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Constraints on extended gravity models through gravitational wave emission

Using recent experimental results of detection of gravitational waves from the binary black hole signals by Advanced LIGO and Advanced Virgo, we investigate the propagation of gravitational waves in the context of fourth order gravity nonminimally coupled to a massive scalar field. In particular, we impose constraints on the free parameters of extended gravity models from the current observational data.


Introduction
The recent observations of gravitational radiation [1][2][3][4][5][6][7][8] from binary black holes by the Advanced LIGO and the Advanced Virgo detectors, offered the ultimate confirmation of Einstein's theory of General Relativity (GR) and a novel opportunity to test all models of extended gravity in a strong field dynamical regime. These alternative models to GR have been widely studied and tested to the regime in which the Newtonian and post-Newtonian contributions describe very well the dynamics. In particular, they have been constrained by Earth tests (e.g., gravitoelectromagnetism and torsion balance experiments), Solar System tests and by astrophysical data on the variation of the orbital period of binary systems. However, direct detection of gravitational waves from coalescing binary black holes, in an extreme gravitational field regime, offers a new test for extended theories of gravity.
On August 2017, the association of gravitational wave GW170817 [9] and gamma-ray burst GRB170817A [10] events allowed to verify experimentally, with deviations smaller than a few 10 −15 [11], that gravitational waves travel at the speed of light c. This measurement has considerable relevance since it provides new insight into fundamental physics and probes the speed of gravitational waves over cosmological distances.
Einstein's theory of GR has been tested to very high precision in the last decade [12]. To date it is the best theory of gravitational interaction, however despite its great success there are still open questions which make the theory incomplete. Several observational data [13][14][15][16][17][18] probe that the Universe is spatially flat and is undergoing a period of accelerated expansion. To understand and describe the current astrophysical data two unrevealed ingredients are needed in order to achieve this phenomenology: dark matter at galactic and extragalactic scales and dark energy at cosmological scales. The extended gravity models have been considered as a viable theoretical mechanism to explain cosmic acceleration and galactic rotation curves. In such models, one extends only the geometric sector, without introducing any exotic matter. In such models, gravity shows a different behaviour either above (infrared modification) or below (ultraviolet modification) a certain length scale, whilst the robust results of General Relativity at local and Solar System scales are preserved [40][41][42][43][44][45][46][47][48][49][50][51][52] (see Refs. [53,54] for a class of models related to Horndeski theory).
Corrections to the gravitational Lagrangian were already considered by several authors [40][41][42][43][44][45][46][47][48][49][50][51][52]. From a conceptual viewpoint, there is no reason a priori to restrict the gravitational Lagrangian to a linear function of the Ricci scalar minimally coupled to matter. In particular, one may consider the generalization of f (R) models, where R is the Ricci scalar, through generic functions containing curvature invariants such as the Ricci squared (R αβ R αβ ) or the Riemann squared (R αβγδ R αβγδ ), which however are not invariant due to the Gauss-Bonnet invariant R 2 − 4R µν R µν + R µνλσ R µνλσ . Note that the same remark applies to the Weyl invariant C αβγδ C αβγδ . Hence, one may add a (massive propagating) scalar field coupled to geometry; this leads to the scalar-tensor fourth order gravity.
Using recent experimental results of detection of gravitational waves from the binary black hole signals by Advanced LIGO and Advanced Virgo, we will study the propagation of gravitational waves in the context of fourth order gravity nonminimally coupled to a massive scalar field, in the weak field limit (for different models see Refs. [55][56][57][58][59]. In particular, we will impose constraints on the free parameters of extended gravity models from the current observational data.
The outline of this paper is the following. In Section 2, we give the action of a scalar tensor fourth order gravity and write down the corresponding field equations, which we then solve in the presence of matter within the weak-field approximation. In Section 3, using the recent observations of gravitational radiation from binary black holes by the Advanced LIGO and the Advanced Virgo detectors we put the constraint on the free parameters of the teories.
In Section 4 ,we analyze some models of extended theory. We summarize our conclusions in Section 5.

Scalar Tensor Fourth Order Gravity
Let us consider the action f is an unspecified function of the Ricci scalar R, the curvature invariant R αβ R αβ ≡ Y (where R µν is the Ricci tensor), and the scalar field φ. The L m denotes the minimally coupled ordinary matter Lagrangian density, ω is a generic function of the scalar field φ, and g stands for the determinant of the metric tensor g µν and X ≡ 8πG. Note that unless otherwise specified, we use natural units such that c = 1.
The field equations (2.6) generalise those of Ref. [22], since in the latter there was no scalar field component. Let us also note that these equations are the weak-field limit of the model discussed in Refs. [26,27].
Applying on the first member of Eq. (2.8) the operators ( η + m Y 2 ) η and ( η + m 2 + )( η + m 2 − ), we get: Considering the associated homogeneous field equation: we note that the dynamics of the perturbation h µν is made by four different modes: a massles one, which is the standard graviton, and three massive modes with momenta m + , m − and m Y . These quantities are free parameters which can be constrained via observational data. In our analysis, we suppose that all momenta are different.

Experimental constraints
Let us define the rest mass M α , where M α = m α /c with α = {+, −, Y }. For the graviton (massless mode) the dispersion relation 3 is E 2 = c 2 p 2 , while for the three massive modes, E 2 = p 2 c 2 + M 2 α c 4 , with α = {+, −, Y }., or equivalently: The group velocity [60], v = p c 2 E associated with the dispersion relation (3.1) is: where v, p and E denote respectively, the velocity, momentum and energy. Note that if the rest mass vanishes, M α = 0, the group velocity is c, and we have a massless particle, i.e. the graviton. Whereas, for M α = 0, the group velocity is less than that of the graviton; we have a massive particle.
Our aim is to test the compatibility of extended gravity models with experimental data. Using the recent experimental results on the detection of gravitational waves by Advanced LIGO and Advanced Virgo [7,61], we impose the constraint c m α > 0.583 × 10 −22 eV, following Ref. [61]. The constraint that we then obtain for the three momenta is Table 1: Here some models of extended theories of gravity.

Scalar tensor fourth order gravity models
Let us consider some extended gravity models studied in the literature: • Case A: f (R) denotes a family of theories, each one defined by a different function f of the Ricci scalar R. The simplest case is for f being equal to Ricci scalar, obtaining just General Relativity. In this case the characteristic scale m R (see, Table 1, case A) depends only on the first and second derivatives of f (R). Therefore, if we consider, for example, a polynomial expression f (R) = R + α R 2 + N n = 3 α n R n , only the terms R and R 2 make contributions. Using the constraint Eq. (3.3), i.e. m R > m 0 , we get the following relation between the derivates of f (R): A particular case of an f (R) theory is the Starobinsky model, f (R) = R − R 2 /R 0 [41], with R 0 a constant. Here the momentum (2.7) reads m 2 R = R 0 /6 and using Eq. (3.3) we obtain However, the current best constraint on R 0 is given by Solar System tests [62] and reads R 0 > 2.5 × 10 8 m −2 .
• Case B: f (R, R αβ R αβ ), namely we also include the curvature invariant R αβ R αβ . In this case there are two characteristic scales m R and m Y . Using Eq. (3.3), m R > m 0 , and m Y > m 0 , we obtain As an illustration, let us consider: where R 0 and R ic are constants: This class of theories includes the case of Weyl squared, i.e. C µνρσ C µνρσ = 2R µν R µν − 2 3 R 2 , where there is only one characteristic scale since m R → ∞.
• Case C: Scalar-tensor models f (R, φ) + ω(φ)φ ;α φ ;α . There there are two effective scales m + and m − generated from the interaction between gravity and the scalar field (see, Table 1, case C). Imposing Eq. (3.3), m + > m 0 and m − > m 0 , we get respectively, or, equivalently, As an illustration, let us consider the the following scalar-tensor theory for which Eq. (4.3) read Equivalently, one gets and In general, the constraints one can obtain are rather weak, see Eq. (4.2), but can in principle be improved once further data of nearby binary black hole signals are available.

Conclusions
We have studied the propagation of gravitational waves in the context of fourth order gravity nonminimally coupled to a massive scalar field, in the weak-field approximation.
We have seen that, in general, extended models of gravity have some massive modes in addition to the massless mode (graviton) of General Relativity. Using the recent observations of gravitational radiation from the binary black hole signals detected by Advanced LIGO and Advanced Virgo, published in the catalog GWTC-1 [6], we have constrained the free parameters of extended gravity models. These constraints are weaker compared to the ones obtained from experimental data of a different origin [26]. Nevertheless, the analysis presented here is important for two reasons. Firstly, there could exist a, yet unknown, screening mechanism operating on Earth and Solar System scales, (for instance the chameleon screening [63][64][65][66]), but could not manifest on larger (astrophysical) scales. In fact, one may expect that the variation of speed of GWs might occur in high density environments, but such a screening effect should be reduced over distances of order 40 Mpc [53,56]. Secondly, the approach we have followed here can be used once further data of nearby binary black hole signals are available, probably leading to stronger constraints. Moreover, further observations over larger distances could provide limits on both screening mechanisms and higher derivative corrections, in particular on the effective model used in the analysis presented here.