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arXiv:1710.06394v1 [astro-ph.CO] 17 Oct 2017 G101)fo h egro eto tr(S bi- (NS) star neutron [ a 4993 NGC of near merger located nary the wav gravitational from of (GW170817) detection interferometer, the announced the recently have and (aLIGO) Observatory tional Introduction: rvttoa ae;w ilfidi ovnett parametri to [ convenient as it this find will we waves; gravitational infrlnaie rvttoa ae nsc eimtak ac- medium a The form: such the in cosmology). waves in gravitational linearized direction for preferred time tion a the de- defines as new that (such configuration the direction a when takes medium freedom a of such gree provide curvature, o to degrees dom extra Altern coupling directly differ. , of can theories speeds tive sponta- propagation is and symmetry Lorentz violated present, neously is medium a when ever, all lcrmgei etraesfcetysrn httespe the that is strong of sufficiently are sector electromagnetic oee,ngtv aus( values negative However, nprinciple, In standard the gravity alternative from by differ provided can media theories in which mass, Planck by defined expanded direction have ε We the be medium. to the chosen time with coordinates η as metric the decomposed have We [ generally more cosmologic (her cosmology on and waves constraints gravity, stringent gravitational place and can we light GW), of after [ time travel merger the NS-NS paring the to counterpart tromagnetic vicinity the the by in observed and was GW170817, seconds, of 1.7 within occurring 170817A), tory Spectrom- the the for for Shield eter Anti-Coincidence the and Monitor, − A αβ 10 ihamplitudes with , ewl sueta osrit nLrnzvoaini the in violation Lorentz on constraints that assume will We aseswvspoaaea h pe flgt How- light. of speed the at propagate waves massless − [ h ikwk erc—b hoiglclyinertial locally choosing by — metric Minkowski the 2 15 , 15 3 yalc fosre gravi- observed of lack a by .Teei togeiec htti vn sa elec- an is event this that evidence strong is There ]. ], c S nentoa am-a srpyisLabora- Astrophysics Gamma-Ray International = α togcntanso omlgclgaiyfo W787a GW170817 from gravity cosmological on constraints Strong h lentvst eea eaiitccosmology. relativistic rang general viable b to the independent alternatives co reduce an severely stringent place constraints place to These us to gravity. allowing us while allows theories, measurement compl tensor this a opens that stars show neutron We two of merger the from (GW170817) T = .I aum oet ymtyipisthat implies symmetry Lorentz , In 1. h dacdLsrItreoee Gravita- Interferometer advanced The h eeto fa lcrmgei onepr GB17081 (GRB counterpart electromagnetic an of detection The ol dp ihrpstv rngtv values. negative or positive either adopt could 1 2 Z 2 h al nttefrCsooia hsc,TeUiest of University The Physics, Cosmological 3 for Institue Kavli d A nttt o hoeia tde,EHZrc,Clausiusst Zurich, ETH Studies, Theoretical for Institute where , 3 1 4 .Baker, T. c d M xdt nvriyo xod ey ikno ulig el Road Keble Building, Wilkinson Denys Oxford, of University EC,FzklíútvAaei vˇed Akademie ústav Fyzikální CEICO, T 2 = c T 1 ∗ 2 < 1 +  A .Agmarybrt(GRB burst ray gamma A ]. h ˙ 1 α = A 2 c h r osrie to constrained are ) T .Bellini, E. − g αβ × . eekvrdainfrom radiation Cerenkov αβ Fermi ˇ c M , T 2 + nplrzto states, in = ( P 6 ∇ . . – η h M 14 c am-a Burst Gamma-ray αβ A T ∗ ) ]. 1 2 stesedof speed the is steeffective the is +  .G Ferreira, G. 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GW of non-linear alter screening to could the fects leads with that of occurs forces) regions as scalar in example, sources, (for GW gravity the strong close behavior exotic some be n upss ewl eefe consider hereafter will we purposes, and (where σ constraints cosmological current σ to this Comparing ( Eq. of limit excess: ric speed tensor theories on bimetric Constraint encompass not does it theories, tational rvddb h e ereo reo)telnaie action linearized the freedom) of ( degree new the medi by the provided than other , empty effectively in propagating rvttoa waves. gravitational omcry [ rays cosmic ev otgtntebudw eieblw Let below. derive only we would bound this sec- but the few duration), tighten a GRB to the be of will serve order this the merger of two NS-NS onds, the a between (for delay some peaks be emission can there photons; and waves egrpa n h ekotclbihns.A ria dela arrival GW An the brightness. optical between of peak times the arrival and peak in merger difference measured the is egrtm dnie ntegaiainlwv ri,and train, wave gravitational the in identified time merger odacrc,te,tetasttm fteG n photon and GW the of time are transit T signals the signal. optical then, the accuracy, in time good brightness peak measured the be p stedsac otesuc.W hnhave then We source. the to distance the is Mpc 1 1 17 α α − ssfcet ti ocial btulkl)ta hr m there that unlikely) (but conceivable is It sufficient. is ) ntergm eaecnieig( rvttoa wave gravitational (a considering are we regime the In Let udo h rvtnms nbmti hoisof theories bimetric in mass the on ound T T ∆ , 1 t ∼ 18 ∼ / ≃ srit ngnrlsaa-esradvector- and scalar-tensor general on nstraints c tl e rn o etn hoiso gravity. of theories testing for arena new etely t ]. 0 s [ 1 T 1 as 7 02Zrc,Switzerland Zurich, 8092 47, rasse . 2 alrepnigti gives this expanding Taylor . [ 1 . etetm feiso o ohtegravitational the both for emission of time the be hcg,Ciao L667 USA 60637, IL Chicago, Chicago, eod mle that implies seconds 7 .Noller, J. 19 A otegaiainlwv signal wave gravitational the to 7A) c 20 T )o oeatcsooia osrit (where constraints cosmological forecast or ]) 16 ( ) hscntan srmral.Fralintents all For remarkable. is constraint this ]), xod X R,UK 3RH, OX1 Oxford, , t T .U onw h nyuprbudo the on bound upper only the now, to Up ]. − fGscmsfo esrn h travel the measuring from comes GWs of 3 t s = ) n .Sawicki I. and | α T 1 1 | dGB170817A. GRB nd d othat so ) svldfrawd ag fgravi- of range wide a for valid is ) . s and 1 × ( 10 t c c − − ecnie h geomet- the consider We T 4 15 t α s side h pe of speed the indeed is . = ) T α ≃ ebgnb con- by begin We T d 2 s ≃ where , ∆ t ( n attempt and 0 / t d c s − α where , t T t T T d < ) s ethe be / ≃ . d 0 avel s . um (3) 40 42 ay ∆ = f- t y o c t 2 equations of motion [21, 22], and is given by S = the scalar fluctuations is equal to that of light. On the other 4 5 L L L d x√ g ∑i=2 i[φ,gµν ]+ M[gµν , ] , where M is hand, the strength of the fifth force, f,φ , is allowed to be simi- theR minimally−  coupled matter action.··· The scalar field la- lar to gravity. grangian is built of four terms: two minimally coupled to In Class (ii), the scalar evolves quickly, X H2M2, and ∼ ∗ gravity, L2 = K and L3 = G3φ and two terms explicitly non-canonical kinetic terms in G2 and G3 play a signifi- − involving the Ricci curvature, R, and the Einstein tensor, Gµν : cant role: they can give rise to self-acceleration, significantly changing the equation of state and the sound speed. Con- 2 µ ν L4 = G4R + G4,X (φ) ∇µ ∇ν φ∇ ∇ φ , straints on the of the Planck mass [31, 32] restrict  − µ ν 1 3 µ ν the strength of coupling to gravity f,φ to be small, since the L5 = G5Gµν ∇ ∇ φ G5 X (∇φ) 3∇ ∇ φ∇µ ∇ν φφ − 6 , − scalar runs during the entire history of the in these ν α µ  models. We reiterate that perturbative control of quantum cor- +2∇ ∇µ φ∇ ∇ν φ∇ ∇α φ . (4) rections in the fast-moving models depends on shift symme- ν Here K and Gi are functions only of φ and X ∇ φ∇ν φ/2, try, which would disallow any dependence on φ in the action, and subscript commas denote derivatives. On≡− a cosmological specifically the conformal coupling f (φ). background, Horndeski models give [23] Horndeski theory is not the most general scalar-tensor the- 2 ory propagating one single extra degree of freedom. New M αT 2X 2G4,X 2G5,φ φ¨ φ˙H G5,X (5) ∗ ≡ − − − terms can be added to construct the “beyond” Horndeski La-    grangian [33, 34] at the price of third derivatives in equations where M2 2 G 2XG + XG φ˙HXG . 4 4,X 5,φ 5,X of motion and new constraints to remove any extra degrees of One way∗ ≡ of satisfying− α 0 is through− a delicate cancel- T freedom naively implied by them. This extension is described lation between G , G and∼G . If G = 0, this cancel- 4,X 5,φ 5,X 5,X by two new free functions, G˜ φ X and G˜ φ X correct- lation is trivial, since it implies that the theory is minimally 4 ( , ) 5 ( , ) ing L and L (see [33] for the complete expressions) and coupled. Any non-trivial cancellation would not only have 4 5 modifying Eq. (5) to to be time-dependent, but also sensitive to the matter con- tent of the universe due to the dependence on H and φ¨. Thus 2 αTM = 4X G4,X G˜ 4,X G5,φ 2φ¨XG5,X even a small change in e.g. the dark matter density, or devia- ∗ − − − ˙ ˜  tions from isotropy and homogeneity, would severely violate + 2φHX G5,X G5,X , (6) − it. Furthermore, any such a cancellation would be accidental,  2 with no symmetry to protect it. Some shift symmetric Horn- where M = 2G4 4X G4,X G˜ 4,X + 2XG5,φ ∗ − − − deski actions (i.e. not dependent on φ) are, to some degree, 2φ˙HX G5 X G˜ 5 X .  , − , stable to radiative corrections. In flat , for K,Gi It is clear from Eq. (6) that one option is to set all the linear in X (galileons [24]), there exists an exact quantum terms contributing to αT to zero, as in the Horndeski case. non-renormalization theorem [25–27] — there are no correc- An intriguing alternative is to choose G5,X = G˜5,X = 0 and tions to these operators. The corrections remain under con- G˜ 4 X = G4 X G5 φ , which indeed leads to αT = 0 but also al- , , − , trol when the galilean symmetry is weakly broken [28], as lows for M = MP and αH = 0, where αH is the additional it must be in curved spacetime. In this case, the Horndeski beyond-Horndeski∗ 6 parameter6 introduced in [33]. Although interactions are suppressed by a scale Λ3, whereas quantum it is beyond the scope of this work to discuss the properties corrections enter suppressed by the parametrically larger scale of this particular model, we should emphasize that this is the 4 3 Λ2 Λ3, which satisfies Λ2 = MPlΛ3 [28]. A typical value only algebraic choice for the Gi functions that ensures αT = 0 ≫ 13 10 is Λ3 10 eV, leading to Λ3/Λ2 10 . With relatively regardless of the underlying cosmology. ∼ − ∼ − mild assumptions on the Gi functions, this can be shown to In our discussion of scalar-tensor theories, we should 4 40 lead to order (Λ3/Λ2) 10− corrections on the Gi [29] briefly mention Degenerate Higher-Order Scalar-Tensor ∼ 15 (and hence on αT as derived here), showing that a 10− tun- (DHOST) theories [35, 36]. DHOST theories are constructed ing of αT unspoiled by radiative corrections can be achieved. to be a further generalization of Horndeski, but have to in- A more natural interpretation of the constraint 3 is that each clude new constraints to avoid Ostrogradsky instabilities. The of the terms (G4,X , G5,φ , G5,X ) contributing to αT is zero, i.e. result is a long list of classes of theories ( 30) having disjoint µ ν ≃ that L5 ∝ Gµν ∇ ∇ φ, vanishing identically as a result of the parameter , but which on a cosmological background Bianchi identity, while L4 = f (φ)R, i.e. the coupling to grav- reduce to just two types [37]. One is unstable and thus irrele- ity can at most be of the Jordan-Brans-Dicke(JBD) type. Such vant here. The other can be transformed to beyond Horndeski a restriction reduces the viable model space for scalar-tensor with a conformal transformation of the formg ˜µν = C(X)gµν . modified gravity to two classes: (i) models in which the scalar Conformal transformations leave null geodesics null. Thus does not evolve significantly on cosmological timescales, and if a DHOST model describes gravity in cosmology, then the (ii) those in which it does. requirements for αT = 0 listed above apply to the beyond- Class (i) is the generalized JBD class, including mod- Horndeski counterpart of the DHOST theory. els such as f (R) gravity. Such models require chameleonic To conclude, if we assume that it is not possible to enforce screening to evade solar-system tests of gravity, and there- precise cancellations for the reasons discussed above, the con- fore cannot have a background evolution significantly differ- straint on αT excludes such models as the quartic and quintic ent from that of concordance cosmology; they do not self- galileon or a generic beyond Horndeski, leaving only models accelerate cosmological expansion [30]. The sound speed of which are conformally coupled to gravity. On the other hand, 3 models where gravity remains minimally coupled remain un- (c1 + c3)F,K ], so the constraint on αT implies c1 = c3. On constrained: fast-moving models such as kinetic gravity braid- , this reduces the theory to the Maxwell− ac- ing [38] can give rise to self-acceleration and admit an inter- tion (with a time-like constraint). On a cosmological back- pretation as the dynamics of a superfluid [39], rather than as a ground, we still allow for modifications as 3M2H2 = (ρ P − modification of gravity. Finally, quintessence models remain F /2)(1 3c2F,K ), whereas the effective Planck mass in − 2 2 unconstrained. Eq. (1), which is generally given by M = MP[1 (c1 + F ∗ − Implications for vector-tensor theories: We now turn to c3) ,K ], will reduce to the GR value. vector tensor theories of gravity, i.e. theories where the A second-class of vector-tensor theories of interest are gen- additional gravitational degree of freedom is given by a eralized Proca theories [42, 43], whose 4Daction is, muchlike 4 L L L 4-vector, Aµ . First, we consider Generalized Einstein- Horndeski theory, given by S = d x√ g( + M), = L 5 L R − Aether gravity, where Aµ is time-like and the action is F + ∑i=2 i , where the vector field Lagrangian is built 2 4 MP µ so that precisely one extra (longitudinal) scalar mode prop- S = d x√ g R + F (K)+ λ(A Aµ + 1) , where λ is − 2 agates in addition to the two usual Maxwell-like transverse R h µ ν i µ 2 a Lagrange multiplier, K = c1∇µ Aν ∇ A + c2(∇µ A ) + polarisations. The individual Li are given by three mini- ν µ F L 1 µν L c3∇µ Aν ∇ A (with ci constants) and (x) is an arbitrary mally coupled terms, F = 4 Fµν F , 2 = G2(X) and µ − function [40, 41]. In this model αT = (c1 + c3)F,K /[1 + L3 = G3(X)∇µ A , and two nontrivial terms given by −

µ 2 ρ σ σ ρ L4 = G4(X)R + G4,X (X) (∇µ A ) + c2∇ρ Aσ ∇ A (1 + c2)∇ρ Aσ ∇ A ,  −  µ ν 1 µ 3 µ ρ σ µ σ ρ L5 = G5(X)Gµν ∇ A G5 X (X)[(∇µ A ) 3d2∇µ A ∇ρ Aσ ∇ A 3(1 d2)∇µ A ∇ρ Aσ ∇ A − 6 , − − − γ ρ σ γ ρ σ +(2 3d2)∇ρ Aσ ∇ A ∇ Aγ + 3d2∇ρ Aσ ∇ A ∇γ A ]. (7) −

As usual, Fµν = ∇µ Aν ∇ν Aµ , c2 and d2 are constants, and only f4 and f5 affect the background evolution and that of lin- − 1 µ G2 3 4 5 are arbitrary functions of X = Aµ A . On a cosmo- ear tensor perturbations, whereas the remaining functions only , , , − 2 µ ~ affect linear vector and scalar perturbations. The αT = 0 con- logical background A = (A,0) and αT is given by 2 3 straint now implies G5 X (HA A˙) 2G4 X = 2 f4A +6 f5HA , , − − , 2 which depends on the new functions f4, f5. Ifwe choosetoset αT = A 2G4,X (HA A˙)G5,X /qT , (8)  − −  all participating functions to zero to ensure αT = 0, this means 2 3 both the background and tensor perturbations will behave ex- where qT = 2G4 2A G4,X + HA G5,X . Analogously to the − actly as in the Generalised Proca case considered above. scalar-tensor case considered above, if αT = 0 we either then have to carefully tune the functional dependence of G4 and Implications for bigravity theories: We now consider models G5 to satisfy this criterion (all the considerations about ra- with two coupled metrics. The only non-linear Lorentz invari- diative stability, time dependence and background symme- ant ghost-free possible interactions are given by the deRham- try we discussed for Horndeski theories hold), or consider Gabadadze-Tolley (dRGT) potential [48–50]. The action is 2 4 2 4 a theory with minimal higher-order interactions by requiring given by S = (Mg /2) d x √ gRg +(Mf /2) d x √ fR f µ ν R − R − − G4,X = G5,X = 0 leading to L4 ∝ R and L5 ∝ Gµν ∇ A . In 2 2 4 4 1 m Mg d x √ g∑n=0 βnen g− f , where we have two the latter case, ghost-freedomfor tensor perturbationsthen en- −   dynamicalR metrics g and fp with their associated Ricci forces G > 0, while ghost and gradient instabilities for vector µν µν 4 scalars R and R , and constant mass scales M and M , re- modes are automatically satisfied. g f g f spectively. Here, β are free dimensionless coefficients, while In Generalised Proca theories the equation of motion for n m is an arbitrary constant mass scale. The dRGT poten- Aµ separates the evolution into two branches, one with a non- tial is defined in terms of the functions e (X), which corre- dynamical scalar degree of freedom and a second one with n spond to the elementary symmetric polynomials of the matrix full dynamics for all three degrees of freedom, which we X 1 will focus on here. Requiring G = G = 0 (and hence = g− f . 4,X 5,X p αT = 0) as above, the modified Friedman equation then be- For simplicity, let us assume that matter fields are coupled 2 minimally to the metric g , and all the parameter βs are of comes 3H = (ρ G2)/(2G4), and thus 2G4 describes a re- µν scaled constant Planck− mass. We note that on the de Sitter order 1 . fixed point of this model [44], in the limit ρ = 0, consistency The bigravity action generally propagates one massive and will enforce G2 < 0, due to the ghost-freedom condition for one massless graviton; and the field gµν will be a combination tensor perturbations G4 > 0. of both modes. The massless mode has a dispersion relation 2 2 2 2 2 One can go a step beyond Generalised Proca theories and given by E0 = k , while the massive mode has Em = k + m consider the "beyond" Generalised Proca model of [45] (also (with omitted factors of βs of order 1) on Minkowski space see [46, 47]). Here six new free functions enter at the level of (and a de-Sitter phase, i.e. late times). the action, denoted G6,g5, f4, f5, f˜5, f˜6. Of the new functions Let us first discuss the restricted case of , 4 when Mf /Mg ∞, and only the massive graviton propagates scales. We would argue that, if such an effect is relevant, then → (while the metric fµν is frozen). In this case, the dispersion the GW would be propagating with a speed which the cosmo- relation of gravitational waves is E2 = k2 + m2. As a result, logical modes will experience when the universe has emptied the speed of GW will be -dependent leading to a out to the same extent as the averaged density along the trajec- phase difference in the waveforms. Bounds from GW150914 tory of the GW. If αT is evolving, we may well have measured 22 led to m 1.2 10− eV [51]. With an EM counterpart to its asymptotic future value. the GWs,≤ the bound× of 1.7 seconds on the time delay also Conclusions: The detection of GW170817, together with its 22 leads to m . 10− eV (note that we have considered a fre- EM counterpart (GRB 170817A), bounds the speed of gravi- quency region of interest of 10 100Hz and ignored the fre- tational waves to deviate from c by no more that one part in quency dependency of the velocity,− which is small) which is 1015. This single fact has profoundrepercussionsfor extended uncompetitive with Solar System fifth force constraints of or- gravity models which are of interest in current cosmology. We 33 der m . 10− eV [52]. In case of massive bigravity, assum- summarize here the key consequences explained in this letter: ing similar amplitudes for both modes, one has a fast oscil- i) Assuming no finely-tuned cancellations between La- lation with a slowly modulated amplitude. The frequency of grangian functions occur, the only viable scalar-tensor theo- the modulated wave is proportional to m and hence negligible ries have a gravitational action of the form ∝ f (φ)R (plus non- compared to the time scale of the NS merger. The dispersion gravitational terms), i.e. conformally coupled theories. This relation of the fast mode is effectively that of a massive gravi- eliminates, for example, the quartic and quintic Galileons. ton E2 = k2 + m2 (omitting again factors or order 1), and thus Quintessence is still allowed as the minimally coupled limit one obtains the same constraint as for massive gravity. of these theories. Unlike for scalar-tensor and vector-tensor theories, in mas- ii) In the conformally coupled class, the only surviving self- sive gravity local constraints from GW propagation have no accelerating theories must have a small (or, indeed, mini- bearing on cosmology. In particular, the existence of scalar mal) coupling strength to gravity, and hence can be treated and tensor instabilities [53, 54] in particular branches of the as generalized fluids. Models in this category include cubic background cosmology will be unconstrained by the measure- Galileons, kinetic gravity braiding and k-essence. ments discussed in this paper. Further discussion on GW con- iii) The “beyond” Horndeski extension of scalar-tensor theo- straints in massive bigravity can be found in [55–57]. Con- ries introduces only one further surviving model, which is also straints in the case where both metrics are coupled to matter conformally coupled to gravity. are discussed in [58]. iv) For vector fields, assuming no finely-tuned cancellations, Caveats: We now address possible caveats. For a start, the (Generalized) Einstein-Aether models are now subject to the source lies at a very low (z 0 01); thus our con- stringent relation c1 = c3. s = . − straint is on the speed of GWs today. It would of course be v) “Beyond” and standard Generalized Proca models, assum- a great coincidence if αT were to vanish now with such pre- ing no finely-tuned cancellations, behave identically at back- cision, but not at other times. However, this is in principle a ground level, with vastly simplified higher order gravitational possibility. interactions, such as a coupling to R, where the proportional- Another uncertainty is the extent to which the effective met- ity constant acts as a rescaled Planck mass in the Friedmann ric relevant for the propagation of perturbations with wave- equations. lengths similar to the size of the universe, as studied in cos- vi) In bimetric theories the mass of the graviton is constrained 22 mology, is the same one experienced by the GW with the to be m . 10− eV, which is weaker than current Solar Sys- wavelength of 3000 km (to which aLIGO/VIRGO are sen- tem bounds but entirely independent of them. This constraint sitive). For cosmological modes with wavelengths of 10– has no bearing on cosmology. 100 Mpc, taking the background— the medium in which fluc- For the first time, powerful and general statements can tuations propagate — to be isotropic and homogeneous is a be made about the structure of (non-)viable gravitational ac- good approximation. Wavelengths probed by aLIGO/VIRGO tions, and some current popular models are ruled out (also are much shorter than the typical size of structures in the uni- see [29, 59–61]). These decisive statements will undoubtedly verse, so the GW should be sensitive to the inhomogeneities. shape the direction of future research into extensions of Gen- Indeed, one can argue that, apart from the initial exit from the eral Relativity. source and the final entrance into the Milky Way, the ACKNOWLEDGMENTS GW was mostly propagating through space with density of We acknowledge conversations with Rob Fender, Filippo matter significantly below the current cosmic average, when Vernizzi and Miguel Zumalacárregui, and the discussions averaged over scales of the order of the GW’s wavelength. made possible by the DARKMOD workshop at IPhT Saclay. Some alternative theories of gravity depend crucially on a TB is supported by All Souls College, University of Oxford. highly non-linear response to the matter density by the extra EB is supported by the ERC and BIPAC. PGF acknowledges degrees of freedom (the need for screening on Solar-System support from STFC, BIPAC, the Higgs Centre at the Univer- scales). This may well mean that the GW speed predicted for sity of Edinburgh and ERC. ML is supported at the University an averaged cosmology, and that for the matter density along of Chicago by the Kavli Institute for Cosmological Physics the particular trajectory this GW took could be different. Thus through an endowment from the Kavli Foundation and its there would not be a simple connection between the time de- founder Fred Kavli. JN acknowledges support from Dr. Max lay observed and the properties of gravity on cosmological Rössler, the Walter Haefner Foundation and the ETH Zurich 5

Foundation. IS is supported by ESIF and MEYS (Project CoGraDS – CZ.02.1.01/0.0/0.0/15_003/0000437).

[1] B. P. A. et. al. (LIGO Scientific Collaboration and Virgo Col- [28] D. Pirtskhalava, L. Santoni, E. Trincherini, and F. Vernizzi, laboration), Phys. Rev. Lett. 119, 161101 (2017). JCAP 1509, 007 (2015), arXiv:1505.00007 [hep-th]. [2] A. G.et. al., The Astrophysical Journal Letters 848, L14 (2017). [29] P. Creminelli and F. Vernizzi, (2017), [3] V.S. et. al., The Astrophysical Journal Letters 848, L15 (2017). arXiv:1710.05877 [astro-ph.CO]. [4] B.P.A.et. al., The Astrophysical Journal Letters 848, L13 (2017). [30] J. Wang, L. Hui, and J. Khoury, [5] B.P.A.et. al., The Astrophysical Journal Letters 848, L12 (2017). Phys. Rev. Lett. 109, 241301 (2012), [6] L. Amendola, M. Kunz, M. Motta, I. D. Saltas, arXiv:1208.4612 [astro-ph.CO]. and I. Sawicki, Phys. Rev. D87, 023501 (2013), [31] J.-P. Uzan, Living Rev. Rel. 14, 2 (2011), arXiv:1210.0439 [astro-ph.CO]. arXiv:1009.5514 [astro-ph.CO]. [7] L. Amendola, G. Ballesteros, and V. Pet- [32] P. A. R. Ade et al. (Planck), torino, Phys. Rev. D90, 043009 (2014), Astron. Astrophys. 580, A22 (2015), arXiv:1405.7004 [astro-ph.CO]. arXiv:1406.7482 [astro-ph.CO]. [8] E. V. Linder, Phys. Rev. D90, 083536 (2014), [33] J. Gleyzes, D. Langlois, F. Piazza, and arXiv:1407.8184 [astro-ph.CO]. F. Vernizzi, Phys. Rev. Lett. 114, 211101 (2015), [9] M. Raveri, C. Baccigalupi, A. Silvestri, and arXiv:1404.6495 [hep-th]. S.-Y. Zhou, Phys. Rev. D91, 061501 (2015), [34] M. Zumalacarregui and J. Garcia-Bellido, arXiv:1405.7974 [astro-ph.CO]. Phys. Rev. D89, 064046 (2014), arXiv:1308.4685 [gr-qc]. [10] I. D. Saltas, I. Sawicki, L. Amendola, and [35] D. Langlois and K. Noui, JCAP 1602, 034 (2016), M. Kunz, Phys. Rev. Lett. 113, 191101 (2014), arXiv:1510.06930 [gr-qc]. arXiv:1406.7139 [astro-ph.CO]. [36] J. Ben Achour et al., (2016), arXiv:1608.08135 [hep-th]. [11] L. Lombriser and A. Taylor, JCAP 1603, 031 (2016), [37] D. Langlois, M. Mancarella, K. Noui, and F. Vernizzi, arXiv:1509.08458 [astro-ph.CO]. JCAP 1705, 033 (2017), arXiv:1703.03797 [hep-th]. [12] J. Beltran Jimenez, F. Piazza, and H. Vel- [38] C. Deffayet, O. Pujolas, I. Sawicki, and A. Vikman, ten, Phys. Rev. Lett. 116, 061101 (2016), JCAP 1010, 026 (2010), arXiv:1008.0048 [hep-th]. arXiv:1507.05047 [gr-qc]. [39] O. Pujolas, I. Sawicki, and A. Vikman, JHEP 11, 156 (2011), [13] D. Bettoni, J. M. Ezquiaga, K. Hinterbichler, and arXiv:1103.5360 [hep-th]. M. Zumalacarregui, Phys. Rev. D95, 084029 (2017), [40] T. Jacobson and D. Mattingly, Phys. Rev. D64, 024028 (2001), arXiv:1608.01982 [gr-qc]. arXiv:gr-qc/0007031 [gr-qc]. [14] I. Sawicki, I. D. Saltas, M. Motta, L. Amen- [41] T. G. Zlosnik, P. G. Ferreira, and G. D. dola, and M. Kunz, Phys. Rev. D95, 083520 (2017), Starkman, Phys. Rev. D75, 044017 (2007), arXiv:1612.02002 [astro-ph.CO]. arXiv:astro-ph/0607411 [astro-ph]. [15] E. Bellini and I. Sawicki, JCAP 1407, 050 (2014), [42] G. Tasinato, JHEP 04, 067 (2014), arXiv:1402.6450 [hep-th]. arXiv:1404.3713 [astro-ph.CO]. [43] L. Heisenberg, JCAP 1405, 015 (2014), [16] G. D. Moore and A. E. Nelson, JHEP 09, 023 (2001), arXiv:1402.7026 [hep-th]. arXiv:hep-ph/0106220 [hep-ph]. [44] A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, [17] D. Blas, M. M. Ivanov, I. Sawicki, and S. Sibiryakov, S. Tsujikawa, and Y.-l. Zhang, JCAP 1606, 048 (2016), JETP Lett. 103, 624 (2016), [Pisma Zh. Eksp. Teor. arXiv:1603.05806 [gr-qc]. Fiz.103,no.10,708(2016)], arXiv:1602.04188 [gr-qc]. [45] L. Heisenberg, R. Kase, and S. Tsujikawa, [18] N. Cornish, D. Blas, and G. Nardini, (2017), Phys. Lett. B760, 617 (2016), arXiv:1605.05565 [hep-th]. arXiv:1707.06101 [gr-qc]. [46] E. Allys, P. Peter, and Y. Rodriguez, JCAP 1602, 004 (2016), [19] E. Bellini, A. J. Cuesta, R. Jimenez, and L. Verde, arXiv:1511.03101 [hep-th]. JCAP 2, 053 (2016), arXiv:1509.07816. [47] J. Beltran Jimenez and L. Heisenberg, [20] D. Alonso, E. Bellini, P. G. Ferreira, and Phys. Lett. B757, 405 (2016), arXiv:1602.03410 [hep-th]. M. Zumalacarregui, Phys. Rev. D95, 063502 (2017), [48] C. de Rham, G. Gabadadze, and A. J. Tolley, arXiv:1610.09290 [astro-ph.CO]. Phys.Rev.Lett. 106, 231101 (2011), arXiv:1011.1232 [hep-th]. [21] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974). [49] C. de Rham and G. Gabadadze, Phys.Rev. D82, 044020 (2010), [22] C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade, arXiv:1007.0443 [hep-th]. Phys. Rev. D84, 064039 (2011), arXiv:1103.3260 [hep-th]. [50] S. Hassan and R. A. Rosen, JHEP 1202, 126 (2012), [23] A. De Felice and S. Tsujikawa, JCAP 1202, 007 (2012), arXiv:1109.3515 [hep-th]. arXiv:1110.3878 [gr-qc]. [51] B. P. Abbott et al. (Virgo, LIGO Sci- [24] A. Nicolis, R. Rattazzi, and E. Trincherini, entific), Phys. Rev. Lett. 116, 221101 (2016), Phys. Rev. D79, 064036 (2009), arXiv:0811.2197 [hep-th]. arXiv:1602.03841 [gr-qc]. [25] M. A. Luty, M. Porrati, and R. Rattazzi, JHEP 09, 029 (2003), [52] C. de Rham, J. T. Deskins, A. J. Tolley, and arXiv:hep-th/0303116 [hep-th]. S.-Y. Zhou, Rev. Mod. Phys. 89, 025004 (2017), [26] G. Goon, K. Hinterbichler, A. Joyce, and M. Trodden, arXiv:1606.08462 [astro-ph.CO]. JHEP 11, 100 (2016), arXiv:1606.02295 [hep-th]. [53] M. Lagos and P. G. Ferreira, JCAP 1412, 026 (2014), [27] I. D. Saltas and V. Vitagliano, JCAP 1705, 020 (2017), arXiv:1410.0207 [gr-qc]. arXiv:1612.08953 [hep-th]. 6

[54] G. Cusin, R. Durrer, P. Guarato, and M. Motta, [57] K. Max, M. Platscher, and J. Smirnov, JCAP 1505, 030 (2015), arXiv:1412.5979 [astro-ph.CO]. Phys. Rev. Lett. 119, 111101 (2017), [55] A. De Felice, T. Nakamura, and T. Tanaka, arXiv:1703.07785 [gr-qc]. PTEP 2014, 043E01 (2014), arXiv:1304.3920 [gr-qc]. [58] P. Brax, A.-C. Davis, and J. Noller, [56] T. Narikawa, K. Ueno, H. Tagoshi, T. Tanaka, N. Kanda, Phys. Rev. D96, 023518 (2017), arXiv:1703.08016 [gr-qc]. and T. Nakamura, Phys. Rev. D91, 062007 (2015), [59] J. Sakstein and B. Jain, (2017), arXiv:1412.8074 [gr-qc]. arXiv:1710.05893 [astro-ph.CO]. [60] J. M. Ezquiaga and M. Zumalacárregui, (2017), arXiv:1710.05901 [astro-ph.CO]. [61] H. Wang et al., (2017), arXiv:1710.05805 [astro-ph.HE].