Gravitational Rainbows: LIGO and Dark Energy at its Cutoff

Claudia de Rham1, 2, ∗ and Scott Melville1, †

1Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2AZ, U.K.2CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, USA

(Dated: December 18, 2018)

The recent direct detection of gravitational waves from a neutron star merger with optical counter-part has been used to severely constrain models of dark energy that typically predict a modificationof the gravitational wave speed. However, the energy scales observed at LIGO, and the particularfrequency of the neutron star event, lie very close to the strong coupling scale or cutoff associatedwith many dark energy models. While it is true that at very low energies one expects gravitationalwaves to travel at a speed different than light in these models, the same is no longer necessarilytrue as one reaches energy scales close to the cutoff. We show explicitly how this occurs in a simplemodel with a known partial UV completion. Within the context of Horndeski, we show how theoperators that naturally lie at the cutoff scale can affect the speed of propagation of gravitationalwaves and bring it back to unity at LIGO scales. We discuss how further missions including LISAand PTAs could play an essential role in testing such models.

Dark Energy after GW170817 and GRB170817A:The recent direct detections of gravitational waves(GWs) have had an unprecedented impact on our un-derstanding of gravity at a fundamental level. The firstevent alone (GW150914 [1]) was already sufficient to putbounds on the graviton with better precision than whatwe know of the photon. Last year, the first detectionof GWs from a neutron star merger (GW170817), some1015 light seconds away, which arrived within one secondof an optical counterpart (GRB170817A), allowed us toconstrain the GW speed with remarkable precision [2–4]

−3× 10−15 ≤ cTcγ− 1 ≤ 7× 10−16 , (1)

with cT the GW phase velocity and cγ the speed of light.

Such a constraint has had far-reaching consequencesfor models of dark energy. Within the context of theEffective Field Theory (EFT) for dark energy [5], it wasrapidly pointed out that (1) was sufficient to suppress theEFT operators that predict non-luminal gravitationalpropagation[6–14]. In particular, within the frameworkof scalar-tensor theories of gravity, Horndeski [15] hasplayed a major part in the past decade as a consistentghost-free EFT in which the scalar degree of freedomcould play the role of dark energy. Yet the interplaybetween the scalar and gravity typically implies thatGWs would not travel luminally. The LIGO constrainton the GW speed only leaves out the generalization ofthe cubic Galileon [16], which is severely constrained byother observations. As a result the Horndeski EFT seemsalmost entirely ruled out as a dark energy candidate [17].

Nevertheless, it should be noted that the recent LIGObound applies to GWs at a frequency of 10 − 100Hz,while the EFT for dark energy is “constructed” asan effective field theory for describing cosmology onscales 20 orders of magnitude smaller. When it comesto constraining such EFT parameters, it is therefore

10-20

10-15

10-10

10-5

100

105

1/3

2/3

1

k / Hz

cS2

H0 Hrec LISA LIGOPTA10

-2

10-1

100

101

α

FIG. 1. Sound speed for δφ fluctuations in the model (6).While subluminal at k � M , luminality is recovered abovethe cutoff (here, M = 10−3Λ). The EFT can safely describecosmology from today H0 to before recombination Hrec, butmay receive order one corrections in the LIGO band.

important to recall that they could in principle dependon scale: generically, the GW speed may depend on thefrequency at which it is measured, cT = cT (k). TheLIGO bound (1) should therefore be read as a constrainton cT (k) at frequencies on the order of k ∼ 10− 100Hz,and from their very construction we expect EFTs suchas Horndeski to break down at a cutoff ∼ 100Hz if notmuch lower. If the theory is to ever admit a Lorentz-invariant (LI) high energy (UV) completion, then thefront velocity [18] must be luminal which implies thatthe sound speed cT (k) will necessarily asymptoteto exactly luminal at high frequencies. While theEFT of dark energy may predict a GW sound speedthat departs from unity at low energy, it is nonethelessnatural to expect a speed arbitrarily close to luminalat higher frequencies. In the case of Horndeski, thescale of the cosmological background generally requiresthat new physics ought to enter at (or parametricallybefore) the energy scales observed at LIGO, where it

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would be natural to observe a luminal velocity. We shallpresent how this would naturally occur in a simple scalarfield model (Fig. 1) before turning to the full-fledgedscalar-tensor theory and discussing the implications ofLI–UV completions to Horndeski.

Scalar EFT: We start by looking at a simple yet repre-sentative scalar EFT example [19]

LM = −1

2(∂φ)2 +

1

2Λ4(∂φ)4 +O

((∂2φ)2

M2

), (2)

where M is the scale of new physics. φ could be aplaceholder for dark energy—for instance, let us set〈φ〉 = αΛ2t as the background and consider fluctuationsφ = 〈φ〉 + δφ. As is well known, on this spontaneouslyLorentz-breaking (sLB) background the sound speed forδφ is

c2S = 1−∆0 = 1− 4α2

1 + 6α2, (3)

leading to an order one deviation from luminality if theparameter α ∼ O(1). At this stage, we may wonder if wecan trust a background configuration close to the strongcoupling scale Λ. This question has been the subjectof extensive work and we refer the reader to [20] forcareful considerations. Here, we take the approach thatthe EFT can be re-organized as a derivative expansionin which, while the field gradient may be “large”, higherderivatives of the field are suppressed. This means thata profile with φ̇ ∼ Λ2 may be considered without goingbeyond the regime of validity of the EFT so long ashigher derivatives are suppressed: ∂nφ�Mn+1 . Λn+1

for any n ≥ 2. Concretely, this implies that backgroundconfigurations with α ∼ O(1) do not necessarily lead toorder one contributions from other irrelevant operators.We follow this approach here as it is the one used in thecontext of Horndeski models of dark energy.

EFT Cutoff: The model (2) predicts a speed of sound(3) which appears to be the same irrespective of the fre-quency of the δφ fluctuations. Yet if we consider δφ-waves at sufficiently high frequencies, they should be in-sensitive to the sLB background. LI should be restored[21] and hence high-frequency δφ waves should be exactlyluminal. The reason this is not manifest in (3) is becausewe are working within the EFT (2), which is only con-sistent at frequencies much smaller than the cutoff, M .Interestingly, in the context of the GW170817 detection,the frequency of the GWs span from 24Hz to a few hun-dred Hz, which is perilously close to the strong couplingscale associated with many Horndeski dark energy mod-els [22],

M . ΛHorndeski ∼ (MPlH20 )1/3 ∼ 260Hz (4)

where H0 is the Hubble parameter today. At thosescales, the EFT (2) can no longer be the appropriate

description for the δφ-waves, as we have neglectedoperators of the form (∂2φ)2/M2, where M is the cutoff[23]. The existence of such higher derivative operatorscannot be ignored—they are mandated by positivitybounds if this theory is to admit a sensible WilsonianUV completion [24, 25].

Sound Speed near the Cutoff: The low-energy EFT(2) is appropriate when considering δφ-waves at frequen-cies k/M � 1, however at higher frequencies one shouldinclude the irrelevant operators that naturally enter theEFT at the scale M and modify the dispersion relation,

c2S(k) = 1−∆0 + ∆2k2

M2+O

(k4

M4

), (5)

where the running ∆2 is controlled by the higher orderoperators. This scale-dependence of the sound speedis unavoidable: not only are the next-to-leading orderoperators required in order to properly renormalizedivergences within the EFT, they also naturally arisefrom a generic UV completion. Of course when reachingthe scale M , we lose control of the EFT and the precisedetails of the UV completion are essential in determiningthe sound speed of δφ-waves (even if—as we haveargued—the background configuration itself may not besensitive to the UV completion).

To give a precise example of how UV physics [26] mayaffect the sound speed at frequencies close to M , considerthe following specific situation where the massless scalarφ couples to a heavy scalar χ via,

LΛ∗ = − 12 (∂φ)2 − 1

2 (∂χ)2 − 12M

2χ2 +χ

Λ∗(∂φ)2 , (6)

where χ becomes dynamical around M and strongly cou-pled at a scale Λ∗. For (6) to represent a (partial) comple-tion of (2) with an extended region of validity, we requirethe scale hierarchy Λ∗ �M implying

M � Λ = (MΛ∗)1/2 � Λ∗ . (7)

Even though (2) only becomes strongly coupled at thescale Λ, its cutoff is in fact even smaller M � Λ (see[27], this hierarchy also appears in the case of Galileons[25] and massive gravity [28, 29], [30]). Integrating out χat tree level gives the EFT (2) with additional irrelevantoperators

L = −1

2(∂φ)2 +

1

2Λ4(∂φ)2 M2

M2 −�(∂φ)2 . (8)

Including these irrelevant operators, we find a dispersionrelation

ω2 = k2 − 4α2

1 + 2α2

ω2M2

M2 − ω2 + k2, (9)

which matches the leading order EFT sound speed (3)at sufficiently small frequencies k � M but leads to

3

luminality at higher frequencies ω2 = k2 (1 + · · ·), wherethe ellipses vanish at high energy and their precise formdepends on the details of the completion. The exactbehavior of the sound speed as a function of frequencyfor various values of α is depicted in Fig. 1. Since theconsistency of the two-field model requires the hierarchyM � Λ, for concreteness we can imagine an examplewhere M = 10−3Λ, so that the partial UV completion (6)remains a valid description up to the scale Λ∗ = 103Λ.In that case if we were to draw an analogy with thefrequencies observed at LIGO (i.e. starting at about24Hz), and considering the scale Λ to be given by about260Hz as in eqn. (4), then kLIGO > 10−1Λ ∼ 102M ,and we clearly see from Fig. 1 that at sufficiently highfrequencies, we expect the sound speed to be arbitrarilyclose to luminal, despite the low-frequency sound speedbeing potentially significantly subluminal. It is worthnoting that these scales should be taken with a grain ofsalt—they are merely provided to illustrate the point inthis simple scalar field model and the precise way thesound speed returns to being luminal at high energydepends on the details of the (partial) UV completion.The purpose of this toy model is simply to illustrate thatthe measured GW speed at LIGO frequencies may besignificantly different than the cosmologically relevantc2T—in practice, the precise numerical running andhierarchy between these two speeds will be determinedby whatever physics UV completes Horndeski.

Horndeski EFT: We now turn to Horndeski as a darkenergy EFT. As is well-known, the scalar field present inHorndeski can play the role of a dark energy fluid drivingthe late-time acceleration of the Universe. In doing so,the Universe is filled with a medium (the dark energycondensate) which in turn affects the GW speed. Forillustration purposes, consider the parts of the Horndeskidark-energy model which affects the sound speed [31],

LH =M2

Pl

2R− 1

2Gab∂aφ∂bφ , (10)

Gab = gab + c2MPl

Λ3Gab + c3

MPl

Λ6Laµbν∇µ∇νφ , (11)

Gab being the Einstein tensor and Laµbν the dual Rie-mann tensor, and we have defined the scale Λ as(H2

0MPl)1/3 as given in (4). The solution 〈φ〉 = αMPlH0t

leads to an accelerated expansion with Hubble parameterH = βH0, where the coefficients α and β are determinedin terms of c2 and c3 and are order one when c2,3 are orderone. There is a region in parameter space where the accel-erated solutions are stable (no ghost nor gradient insta-bilities). In order to exhibit the scales involved, it is use-ful to normalize metric fluctuations gµν = γµν+hµν/MPl,so that the c2,3 terms enter at the scale Λ,

LH ⊃ (∂h)2 + (∂δφ)2 + (∂δφ)(∂δh) +1

Λ3∂2h(∂δφ)2.(12)

At first sight the c2,3 terms in (10) would also seem togenerate operators at a much lower scale, for instance〈φ̇〉∂2h∂δφ/Λ3 ∼ ∂2h∂δφ/H0, however all those opera-tors are total derivatives.

At low frequencies with respect to the cutoff M of theHorndeski EFT, tensor modes have a subluminal speed,

c2T (k) = 1− 2c2α2β2 + 6c3α

3β3

2 + c2α2β2 + 6c3α3β3+O

(k2

M2

), (13)

where M is at most the strong coupling scale of the EFT[32], but it may be lower, M . Λ [33].

As was the case for the scalar field theory (2), theexistence of a UV completion mandates the existence ofother irrelevant operators in addition to the Horndeskiones. Precisely which operators would enter depends onthe UV completion and within an EFT approach oneshould allow for all operators to be present. However,for concreteness, we present here a class of operators thatwould typically enter the Horndeski EFT at a scale M .Λ,

L(n)higher−der =

(M2

PlGµν) �n

M2n+4n

∂µφ∂νφ , (14)

with n ≥ 2 and appropriate scales Mn, which we nowstudy. First, notice that such operators affect the back-ground solutions by an amount proportional to

E(n)

EH∼ H

2(n−1)0 Λ6

M2n+4n

, (15)

where symbolically E(n) is the contribution from L(n)

to the background equations of motion and EH thatfrom the Horndeski Lagrangian (10). Trusting the back-ground provided by the Horndeski EFT (10) requiresthis ratio to be small. So in principle the scale ofthe higher derivative operator L(n) could be as smallas say M2n+4

n ∼ H2n−40 Λ8 ≪ Λ2n+4 and these opera-

tors would still not significantly affect the background.Furthermore, on this background the higher derivativeterms (14) lead to operators that scale at worst as(∂n+1h)2∂δφ/

(H0M

2n+4n Λ−3

), (for n ≥ 2), so if those

were at all representative of the types of operators wewould expect from the UV completion, it would meanthat the Horndeski EFT (10) can be trusted until thestrong coupling scale Λ∗,

Λ∗ = minn(M2n+4n H0Λ−3

)1/(2n+2). (16)

It will depend on the precise UV completion whetherall the Mn are the same order (maybe all set to Λ ora lower scale M) or whether they scale so that Λ∗ >Λ. For now we simply point out that we have a greatdeal of flexibility in the scales Mn which do not alter the

4

background evolution, yet do affect the GW speed. Forinstance

L = LH +∑n≥2

cnL(n)higher−der (17)

modifies the GW dispersion relation (symbolically) as,

ω2 ∼ c2T (0)k2 +O(H20 ) (18)

+∑n≥2

cnΛ6

3M2n+4n

(−ω2 + k2

)n−1(ω2 +O(k2, H2

0 )) ,

where at frequencies close to Mn the(−ω2 + k2

)n−1

terms push the GW speed arbitrarily close to unity. Therate at which the low energy sound speed asymptotes toluminal depends on the scales Mn, and is thus rathersensitive to details of the underlying UV completion. Ifone imagines a running in the form of a power law, 1/k2,then one requires that the cutoff of the theory is someorders of magnitude below the LIGO band if one is toaccommodate cs � 1 at low energies, but in principlethe rate could be exponential or arbitrarily fast withoutaffecting the low energy EFT.

Conspiracy vs Lorentz-invariant UV completion:The fact that the Horndeski cutoff is close to theLIGO band (and particularly the GW170817 event) wasnoticed in [8], who pointed out that from a bottom-upapproach it would seem unlikely that order one effectsentering at the cutoff would conspire to precisely cancelc2T − 1 within an accuracy of one part in the 1015.However from a top-down approach, it is very unlikelythat the UV completion knows anything about thespecial structure of the sLB background. Quite theopposite, we expect that at sufficiently large energiesmodes should be insensitive to the sLB cosmologicalsolution and we would naturally expect a return toluminality. Indeed the operators presented in (14) (and(6)) have in no way been tuned so as to precisely cancelc2T − 1. Rather the operators simply satisfy LI and atsufficiently high energy that symmetry is restored. It isimportant to note that for the GW speed to be unity atLIGO frequencies, the EFT must breakdown at scaleslower that Λ.

Modified Gravity: One motivation for studyingHorndeski is that these scalar-tensor theories can mimicthe behaviour of some modified gravity models [34]: forinstance the decoupling limit of DGP [35], cascadinggravity [36] and massive gravity [37]. Since some Horn-deski EFTs arise from the decoupling limit of varioustheories of modified gravity, it is clear that Horndeskican be seen as an EFT with an infrared cutoff (of theorder of the Hubble parameter today), as well as a UVcutoff and we could take the perspective that thesemodels of modified gravity are in fact what (partially)“completes” those Horndeski theories. Interestingly in

all these models of modified gravity, while the dispersionrelation is modified at very low frequencies (of the orderof the effective graviton mass), the sound speed remainsluminal independently of the background configuration.This suggests that Horndeski EFTs could very easilybe implemented within some completion for which theGW speed at LIGO frequencies is luminal to impeccableprecision. All such EFTs may remain viable in the wakeof GW170817.

Gravitational Rainbows: Throughout this work, wehave raised the possibility that the frequencies observedat LIGO are at the edge of (or even beyond) the regimeof validity of the Horndeski EFT and shown how thespeed of GWs could be close to unity at those scales eventhough the low-energy EFT may predict a subluminalpropagation. By no means do we suggest that everytime an observation is performed, one should simplyshield the EFT from constraints by invoking a lowercutoff. However, within the context of Horndeski andcurrent LIGO observations, the frequencies observedare dangerously close to the cutoff if the EFT is todescribe dark energy and in a standard EFT approachnew physics is required to enter at or below that scale.

Turning towards future surveys, the upcoming LISAmission will have peak sensitivity near 10−3 Hz, at whichscale k/Λ ∼ 10−5. If LISA were to bound the speed ofGWs [38] with a similar precision as LIGO but at suchlow frequencies, it would be very hard for a HorndeskiEFT to remain viable as a model of dark energy andstill have an interesting regime of predictability. Suchobservations would be complementary to those fromfuture ground-based interferometers like the EinsteinTelescope [39] that may help distinguish between variousdark energy models [40, 41].

Interestingly, in the case where M is not muchsmaller than Λ, the running of c2T (k) induced by EFTcorrections may be sufficiently large to rule out thesemodels without the need for an optical counterpart.The modification to the dispersion relation within theLIGO window would be dramatic, unless the transitionbetween the low-energy and high-energy values of c2T (k)happens extremely fast. If not, then for the exampleprovided for Horndeski, it would require the higherderivative operators to enter at a scale at least 9 ordersof magnitude below the observed scale so that we havecompletely transitioned between the low energy andhigh energy speed before LIGO starts taking data.

Outlook for the EFT of Dark Energy: In one ofits simplest formulations [42], the EFT of dark energyhas only four free functions of time [43]. One of thosefree functions (m4) is directly related to the GW speed.While recent observations have been very successful at

5

reducing the large parameter space, through this work westress that those quantities are typically scale-dependent(in addition to their time dependence) and the currentconstraints (m4(kLIGO) ≈ 0) may not necessarily implym4(k ∼ H0 . 10−20kLIGO) = 0.

In particular, we have focused on a picture wherenew physics enters the low-energy EFT at a scale belowΛ = 260Hz so as to restore perturbative unitarity.We should stress that even if the UV completionwere to be manifestly Lorentz-violating, one wouldnot expect the scale of Lorentz breaking at high en-ergy to be linked to the scale of sLB at low energy andthus we would still expect a running of the speed of GWs.

We emphasize that the aim of this work is not to re-vive Horndeski or any specific EFT as a particular modelfor dark energy. Rather the aim is to bring across thesubtleties related with measurements such as the soundspeed when dealing with EFTs, especially when the effec-tive cutoff may be relatively low and comparable to thescale associated with the measurement. In the comingage of precision cosmology, correctly interpreting whatEFT corrections mean for these measurements will bemore important than ever before and crucial for discrim-inating between different classes of models.

Acknowledgments: We thank Paolo Creminelli,Lavinia Heisenberg, Atsushi Naruko, Andrew Tolley, Fil-ippo Vernizzi and Toby Wiseman for useful discussions.CdR would like to thank the Graduate Program onPhysics for the Universe at Tohoku University for its hos-pitality during the latest stages of this work. The workof CdR is supported by an STFC grant ST/P000762/1.CdR thanks the Royal Society for support at ICL througha Wolfson Research Merit Award. CdR is also sup-ported in part by the European Union’s Horizon 2020 Re-search Council grant 724659 MassiveCosmo ERC-2016-COG and in part by a Simons Foundation award ID555326 under the Simons Foundation’s Origins of theUniverse initiative, ‘Cosmology Beyond Einstein’s The-ory’. SM is funded by the Imperial College President’sFellowship.

∗ c.de-rham@imperial.ac.uk† s.melville16@imperial.ac.uk

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JCAP 1010, 026 (2010), arXiv:1008.0048 [hep-th].[17] The impact of (1) is not limited to scalar-tensor the-

ories of gravity—other models of dark energy, such asvector-tensor gravity or scalar-vector-tensor gravity, havealso seen their parameter space remarkably affected (seehowever [13, 14, 44] for models that survive the bound).Lorentz-violating theories have also been profoundly con-strained [45–47], although we shall focus on theorieswhich are fundamentally LI here.

[18] Strictly speaking, the front velocity is defined as thespeed of the front of a disturbance [48, 49]. In practice,the front velocity is defined as the high frequency limitof the phase velocity [48–50],

cfront = limk→∞

cT (k) . (19)

[19] The argument goes through essentially unaffected ifinstead of (2) we had chosen an arbitrary functionΛ4P ((∂φ)2/Λ4). We chose the particular form of (2) forconcreteness.

[20] C. de Rham and R. H. Ribeiro, JCAP 1411, 016 (2014),arXiv:1405.5213 [hep-th].

[21] Even if the UV completion was not LI, it would be sur-prising that high energy physics knows about the scaleof the sLB background.

[22] In particular, this is the largest strong coupling scalewhich would allow the second derivative operators, ∂2φ,to have an order unity effect on cosmological perturba-

tions. For instance, (∂φ)2 �φΛ3 ∼ (∂δφ)2MPH

20

Λ3 remember-

ing that φ̇2 ∼M2PH

20 if an approximately shift symmetric

(φ ∝ t) scalar is to drive the late time expansion of theuniverse.

[23] Note that the existence of higher derivative operators inthis EFT should not be confused with the existence ofan Ostrogradsky ghost. Indeed, higher derivative opera-tors naturally enter from integrating out heavy degrees offreedom, and just manifest the fact that the EFT breaksdown at the cutoff scale.

[24] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis,and R. Rattazzi, JHEP 0610, 014 (2006), arXiv:hep-th/0602178 [hep-th].

6

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[26] P (X) theories such as (2) can be seen as the low en-ergy EFT of a high energy U(1) theory, broken to noth-ing as one integrates over the (massive) radial compo-nent. In that case the completion could be renormaliz-able and could then be a complete (rather than partial)UV completion of (2). For the case of Horndeski, it willcertainly not be our aim to find a renormalizable comple-tion and for simplicity we shall consider (2) as a potentialcompletion here. We thank Paolo Creminelli and FilippoVernizzi for pointing this out.

[27] B. Bellazzini, JHEP 02, 034 (2017), arXiv:1605.06111[hep-th].

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[29] C. de Rham, S. Melville, and A. J. Tolley, JHEP 04, 083(2018), arXiv:1710.09611 [hep-th].

[30] Note that [28] included some confusions in interpretingthe context in which EFTs should be valid that wereclarified in [29].

[31] Horndeski models of dark energy involve many othertypes of other operators which do not affect the soundspeed. Including those would not affect our conclusionsabout cT , namely that it is naturally of order unity nearLIGO frequencies. However, these other operators mayhave other effects on the gravitational waveform, for ex-ample through a time-dependent Planck mass. If thespontaneous symmetry breaking is solely responsible forthe time dependence of the Planck mass, then this willalso asymptote to a constant in the UV as Lorentz invari-ance is restored. Interestingly, it was recently pointed outin [51] that most Horndeski models of dark energy thatdo not predict a luminal GW sound speed would lead toa decay of GW into the dark energy field and are alsoseverally constrained.

[32] A skeptic reader may worry about an EFT with such alow cutoff of the order of 10−13eV when GR is clearlyvalid and predictive over a much broader set of scales.Yet we should bear in mind that such a theory is typi-cally introduced to tackle dark energy and would be validfrom a scale of the order 10−33eV, that is at 20 orders ofmagnitude lower than that cutoff.

[33] In theories that admit a Vainshtein mechanism [52, 53],we may hope to be able to trust the theory at scales oforder Λ and to invoke a Vainshtein redressing to push theregime of validity of the theory to higher scales, howeverthe Vainshtein redressing is negligible for the physicalsetup considered here.

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[37] C. de Rham and L. Heisenberg, Phys. Rev. D84, 043503(2011), arXiv:1106.3312 [hep-th].

[38] Interestingly gas bounds to Black Holes in the LISA bandmay produce an X-ray signal [54].

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