H⃛. parameter, not to be confused with the frequency ω of the GWs.

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particles are charged and whether they are pseudoscalar are irrelevant for this discussion as all that matters here is the coupling to gravity. The considerations laid here could then potentially impact some of the following scalar models of dark matter such as axion or axionlike dark matter as well as a fuzzy cold dark matter. In particular axion dark matter [89] (see [90] for a review) has lead to a quest for a multitude of experimental searches [91]. The theoretical mass window for axion or axionlike dark matter spans over many orders of magnitude, but the open range is typically considered to be around 10−6–10−2 eV, while it is suggested that string theory may favor lower masses [91].As for fuzzy cold dark matter [92] their postulated mass could be as low as 10−22 eV and within this logic would require equally low-mass spinors or vectors to avoid a small FIG. 3. Regions of infinitesimal sub- and superluminal GW superluminal speed for GWs. speed on FLRW with a constant equation of state parameter ϖ, from integrating out scalars, vectors, or spinors. Speed of GWs as a discriminator redux.—It is, however, clear from the discussions related to positivity bounds in integrating out heavier fields appears to lead to a sub- previous sections for the curvature-squared terms (see luminal speed for gravitational during the radiation era Sec. IV) that one should take great caution prior to jumping when ϖ ¼ 1=3. to any conclusion and using the presence, or absence, of superluminal speed for gravitational waves as a discrimi- Speed of GWs as a discriminator.—In the past literature, an nator. Indeed, as we have seen in Sec. IV, a superluminal even small (unobservable) superluminal speed of gravita- speed for gravitational waves at low energy may not tional waves has been commonly used as a discriminator necessarily be in conflict with a causal UV completion between models or as a way to constrain parameters within and may sometimes even be favored. Perhaps a more an EFT (for instance, related to the EFT of inflation or dark appropriate criterion would be to require that gravitational energy introduced in Refs. [12,13]). In constructing the EFT for gravity, we are free to choose at what scale we wish waves propagate faster than the light cone to which matter to consider M to be, provided we only consider back- minimally couples. This would then be the case for most of grounds with H ≪ M. With this in mind, we are free to the recent cosmological history of our Universe if the integrate out Standard Model particles, including the curvature-squared corrections are included or if fields of electron and neutrino. In this family of EFTs, it will be spin-0 dominate the contributions to the curvature-cubed that EFT defined with the lowest mass M for which the corrections. However, applying such a criterion would also ϖ ¼ 1 3 dimension-six operators that scale 1=M2 will have the remain puzzling during the radiation era = , where largest effect. If we were to demand the gravitational waves modes are always subluminal, unless one relies on the to be subluminal, it would “rule out” any model that curvature-squared corrections. It may be possible that it is postulates the existence of a scalar field with mass between not consistent in the given EFT to tune curvature-squared the Hubble parameter and the neutrino mass, unless that terms to zero, and that their positive contribution should particle was also accompanied with either vectors or dominate over the negative contribution from the dimen- spinors of comparable or lower mass. Applying this logic sion-six curvature operator. For this to be true we need to current late-time cosmology with H ∼ 10−33 eV, this jH˙j jH˙ jH2 would discriminate against any model that carries a scalar 2 ≫ ð Þ −30 −3 CW 2 N 2 2 ; 5:12 field with mass m between say 10 eV and 10 eV, MPl M MPl unless other fermions or vectors of comparably low mass were also included. For concreteness, in the absence of essentially at all scales M as long as H ≪ M. In other ≲ 0 2 vectors, one should have N0 . N1=2 to avoid cs being words superluminal as ϖ approaches −1 (from above). For one spinor and one scalar field, this requires the mass of the H2 C 2 ≫ N : ð5:13Þ scalar to be at least twice that of the spinor. W M2

— 2 Implications for light scalar dark matter. The search for Given the expectation that CW scales with the total number dark matter has inspired the development of many light or of species, and the EFT requirement that H ≪ M, this even ultralight scalar field models. Whether those scalar condition is easily satisfied. Nevertheless from a low

063518-25 CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020) energy perspective there appeared to be nothing wrong with pffiffiffiffiffiffi M2 1 L ¼ − Pl þ ð ð 2 Þ2 þ 2Þ ð Þ 2 ¼ 0 g R 4 c1 Rμναβ c2GB ; 5:14 tuning CW . 2 M

B. Dimension-eight curvature operators where GB is the Gauss-Bonnet term defined in (3.15). Then Following from the previous logic, we may also inspect following the same procedure as highlighted in Sec. III A as the effect of dimension-eight curvature operators (fourth applied for curvature-squared operators in Sec. III B and for power of curvature) on the graviton speed of sound. Such curvature-cubed operators in Sec. VA, we find that on operators are known to arise in the effective action for string introducing a minimally coupled matter field (e.g., dilaton) theory from tree level α0 corrections [44,93]. These were which sources an FLRW (or equivalently on a space considered in [11] for Ricci flat backgrounds with the dependent) background, we find a departure of the gravi- following effective Lagrangian: tational waves sound speed given by

384 2 ¼ 1 þ ½ ð−8 ˙ 4 þ 3 ̈ 3 þ 6 ˙ 2 2 þ ⃛ 2 þ 5 ˙ ̈þ ˙ ⃛ þ ̈2Þ cs 2 4 c1 HH HH H H HH HH H HH H MPlM ˙ 4 ̈ 3 ˙ 2 2 ⃛ 2 ˙ ̈ ̈2 þ c2ð−4HH þ 3HH þ 10H H þ HH þ 6HH H þH Þ: ð5:15Þ For constant equation of state parameter ϖ, this leads to a sound speed

144 6 2 ¼ 1 þ H ð1 þ ϖÞð ð19 þ 15ϖÞð5 þ 6ϖ þ 9ϖ2Þ − 4 ð1 þ 3ϖÞð17 þ 15ϖÞÞ ð Þ cs 2 4 c1 c2 : 5:16 MPlM

It is clear that if the quadratic and cubic curvature operators Sec. III A, is inevitably different than unity on a back- were set to zero, then demanding the sound speed to be ground which spontaneously breaks Lorentz invariance, subluminal for NEC preserving backgrounds (i.e., for such as FLRW. The precise origin of this effect could be ϖ > −1) would require c1 ≤ 0 which is in tension with loop corrections from matter fields, or from higher spin the requirement found in [11] for Ricci-flat backgrounds particles s ≥ 2 that may arise in a given UV completion. (unless we restricted ourselves to c1 ¼ c2 ≡ 0). Indeed in The former are effects that can never be turned off and arise Ref. [11] it was found that subluminality on Ricci-flat in any UV completion. The latter are known to already arise backgrounds imposed c1, c2 ≥ 0. in the low energy effective action for strings as discussed in If, on the other hand, one required that gravitational Sec. VB. Perhaps surprisingly, for natural expectations of waves propagate faster than the light cone to which signs of Wilson (interaction) coefficients, the gravitational matter minimally couples, this would require c1 ≥ 0 and waves are generically found to be superluminal with −1.49c1 ≲ c2 ≲ 1.09c1. Interestingly, it was argued in [11] respect to the metric with which the matter driving the that considering the explicit quartic Riemann corrections background expansion is minimally coupled. This effect is from the string theory example proposed in [93] and in some sense a pure gravitational analogue of the well- compactifying on a four-dimensional flat manifold would known property of the low energy EFT for QED on a lead to c1 ¼ c2 > 0, which in the analysis provided on curved spacetime [33,34]. FLRW would be compatible with a superluminal sound As is well known, low energy superluminal group or speed for gravitational waves. phase velocities are themselves not direct signs of acausality. A quite different result would arise if we first perform a The front velocity, corresponding to the large frequency limit field redefinition to remove any Rμν and R terms in (5.14) of the phase velocity, is the speed at which information and then couple matter minimally to that new metric. In this travels, and a superluminal group velocity does not imply case (5.14) reduces to a quartic Weyl operator which will superluminal propagation of information. A discussion of not affect the speed of gravitational waves by virtue of the this point is presented in [69,94,95] and has been empha- vanishing of the Weyl tensor on FLRW. This is an example sized in discussions of QED in curved spacetime [35–42].In of how the specific coupling to the light fields that source the present context, the ambiguity of metric field redefini- the background expansion is crucial in the analysis. tions means that there is no absolute definition of low energy speeds, only relative ones. The ratio of the speed of gravitational waves to an individual species of matter is VI. DISCUSSION invariant under field redefinitions. We have shown that in the standard effective field theory Causality is likely better implemented by demanding treatment of general relativity coupled to matter, the low S-matrix analyticity. At low energies, this criterion can be energy speed of gravitational waves, defined precisely in used to derive various positivity bounds that fix the signs of

063518-26 SPEED OF GRAVITY PHYS. REV. D 101, 063518 (2020) coefficients in the effective Lagrangian [10,25–28].Ifwe to assume that all fields (including tensor modes) are (sub) assume that these bounds may be applied in the context of luminal in constraining the form of the effective action. gravitational systems—assuming following [32] that it is Generically such a criterion is not well founded, and indeed possible to neglect the massless t-channel graviton pole— it is not even invariant under field redefinitions. At best it can then we have derived precise positivity bounds in Sec. IV be implemented in some field frame, but as we have seen this that enforce that regardless of field redefinitions, gravita- is not the natural one to which we expect Standard Model tional waves travel faster than allowed by the metric to fields to couple. Already in the case of the low energy which matter minimally couples. This apparent acausality effective theory for QED, it is known that backgrounds can is in fact seen after a field redefinition to be equivalent to be found in which different photon polarizations travel both the more prosaic requirement that matter fluctuations are superluminally and subluminally, undermining the existence (sub)luminal, and is thus usually regarded as a requirement of a single preferred field frame. What is needed is a better of causality. understanding of how causality, likely through S-matrix In our analysis we have focused mainly on curvature analyticity, could be used to constrain cosmological EFTs. corrections in the EFT, keeping the matter that sources the There have been some attempts in the recent literature background relatively minimal. A more complete analysis [46,96–98] although a clear understanding is absent due could for example focus on the full effective theory of the to the particular challenge of dealing with a massless Standard Model coupled to GR (see [8] for an explicit graviton and the unclear meaning of analyticity on a curved discussion), but the examples considered in Sec. IV D spacetime. From our analysis it is clear that a significant role support the universality of the connection between pos- is played by the fully pole subtracted, elastic scattering itivity bounds and superluminal gravitational wave propa- amplitudes for matter fields. This is not so surprising given gation. Furthermore, in the case of low energy QED [or the similar role in nongravitational theories [10,25–28], more generally the EFT of a Uð1Þ gauge field coupled to nevertheless the gravitational extension has yet to be fully gravity], for which the general leading order EFT treatment understood. is well known, positivity bounds have been applied recently in [32], and the signs are consistent with expectations when ACKNOWLEDGMENTS applied in FLRW. Assuming no fine-tuning of the EFT (no special proper- We thank John Donoghue and Subodh Patil for useful ties of the UV completion), the magnitude of the departure discussions and comments. A. J. T. and C.d.R. thank the of the propagation speed from unity is of order Perimeter Institute for Theoretical Physics for its hospitality during part of this work and for support from the Simons H˙ Emmy Noether program. The work of A. J. T. and C.d.R. is c2 − 1 ∼ ; ð : Þ s Λ2 6 1 supported by STFC Grant No. ST/P000762/1. C.d.R. thanks the Royal Society for support at ICL through a Wolfson Λ where is the EFT cutoff, i.e., either the mass scale of any Research Merit Award. C.d.R. is supported by the European heavy state or the strong coupling scale of the theory Union’s Horizon 2020 Research Council Grant No. 724659 ˙ ¼ 0 ¼ 1 integrated out. On de Sitter H and so cs as MassiveCosmo ERC-2016-COG and by Simons Foundation required by de Sitter invariance. Hence during inflation Grant No. 555326 under the Simons Foundation’s Origins of this effect is further slow roll suppressed. We have found the Universe initiative, Cosmology Beyond Einstein’s that quite generally the magnitude and sign of the effect is Theory. A. J. T. thanks the Royal Society for support at controlled by the coefficient of the Weyl squared term in the ICL through a Wolfson Research Merit Award. A. J. T. effective Lagrangian. Given this, it is worth looking for expresses special thanks to the Mainz Institute for independent arguments that may be used to constrain its Theoretical Physics (MITP) off the DFG Cluster of sign and magnitude (see for example [46,84]). Excellence PRISMA+ (Project ID No. 39083149) for its If the leading curvature-squared terms in the EFT hospitality and support. expansion are tuned to zero, as may be implied by a specific UV completion, then the dominant effect will come APPENDIX A: ONE-LOOP EFFECTIVE ACTION from higher order, whether dimension-six or dimension- eight curvature operators. When this is the case, not only Here we show how to recover the one-loop effective does the sign of the effect switch at the onset of NEC action derived in [85] following a perturbative diagram- violation but also it becomes epoch dependent, a function matic approach in the case of scalar fields; see Fig. 4. The of the precise equation of state of the Universe. The attempt results are in complete agreements with those provided to demand either universal subluminality or superlumin- in [85]. ality of gravitational waves would in turn place strong As we shall see below, by integrating (perturbatively) a constraints on the particle content in the Universe. massive scalar field minimally coupled to gravity, we Our results have clear implications for cosmological recover the one-loop effective action (5.1) with the precise effective field theory model building where it is common same coefficients for the case of scalar fields. In Ref. [85]

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FIG. 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 quantum field theory in Minkowski spacetime (as we show below), giving identical results to covariant approaches. the one-loop effective action for the spin-1=2 and -1 fields fluctuation as in what follows it will also carry the role were also derived, and depending on the spin of the particle of the background field. To third order in perturbations for integrated out, the coefficients cn are given in Table I below. hμν (i.e., third order in ς), one has pffiffiffiffiffiffi 1 1. Minimally coupled scalar field abcd a0b0c0d0 −g ¼ ε ε ðηaa0 þ ςhaa0 Þðηbb0 þ ςhbb0 Þ For concreteness, we shall integrate a scalar field φ of 4! mass M minimally coupled to gravity, × ðηcc0 þ ςhcc0 Þðηdd0 þ ςhdd0 ÞðA3Þ pffiffiffiffiffiffi 1 1 ς2 L½φ¼ − − μν∂ φ∂ φ − 2φ2 ð Þ ¼ 1 þ ς þ ð½ 2 − ½ 2Þ g 2 g μ ν 2 M : A1 h 2 h h ς3 þ ð½ 3 − 3½ ½ 2þ2½ 3Þ þ Oðς4Þ ð Þ We work perturbatively in the metric perturbations about 3! h h h h ; A4 flat spacetime defined as gμν ¼ ημν − 2ςhμν þ 3ς2h2μν − 4ς3h3μν þ Oðϵ4Þ; ðA5Þ 2 2 αβ gμν ¼ðημν þ ςhμνÞ ¼ ημν þ 2ςhμν þ ς hμαhνβη ; ðA2Þ where square brackets represent the trace (with respect to where the parameter ς has been introduced for bookkeeping Minkowski) of a tensor. and we do not yet commit to hμν being the tensor The relevant graviton-scalar vertices are therefore

ðA6Þ

ðA7Þ

ðA8Þ

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With those vertices in mind we can directly compute Il ¼ Il−1 − Il−1; ðA15Þ the relevant one-loop scalar field contributions to the two n n−1 n and three point gravitons. We do so using dimensional- ¯ regularization and summarize our conventions in what where we emphasize that both If and Ii are finite but I is follows. not in the limit ϵ → 0. We will also make use of the following relations: 2. Dimensional regularization Z We work in d ¼ 4 − ϵ dimensions in what follows. μϵ d ð 2Þl−1 1 l d k kμkν k l Defining the following integrals I as ¼ ημνI ; ðA16Þ n M4þ2ðl−nÞ ð2πÞd=2 ðk2 þ M2Þn d n Z ϵ d 2 l l μ d k ðk Þ ¼ ð Þ Z 2 l−2 In 4þ2ðl− Þ 2 2 2 ; A9 μϵ d ð Þ M n ð2πÞd= ðk þ M Þn d k kμkνkαkβ k M4þ2ðl−nÞ ð2πÞd=2 ðk2 þ M2Þn we have in dimensional regularization, 1 l ¼ ðημνηαβ þ ημαηνβ þ ηναημβÞIn; ðA17Þ l dðd þ 2Þ I0 ¼ 0 ∀ l; ðA10Þ and similarly for higher order integrals. 1 0 ≡ ¯ ≡ þ ð Þ I1 I ϵ Ii If; A11 3. Perturbative one-loop contributions 0 0 0 I2 ¼ −I1 − 2I3; ðA12Þ a. Graviton one-point function 1 For consistency, we start with the one-loop massive 0 ¼ − ð Þ I3 4 If; A13 scalar field contribution to the one-point function, which is the leading term of the cosmological constant (and of 0 2ðn − 3Þ! 0 In ¼ I3n ≥ 4; ðA14Þ course at the origin of the cosmological constant problem). ðn − 1Þ! The related Feynman diagram and amplitudes are

ðA18Þ this is a diverging contribution and leads to a contribution to the effective cosmological constant that scales as M4. The order of magnitude of this cosmological constant is of course well above that of the classical value we have considered so far. Tackling the cosmological constant problem is well beyond the scope of this work, and for now we shall simply put this problem aside. b. Graviton two-point function ðaÞ Two types of diagrams contribute to the two–point function. The first one, referred to below as A2 and coming from the vertex (A7), is simply

ðA19Þ

ðbÞ As for the second one, referred to as A2 , it involves a product of Feynman propagators, which, for the context of this work we simply deal with by expansion in the power of derivatives p of the external legs,

1 1 2k · p 4ðk · pÞ2 p2 ¼ þ 2 þ − þ; ðk2 þ M2Þððk − pÞ2 þ M2Þ ðk2 þ M2Þ2 ðk2 þ M2Þ3 ðk2 þ M2Þ4 ðk2 þ M2Þ3 and work up to sixth order in the external leg derivative.

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Then as a perturbative expansion, the contribution of the second diagrams to the graviton two-point function is

ðA20Þ

where we immediately recognizepffiffiffiffiffiffi the Lichnerowicz operator on the second line, corresponding to the leading expansion of the Einstein-Hilbert term −gR. Adding the one- and two-point functions, we see that to the order we are working this is precisely equivalent to the following effective action: 4 pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 1 pffiffiffiffiffiffi 1 A þ A ≡ M ¯ þ Ii − þ M ¯ − þ ð¯ − Þ − 2 þ 2 1 2 4 I 4ϵ g 6 I g R 240 I Ii g 2 R Rμν 8 Ii pffiffiffiffiffiffi 2 2 3 ∂ þ −gð5ð∂RÞ þ 2ðDμRαβÞ ÞþO ς ; ; ðA21Þ 6720M2 M4 where so far the right-hand side should be understood as being only up to second order in the metric fluctuation. We see that all the terms on the first line diverge and can in principle be fully removed by an appropriate renormalization procedure while the term on the second line is entirely finite.

c. Graviton three-point function We can follow the same procedure for the three-point function. Three types of diagrams contribute at that level. The first one is a tadpole type and contains no derivatives acting on the external legs, hence solely leading to a contribution toward the cosmological constant at cubic order in ς,

ðA22Þ

The next diagram, involving both a hhφφ and a hφφ vertex, leads to contributions that are again computed performing a derivative expansion. Below we only present explicitly the leading order contributions, but they have been explicitly computed up to sixth order in derivative,

ðA23Þ

where the first line is exact while the second line is of course only symbolic but accounts for the correct split between finite and divergent pieces.

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Finally the diagram involving three hφφ vertices is the most challenging to account for, and once again we only give its symbolic form in what follows (apart from the first line which is exact) even though it has been computed explicitly to sixth order in derivatives,

ðA24Þ

Combining the one, two, and three point functions explicitly, one can check that up to cubic order in the metric perturbation and up to sixth order in derivatives, we obtain the following effective action (up to integrations by parts):

4 pffiffiffiffiffiffi 2 pffiffiffiffiffiffi 1 pffiffiffiffiffiffi 1 A þ A þ A ≡ M ¯ þ Ii − þ M ¯ − þ ð¯ − Þ − 2 þ 2 1 2 3 4 I 4ϵ g 6 I g R 240 I Ii g 2 R Rμν

Ii pffiffiffiffiffiffi 2 2 þ −gð5ð∂RÞ þ 2ðDμRαβÞ Þ 6720M2 pffiffiffiffiffiffi Ii 3 3 2 μν ab − −gð47R − 80Rμν − 60RRμν þ 72R R Rμ ν 241920 2 a b M ∂8 αaβb μν αaβb μ ν 2 4 þ 56RαμβνR R − 40RαμβνR R þ 114RR ÞþO ς ; ; ðA25Þ ab a b abcd M4

¯ ¯ where again only the terms on the first two lines include where the quantities CR2 and CW2 include the UV counter- running and divergent pieces and could be entirely removed terms and with the contribution from integrating out the via an appropriate renormalization procedure while the massive particles going as ¯ remainder is finite. In particular I contains running and divergent pieces, Ii=ϵ is divergent, but Ii itself is finite and 8 1 1 δ 2 ð Þ¼4δ 2 ð Þ¼ does not run. One can check that they precisely match the 5 μCR M μCW M 240 8π2ϵ coefficients given in Table I for scalars as derived by [85], 1 M2 up to appropriate integrations by parts. − −1 þ γ þ log : ðA28Þ It is worth noting that up to integrations by parts (i.e., up 16π2 4πμ2 to the Gauss-Bonnet term which is irrelevant in four dimensions), the R2 terms can be expressed in terms of Had this been calculated directly in a cutoff scheme, with Λ the Weyl term Wabcd as follows: cutoff , we would have found 2 1 pffiffiffiffiffiffi 1 8 1 1 Λ δ 2 ð Þ¼4δ 2 ð Þ ∼ ð¯ − Þ − 2 þ 2 ΛCR M ΛCW M 2 log 2 ; 240 I Ii g 2 R Rμν 5 240 16π M 1 pffiffiffiffiffiffi 5 1 ðA29Þ ¼ ð¯ − Þ − 2 þ 2 ð Þ 240 I Ii g 8 R 4 W : A26 ≪ Λ δ 2 0 which given the requirement M implies ΛCW > . Of course neither (A28) nor (A29) can be taken as an This implies that the low energy EFT defined at a scale answer because they are UV divergent and the 1=ϵ and Λ M0

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ˆ for instance, if we considered the coefficient of the Weyl where Odim -6 is a fourth order operator similar in spirit to square term in a low energy EFT endowed with a UV the one found in (3.23) but with slightly different coef- completion where the mass of the particle we integrate out ficients, is M1 and compared it with the coefficient of another low X 1 1 ˆ ¼ ðsÞ□2 þ ðsÞ□ ∇⃗2 energy EFT that now enjoys another UV completion with a Odim -6 Ns 4 f1 η 4 f2 η massive state M2 then regardless of renormalization s¼0;1=2;1 a a 21 scheme we would have 1 1 þ ðsÞ□ ∂ þ ðsÞ□ 3 f3 η η 2 f4 η 2 a a 1 1 M2 2 ð Þ − 2 ð Þ¼ 0 CW M1 CW M2 2 log 2 > ; 1 ðsÞ 1 ðsÞ ⃗2 1 ðsÞ ⃗2 ðsÞ 960 16π M1 þ f ∂η þ f ∇ ∂η þ f ∇ þ f ; a 5 a3 6 a2 7 8 M1

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˜ ðsÞ ðsÞ ðsÞ 3 ˙ ̈ ðsÞ 2 ˙ ð1Þ ̈ ˙ f5 ¼ f5 þ 2f1 ð4H þ 6HH þ HÞþf3 ð−2H − HÞ; f6 ¼ 446H − 892HH; ðB17Þ ðB30Þ ð1Þ ˙ 2 ⃛ ̈ ˙ 2 f7 ¼ −362ðHÞ þ 166H − 1286HH þ 872HH ; ðB18Þ ˜ ðsÞ ðsÞ f6 ¼ f6 ; ðB31Þ ð1Þ 6 ̈ 2 ˙ 2 2 ⃛ 2 f8 ¼ −1016H þ 618ðHÞ þ 11253ðHÞ H þ 109HH þ 2687 ̈ 3 þ 1714ð ˙Þ3 þ 2 ˙ð309 ⃛ þ 2441 ̈ ˜ ðsÞ ðsÞ 2 ˙ HH H H H HH f7 ¼ f7 þ f2ð−2H − HÞ; ðB32Þ þ 1528H4Þ; ðB19Þ ð Þ ð Þ ð Þ f˜ s ¼ f s þ f s ð16H4 þ 34H2H˙ þ 9HḦþ H⃛ þ 7ðH˙ Þ2Þ and finally for spinors 8 8 1 ðsÞ 3 ˙ ̈ ðsÞ 2 ˙ þ f3 ð−4H − 6HH − HÞþf4 ð−2H − HÞ: 1 ð1=2Þ ¼ ð−21 2 þ 104 ˙Þ ð Þ ð Þ f1 2 H H ; B20 B33

ð1=2Þ ˙ The original normalization of the tensor modes gij ¼ f2 ¼ −76H; ðB21Þ ahij was chosen precisely so as to remove the friction term that would otherwise have arisen from the standard ð1=2Þ ¼ −42 3 − 104 ̈þ 250 ˙ ð Þ f3 H H HH; B22 Einstein-Hilbert term. We now perform a subleading rescaling of the tensor modes so as to absorb the subleading ð1=2Þ 1 4 2 ðsÞ −2 ðsÞ ⃗2 ¼ ð−113 þ 359ð ˙ Þ − 104 ⃛ þ 450 ̈ friction term ðf˜ þ a f˜ ∇ Þ present in (B29). For this it f4 2 H H H HH 5 6 is easier to move to momentum space and perform the − 284 ˙ 2Þ ð Þ HH ; B23 rescaling

ð1=2Þ ˙ 2 ⃛ ̈ 2 f5 ¼ −92ðHÞ H − 48HH − 46HH Ωðk; ηÞ h ðηÞ → 1 þ h ðηÞ; ðB34Þ ˙ ̈ 3 k 12 420ð2πÞ4 2 2 k − Hð157H − 310H Þ; ðB24Þ × M MPl

ð1=2Þ ̈ ˙ with f6 ¼ 76H − 152HH; ðB25Þ 2 ð1=2Þ ˙ 2 ⃛ ̈ ˙ 2 ˙ k f7 ¼ −90ðHÞ þ 48H − 160HH þ 2HH ; ðB26Þ Ωðk; ηÞ¼− 4ð9N þ 38N þ 223N ÞH 0 1=2 1 a2 1 − ð592 − 71 − 72 Þ 4 ð1=2Þ 6 ̈ 2 ˙ 2 2 N0 N1=2 N1 H f8 ¼ ð−288H þ 100ðHÞ þ 2811ðHÞ H 2 − 14ð74 − 27 − 226 Þ 2 ˙ N0 N1=2 N1 H H þ 53HH⃛ 2 þ 755HḦ 3 þ 391ðH˙Þ3 þ H˙ð100H⃛ − ð298 − 99 þ 1430 Þð ˙ Þ2 N0 N1=2 N1 H þ 1088HḦ þ 753H4ÞÞ: ðB27Þ − ð92 þ 96 þ 332 Þ ̈ ð Þ N0 N1=2 N1 HH; B35 Working perturbatively in the dimension-six operators, we may substitute the relation for □ηh in terms of h as leading to the following equation of motion for the tensor derived from (B1), modes,

2 2 ˙ □ηh ¼ a ð−2H − HÞh: ðB28Þ 00 2 2 2 h þ cs k h ¼ m0h; ðB36Þ This perturbative substitution can be performed on the first ˆ with line of the operator Odim -6 defined in (B3) so that only the second line remains which slightly altered coefficients, 1 2 ¼ 1 − cs 12 105ð2πÞ4 2 2 X × M MPl 1 ðsÞ 1 ðsÞ 2 ˆ ¼ ˜ ∂ þ ˜ ∇⃗ ∂ 2 Odim -6 Ns f5 η 3 f6 η ð2ð163 − 39 − 659 Þ ˙ a × N0 N1=2 N1 H H s¼0;1=2;1 a − ð46 þ 62 þ 530 Þ ˙ 2 1 N0 N1=2 N1 H þ ˜ ðsÞ∇⃗2 þ ˜ ðsÞ ð Þ 2 f7 f8 ; B29 þð−93 þ 46 þ 617 Þ ̈ a N0 N1=2 N1 HH þð−37N − 10N þ 57NÞH⃛Þ; ð Þ with 0 1=2 1 B37

063518-33 CLAUDIA DE RHAM and ANDREW J. TOLLEY PHYS. REV. D 101, 063518 (2020) while the IR part in the dispersion relation is derivatives are canceled by similar spatial derivative terms by virtue of the leading order equations of motion. Field H6 redefining the equation of motion into standard two time m2 ¼ a2 −2H2 − H˙ þ O : ðB38Þ 0 2 2 derivative form (3.7) is helpful so that we may define the MPlM retarded propagator in the standard way via time order- ð 0Þ¼ θð − 0ÞΔð 0Þ¼ h ˆ ð Þ ð 0Þi − ing, i.e., Gret x; x i t t x; x i Th x h x APPENDIX C: RESUMMING GREEN’S ihhðx0ÞhðxÞi with Δðx; x0Þ the commutator function. But FUNCTION SECULAR BEHAVIOR perhaps more importantly the resummation removes secular behaviors that would have been obtained otherwise. Rather than following the logic presented in Sec. III A 2 to Specifically the resummed WKB modes can be rewritten determine the speed, one could have attempted to define perturbatively in terms of the unresummed ones, Green’s function iteratively around the GR one. Following R such an approach, one would be lured into the belief that the η 0 ikx−i dη k=c ðηÞ causal structure appears to be the standard one at any finite e s R Z order. A similar argument can be applied to any theory with a η 0 η ¼ ikx−i dη k 1 þ η0ð ðη0Þ − 1Þþ ð Þ small departure of the sound speed. The reason why such a e ik d cs : C2 procedure is not correct in general is because it relies on perturbations that carry secular effects, whose growth can in The secular perturbative growth of the right-hand side turn undermine the perturbative expansion. would become large for jkηj ∼ 1=ðcs − 1Þ, invalidating this Indeed in rewriting in the form (3.7) we have implicitly perturbative approach of solving (3.1) by directly iterating resummed effects, similar to a self-energy resummation, about the GR result. Instead, in Sec. III A 2,wehave which contribute already at two-derivative order. Super- identified the retarded Green’s function by iterating the ficially higher order time derivatives generate second order resummed counterpart given in Eqs. (3.8) and (3.9).In time derivatives by partly acting on the background, e.g., doing so, we have effectively resummed an infinite number of contributions, leading to a better behaved perturbative n! ∂nð ðηÞm Þ¼ ð∂n−2 ðηÞmÞ∂2 þ expansion. This is the approach we follow in identifying the a h 2!ð − 2Þ! a h ; n speed throughout this work. ðC1Þ Ultimately the causal structure is determined by the front velocity, i.e., the high energy limit of the phase velocity. hence giving a contribution to the low energy sound speed. As we have argued we expect this to be luminal in a The superficially larger contributions from higher time fundamentally Lorentz invariant theory.

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