From amplitudes to gravitational radiation with cubic interactions and tidal effects
Manuel Accettulli Huber, Andreas Brandhuber, Stefano De Angelis, Gabriele Travaglini
QQMUL-PH-20-19SAGEX-20-19-E
From amplitudes to gravitational radiationwith cubic interactions and tidal effects
Manuel Accettulli Huber, Andreas Brandhuber, Stefano De Angelis and Gabriele Travaglini
Centre for Research in String TheorySchool of Physics and AstronomyQueen Mary University of LondonMile End Road, London E1 4NS, United Kingdom
Abstract
We study the effect of cubic and tidal interactions on the spectrum of gravitational waves emittedin the inspiral phase of the merger of two non-spinning objects. There are two independentparity-even cubic interaction terms, which we take to be I = R αβµν R µν ρσ R ρσαβ and G = I − R α µ β ν R µ ρ ν σ R ρ α σ β . The latter has vanishing pure graviton amplitudes but modifiesmixed scalar/graviton amplitudes which are crucial for our study. Working in an effective fieldtheory set-up, we compute the modifications to the quadrupole moment due to I , G and tidalinteractions, from which we obtain the power of gravitational waves radiated in the process tofirst order in the perturbations and leading order in the post-Minkowskian expansion. The I predictions are novel, and we find that our results for G are related to the known quadrupolecorrections arising from tidal perturbations, although the physical origin of the G coupling isunrelated to the finite-size effects underlying tidal interactions. We show this by recomputingsuch tidal corrections and by presenting an explicit field redefinition. In the post-Newtonianexpansion our results are complete at leading order, which for the gravitational-wave flux is5PN for G and tidal interactions, and 6PN for I . Finally, we compute the correspondingmodifications to the waveforms. {m.accettullihuber, a.brandhuber, s.deangelis, g.travaglini}@qmul.ac.uk a r X i v : . [ h e p - t h ] D ec ontents The first direct detection of gravitational waves and the first observation of a binary black holemerger by the LIGO/Virgo collaboration [1] has opened a new observational window potentiallychallenging our understanding of gravity. Anticipating improved experimental sensitivity in thefuture, high-precision theoretical predictions from general relativity will be required and in therecent few years much effort went into developing new theoretical tools using traditional andnovel approaches. This includes important calculations of the effective gravitational potential atsecond [2, 3], third [4–7], fourth [8–20], fifth [21–23] and sixth [24, 25] post-Newtonian order , aswell as in the post-Minkowskian expansion [28–30] and formal developments in computing clas-sical observables from scattering amplitudes [31–50]. A related, ambitious question is whethergravitational waves can, now or in the near future, provide feasible tests of modifications of gen-eral relativity as implied by string theory or other extensions of Einstein-Hilbert (EH) gravity.Even if the experimental precision has not been reached today, one can entertain this tantalisingpossibility.An effective field theory (EFT) framework for gravity was advocated in [51], and is ideallysuited to study systematically higher-derivative corrections to the EH theory. In [52], this ap-proach was followed to compute the corrections to the gravitational potential between compactobjects and their effective mass and current quadrupoles due to perturbations quartic in theRiemann tensor, and the corresponding modifications to the waveforms were then analysedin [53]. Modifications to the gravitational potential due to cubic interactions in the Riemann For a recent review of the EFT approach to the binary problem [26] in the PN expansion see [27]. , F F R [56]. Termsthat are quadratic in the Riemann tensor do not contribute to the classical scattering of particlesin four dimensions [57]. In this paper we wish to describe dissipative effects in the dynamics ofbinaries, that is gravitational-wave radiation, from appropriate five-point amplitudes with fourmassive scalars and one radiation graviton. We perform this study in the presence of cubic mod-ifications to the EH action and tidal effects. Interestingly, we will see that there is an overlapbetween these two types of corrections, which are linked by appropriate field redefinitions [58,59]which we construct explicitly. We note however that the physical origin of these interactions isvery different – for instance, I and G appear in the low-effective action of bosonic strings, orcan be induced by integrating out massive matter [60, 61].In the presence of scalars and restricting our focus to parity-even interactions, there are twoindependent cubic terms: I := R αβµν R µν ρσ R ρσαβ and I := R α µ β ν R µ ρ ν σ R ρ α σ β . A morenatural combination is in fact G := I − I , which, as is well known, is topological in sixdimensions [62] and has vanishing graviton amplitudes. In [63], it was argued from studyingthe scattering of polarised gravitons that I potentially leads to superluminal effects/causalityviolation in the propagation of gravitons for impact parameter b (cid:46) α . Here α ∼ Λ − is thecoupling constant of the I interaction, and Λ is the cutoff of the theory. In that paper, α waschosen to be much larger than G ∼ M − . This allows to treat the gravitational scatteringin a semiclassical set-up, where predictions can be trusted up to M Planck ( > Λ). This questionwas reinvestigated in an EFT framework in [56], where it was found that the I interactionleads to a time advance in the propagation of gravitons (but not photons and scalars) when b (cid:46) α . Finally, G does not lead to any time advance/delay for massless particles [56], whilestill correcting the gravitational potential [54, 55]. An identical conclusion for the propagationof massless particles in the background of a black hole was reached in [64], both for the I and G interactions .In this respect, an important observation was made in [65], namely that such superluminalityeffects (and those observed earlier on in [66–68]) are unresolvable within the regime of validityof the EFT, and do not lead to violations of causality. In our set-up such violations wouldindeed occur at b (cid:46) Λ − , which is at the boundary of the regime of validity of our EFT, whilethe processes we are interested in only probe the regime where the EFT is valid. Above Λ, theonly known way to restore causality is to introduce an infinite tower of massive particles [63].In conclusion, these observations do not rule out cubic interactions for our EFT computation,although they may impose constraints on the cutoff – it needs to be such that possible effectsdue to the massive modes, required to ensure causality, cannot be resolved with current-dayexperiments. We also note that, assuming that these interactions can contribute to any classicalgravitational scattering (Λ < M Planck ), then we have α > G , independently of precise estimatesof the cutoff Λ.In the following we work in an effective theory containing cubic and tidal perturbations, andcompute a five-point amplitude with four massive scalars (representing the black holes) and oneradiated soft graviton. From this, one can in principle extract all radiative multipole momentsto this order, but for the sake of our applications we will only focus on the quadrupole momentinduced by the cubic and tidal interactions, from which we then derive the correspondingchanges to the power radiated by gravitational waves and to the waveforms. Our results forthe quadrupole correction are exact to leading order in the perturbations and in the post- Note that for G the coefficient 2 d + d in Eq. (2.24) of [64] vanishes. G and tidal interaction corrections, and at 6PN orderfor the I corrections. We find that the corrections due to G have the same form as thosegenerated by a particular type of tidal interaction (although the corresponding coefficients inthe EFT action are independent). We also explain this result by constructing an explicit fieldredefinition relating the two couplings. For the PN-expanded result of the tidal corrections tothe mass quadrupole we find agreement with [69–71]. The remaining tasks consist in usingthe corrected quadrupole moment to compute the modifications compared to EH gravity tothe power emitted by the radiated gravitational waves, and the corresponding corrections tothe waveforms in the Stationary Phase Approximation (SPA) . Here we follow closely [53], andalso present a comparison with their result obtained with perturbations that are quartic in theRiemann tensor.The rest of the paper is organised as follows. In Section 2 we introduce the EFT we arediscussing, reviewing some of the relevant results, including the corrections to the gravitationalpotential from cubic [54, 55] and tidal interactions [74–76]. Furthermore, we point out thevanishing of all graviton amplitudes in the pure gravity plus G theory, and explicitly constructa field redefinition that maps G into a tidal perturbation. Section 3 contains the calculationof the relevant four-scalar, one soft graviton amplitude in our EFT, from which we extract theperturbations to the quadrupole moment. In Section 4 we compute the power radiated by thegravitational waves, and finally in Section 5 the corrections to the waveforms in the SPA. Inan Appendix we present some details on the modifications to the circular orbits due to theperturbations. We consider an EFT describing EH gravity with higher-derivative couplings interacting withtwo massive scalars. These model spinless heavy objects, and we also include the leading tidalinteractions in our description which describe finite size effects of the heavy objects. Specifically,the EFT action we consider is S = S eff + S φ φ + S tidal , (2.1)where S eff = Z d x √− g (cid:20) − κ R − κ L − · · · (cid:21) (2.2)is the effective action for gravity, with L = α I + α G . (2.3) I and G are the parity-even cubic couplings defined as I := R αβµν R µν ρσ R ρσαβ , G := I − I , (2.4) See e.g. [72, 73] for details of this approximation. I := R α µ β ν R µ ρ ν σ R ρ α σ β . (2.5)The dots in (2.2) stand for higher-derivative interactions that we will not consider here. Thetwo scalars, with masses m and m , couple to gravity with an action S φ φ = Z d x √− g X i =1 , (cid:16) ∂ µ φ i ∂ µ φ i − m i φ i (cid:17) , (2.6)and in addition we include higher-derivative couplings describing tidal effects of extended heavyobjects, S tidal = Z d x √− g R µανβ R ρασβ X i =1 , (cid:16) λ i φ i δ µρ δ νσ + η i m i ∇ µ ∇ ν φ i ∇ ρ ∇ σ φ i (cid:17) + · · · . (2.7)These tidal interactions were recently studied in [76], and the dots stand for the (Hilbert) seriesof higher-dimensional operators classified in [59, 77], which will not play any role in this work.We now briefly discuss some properties of the interactions we consider. The I and G interactions naturally arise in the low-energy effective description of bosonicstring theory, whose terms cubic in the curvature can be obtained by making the replacement α = α → α e − (2.8)in (2.3), where Φ is the dilaton. These interactions are also produced in the process of integratingout massive matter [60, 61]. In pure gravity only one of them is independent in four dimensions[78, 79], while in the presence of matter coupled to gravity they become independent. For thesake of the computation of the power radiated by the gravitational waves performed in latersections we need the correction induced by the cubic interactions to the gravitational potential.The full 2PM computation of this quantity was performed in [54,55], and expanding their resultone obtains V ( ~r, | ~p | ) = − Gm m r + 38 α G r ( m + m ) m m ~p − α G r m m ( m + m ) (cid:18) − m + m m m ~p (cid:19) + · · · , (2.9)where the dots indicate higher PN corrections which we do not consider here. Note that theterms proportional to α and α are the result of a one-loop computation. In the PN expansion,the term proportional to α (from the I interaction) is suppressed by a factor of ~p /m , compared to the dominant correction proportional to α (from G ). Amplitudes from the G interaction It is well known that, unlike I , the G interaction has a vanishing three-graviton amplitude anddoes not contribute to graviton scattering up to four particles [62,80] – and in fact to any numberof gravitons. This can be understood by the fact that G is topological in six dimensions [62],and therefore computing tree-level four-dimensional graviton amplitudes from dimensionallyreducing the six-dimensional ones automatically gives zero. Combining this observation with4nitarity techniques leads to M EH+ G ( h , . . . , h n ) (cid:12)(cid:12) d< = M EH ( h , . . . , h n ) (cid:12)(cid:12) d< , (2.10)for any n . Hence the G interaction does not affect the perturbative dynamics in theories ofpure gravity. However, if we consider a theory of gravity with matter, e.g. massive scalarsmimicking black holes or neutron stars, the presence of a G coupling alters their dynamics. Inparticular the four-point amplitude with two gravitons and two scalars becomes [54, 55] M (0)EH+ G ( φ , φ , h ++3 , h ++4 ) = M (0)EH ( φ , φ , h ++3 , h ++4 ) + i α (cid:18) κ (cid:19) [34] (2 m + s ) . (2.11)The non-trivial contribution to the scattering amplitude of two massive scalars and two gravitonsfrom the G interactions modifies the classical potential in the two-body system, as shownin [54, 55]. As we will show below, both G and I produce corrections to the quadrupolemoment already at tree level. Specifically we find that the G quadrupole correction is dominantin the post-Newtonian (PN) expansion, which parallels the results found for the correspondingcorrections to the gravitational potential quoted earlier in (2.9). The G interaction as a tidal effect It is easy to show that the contact term proportional to [34] (2 m + s ) in the amplitude (2.11)is (up to a numerical coefficient) the amplitude arising from a particular tidal interactions ofthe form R µνρσ R µνρσ m φ − ∇ α R µνρσ ∇ α R µνρσ φ . This suggests that there should exist a four-dimensional field redefinition mapping the G interaction into a tidal effect, as already noticedin [58, 59] . In this section we construct this field redefinition explicitly.We begin by rewriting G in a more convenient form, making use of two identities in fourdimensions [83]: R αβ [ αβ R µν µν R ρ ρ ] = 0 , (2.12)which translates into R α β R βρ µν R µν αρ = 14 R − RR α β R β α + 2 R α β R µ ν R βν αµ − R α β R β µ R µ α + 14 RR αβ µν R µν αβ , (2.13)and R αβ [ αβ R µν µν R ρσ ρσ ] = 0 , (2.14)which, in combination with (2.13), leads to R α µ β ν R µ ρ ν σ R ρ α σ β = 12 R αβ µν R µν ρσ R ρσ αβ − R + 92 RR α β R β α − RR αβ µν R µν αβ − R α β R µ ν R βν αµ + 4 R α β R β µ R µ α . (2.15) We also observe that black holes in four dimensions have non-vanishing Love numbers when higher-derivativeinteractions are considered [81, 82]. G can be rewritten as G | d =4 = 34 R R αβ µν R µν αβ + 54 R − R R α β R β α − R α β R β µ R µ α + 6 R α β R µ ν R βν αµ ∼ R R αβ µν R µν αβ , (2.16)where in the second line we have dropped all terms involving more than one Ricci scalar/tensor.These terms can be traded, via a further field redefinition, for a contact term of the form R µνρσ ∂ µ φ ∂ ν φ ∂ ρ φ ∂ σ φ , which only contributes to quantum corrections to the quadrupolemoment. Thus S eff = Z d x √− g (cid:20) − κ R − α κ G (cid:21) + S φ ,φ = Z d x √− g (cid:20) − κ R − α κ R ( R αβµν ) + · · · (cid:21) + S φ ,φ → Z d x √− g (cid:20) − κ R + α
64 ( R αβµν ) X i =1 , (cid:16) m i φ i − ∂ µ φ i ∂ µ φ i (cid:17) + O ( α ) (cid:21) + S φ ,φ , (2.17)where in the last line we have used the field redefinition g αβ → g αβ − α g αβ R µν ρσ R ρσ µν . (2.18)Finally, integrating by parts and discarding boundary contributions, we can rewrite the newinteraction term in (2.17) as( R αβµν ) (cid:16) m φ − ∂ µ φ∂ µ φ (cid:17) = R µνρσ R µνρσ m φ − ∇ α R µνρσ ∇ α R µνρσ φ , (2.19)where the second term does not give any classical contribution to the scattering amplitude.Hence, for the sake of computing classical contributions to amplitudes, we can replace S eff → Z d x √− g (cid:20) − κ R + α
64 ( R αβµν ) X i =1 , m i φ i + O ( α ) (cid:21) + S φ ,φ , (2.20)thereby explicitly showing that the G interaction can be absorbed into the first of the two tidalinteractions in (2.7). During the inspiral phase of binary systems involving at least one extended heavy object like aneutron star, corrections due to the finite size of the object(s) increase as the distance betweenthe objects decreases. These effects can be included systematically using a tidal expansion, i.e. amultipole expansion dominated by the mass quadrupole moment. Finite-size effects are boundto become of ever increasing importance in the light of future gravitational-wave experiments,and will likely play a key role in a deeper understanding of the internal structure of compactobjects. The computation of tidal effects has been addressed in the past by a wide variety ofmethods, recently including complete PM results [74–76] for the conservative dynamics.In order to compute the modifications to the waveform coming from the tidal interactions in62.7) we need to expand the 2PM potential in the conservative Hamiltonian computed in [74–76]up to O ( ~p ), with the result V tidal ( ~r, ~p ) = − G r m m " − m + m m m ~p ! λ + m + 2 m + 5 m m m m ~p ! η + 1 ↔ · · · , (2.21)where the dots indicate higher PN terms. In the PN framework, the conservative and dissipative dynamics of two objects of mass m and m , coupled to the gravity effective action (2.2) is described by the following point-particleeffective action [26, 52]: S pp = Z dt (cid:20) µ ˙ ~r − V ( ~r, ~p ) + 12 Q ij ( ~r, ~p ) R i j + · · · (cid:21) , (3.1)where µ := m m m + m (3.2)is the reduced mass, and ~r ( t ) is the relative position of the two objects. V (cid:0) ~r, ~p (cid:1) denotes thepotential, whose explicit expression to first order in α , α [54, 55], and λ , , η , [74–76] isobtained by summing (2.9) and (2.21), and Q ij (cid:0) ~r, ~p (cid:1) is the quadrupole moment, to be computedbelow. The dots represent higher-order terms that will be irrelevant in our analysis. This actioncan be trusted in the inspiral phase before the objects reach relativistic velocities.We now present the computation of the five-point amplitude φ φ → φ φ + h ( k ) with fourscalars and one radiated soft graviton h ( k ). Its momentum k µ is on shell, while the momentumof the graviton exchanged between the two objects is purely spacelike (corresponding to aninstantaneous interaction), and in our set-up is given by q µ = − p µ − p µ = (0 , ~q ). Furthermore,the energy of the radiated graviton is such that k (cid:28) | ~q | , so that k µ can be ignored for practicalpurposes, and the radiated graviton enters the amplitude only through its associated Riemanncurvature tensor R αβµν . Finally, because we are only interested in classical contributions ( i.e. O ( (cid:126) )), we keep only the leading terms in ~q .In the following we first compute fully relativistic scattering amplitudes and then perform thePN expansion to extract the correction to the quadrupole term in the effective action (3.1). Inthe centre-of-mass frame, the momenta of the particles can be parametrised as p µ = − (cid:16) E , ~p − ~q (cid:17) , p µ = − (cid:16) E , − ~p + ~q (cid:17) ,p µ = (cid:16) E , ~p + ~q (cid:17) , p µ = (cid:16) E , − ~p − ~q (cid:17) , (3.3)with p = p = m , p = p = m . Furthermore, we have E = E = q m + ~p + ~q / , E = E = q m + ~p + ~q / , (3.4)where ~p · ~q = 0 because of momentum conservation. In our all-outgoing convention for the7 O ≡ φ ( p ) φ ( p ) µνρσφ ( p ) φ ( p ) O × R µνρσ ( k ) (cid:12)(cid:12)(cid:12) k → q Figure 1:
The single diagram contributing to the radiation process with an insertion of the operators O = I , I . All momenta are treated as outgoing and the radiated graviton is taken to be soft. external lines, the four-momenta p and p correspond to the incoming particles, and hencetheir energies are negative. Our next task is to compute the five-point amplitude A O shown in Figure 1, with O = I , I (which we can then combine to obtain A G ). We first obtain its relativistic expression, factoringout a single Riemann tensor associated with the radiated graviton, and then split the Lorentzindices into time and spatial components and isolate the terms contracted into R i j . UponFourier transforming to position space, these components will allow to directly read off Q ij bymatching to the Hamiltonian density associated to the point particle effective action (3.1). Theclassical relativistic results are, for I : A I = i ( α + 2 α ) (cid:18) κ (cid:19) q µ q ρ q h m p ν p σ + m p ν p σ − p · p ) p ν p σ i R µνρσ , (3.5)while for I : A I = i α (cid:18) κ (cid:19) q µ q ρ q (cid:0) m p ν p σ + m p ν p σ (cid:1) R µνρσ . (3.6)Note that the result for the G interaction introduced in (2.3) can be obtained as A G := ( A I + A I ) (cid:12)(cid:12) α =0 . (3.7)The terms in the amplitude contributing to the quadrupole radiation are then A I ( q ) = − i ( α + 2 α ) (cid:18) κ (cid:19) (cid:16) m E + m E − E E − ~p E E (cid:17) q i q j ~q R i j + · · · , (3.8)and A I ( q ) = − i α (cid:18) κ (cid:19) (cid:16) m E + m E (cid:17) q i q j ~q R i j + · · · , (3.9)where we have used that E = E in order to write the result as a function of the energies andmomenta of the incoming particles p and p . The dots stand for additional terms proportionalto R ijk and R ijkl , which can also be extracted from our result.8 O µνρσ ≡ φ ( p ) φ ( p ) µνρσφ ( p ) φ ( p ) O q + φ ( p ) φ ( p ) µνρσφ ( p ) φ ( p ) O q Figure 2:
The two diagrams contributing to the gravitational radiation, where O denotes any of thetwo tidal interactions in (2.7). An overall Riemann tensor of the radiated graviton is factored out, sothat A O = A O µνρσ R µνρσ ( k → A calculation similar to the one outlined in the previous section leads to the fully relativisticresult A tidal ( q ) = i (cid:18) κ (cid:19) q µ q ρ q (cid:26) λ p ν p σ + 8 λ p ν p σ + 12 h ( m + m − t ) − m m i (cid:18) η m p ν p σ + η m p ν p σ (cid:19) (cid:27) R µνρσ , (3.10)which, upon expanding in the spatial and time components, reads A tidal ( q ) = − i (cid:18) κ (cid:19) (cid:26) λ E + 8 λ E + h E E + ~p ) − m m i η E m + η E m ! (cid:27) q i q j ~q R i j + · · · , (3.11)where the ellipses stand once again for terms proportional to R ijk and R ijkl which we will notneed in the remainder of this paper. Next we extract the corrections to the mass quadrupole moment Q ij from (3.8), (3.9) and(3.11). To do so we simply match the appropriately normalised and Fourier-transformed A O ,as defined in (3.13) below, to the quadrupole contribution in (3.1) . To begin with, we performthe relevant Fourier transforms using Z dt Z d q (2 π ) q i q j | ~q | e i~q · ~r R i j = − π Z dt r (cid:18) x i x j − r δ ij (cid:19) R i j . (3.12) For further details on the procedure see for example [26, 52]. − i/ E E , we arrive at thequadrupole-like terms e A quad O ( r ) : = − i A quad O ( r )4 E E = 12 C O ( E i , m i , ~p ) Z dt r (cid:18) x i x j − r δ ij (cid:19) R i j , (3.13)where C O are coefficients depending on the energies and masses as well as ~p of the heavyparticles, with C I ( E i , m i , ~p ) = 38 π ( α + 2 α ) (cid:18) κ (cid:19) (cid:18) m E E + m E E − E E − ~p (cid:19) ,C I ( E i , m i , ~p ) = 316 π α (cid:18) κ (cid:19) (cid:18) m E E + m E E (cid:19) ,C tidal ( E i , m i , ~p ) = 38 π (cid:18) κ (cid:19) (cid:26) λ E E + 8 λ E E + h (cid:0) E E + ~p (cid:1) − m m i (cid:18) η E E m + η E E m (cid:19)(cid:27) . (3.14)Comparing (3.13) with the Hamiltonian density obtained from the action (3.1), we concludethat the modifications to the quadrupole moment arising from the cubic and tidal couplings aregiven by Q ij O = C O µ r Q ijN , (3.15)where we have introduced the leading-order quadrupole moment in the EH theory for a binarysystem with masses m and m , Q ijN = µ (cid:18) x i x j − r δ ij (cid:19) , (3.16)with µ being the reduced mass defined in (3.2). Combining the various correction terms, wearrive at Q ij = Q ijN + Q ijI + Q ijI + Q ij tidal = (cid:18) C I µ r + C I µ r + C tidal µ r (cid:19) Q ijN . (3.17)It is interesting to write the three coefficients C I , C I and C tidal in the PN expansion. Keepingterms up to first order in ~p one has C PN I = − G (cid:0) α + 2 α (cid:1) M ~p µ ,C PN I = 3 G α m m ,C PNtidal = 3 G " λ + η + 12 M (cid:16) m − m ) λ + (3 m + 5 m ) η (cid:17) ~p µ m m + 1 ↔ , (3.18)where M := m + m , (3.19)10nd, as usual, κ := 32 π G . For convenience we also quote the contribution due to the G interaction alone – this is given by Q ijG = (cid:16) Q ijI + Q ijI (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) α =0 = 3 G α Mr − ~p µ ! Q ijN . (3.20) We can now compute the power radiated by the gravitational waves in the approximation ofcircular orbits. In the EH theory, the radius of the circular orbit is given by the well-knownformula r N = (cid:18) GM Ω (cid:19) . (4.1)In the presence of the cubic and tidal interactions, this quantity gets modified as r ◦ = r N + δr ,δr = Ω (cid:20) − α v + (cid:18) α λ (cid:19) (cid:18) v + v (2 ν − (cid:19) + η (cid:18) v + v ( ν + 2) (cid:19)(cid:21) + O ( g i ) , (4.2)where g i stands for any of the coupling constants of the cubic and tidal perturbations. We alsointroduced the symmetric mass ratio ν defined as ν := m m M , (4.3)and the parameter v := r N Ω = ( GM Ω) , (4.4)as well as the following combinations of the couplings λ := µ (cid:18) λ m + λ m (cid:19) , η := µ (cid:18) η m + η m (cid:19) . (4.5)Finally, Ω denotes the angular velocity on the circular orbit, and the value δr has been computedusing (4.2) and (A.5), where the potentials entering (A.5) are given in (2.9) and (2.21). Thetotal energy per unit mass M of the system, to first order in the couplings, is then given by E ( v ) = − ν v + 94 v ( GM ) ν ( α + 16 λ + 2 η ) + 118 v ( GM ) h − να + ν (2 ν −
1) ( α + 16 λ ) + 4 ν ( ν + 2) η i . (4.6)The above formula is complete at leading order in all of the perturbations (that is O ( v ) andat O ( v ) for the α correction only. The remaining O ( v ) terms have been obtained from asmall-velocity expansion of our 2PM result, and in order to get a complete result at that PNorder one would need to include also the 3PM corrections to the potential generated by cubicand tidal interactions . We have also compared the contribution to the energy from the η , Similar considerations apply to our results for the flux in (4.11). µ (2) A to ours) .Next, we compute the leading-order gravitational-wave flux using the quadrupole formula F ( v ) = G h ... Q ij ... Q ij i , (4.7)using the result of our computation for Q ij in (3.17). To first order in the couplings α and α the flux becomes F ( v ) = G h ... Q ijN ... Q ijN i (cid:20) µr (cid:16) C PN I + C PN I + C PNtidal (cid:17)(cid:21) + O ( α i ) , (4.8)where the PN-expanded coefficients C PN O are explicitly given in (3.18).Two comments are in order here. First, we note that the prefactor h ... Q ijN ... Q ijN i is evaluated onthe radius r ◦ of the circular orbit in the presence of the cubic and tidal interactions, as given in(4.2). Furthermore, the quantity ~p := p r + p φ /r can be obtained using the fact that p r = 0on the circular orbit while p φ := l is a constant, which can be determined from Hamilton’sequations, with the result l := µr ◦ Ω1 + 2 µU ( r ◦ ) , (4.9)where r ◦ is given in (4.2) and U ( r ) is the part of the potential proportional to ~p , followingthe conventions of Appendix A. Using these relations, ~p is re-expressed as a function of Ω, themasses, and the couplings.Factoring out the standard power radiated by the gravitational wave in EH, F N ( v ) := G h ... Q ijN ... Q ijN i (cid:12)(cid:12)(cid:12) r = r N = 325 Gµ r N Ω = 325 ν v G , (4.10)we can rewrite the expression for the flux as F ( v ) = 325 ν v G " v ( GM ) (cid:0) α + 144 λ + 48 λ + 18 η + 6 η (cid:1) + v ( GM ) h − α + 2(2 ν − α + 8(8 ν − λ + 24 λ + (8 ν + 31) η + 9 η i , (4.11)with λ and η defined in (4.5) and λ := 1 M (cid:18) λ m + λ m (cid:19) , η := 1 M (cid:18) η m + η m (cid:19) . (4.12)Similarly to (4.6), the first line and the α term in the second line of (4.11) are complete. Wealso note that the η , part of the tidal flux is in agreement with [71]. For further details on mapping field-theory to point-particle actions see e.g. [36, 84] Waveforms in EFT of gravity
Following [53] we can also compute the correction induced by the cubic and tidal interactionsto the gravitational phase in the saddle point approximation. In this approach, the waveformin the frequency domain is written as ˜ h SPA ( f ) ∼ exp h i (cid:16) ψ f ( t f ) − π (cid:17)i , (5.1)where ψ ( t ) := 2 πf t − φ ( t ) . (5.2)Here φ ( t ) is the orbital phase, while ˙ φ ( t ) = πF ( t ) defines the instantaneous frequency F ( t ) ofthe gravitational wave. t f is defined as the time where˙ ψ ( t ) (cid:12)(cid:12)(cid:12) t = t f = 0 , (5.3)implying that F ( t f ) = 2 f . In the adiabatic approximation, the work of [73, 72] provides explicitformulae for ψ SPA ( t f ) and t f : ψ SPA ( t f ) = 2 πf t ref − φ ref + 2 G Z v ref v f dv ( v f − v ) E ( v ) F ( v ) , (5.4) t f = t ref + M Z v ref v f dv E ( v ) F ( v ) , (5.5)where v ref = v ( t ref ) and t ref are integration constants, v f := ( πGM f ) , and E ( v ) and F ( v ) werecomputed to lowest order in the cubic and tidal perturbations in (4.6) and (4.8), respectively.We can now compute the correction to ψ SPA ( t f ) due to the presence of the perturbations,expanding the ratio E ( v ) / F ( v ) at consistent PN order and performing the integration in (5.4).Doing so we arrive at ψ SPA ( t f ) = ψ EHSPA ( t f ) + ψ I + I SPA ( t f ) + ψ tidalSPA ( t f ) . (5.6)Here ψ EHSPA ( t f ) = 2 πf t ref − φ ref + 3128 ν v f (5.7)is the EH contribution, where we have also included the reference time and phase t ref and φ ref ,which have been redefined in order to absorb terms that depend on v ref ; and ψ I + I SPA ( t f ) = − ν v f (cid:20) α ( GM ) v f − α + (665 − ν ) α GM ) v f (cid:21) ,ψ tidalSPA ( t f ) = − ν v f ( v f ( GM ) (cid:0) λ + λ ) + 12 η + η (cid:1) − v f ( GM ) h − ν ) λ − λ ) − ν + 37) η − η i) , (5.8) See for example Section III F of [73] for a detailed derivation. L = β C + β C e C + β e C , (5.9)where C := R µνρσ R µνρσ , e C := 12 R µναβ (cid:15) αβ γδ R γδµν . (5.10)The modifications to ψ SPA ( t f ) due to quartic interactions as found in [53] are (reinstating powersof G in the result of that paper, and converting their d Λ into our β as defined in (5.9)), ψ quarticSPA ( t f ) = ψ EHSPA ( t f ) + 3128 ν v f (cid:20)(cid:18) − ν (cid:19) β ( GM ) v f (cid:21) . (5.11)Note the different dependence on v f in the correction terms in (5.8) and (5.11), which are of O ( v f ) and O ( v f ) in the leading cubic and tidal, and quartic cases, respectively. Finally, it willbe interesting to perform a comparison of our result in (5.6) to experimental data, as performedin [53] for the case of quartic perturbations in the Riemann tensor. Acknowledgements
We would like to thank Alessandra Buonanno and Jung-Wook Kim for very useful discussions.This work was supported by the Science and Technology Facilities Council (STFC) ConsolidatedGrants ST/P000754/1 “String Theory, Gauge Theory and Duality” and ST/T000686/1 “Ampli-tudes, Strings and Duality” , and by the European Union’s Horizon 2020 research and innovationprogramme under the Marie Skłodowska-Curie grant agreement No. 764850 “SAGEX” . A Hamiltonians with momentum-dependent potentials
Consider a momentum-dependent Hamiltonian of the form H = ~p µ h µ U ( r ) i + V ( r ) , (A.1)where ~p = p r + p φ r . From Hamilton’s equations we learn that p φ := l is constant, as well as˙ φ = lµr h µU ( r ) i . The latter equation can be used to re-express l as a function of Ω. Wealso have ˙ r = p r µ h µ U ( r ) i , (A.2)and, for circular orbits, we see that p r = 0 and hence ˙ p r = 0. In this case, the Hamilton equation˙ p r = − ∂H∂r simplifies to V ( r ◦ ) − l µr ◦ (cid:2) µU ( r ◦ ) (cid:3) + l r ◦ U ( r ◦ ) = 0 , (A.3)14here r ◦ is the radius of the circular orbit. We will also set Ω := ˙ φ ( r = r ◦ ), orΩ := lµr ◦ (cid:2) µU ( r ◦ ) (cid:3) . (A.4)Using this to eliminate l in favour of Ω, we finally get V ( r ◦ ) − µr ◦ Ω µU ( r ◦ ) h − µr ◦ U ( r ◦ )1 + 2 µU ( r ◦ ) i = 0 . (A.5)This equation determines r ◦ as a function of Ω. In the absence of a perturbation, we haveΩ N = lµr N , (A.6)where r N is the radius of the circular orbit in the EH theory, given in (4.1). References [1]
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