From amplitudes to gravitational radiation with cubic interactions and tidal effects

We study the effect of cubic and tidal interactions on the spectrum of gravitational waves emitted in the inspiral phase of the merger of two non-spinning objects. There are two independent parity-even cubic interaction terms, which we take to be $I_1 = {R^{\alpha \beta}}_{\mu \nu} {R^{\mu \nu}}_{\rho \sigma} {R^{\rho \sigma}}_{\alpha \beta}$ and $G_3 = I_1-2 R^{\alpha}\,_{\mu}\,^{\beta}\,_{\nu} R^{\mu}\,_{\rho}\,^{\nu}\,_{\sigma} R^{\rho}\,_{\alpha}\,^{\sigma}\,_{\beta}$. The latter has vanishing pure graviton amplitudes but modifies mixed scalar/graviton amplitudes which are crucial for our study. Working in an effective field theory set-up, we compute the modifications to the quadrupole moment due to $I_1$, $G_3$ and tidal interactions, from which we obtain the power of gravitational waves radiated in the process to first order in the perturbations and leading order in the post-Minkowskian expansion. The $I_1$ predictions are novel, and we find that our results for $G_3$ are related to the known quadrupole corrections arising from tidal perturbations, although the physical origin of the $G_3$ coupling is unrelated to the finite-size effects underlying tidal interactions. We show this by recomputing such tidal corrections and by presenting an explicit field redefinition. In the post-Newtonian expansion our results are complete at leading order, which for the gravitational-wave flux is 5PN for $G_3$ and tidal interactions, and 6PN for $I_1$. Finally, we compute the corresponding modifications to the waveforms.

An effective field theory (EFT) framework for gravity was advocated in [51], and is ideally suited to study systematically higher-derivative corrections to the EH theory. In [52], this approach was followed to compute the corrections to the gravitational potential between compact objects and their effective mass and current quadrupoles due to perturbations quartic in the Riemann tensor, and the corresponding modifications to the waveforms were then analysed in [53]. Modifications to the gravitational potential due to cubic interactions in the Riemann tensor were computed in [54,55] using amplitude techniques, and the deflection angle and time delay/advance of massless particles of spin 0, 1 and 2 were derived in [56] for cubic and quartic perturbations in the Riemann tensor as well as for interactions of the type F F R [56]. Terms that are quadratic in the Riemann tensor do not contribute to the classical scattering of particles in four dimensions [57]. In this paper we wish to describe dissipative effects in the dynamics of binaries, that is gravitational-wave radiation, from appropriate five-point amplitudes with four massive scalars and one radiation graviton. We perform this study in the presence of cubic modifications to the EH action and tidal effects. Interestingly, we will see that there is an overlap between 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 is very different -for instance, I 1 and G 3 appear in the low-effective action of bosonic strings, or can be induced by integrating out massive matter [60,61].
In the presence of scalars and restricting our focus to parity-even interactions, there are two independent cubic terms: A more natural combination is in fact G 3 := I 1 − 2 I 2 , which, as is well known, is topological in six dimensions [62] and has vanishing graviton amplitudes. In [63], it was argued from studying the scattering of polarised gravitons that I 1 potentially leads to superluminal effects/causality violation in the propagation of gravitons for impact parameter b α 1 4 . Here α ∼ Λ −4 is the coupling constant of the I 1 interaction, and Λ is the cutoff of the theory. In that paper, α was chosen to be much larger than Planck . This allows to treat the gravitational scattering in a semiclassical set-up, where predictions can be trusted up to M Planck (> Λ). This question was reinvestigated in an EFT framework in [56], where it was found that the I 1 interaction leads to a time advance in the propagation of gravitons (but not photons and scalars) when b α 1 4 . Finally, G 3 does not lead to any time advance/delay for massless particles [56], while still correcting the gravitational potential [54,55]. An identical conclusion for the propagation of massless particles in the background of a black hole was reached in [64], both for the I 1 and G 3 interactions 2 .
In this respect, an important observation was made in [65], namely that such superluminality effects (and those observed earlier on in [66][67][68]) are unresolvable within the regime of validity of the EFT, and do not lead to violations of causality. In our set-up such violations would indeed occur at b Λ −1 , which is at the boundary of the regime of validity of our EFT, while the processes we are interested in only probe the regime where the EFT is valid. Above Λ, the only 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 effects due to the massive modes, required to ensure causality, cannot be resolved with current-day experiments. We also note that, assuming that these interactions can contribute to any classical gravitational scattering (Λ < M Planck ), then we have α > G 2 , independently of precise estimates of the cutoff Λ.
In the following we work in an effective theory containing cubic and tidal perturbations, and compute a five-point amplitude with four massive scalars (representing the black holes) and one radiated soft graviton. From this, one can in principle extract all radiative multipole moments to this order, but for the sake of our applications we will only focus on the quadrupole moment induced by the cubic and tidal interactions, from which we then derive the corresponding changes to the power radiated by gravitational waves and to the waveforms. Our results for the quadrupole correction are exact to leading order in the perturbations and in the post-Minkowskian expansion. We also take the post-Newtonian expansion of our results, which are complete at 5PN order for the G 3 and tidal interaction corrections, and at 6PN order for the I 1 corrections. We find that the corrections due to G 3 have the same form as those generated by a particular type of tidal interaction (although the corresponding coefficients in the EFT action are independent). We also explain this result by constructing an explicit field redefinition relating the two couplings. For the PN-expanded result of the tidal corrections to the mass quadrupole we find agreement with [69][70][71]. The remaining tasks consist in using the corrected quadrupole moment to compute the modifications compared to EH gravity to the power emitted by the radiated gravitational waves, and the corresponding corrections to the waveforms in the Stationary Phase Approximation (SPA) 3 . Here we follow closely [53], and also present a comparison with their result obtained with perturbations that are quartic in the Riemann tensor.
The rest of the paper is organised as follows. In Section 2 we introduce the EFT we are discussing, reviewing some of the relevant results, including the corrections to the gravitational potential from cubic [54,55] and tidal interactions [74][75][76]. Furthermore, we point out the vanishing of all graviton amplitudes in the pure gravity plus G 3 theory, and explicitly construct a field redefinition that maps G 3 into a tidal perturbation. Section 3 contains the calculation of the relevant four-scalar, one soft graviton amplitude in our EFT, from which we extract the perturbations to the quadrupole moment. In Section 4 we compute the power radiated by the gravitational waves, and finally in Section 5 the corrections to the waveforms in the SPA. In an Appendix we present some details on the modifications to the circular orbits due to the perturbations.

The EFT action
We consider an EFT describing EH gravity with higher-derivative couplings interacting with two massive scalars. These model spinless heavy objects, and we also include the leading tidal interactions in our description which describe finite size effects of the heavy objects. Specifically, the EFT action we consider is is the effective action for gravity, with I 1 and G 3 are the parity-even cubic couplings defined as with The dots in (2.2) stand for higher-derivative interactions that we will not consider here. The two scalars, with masses m 1 and m 2 , couple to gravity with an action and in addition we include higher-derivative couplings describing tidal effects of extended heavy objects, These tidal interactions were recently studied in [76], and the dots stand for the (Hilbert) series of 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.

Cubic interactions
The I 1 and G 3 interactions naturally arise in the low-energy effective description of bosonic string theory, whose terms cubic in the curvature can be obtained by making the replacement where Φ is the dilaton. These interactions are also produced in the process of integrating out 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 the sake of the computation of the power radiated by the gravitational waves performed in later sections 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 result one obtains where the dots indicate higher PN corrections which we do not consider here. Note that the terms proportional to α 1 and α 2 are the result of a one-loop computation. In the PN expansion, the term proportional to α 1 (from the I 1 interaction) is suppressed by a factor of p 2 /m 2 1,2 compared to the dominant correction proportional to α 2 (from G 3 ).

Amplitudes from the G 3 interaction
It is well known that, unlike I 1 , the G 3 interaction has a vanishing three-graviton amplitude and does not contribute to graviton scattering up to four particles [62,80] -and in fact to any number of gravitons. This can be understood by the fact that G 3 is topological in six dimensions [62], and therefore computing tree-level four-dimensional graviton amplitudes from dimensionally reducing the six-dimensional ones automatically gives zero. Combining this observation with unitarity techniques leads to for any n. Hence the G 3 interaction does not affect the perturbative dynamics in theories of pure gravity. However, if we consider a theory of gravity with matter, e.g. massive scalars mimicking black holes or neutron stars, the presence of a G 3 coupling alters their dynamics. In particular the four-point amplitude with two gravitons and two scalars becomes [54,55] M (0) The non-trivial contribution to the scattering amplitude of two massive scalars and two gravitons from the G 3 interactions modifies the classical potential in the two-body system, as shown in [54,55]. As we will show below, both G 3 and I 1 produce corrections to the quadrupole moment already at tree level. Specifically we find that the G 3 quadrupole correction is dominant in the post-Newtonian (PN) expansion, which parallels the results found for the corresponding corrections to the gravitational potential quoted earlier in (2.9).

The G 3 interaction as a tidal effect
It is easy to show that the contact term proportional to [34] 4 (2m 2 + s) in the amplitude (2.11) is (up to a numerical coefficient) the amplitude arising from a particular tidal interactions of This suggests that there should exist a fourdimensional field redefinition mapping the G 3 interaction into a tidal effect, as already noticed in [58,59] 4 . In this section we construct this field redefinition explicitly.
We begin by rewriting G 3 in a more convenient form, making use of two identities in four dimensions [83]: and which, in combination with (2.13), leads to (2.15) 4 We also observe that black holes in four dimensions have non-vanishing Love numbers when higher-derivative interactions are considered [81,82].
The latter identity implies that, in four dimensions, G 3 can be rewritten as 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 which only contributes to quantum corrections to the quadrupole moment. Thus where in the last line we have used the field redefinition Finally, integrating by parts and discarding boundary contributions, we can rewrite the new interaction term in (2.17) as 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 20) thereby explicitly showing that the G 3 interaction can be absorbed into the first of the two tidal interactions in (2.7).

Tidal effects
During the inspiral phase of binary systems involving at least one extended heavy object like a neutron star, corrections due to the finite size of the object(s) increase as the distance between the objects decreases. These effects can be included systematically using a tidal expansion, i.e. a multipole expansion dominated by the mass quadrupole moment. Finite-size effects are bound to 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 compact objects. The computation of tidal effects has been addressed in the past by a wide variety of methods, recently including complete PM results [74][75][76] for the conservative dynamics.
In order to compute the modifications to the waveform coming from the tidal interactions in (2.7) we need to expand the 2PM potential in the conservative Hamiltonian computed in [74][75][76] up to O( p 2 ), with the result where the dots indicate higher PN terms.

Quadrupole moments in EFTs of gravity
In the PN framework, the conservative and dissipative dynamics of two objects of mass m 1 and m 2 , coupled to the gravity effective action (2.2) is described by the following point-particle effective action [26,52]: is the reduced mass, and r(t) is the relative position of the two objects. V r, p denotes the potential, whose explicit expression to first order in α 1 , α 2 [54,55], and λ 1,2 , η 1,2 [74][75][76] is obtained by summing (2.9) and (2.21), and Q ij r, p is the quadrupole moment, to be computed below. The dots represent higher-order terms that will be irrelevant in our analysis. This action can be trusted in the inspiral phase before the objects reach relativistic velocities.
We now present the computation of the five-point amplitude φ 1 φ 2 → φ 1 φ 2 + h(k) with four scalars and one radiated soft graviton h(k). Its momentum k µ is on shell, while the momentum of the graviton exchanged between the two objects is purely spacelike (corresponding to an instantaneous interaction), and in our set-up is given by q µ = −p µ 1 − p µ 2 = (0, q). Furthermore, the energy of the radiated graviton is such that k 0 | q |, so that k µ can be ignored for practical purposes, and the radiated graviton enters the amplitude only through its associated Riemann curvature tensor R αβµν . Finally, because we are only interested in classical contributions (i.e. O( 0 )), we keep only the leading terms in q 2 .
In the following we first compute fully relativistic scattering amplitudes and then perform the PN expansion to extract the correction to the quadrupole term in the effective action (3.1). In the centre-of-mass frame, the momenta of the particles can be parametrised as with p 2 1 = p 2 2 = m 2 1 , p 2 3 = p 2 4 = m 2 2 . Furthermore, we have where p · q = 0 because of momentum conservation. In our all-outgoing convention for the Figure 1: The single diagram contributing to the radiation process with an insertion of the operators O = I 1 , I 2 . All momenta are treated as outgoing and the radiated graviton is taken to be soft. external lines, the four-momenta p 1 and p 4 correspond to the incoming particles, and hence their energies are negative.

The amplitude with cubic interactions
Our next task is to compute the five-point amplitude A O shown in Figure 1, with O = I 1 , I 2 (which we can then combine to obtain A G 3 ). We first obtain its relativistic expression, factoring out a single Riemann tensor associated with the radiated graviton, and then split the Lorentz indices into time and spatial components and isolate the terms contracted into R 0i0j . Upon Fourier transforming to position space, these components will allow to directly read off Q ij by matching to the Hamiltonian density associated to the point particle effective action (3.1). The classical relativistic results are, for I 1 : while for I 2 : Note that the result for the G 3 interaction introduced in (2.3) can be obtained as The terms in the amplitude contributing to the quadrupole radiation are then (3.8) and where we have used that E 3 = E 4 in order to write the result as a function of the energies and momenta of the incoming particles p 1 and p 4 . The dots stand for additional terms proportional to R 0ijk and R ijkl , which can also be extracted from our result.
O q Figure 2: The two diagrams contributing to the gravitational radiation, where O denotes any of the two tidal interactions in (2.7). An overall Riemann tensor of the radiated graviton is factored out, so that

The amplitude with tidal effects
A calculation similar to the one outlined in the previous section leads to the fully relativistic result which, upon expanding in the spatial and time components, reads where the ellipses stand once again for terms proportional to R 0ijk and R ijkl which we will not need in the remainder of this paper.

The quadrupole corrections
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) 5 . To begin with, we perform the relevant Fourier transforms using (3.12) 5 For further details on the procedure see for example [26,52].
Taking into account the non-relativistic normalisation factor of −i/4E 1 E 4 , we arrive at the quadrupole-like terms where C O are coefficients depending on the energies and masses as well as p 2 of the heavy particles, with (3.14) Comparing (3.13) with the Hamiltonian density obtained from the action (3.1), we conclude that the modifications to the quadrupole moment arising from the cubic and tidal couplings are given by where we have introduced the leading-order quadrupole moment in the EH theory for a binary system with masses m 1 and m 2 , with µ being the reduced mass defined in (3.2). Combining the various correction terms, we arrive at It is interesting to write the three coefficients C I 1 , C I 2 and C tidal in the PN expansion. Keeping terms up to first order in p 2 one has where 19) and, as usual, κ 2 := 32π G. For convenience we also quote the contribution due to the G 3 interaction alone -this is given by (3.20)

Power radiated by the gravitational waves
We can now compute the power radiated by the gravitational waves in the approximation of circular orbits. In the EH theory, the radius of the circular orbit is given by the well-known formula In the presence of the cubic and tidal interactions, this quantity gets modified as where g i stands for any of the coupling constants of the cubic and tidal perturbations. We also introduced the symmetric mass ratio ν defined as (4.5) Finally, Ω denotes the angular velocity on the circular orbit, and the value δr has been computed using (4.2) and (A.5), where the potentials entering (A.5) are given in (2.9) and (2.21). The total energy per unit mass M of the system, to first order in the couplings, is then given by (4.6) The above formula is complete at leading order in all of the perturbations (that is O(v 12 ) and at O(v 14 ) for the α 1 correction only. The remaining O(v 14 ) terms have been obtained from a small-velocity expansion of our 2PM result, and in order to get a complete result at that PN order one would need to include also the 3PM corrections to the potential generated by cubic and tidal interactions 6 . We have also compared the contribution to the energy from the η 1,2 corrections to [71], finding agreement (after mapping their coefficients µ (2) A to ours) 7 . Next, we compute the leading-order gravitational-wave flux using the quadrupole formula using the result of our computation for Q ij in (3.17). To first order in the couplings α 1 and α 2 the flux becomes ... Q ij N is evaluated on the radius r • of the circular orbit in the presence of the cubic and tidal interactions, as given in (4.2). Furthermore, the quantity p 2 := p 2 r + p 2 φ /r 2 can be obtained using the fact that p r = 0 on the circular orbit while p φ := l is a constant, which can be determined from Hamilton's equations, with the result where r • is given in (4.2) and U (r) is the part of the potential proportional to p 2 , following the conventions of Appendix A. Using these relations, p 2 is re-expressed as a function of Ω, the masses, and the couplings.

Waveforms in EFT of gravity
Following [53] we can also compute the correction induced by the cubic and tidal interactions to the gravitational phase in the saddle point approximation. In this approach, the waveform in the frequency domain is written as 8 Here φ(t) is the orbital phase, whileφ(t) = πF (t) defines the instantaneous frequency F (t) of the gravitational wave. t f is defined as the time wherė implying that F (t f ) = 2f . In the adiabatic approximation, the work of [73,72] provides explicit formulae for ψ SPA (t f ) and t f : , and E(v) and F(v) were computed 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 EH SPA (t f ) + ψ I 1 +I 2 SPA (t f ) + ψ tidal SPA (t f ) . are the new contributions due to cubic and tidal perturbations. Similarly to our comment after (4.6), we note that all the terms at leading order in velocity in (5.8) are complete, while the remaining ones would also receive further modifications from a 3PM computation of the potential and a 2PM computation of the quadrupole.
Finally, it is interesting to compare our results with those of [53]. The perturbations considered in that paper have the form where C := R µνρσ R µνρσ , C := 1 2 R µναβ αβ γδ R γδµν . (5.10) The modifications to ψ SPA (t f ) due to quartic interactions as found in [53] are (reinstating powers of G in the result of that paper, and converting their d Λ into our β 1 as defined in (5.9)), Note the different dependence on v f in the correction terms in (5.8) and (5.11), which are of O(v 10 f ) and O(v 16 f ) in the leading cubic and tidal, and quartic cases, respectively. Finally, it will be interesting to perform a comparison of our result in (5.6) to experimental data, as performed in [53] for the case of quartic perturbations in the Riemann tensor.
where r • is the radius of the circular orbit. We will also set Ω :=φ(r = r • ), or Ω := l µr 2 • 1 + 2µU (r • ) . (A.4) Using this to eliminate l in favour of Ω, we finally get This equation determines r • as a function of Ω. In the absence of a perturbation, we have where r N is the radius of the circular orbit in the EH theory, given in (4.1).