Constraints on Scalar–Tensor Theory of Gravity by the Recent Observational Results on Gravitational Waves

Eur. Phys. J. C (2018) 78:738 https://doi.org/10.1140/epjc/s10052-018-6227-9

Regular Article - Theoretical Physics

Constraints on scalar–tensor theory of gravity by the recent observational results on gravitational waves

Yungui Gong1,a , Eleftherios Papantonopoulos2,b,ZhuYi1,c 1 School of Physics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China 2 Department of Physics, National Technical University of Athens, Zografou Campus, 157 73 Athens, Greece

Received: 27 April 2018 / Accepted: 7 September 2018 / Published online: 15 September 2018 © The Author(s) 2018

Abstract The speed of gravitational waves provides us a the faster speed of GWs, then we get an upper bound on the −16 new tool to test alternative theories of gravity. The constraint speed of GWs cgw/c − 1 ≤ 7 × 10 [9]. If we assume on the speed of gravitational waves from GW170817 and that the GRB signal was emitted 10s after the GW signal, −15 GRB170817A is used to test some classes of Horndeski the- then we get a lower bound cgw/c − 1 > −3 × 10 [9]. ory. In particular, we consider the coupling of a scalar field to The precise measurement of the propagation speed of GWs Einstein tensor and the coupling of the GaussÐBonnet term is a very powerful tool to test alternative theories of gravity to a scalar field. The coupling strength of the GaussÐBonnet [10Ð20]. coupling is constrained to be in the order of 10−15.Inthe Recently there is a lot of activity studying scalarÐtensor Horndeski theory we show that in order for this theory to theories [21] and one of them is the gravitational theory which satisfy the stringent constraint on the speed of GWs the mass is the result of the Horndeski Lagrangian [22]. Horndeski scale M introduced in the non-minimally derivative coupling theories because they lead to second-order field equations is constrained to be in the range 1015 GeV M 2×10−35 can be technically simple, and they prove consistent without GeV taking also under consideration the early times upper ghost instabilities [23]. In Horndeski theory the derivative bound for the mass scale M. The large mass ranges require self-couplings of the scalar field screen the deviations from no fine-tuning because the effect of non-minimally derivative GR at high gradient regions (small scales or high densities) coupling is negligible at late times. through the Vainshtein mechanism [24], thus satisfying solar system and early universe constraints [25Ð32]. A subclass of Horndeski theories includes the coupling of 1 Introduction the scalar field to Einstein tensor. This term introduces a new mass scale in the theory which on short distances allows to The detection of gravitational waves (GWs) by the Laser find black hole solutions [33Ð37], while a black hole can be Interferometer Gravitational-Wave Observatory (LIGO) Sci- formed if one considers the gravitational collapse of a scalar entific Collaboration and Virgo Collaboration opens the win- field coupled to the Einstein tensor [38]. On large distances dow to study strong field gravitational physics and test alter- the presence of the derivative coupling acts as a friction term native theories of gravity [1Ð6]. In particular, the recent detec- in the inflationary period of the cosmological evolution [39Ð tion of the GW170817 from the merger of a binary neu- 44]. Also, the preheating period at the end of inflation was tron star [6] and the electromagnetic counterparts starts a studied, and it was found that there is a suppression of heavy new era of multi-messenger GW astronomy. A gamma ray particle production as the derivative coupling is increased. burst GRB170817A was observed 1.74 ± 0.05 s later by This was attributed to the fast decrease of kinetic energy Fermi Gamma-Ray Burst Monitor [7] and the International of the scalar field because of its wild oscillations [45]. A Gamma-Ray Astrophysics Laboratory [8]. If we assume that holographic application was performed in [46] where it was the peak of the GW signal and the first photons were emitted shown that the change of the kinetic energy of the scalar field simultaneously, and the 1.74 s time difference is caused by coupled to Einstein tensor allowed to holographically simu- late the effects of a high concentration of impurities in a real a e-mail: [email protected] material. The above discussion indicates that the coupling b e-mail: [email protected] of the scalar field to Einstein tensor alters the kinematical c e-mail: [email protected] properties of the scalar field. 123 738 Page 2 of 7 Eur. Phys. J. C (2018) 78 :738

Assuming that the scalar field coupled to Einstein tensor scale introduced in derivative coupling of the scalar field to plays the role of dark energy and drives the late cosmological Einstein tensor. expansion it was found [47,48] that the propagation speed The action of the Horndeski theory is given by [22], of the tensor perturbations around the cosmological background with FriedmannÐRobertsonÐWalker (FRW) metric is √ S = d4x −g(L + L + L + L ), (1) different from the speed of light c, so the measurement of 2 3 4 5 the current speed of GWs can be used to test the applica- bility of this and Horndeski theories to explain the late time where accelerated cosmological expansion [13Ð16,49Ð51]. Due to L = K (φ, X), L =−G (φ, X)φ, small deviation (on the order of 10−15) of the current speed 2 3 3 2 μ ν of GWs from the speed of light, in Refs. [13,14,16]itwas L4 = G4(φ, X)R + G4,X (φ) − (∇μ∇νφ)(∇ ∇ φ) , argued that dark energy models that predict cgw = c at late μ ν 1 3 cosmological times are ruled out, and in Horndeski theory L5 = G5(φ, X)Gμν∇ ∇ φ − G5,X [(φ) 6 the only viable nonminimal coupling to gravity has the con- μ ν −3(φ)(∇μ∇νφ)(∇ ∇ φ) formal form f (φ)R. However, with the help of derivative μ α β +2(∇ ∇αφ)(∇ ∇β φ)(∇ ∇μφ)], conformal or disformal transformations, some Horndeski and μ μ beyond Horndeski theories can survive the speed constraint with X =−∇μφ∇ φ/2, φ =∇μ∇ φ, the functions [16,36,52,53]. K , G3, G4 and G5 are arbitrary functions of φ and X, and In this work, we will perform a detailed analysis on G j,X (φ, X) = ∂G j (φ, X)/∂ X with j = 4, 5. the effect of the latest observational results on the current This action is the most general one for scalarÐtensor theory speed of GWs cgw to the Horndeski theories with the non- with at most second-order field equations. If we take K = minimally derivative coupling. From the early cosmological G3 = G5 = 0 and G4 = MPl/2, then we obtain Einstein’s evolution we know that the derivative coupling of the scalar general relativity. If we take G3 = G5 = 0, K = X − V (φ), field to Einstein tensor alters the kinetic energy of the scalar and G4 = f (φ), then we get scalarÐtensor f (φ)R theories. field [40] influencing in this way the dynamical evolution of = 2 / + /( 2) = 2 / If we take G4 MPl 2 X 2M or G4 MPl 2 and the Universe giving an upper bound to the mass scale cou- 2 G5 =−φ/(2M ), then we get the non-minimally derivative μ ν pling. For the late cosmological evolution we will perturb the coupling Gμν∇ φ∇ φ/(2M2) with the mass scale M [54]. FRW metric under tensor perturbations and we will show that The stability of the Horndeski theory in the FRW back- a subclass of Horndeski theory consisting of the usual kinetic ground was studied in [55]. General conditions on the func- term and the coupling of the scalar field to Einstein tensor tions appearing in the Horndeski Lagrangian were given for is still viable provided that the mass scale M introduced in the theory to be ghost free and stable under tensor perturba- the non-minimally derivative coupling is highly constrained tions. While in the flat background, the propagation speed of from the recent results on the speed of GWs. The result also tensor perturbations is the same as the speed of light [56], in shows that no tuning on the parameter is needed to satisfy the the cosmological FRW background the propagation speed of stringent constraints on the speed of GWs. For comparison tensor perturbations in the Horndeski theory was found to be we also discuss bounds on the GaussÐBonnet coupling from [54] the observational bounds on c w. g The paper is organized as follows. In Sect. 2 we discuss G − X φ¨G , + G ,φ 2 = 4 5 X 5 . the Horndeski theory, and studying the speed of tensor per- cgw ˙ (2) G − 2XG , − X HφG , − G ,φ turbations we obtain bounds on the mass scale introduced 4 4 X 5 X 5 by the presence of the derivative coupling of the scalar field We are interested in the propagation speed of tensor pertur- to Einstein tensor. In Sect. 3 we discuss the bounds on cou- bations of the subclass of the Horndeski theory that consists pling α of the GaussÐBonnet theory coupled to a scalar field. of the usual kinetic scalar field term and the coupling of the Finally, in Sect. 4 are our conclusions. scalar field to Einstein tensor given by the action

√ 2 4 MPl 1 1 μ ν 2 The effect of the speed of gravitational waves in the S = d x −g R − gμν − Gμν ∇ φ∇ φ − V (φ) . 2 Horndeski theory 2 2 M (3) In this section we will briefly review the Horndeski theory, we will discuss the speed of GWs in this theory and assuming Perturbing the FRW metric as that the scalar field present in the Horndeski Lagrangian plays 2 2 2 i j the role of dark energy we will find a lower bound on the mass ds =−dt + a (t) δij + hij dx dx , (4) 123 Eur. Phys. J. C (2018) 78 :738 Page 3 of 7 738 and expanding the action (3) to the second order of the tensor This constraint is much less stringent than the classical con- −20 perturbations hij, we obtain the quadratic action [44,48] straint Γ ≤ 2/3 × 10 derived from the constraint on α = the Parameterized Post-Newtonian (PPN) parameter 3 2 Γ< × −20 MPl 3 3 ˙2 1 2 6 4 10 [57]. S = d xdta (1 − Γ ) h − (1 + Γ ) ∂k hij , 8 ij a2 Using the constraint (11) we will obtain a lower bound (5) on the mass scale M of the derivative coupling. If we take the scalar field as dark energy and use the observational con- ˙ ˙ 2 2 Γ = φ˙2/( 2 2 ) straint 1 + w = φ2/ρφ = φ2/(3M H Ωφ) ∼ 0.2 and where 2M MPl . From the action (5) we derive Pl 0 the equation of motion for GWs Ωm = 0.3, then we get the contribution of the canonical kinetic energy as Γ˙ + Γ ∇2 ¨ ˙ ˙ 1 hij hij + 3Hhij − hij − = 0. (6) ˙2 2 1 − Γ 1 − Γ a2 (1 + w)(1 − Ωm) φ Γ M = = ∼ 0.07. (12) 2 6M2 H 2 3 H 2 Under the transverse-traceless gauge, the Fourier compo- Pl 0 0 ( , ) nents of tensor perturbations hij x t is Combining Eqs. (11) and (12), we get the constraint on the coupling constant M + + × × h (x, t) = d3k h (t) ε + h (t) ε exp (ik · x) , ij k ij k ij 2 H − 0 3.3 × 10 15. (13) (7) M2 i εs = εs = εs εs = δ where k ij ii 0, ij ij 2 ss , and the superscript For the theory with the coupling of a scalar field to Einstein “s” stands for the “+”or“×” polarizations. Substituting Eqs. tensor, the PPN parameters are [57] (7)into(6), we get β = + Γ, γ = + Γ, α = Γ, α = Γ, 1 6 1 3 1 12 2 3 ˙ 2 2 (14) Γ cgwk α = Γ, ζ = Γ, ζ = Γ, ξ = ζ = ζ = . h¨s + 3Hh˙s 1 − + hs = 0. (8) 3 6 2 15 3 3 1 4 0 k k 3H (1 − Γ ) a2 k −20 The stringent constraint coming from α3 = 6Γ<4×10 The propagation speed for both polarization states is [58,59], substituting this result into Eq. (12), we get 1 + Γ 2 = . 2 cgw (9) H − 1 − Γ 0 3.2 × 10 19. (15) M2 This result can also be obtained from the general formula This constraint is much stronger than Eq. (13). If we use the of the Horndeski theory given in Eq. (2) choosing G4 = M2 /2 + X/(2M2) or G = M2 /2 and G =−φ/(2M2). constraint (13), we get the lower bound on the coupling M Pl 4 Pl 5 × −35 For the Horndeski theory, it was argued that the precise as M 2 10 GeV. In the New Higgs inflation [47], the non-minimally derivative coupling enhances the friction of measurement on the speed of GWs cgw = 1 requires the expansion and the high friction limit requires M 1015 G4,X = G5,φ = G5,X = 0, and only the conformal coupling f (φ)R is allowed [13,14,16]. At late times, since the effect Gev [44] while in [60] limits on M are also discussed during μ ν of the non-minimally derivative coupling Gμν∇ φ∇ φ ∼ the reheating period. Therefore, the mass scale introduced 15 × −35 H 2φ˙2/M2 is negligible compared with the canonical kinetic in the derivative coupling is 10 GeV M 2 10 1 term φ˙2 due to the decrease of the Hubble parameter H as the GeV. Universe expands, the speed given by Eq. (9) can be close to This result is interesting and it shows that the coupling of the scalar field to Einstein tensor has a complete differ- 1. Instead of requiring that G4,X = G5,φ = G5,X = 0, we show that a large mass range for the coupling M is allowed ent behaviour compared to a scalar field minimally coupled to satisfy the stringent constraint on the speed of GWs with to gravity. While the kinetic energy of a minimally coupled negligible but nonzero deviation. scalar field practically does not understand the cosmolog- Using the the upper bound on the speed of of GWs [9] ical evolution, the kinetic energy of scalar field coupled to Einstein tensor changes as the Universe expands. At the infla- cgw − tionary epoch it can drive inflation with steep potentials while − 1 ≤ 7 × 10 16, (10) c 1 This lower bound of the mass M allows the sound speed squared of we obtain, the tensor perturbations to reach the observational bounds of the GWs in the model discussed in [61] making in this way the model of unification − 0 <Γ ≤ 7 × 10 16. (11) of dark matter with dark energy in Horndeski theory viable. 123 738 Page 4 of 7 Eur. Phys. J. C (2018) 78 :738 √ as the Universe expands its contribution to the cosmological β 1 −φ/ β V (φ) = − + (3 − 8α) g(φ) − ρ0e , (23) evolution is less important and at the late cosmological epoch 2 9 is negligible, so GWs propagate at the speed of light at late α β times. where and are model parameters. An√ exact solution of the gravitational field equations φ = 3 β ln a was found with the coupling function of the scalar field to be 3 The effect of the speed of gravitational waves in the 8(3 − 4α)dφ Gauss–Bonnet theory f (φ) = α √ , (24) 3 βg(φ)

Another non-trivial extension of GR which gives second where order differential equations is the Lovelock theory [62], (3 − 4α)A (8α + 27β)φ which apart from the EinsteinÐHilbert term also includes g(φ) = exp √ 3 3 β(4α − 3) higher order curvature terms. The simplest case is the GaussÐ 9(4α − 3)ρ √ Bonnet theory which is a second order Lovelock theory which + 0 e−φ/ β , α + β − (25) however in four dimensions is a topological invariance. If 20 27 9 however it is coupled to a scalar field then a scalarÐtensor and A is an integration constant, ρ0 is the present value of the theory is generated from the action [63], energy density for matter. Note that if the coupling strength α = 0 we do not have the GaussÐBonnet coupling. From the √ 2 (φ) solution, we get the current value of the ratio of the energy 4 MPl f S = −gd x R + X − V (φ) + RGB , densities between dark energy and matter, 2 8 (16) Ωde A(20α + 27β − 9) − 3ρ0(20α + 27β) = , (26) Ωm 3ρ0(20α + 27β − 9) where and the current equation of state parameter for dark energy 2 μν μναβ R = R − 4Rμν R + Rμναβ R , (17) GB ( α + β − )2 w =− A 20 27 9 . de ( α − ) ( ( α+ β − )− ρ ( α + β)) and we have also included a scalar potential V (φ). 3 4 3 c 20 27 9 3 0 20 27 It is interesting to notice that the GaussÐBonnet theory (27) coupled to a scalar field in four dimensions can be generated If we take the current value of the ratio Ω /Ω to be 7/3 from the general Horndeski action (1)[54,64] making the de m [78], then we obtain the integration constant A, following identifications of the functions involved

(φ, ) = − (φ) + (φ) 2 ( − ), (200α + 270β − 63)ρ0 K X X V f X 3 ln X (18) A = , (28) α + β − f (φ) 20 27 9 G3 (φ, X) = X (7 − 3lnX), (19) 2 and the current equation of state parameter for dark energy M2 f (φ) G (φ, X) = Pl + X (2 − ln X), (20) 4 2 2 63 − 200α − 270β wde =− . (29) f (φ) 63 − 84α G (φ, X) =− ln X. (21) 5 2 Using Eq. (22) the speed of GWs is Then we can use the general formula for the propagation of gravitational waves Eq. (2) for the functions (18)Ð(21) and α2 + α( β − ) 2 320 6 90 11 c w = 1 − . (30) we get g 45 + 160α2 − 180α

2 + (φ) + φ¨ (φ) Using the bound on the speed of GWs [9] 2 MPl 2Xf f c w = . (22) g M2 + Hφ˙ f (φ) Pl − cgw − −3 × 10 15 ≤ − 1 ≤ 7 × 10 16, (31) c The role that the GaussÐBonnet tern coupled to a scalar ﬁeld plays in the late cosmological evolution has been exten- and the constraint −1.1

Acknowledgements We thank K. Ntrekis, S. Tsujikawa, I. Dalianis, J. Sakstein, L. Lombriser, M. Zumalacarregui and D. Shantanu for useful discussions. E.P acknowledges the hospitality of School of Physics of Huazhong University of Science and Technology where part of this work was carried out. This research was supported in part by the Major Program of the National Natural Science Foundation of China under Grant no. 11690021 and the National Natural Science Foundation of China under Grant nos. 11875136 and 11475065.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative α Fig. 1 The constraint on the coupling constant and the model param- Commons license, and indicate if changes were made. β eter Funded by SCOAP3. term is in the order of 10−15. For the inflationary model with References V (φ) ∼ φ2/2 and f (φ) =−8αφ2/2, the absolute value of the coupling strength α is constrained to be less than the 1. B.P.Abbott, Observation of gravitational waves from a binary black . hole merger. Phys. Rev. Lett. 116(6), 061102 (2016) order of 0 01 [72]. For the power-law inflation with both 2. B.P. Abbott, GW151226: observation of gravitational waves from exponential coupling and potential, the coupling strength is a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. constrained to be −1×10−4 <α<4×10−4 [73]. Therefore, 116(24), 241103 (2016) the constraint from the speed of GWs is much stronger. 3. B.P. Abbott, GW170104: observation of a 50-solar-mass binary black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118(22), 221101 (2017) 4. B.P. Abbott, GW170608: observation of a 19-solar-mass binary 4 Conclusions black hole coalescence. Astrophys. J. 851(2), L35 (2017) 5. B.P. Abbott, GW170814: a three-detector observation of gravita- The first measurement of the speed of GWs by GW170817 tional waves from a binary black hole coalescence. Phys. Rev. Lett. and GRB170817A bounds the deviation of the speed of GWs 119(14), 141101 (2017) 15 6. B.P. Abbott, GW170817: observation of gravitational waves from from the speed of light to be no more than one part in 10 , a binary neutron star inspiral. Phys. Rev. Lett. 119(16), 161101 so it provides the evidence that cgw = c. Using these obser- (2017) vational result we can test alternative theories of gravity for 7. A. Goldstein, An ordinary short gamma-ray burst with extraordi- their validity to describe the cosmological evolution at late nary implications: fermi-GBM detection of GRB 170817A. Astro- phys. J. Lett. 848(2), L14 (2017) times. 8. V. Savchenko, INTEGRAL detection of the first prompt gamma- We used these bounds on cgw to constrain first a subclass ray signal coincident with the gravitational-wave event GW170817. of the Horndeski theory in which a scalar field except its Astrophys. J. Lett. 848(2), L15 (2017) minimal coupling is also coupled to Einstein tensor. Assum- 9. B.P. Abbott, Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys. ing that the scalar field plays the role of dark energy we J. 848(2), L13 (2017) found a lower bound on the mass scale introduced by this 10. S. Mirshekari, N. Yunes, C.M. Will, Constraining generic lorentz coupling and combining the constraints from inflation the violation and the speed of the graviton with gravitational waves. energy scale of the derivative coupling is bounded to be Phys.Rev.D85, 024041 (2012) 11. J. Beltran Jimenez, F. Piazza, H. Velten, Evading the Vainshtein 15 × −35 10 GeV M 2 10 GeV. This result requires mechanism with anomalous gravitational wave speed: constraints no fine-tuning and shows that it is possible to get cgw ≈ c on modified gravity from binary pulsars. Phys. Rev. Lett. 116(6), 061101 (2016) from the terms with G4,X = 0 and G5,φ = 0 if their effects are negligible at late times. 12. P.M. Chesler, A. Loeb, Constraining relativistic generalizations of modified Newtonian dynamics with gravitational waves. Phys. Rev. We also studied the GaussÐBonnet theory in four dimen- Lett. 119(3), 031102 (2017) sions coupled to a scalar field. The coupling of the GaussÐ 13. T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller, I. Sawicki, Bonnet term to scalar field not only gives successful infla- Strong constraints on cosmological gravity from GW170817 and tion, but also provides late time cosmic acceleration. Using GRB 170817A. Phys. Rev. Lett. 119(25), 251301 (2017) 14. P. Creminelli, F. Vernizzi, Dark energy after GW170817 and a particular model with a specific form of the coupling func- GRB170817A. Phys. Rev. Lett. 119(25), 251302 (2017) tion f (φ) which allows an exact solution of the gravitational 15. J. Sakstein, B. Jain, Implications of the neutron star merger equations, we found that the bounds on cgw constrains the GW170817 for cosmological scalarÐtensor theories. Phys. Rev. GaussÐBonnet coupling strength to be α 10−15, a con- Lett. 119(25), 251303 (2017) 16. J.M. Ezquiaga, M. Zumalacárregui, Dark energy after GW170817: − × −4 < straint much stronger than the coupling strength 1 10 dead ends and the road ahead. Phys. Rev. Lett. 119(25), 251304 − α<4×10 4 resulted from models with power-law inflation. (2017) 123 738 Page 6 of 7 Eur. Phys. J. C (2018) 78 :738

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