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

## Abstract

The speed of gravitational waves provides us a new tool to test alternative theories of gravity. The constraint on the speed of gravitational waves from GW170817 and GRB170817A is used to test some classes of Horndeski theory. In particular, we consider the coupling of a scalar field to Einstein tensor and the coupling of the Gauss–Bonnet term to a scalar field. The coupling strength of the Gauss–Bonnet coupling is constrained to be in the order of \(10^{-15}\). In the Horndeski theory we show that in order for this theory to satisfy the stringent constraint on the speed of GWs the mass scale *M* introduced in the non-minimally derivative coupling is constrained to be in the range \(10^{15} \,\,\text {GeV}\gg M \gtrsim 2\times 10^{-35}\) GeV taking also under consideration the early times upper bound for the mass scale *M*. The large mass ranges require no fine-tuning because the effect of non-minimally derivative coupling is negligible at late times.

## 1 Introduction

The detection of gravitational waves (GWs) by the Laser Interferometer Gravitational-Wave Observatory (LIGO) Scientific Collaboration and Virgo Collaboration opens the window to study strong field gravitational physics and test alternative theories of gravity [1, 2, 3, 4, 5, 6]. In particular, the recent detection of the GW170817 from the merger of a binary neutron star [6] and the electromagnetic counterparts starts a new era of multi-messenger GW astronomy. A gamma ray burst GRB170817A was observed \(1.74\pm 0.05\) s later by Fermi Gamma-Ray Burst Monitor [7] and the International Gamma-Ray Astrophysics Laboratory [8]. If we assume that the peak of the GW signal and the first photons were emitted simultaneously, and the 1.74 s time difference is caused by the faster speed of GWs, then we get an upper bound on the speed of GWs \(c_{gw}/c-1\le 7\times 10^{-16}\) [9]. If we assume that the GRB signal was emitted 10s after the GW signal, then we get a lower bound \(c_{gw}/c-1>-3\times 10^{-15}\) [9]. The precise measurement of the propagation speed of GWs is a very powerful tool to test alternative theories of gravity [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

Recently there is a lot of activity studying scalar–tensor theories [21] and one of them is the gravitational theory which is the result of the Horndeski Lagrangian [22]. Horndeski theories because they lead to second-order field equations can be technically simple, and they prove consistent without ghost instabilities [23]. In Horndeski theory the derivative self-couplings of the scalar field screen the deviations from GR at high gradient regions (small scales or high densities) through the Vainshtein mechanism [24], thus satisfying solar system and early universe constraints [25, 26, 27, 28, 29, 30, 31, 32].

A subclass of Horndeski theories includes the coupling of the scalar field to Einstein tensor. This term introduces a new mass scale in the theory which on short distances allows to find black hole solutions [33, 34, 35, 36, 37], while a black hole can be formed if one considers the gravitational collapse of a scalar field coupled to the Einstein tensor [38]. On large distances the presence of the derivative coupling acts as a friction term in the inflationary period of the cosmological evolution [39, 40, 41, 42, 43, 44]. Also, the preheating period at the end of inflation was studied, and it was found that there is a suppression of heavy particle production as the derivative coupling is increased. This was attributed to the fast decrease of kinetic energy of the scalar field because of its wild oscillations [45]. A holographic application was performed in [46] where it was shown that the change of the kinetic energy of the scalar field coupled to Einstein tensor allowed to holographically simulate the effects of a high concentration of impurities in a real material. The above discussion indicates that the coupling of the scalar field to Einstein tensor alters the kinematical properties of the scalar field.

Assuming that the scalar field coupled to Einstein tensor plays the role of dark energy and drives the late cosmological expansion it was found [47, 48] that the propagation speed of the tensor perturbations around the cosmological background with Friedmann–Robertson–Walker (FRW) metric is different from the speed of light *c*, so the measurement of the current speed of GWs can be used to test the applicability of this and Horndeski theories to explain the late time accelerated cosmological expansion [13, 14, 15, 16, 49, 50, 51]. Due to small deviation (on the order of \(10^{-15}\)) of the current speed of GWs from the speed of light, in Refs. [13, 14, 16] it was argued that dark energy models that predict \(c_{gw}\ne c\) at late cosmological times are ruled out, and in Horndeski theory the only viable nonminimal coupling to gravity has the conformal form \(f(\phi )R\). However, with the help of derivative conformal or disformal transformations, some Horndeski and beyond Horndeski theories can survive the speed constraint [16, 36, 52, 53].

In this work, we will perform a detailed analysis on the effect of the latest observational results on the current speed of GWs \(c_{gw}\) to the Horndeski theories with the non-minimally derivative coupling. From the early cosmological evolution we know that the derivative coupling of the scalar field to Einstein tensor alters the kinetic energy of the scalar field [40] influencing in this way the dynamical evolution of the Universe giving an upper bound to the mass scale coupling. For the late cosmological evolution we will perturb the FRW metric under tensor perturbations and we will show that a subclass of Horndeski theory consisting of the usual kinetic term and the coupling of the scalar field to Einstein tensor is still viable provided that the mass scale *M* introduced in the non-minimally derivative coupling is highly constrained from the recent results on the speed of GWs. The result also shows that no tuning on the parameter is needed to satisfy the stringent constraints on the speed of GWs. For comparison we also discuss bounds on the Gauss–Bonnet coupling from the observational bounds on \(c_{gw}\).

The paper is organized as follows. In Sect. 2 we discuss the Horndeski theory, and studying the speed of tensor perturbations we obtain bounds on the mass scale introduced by the presence of the derivative coupling of the scalar field to Einstein tensor. In Sect. 3 we discuss the bounds on coupling \(\alpha \) of the Gauss–Bonnet theory coupled to a scalar field. Finally, in Sect. 4 are our conclusions.

## 2 The effect of the speed of gravitational waves in the Horndeski theory

In this section we will briefly review the Horndeski theory, we will discuss the speed of GWs in this theory and assuming that the scalar field present in the Horndeski Lagrangian plays the role of dark energy we will find a lower bound on the mass scale introduced in derivative coupling of the scalar field to Einstein tensor.

*K*, \(G_3\), \(G_4\) and \(G_5\) are arbitrary functions of \(\phi \) and

*X*, and \(G_{j,X}(\phi ,X)=\partial G_j(\phi ,X)/\partial X\) with \(j=4,5\).

This action is the most general one for scalar–tensor theory with at most second-order field equations. If we take \(K=G_3=G_5=0\) and \(G_4=M_{\text {Pl}}/2\), then we obtain Einstein’s general relativity. If we take \(G_3=G_5=0\), \(K=X-V(\phi )\), and \(G_4=f(\phi )\), then we get scalar–tensor \(f(\phi ) R\) theories. If we take \(G_4=M_{\text {Pl}}^2/2+X/(2M^2)\) or \(G_4=M_{\text {Pl}}^2/2\) and \(G_5=-\phi /(2M^2)\), then we get the non-minimally derivative coupling \(G_{\mu \nu }\nabla ^\mu \phi \nabla ^\nu \phi /(2M^2)\) with the mass scale *M* [54].

*s*” stands for the “\(+\)” or “\(\times \)” polarizations. Substituting Eqs. (7) into (6), we get

*H*as the Universe expands, the speed given by Eq. (9) can be close to 1. Instead of requiring that \(G_{4,X}=G_{5,\phi }=G_{5,X}=0\), we show that a large mass range for the coupling

*M*is allowed to satisfy the stringent constraint on the speed of GWs with negligible but nonzero deviation.

*M*of the derivative coupling. If we take the scalar field as dark energy and use the observational constraint \(1+w=\dot{\phi }^2/\rho _\phi =\dot{\phi }^2/(3M^2_{\text {Pl}}H^2_0\varOmega _\phi )\sim 0.2\) and \(\varOmega _m=0.3\), then we get the contribution of the canonical kinetic energy as

*M*

*M*as \(M \gtrsim 2\times 10^{-35}\) GeV. In the New Higgs inflation [47], the non-minimally derivative coupling enhances the friction of the expansion and the high friction limit requires \(M\ll 10^{15}\) Gev [44] while in [60] limits on

*M*are also discussed during the reheating period. Therefore, the mass scale introduced in the derivative coupling is \(10^{15}\,\,\text {GeV}\gg M \gtrsim 2\times 10^{-35}\) GeV.

^{1}

This result is interesting and it shows that the coupling of the scalar field to Einstein tensor has a complete different behaviour compared to a scalar field minimally coupled to gravity. While the kinetic energy of a minimally coupled scalar field practically does not understand the cosmological evolution, the kinetic energy of scalar field coupled to Einstein tensor changes as the Universe expands. At the inflationary epoch it can drive inflation with steep potentials while as the Universe expands its contribution to the cosmological evolution is less important and at the late cosmological epoch is negligible, so GWs propagate at the speed of light at late times.

## 3 The effect of the speed of gravitational waves in the Gauss–Bonnet theory

*A*is an integration constant, \(\rho _0\) is the present value of the energy density for matter. Note that if the coupling strength \(\alpha =0\) we do not have the Gauss–Bonnet coupling. From the solution, we get the current value of the ratio of the energy densities between dark energy and matter,

*A*,

## 4 Conclusions

The first measurement of the speed of GWs by GW170817 and GRB170817A bounds the deviation of the speed of GWs from the speed of light to be no more than one part in \(10^{15}\), so it provides the evidence that \(c_{gw}=c\). Using these observational result we can test alternative theories of gravity for their validity to describe the cosmological evolution at late times.

We used these bounds on \(c_{gw}\) to constrain first a subclass of the Horndeski theory in which a scalar field except its minimal coupling is also coupled to Einstein tensor. Assuming that the scalar field plays the role of dark energy we found a lower bound on the mass scale introduced by this coupling and combining the constraints from inflation the energy scale of the derivative coupling is bounded to be \(10^{15}\,\,\text {GeV}\gg M \gtrsim 2\times 10^{-35}\) GeV. This result requires no fine-tuning and shows that it is possible to get \(c_{gw}\approx c\) from the terms with \(G_{4,X}\ne 0\) and \(G_{5,\phi }\ne 0\) if their effects are negligible at late times.

We also studied the Gauss–Bonnet theory in four dimensions coupled to a scalar field. The coupling of the Gauss–Bonnet term to scalar field not only gives successful inflation, but also provides late time cosmic acceleration. Using a particular model with a specific form of the coupling function \(f(\phi )\) which allows an exact solution of the gravitational equations, we found that the bounds on \(c_{gw}\) constrains the Gauss–Bonnet coupling strength to be \(\alpha \lesssim 10^{-15}\), a constraint much stronger than the coupling strength \(-1\times 10^{-4}<\alpha <4\times 10^{-4}\) resulted from models with power-law inflation.

## Footnotes

## Notes

### 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.

## References

- 1.B.P. Abbott, Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett.
**116**(6), 061102 (2016)ADSMathSciNetCrossRefGoogle Scholar - 2.B.P. Abbott, GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett.
**116**(24), 241103 (2016)ADSCrossRefGoogle Scholar - 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)ADSCrossRefGoogle Scholar - 4.B.P. Abbott, GW170608: observation of a 19-solar-mass binary black hole coalescence. Astrophys. J.
**851**(2), L35 (2017)ADSCrossRefGoogle Scholar - 5.B.P. Abbott, GW170814: a three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett.
**119**(14), 141101 (2017)ADSCrossRefGoogle Scholar - 6.B.P. Abbott, GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett.
**119**(16), 161101 (2017)ADSCrossRefGoogle Scholar - 7.A. Goldstein, An ordinary short gamma-ray burst with extraordinary implications: fermi-GBM detection of GRB 170817A. Astrophys. J. Lett.
**848**(2), L14 (2017)ADSCrossRefGoogle Scholar - 8.V. Savchenko, INTEGRAL detection of the first prompt gamma-ray signal coincident with the gravitational-wave event GW170817. Astrophys. J. Lett.
**848**(2), L15 (2017)ADSCrossRefGoogle Scholar - 9.B.P. Abbott, Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys. J.
**848**(2), L13 (2017)ADSCrossRefGoogle Scholar - 10.S. Mirshekari, N. Yunes, C.M. Will, Constraining generic lorentz violation and the speed of the graviton with gravitational waves. Phys. Rev. D
**85**, 024041 (2012)ADSCrossRefGoogle Scholar - 11.J. Beltran Jimenez, F. Piazza, H. Velten, Evading the Vainshtein mechanism with anomalous gravitational wave speed: constraints on modified gravity from binary pulsars. Phys. Rev. Lett.
**116**(6), 061101 (2016)ADSCrossRefGoogle Scholar - 12.P.M. Chesler, A. Loeb, Constraining relativistic generalizations of modified Newtonian dynamics with gravitational waves. Phys. Rev. Lett.
**119**(3), 031102 (2017)ADSCrossRefGoogle Scholar - 13.T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller, I. Sawicki, Strong constraints on cosmological gravity from GW170817 and GRB 170817A. Phys. Rev. Lett.
**119**(25), 251301 (2017)ADSCrossRefGoogle Scholar - 14.P. Creminelli, F. Vernizzi, Dark energy after GW170817 and GRB170817A. Phys. Rev. Lett.
**119**(25), 251302 (2017)ADSCrossRefGoogle Scholar - 15.J. Sakstein, B. Jain, Implications of the neutron star merger GW170817 for cosmological scalar–tensor theories. Phys. Rev. Lett.
**119**(25), 251303 (2017)ADSCrossRefGoogle Scholar - 16.J.M. Ezquiaga, M. Zumalacárregui, Dark energy after GW170817: dead ends and the road ahead. Phys. Rev. Lett.
**119**(25), 251304 (2017)ADSCrossRefGoogle Scholar - 17.M.A. Green, J.W. Moffat, V.T. Toth, Modified gravity (MOG), the speed of gravitational radiation and the event GW170817/GRB170817A. Phys. Lett. B
**780**, 300 (2018)ADSCrossRefGoogle Scholar - 18.A. Nishizawa, Generalized framework for testing gravity with gravitational-wave propagation. I. Formulation. Phys. Rev. D
**97**(10), 104037 (2018)ADSCrossRefGoogle Scholar - 19.S. Arai, A. Nishizawa, Generalized framework for testing gravity with gravitational-wave propagation. II. Constraints on Horndeski theory. Phys. Rev. D
**97**(10), 104038 (2018)ADSCrossRefGoogle Scholar - 20.R.A. Battye, F. Pace, D. Trinh, Gravitational wave constraints on dark sector models. Phys. Rev. D
**98**(2), 023504 (2018)ADSCrossRefGoogle Scholar - 21.Y. Fujii, K. Maeda,
*The scalar–tensor theory of gravitation*(Cambridge University Press, 2007). http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521811597 - 22.G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space. Int. J. Theor. Phys.
**10**, 363 (1974)MathSciNetCrossRefGoogle Scholar - 23.M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg
**6**(4), 385 (1850)Google Scholar - 24.A.I. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett.
**39B**, 393 (1972)ADSCrossRefGoogle Scholar - 25.A. Nicolis, R. Rattazzi, E. Trincherini, The Galileon as a local modification of gravity. Phys. Rev. D
**79**, 064036 (2009)ADSMathSciNetCrossRefGoogle Scholar - 26.C. Deffayet, G. Esposito-Farese, A. Vikman, Covariant Galileon. Phys. Rev. D
**79**, 084003 (2009)ADSCrossRefGoogle Scholar - 27.N. Chow, J. Khoury, Galileon cosmology. Phys. Rev. D
**80**, 024037 (2009)ADSCrossRefGoogle Scholar - 28.A. De Felice, R. Kase, S. Tsujikawa, Vainshtein mechanism in second-order scalar–tensor theories. Phys. Rev. D
**85**, 044059 (2012)ADSCrossRefGoogle Scholar - 29.E. Babichev, C. Deffayet, G. Esposito-Farese, Improving relativistic MOND with Galileon k-mouflage. Phys. Rev. D
**84**, 061502 (2011)ADSCrossRefGoogle Scholar - 30.S. Chakraborty, S. SenGupta, Solar system constraints on alternative gravity theories. Phys. Rev. D
**89**(2), 026003 (2014)ADSCrossRefGoogle Scholar - 31.G. Lambiase, M. Sakellariadou, A. Stabile, A. Stabile, Astrophysical constraints on extended gravity models. JCAP
**1507**(07), 003 (2015)ADSMathSciNetCrossRefGoogle Scholar - 32.S. Bhattacharya, S. Chakraborty, Constraining some Horndeski gravity theories. Phys. Rev. D
**95**(4), 044037 (2017)ADSMathSciNetCrossRefGoogle Scholar - 33.T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos, G. Siopsis, Scalar hair from a derivative coupling of a scalar field to the Einstein tensor. Class. Quant. Grav.
**29**, 205011 (2012)ADSMathSciNetCrossRefGoogle Scholar - 34.M. Rinaldi, Black holes with non-minimal derivative coupling. Phys. Rev. D
**86**, 084048 (2012)ADSCrossRefGoogle Scholar - 35.T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos, G. Siopsis, Phase transition to a hairy black hole in asymptotically flat spacetime. JHEP
**11**, 133 (2013)ADSMathSciNetCrossRefGoogle Scholar - 36.E. Babichev, C. Charmousis, Dressing a black hole with a time-dependent Galileon. JHEP
**08**, 106 (2014)ADSCrossRefGoogle Scholar - 37.C. Charmousis, T. Kolyvaris, E. Papantonopoulos, M. Tsoukalas, Black holes in bi-scalar extensions of Horndeski theories. JHEP
**07**, 085 (2014)ADSCrossRefGoogle Scholar - 38.G. Koutsoumbas, K. Ntrekis, E. Papantonopoulos, M. Tsoukalas, Gravitational collapse of a homogeneous scalar field coupled kinematically to Einstein tensor. Phys. Rev. D
**95**(4), 044009 (2017)ADSMathSciNetCrossRefGoogle Scholar - 39.L. Amendola, Cosmology with nonminimal derivative couplings. Phys. Lett. B
**301**, 175 (1993)ADSMathSciNetCrossRefGoogle Scholar - 40.S.V. Sushkov, Exact cosmological solutions with nonminimal derivative coupling. Phys. Rev. D
**80**, 103505 (2009)ADSCrossRefGoogle Scholar - 41.C. Germani, A. Kehagias, UV-protected inflation. Phys. Rev. Lett.
**106**, 161302 (2011)ADSCrossRefGoogle Scholar - 42.E.N. Saridakis, S.V. Sushkov, Quintessence and phantom cosmology with non-minimal derivative coupling. Phys. Rev. D
**81**, 083510 (2010)ADSCrossRefGoogle Scholar - 43.Y. Huang, Q. Gao, Y. Gong, The phase-space analysis of scalar fields with non-minimally derivative coupling. Eur. J. Phys. C
**75**, 143 (2015)ADSCrossRefGoogle Scholar - 44.N. Yang, Q. Fei, Q. Gao, Y. Gong, Inflationary models with non-minimally derivative coupling. Class. Quant. Grav.
**33**(20), 205001 (2016)ADSMathSciNetCrossRefGoogle Scholar - 45.G. Koutsoumbas, K. Ntrekis, E. Papantonopoulos, Gravitational particle production in gravity theories with non-minimal derivative couplings. JCAP
**08**, 027 (2013)ADSCrossRefGoogle Scholar - 46.X.M. Kuang, E. Papantonopoulos, Building a holographic superconductor with a scalar field coupled kinematically to Einstein tensor. JHEP
**08**, 161 (2016)ADSMathSciNetCrossRefGoogle Scholar - 47.C. Germani, A. Kehagias, New model of inflation with non-minimal derivative coupling of standard model Higgs Boson to gravity. Phys. Rev. Lett.
**105**, 011302 (2010)ADSCrossRefGoogle Scholar - 48.C. Germani, Y. Watanabe, UV-protected (natural) inflation: primordial fluctuations and non-Gaussian features. JCAP
**1107**, 031 (2011). [Addendum: JCAP1107,A01(2011)]Google Scholar - 49.L. Lombriser, A. Taylor, Breaking a dark degeneracy with gravitational waves. JCAP
**1603**(03), 031 (2016)ADSCrossRefGoogle Scholar - 50.L. Lombriser, N.A. Lima, Challenges to self-acceleration in modified gravity from gravitational waves and large-scale structure. Phys. Lett. B
**765**, 382 (2017)ADSCrossRefGoogle Scholar - 51.D. Bettoni, J.M. Ezquiaga, K. Hinterbichler, M. Zumalacárregui, Speed of gravitational waves and the fate of scalar–tensor gravity. Phys. Rev. D
**95**(8), 084029 (2017)ADSMathSciNetCrossRefGoogle Scholar - 52.E. Babichev, C. Charmousis, G. Esposito-Farèse, A. Lehébel, Stability of black holes and the speed of gravitational waves within self-tuning cosmological models. Phys. Rev. Lett.
**120**(24), 241101 (2018)ADSCrossRefGoogle Scholar - 53.E. Babichev, C. Charmousis, G. Esposito-Farèse, A. Lehébel, Hamiltonian vs stability and application to Horndeski theory. arXiv:1803.11444
- 54.T. Kobayashi, M. Yamaguchi, J. Yokoyama, Generalized G-inflation: inflation with the most general second-order field equations. Prog. Theor. Phys.
**126**, 511 (2011)ADSCrossRefGoogle Scholar - 55.A. De Felice, S. Tsujikawa, Conditions for the cosmological viability of the most general scalar–tensor theories and their applications to extended Galileon dark energy models. JCAP
**1202**, 007 (2012)CrossRefGoogle Scholar - 56.S. Hou, Y. Gong, Y. Liu, Polarizations of gravitational waves in Horndeski theory. Eur. Phys. J. C
**78**, 378 (2018)ADSCrossRefGoogle Scholar - 57.Z. Yi, Y. Gong, PPN parameters in gravitational theory with nonminimally derivative coupling. Int. J. Mod. Phys. D
**26**, 1750005 (2017)ADSMathSciNetCrossRefGoogle Scholar - 58.J.F. Bell, T. Damour, A New test of conservation laws and Lorentz invariance in relativistic gravity. Class. Quant. Grav.
**13**, 3121 (1996)ADSMathSciNetCrossRefGoogle Scholar - 59.I.H. Stairs, Discovery of three wide-orbit binary pulsars: implications for binary evolution and equivalence principles. Astrophys. J.
**632**, 1060 (2005)ADSCrossRefGoogle Scholar - 60.I. Dalianis, G. Koutsoumbas, K. Ntrekis, E. Papantonopoulos, Reheating predictions in gravity theories with derivative coupling. JCAP
**1702**(02), 027 (2017)ADSCrossRefGoogle Scholar - 61.G. Koutsoumbas, K. Ntrekis, E. Papantonopoulos, E.N. Saridakis, Unification of dark matter–dark energy in generalized Galileon theories. JCAP
**1802**(02), 003 (2018)ADSCrossRefGoogle Scholar - 62.D. Lovelock, The Einstein tensor and its generalizations. J. Math. Phys.
**12**, 498 (1971)ADSMathSciNetCrossRefGoogle Scholar - 63.J. Rizos, K. Tamvakis, On the existence of singularity free solutions in quadratic gravity. Phys. Lett. B
**326**, 57 (1994)ADSMathSciNetCrossRefGoogle Scholar - 64.A. De Felice, S. Tsujikawa, Inflationary non-Gaussianities in the most general second-order scalar–tensor theories. Phys. Rev. D
**84**, 083504 (2011)ADSCrossRefGoogle Scholar - 65.S. Nojiri, S.D. Odintsov, M. Sasaki, Gauss-Bonnet dark energy. Phys. Rev. D
**71**, 123509 (2005)ADSCrossRefGoogle Scholar - 66.T. Koivisto, D.F. Mota, Cosmology and astrophysical constraints of Gauss–Bonnet dark energy. Phys. Lett. B
**644**, 104 (2007)ADSMathSciNetCrossRefGoogle Scholar - 67.T. Koivisto, D.F. Mota, Gauss–Bonnet quintessence: background evolution, large scale structure and cosmological constraints. Phys. Rev. D
**75**, 023518 (2007)ADSCrossRefGoogle Scholar - 68.G. Calcagni, B. de Carlos, A. De Felice, Ghost conditions for Gauss–Bonnet cosmologies. Nucl. Phys. B
**752**, 404 (2006)ADSMathSciNetCrossRefGoogle Scholar - 69.B.M.N. Carter, I.P. Neupane, Towards inflation and dark energy cosmologies from modified Gauss–Bonnet theory. JCAP
**0606**, 004 (2006)Google Scholar - 70.E.J. Copeland, M. Sami, S. Tsujikawa, Dynamics of dark energy. Int. J. Mod. Phys. D
**15**, 1753 (2006)ADSMathSciNetCrossRefGoogle Scholar - 71.B.M. Leith, I.P. Neupane, Gauss–Bonnet cosmologies: crossing the phantom divide and the transition from matter dominance to dark energy. JCAP
**0705**, 019 (2007)ADSMathSciNetCrossRefGoogle Scholar - 72.K. Nozari, N. Rashidi, Perturbation, non-Gaussianity, and reheating in a Gauss–Bonnet \(\alpha \)-attractor model. Phys. Rev. D
**95**(12), 123518 (2017)ADSMathSciNetCrossRefGoogle Scholar - 73.Z.K. Guo, D.J. Schwarz, Power spectra from an inflaton coupled to the Gauss–Bonnet term. Phys. Rev. D
**80**, 063523 (2009)ADSCrossRefGoogle Scholar - 74.S. Koh, B.H. Lee, W. Lee, G. Tumurtushaa, Observational constraints on slow-roll inflation coupled to a Gauss–Bonnet term. Phys. Rev. D
**90**(6), 063527 (2014)ADSCrossRefGoogle Scholar - 75.P. Kanti, R. Gannouji, N. Dadhich, Early-time cosmological solutions in Einstein–Scalar–Gauss–Bonnet theory. Phys. Rev. D
**92**(8), 083524 (2015)ADSMathSciNetCrossRefGoogle Scholar - 76.S. Koh, B.H. Lee, G. Tumurtushaa, Reconstruction of the scalar field potential in inflationary models with a Gauss–Bonnet term. Phys. Rev. D
**95**(12), 123509 (2017)ADSCrossRefGoogle Scholar - 77.M. Heydari-Fard, H. Razmi, M. Yousefi, Scalar–Gauss–Bonnet gravity and cosmic acceleration: comparison with quintessence dark energy. Int. J. Mod. Phys. D
**26**(02), 1750008 (2016)ADSMathSciNetCrossRefGoogle Scholar - 78.P.A.R. Ade, Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys.
**594**, A13 (2016)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.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 Commons license, and indicate if changes were made.

Funded by SCOAP^{3}