Constraints on Scalar-Tensor Theory of Gravity by Solar System Tests

We study the motion of particles in the background of a scalar-tensor theory of gravity in which the scalar field is kinetically coupled to Einstein tensor. We constrain the value of the derivative parameter $z$ through solar system tests. By considering the perihelion precession we obtain the constrain $\sqrt{z}/m_p= 2.6\times 10^{9}$ Km., the gravitational red-shift $\frac{\sqrt{z}}{m_{p}}>2.7\times10^{\,7}$ Km., the deflection of light $\sqrt{z}/m_p>1.6 \times 10^8$ Km., and the gravitational time delay $\sqrt{z}/m_p>7.9 \times 10^9$ Km. Our results show that it is possible to constrain the value of the $z$ parameter in agreement with the following observational tests: gravitational red-shift, the deflection of light, and the gravitational time delay or with the gravitational red-shift, the deflection of light and the perihelion precession. However, it is not possible to constraint the $z$ parameter according with all the observational test that have been considered.


I. INTRODUCTION
Modified theories of gravity were recently introduced in a attempt to cure certain inconsistencies of General Relativity (GR) and to explain certain observational results on dark matter and dark energy. These theories introduce modifications of GR on short and large distances in a attempt to provide a viable theory of Gravity. The recent observational results on the Gravitational Waves (GWs) [1][2][3][4][5] provide a new area of testing alternative gravity theories and differentiating them from GR. Therefore, it is very important to study the compact objects predicted by different modified theories of gravity and possible GWs observational signatures they might give. It would also be very interesting to study if classical solar system tests on these objects such as light deflection, the perihelion shift of planets, the gravitational time-delay can give any discrepancy between the GR and the observations. Some of the simplest and extensively studied modifications of GR are the scalar-tensor theories [6]. The presence of a scalar field coupled to gravity results to black holes and compact objects dressed with a hairy matter distribution. The Horndeski Lagrangian [7] provides one of the most well studied scalar-tensor theories. This is because the Horndeski theories lead to second-order field equations and they result to consistent theories without ghost instabilities [8] and they preserve a classical Galilean symmetry [9,10]. The Horndeski theory has been studied on short and large distances. Local black hole solutions were found in a subclass of Horndeski theories which are characterized by the presence of scalar field which is kinetically coupled to Einstein tensor [11][12][13][14][15]. On large distances the presence of the derivative coupling acts as a friction term in the inflationary period of the cosmological evolution [16][17][18][19][20][21][22]. This derivative coupling introduces a mass scale in the theory which on large distances it can be constrainted by the recent results on GWs.
If we assume that the dark energy drives the late cosmological expansion and it is parameterized by a scalar field coupled to Einstein tensor it was found [23,24] that the propagation speed of the tensor perturbations around the cosmological Friedmann-Robertson-Walker (FRW) background is different from the speed of light c. Therefore the measurement of the speed of GWs can be used to constraint the value of the derivative coupling and test in general the applicability of the Horndeski theories at large distances [25][26][27][28][29][30][31].
The measurement of the speed of GWs by GW170817 and GRB170817A showed that the difference of the speed of GWs from the speed of light is not more than one part in 10 15 so we can safely conclude that c gw = c. In [32] this result was used to constraint the mass scale which is introduced in the Horndeski theory by the coupling of the scalar field to Einstein tensor. It was assumed that the scalar field plays the role of dark energy and a lower bound on the mass scale was found and combining the constraints from inflation the energy scale of the derivative coupling is bounded to be 10 15 GeV ≫ M 2 × 10 −35 GeV.
Modified gravity theories can also be compared to GR predictions at relative small scales. Solar system observations, such as light deflection, the perihelion shift of planets, the gravitational time-delay among other are described within GR. To study such effects you have to calculate the geodesics for the motion of particles around a black hole background. In [33] the perihelion precession of planetary orbits and the bending angle of null geodesics are estimated for different gravity theories in string-inspired models. The solar system effects have been studied in black hole AdS geometries by calculating the motion of particles on AdS spacetime [34][35][36][37][38][39][40][41]. The motion of massless and massive particles in the background of four-dimensional asymptotically AdS black holes with scalar hair [42] were studied in [43]. The geodesics are studied numerically and the differences in the motion of particles between the four-dimensional asymptotically AdS black holes with scalar hair and their no-hair limit were discussed. In the context of solar system and astrophysical scenarios spherically symmetric solutions resulting from the coupling of the Gauss-Bonnet with a scalar field were discussed in [44].
Care should be taken when one studies specific scalar-tensor theories and compare their predictions with GR. In general scalar fields, depending on their coupling to gravity, mediate fifth forces. Therefore in these theories scalar fields should accommodate a mechanism to suppress the scalar interaction on small scales and make sure that precision tests of gravity at solar system scales are applicant. There are various screening mechanisms to suppress scalar interactions on small scales. One of the basic screening mechanism is the Vainshtein mechanism [45] which was developed for the massive gravity (for an extensive review on the Vainshtein mechanism in massive gravity see [46]). The Vainshtein screening mechanism applies also to Galileon-like models [9] and to nonlinear massive gravity [47] in which the presence of nonlinear derivative scalar fields φ's self-interactions can suppress the propagation of fifth forces through the Vainshtein mechanism. In [48,49] the consequences of the Vainshtein mechanism was studied in scalar-tensor theories taking into account the nonlinear effect. Therefore, for models of modified gravity we need to clarify the behavior of gravity around and below the scale at which the relevant nonlinearities appear in order to test them against experiments and cosmological observations. An extensive study of the Vainshtein mechanism was carried out in [50] in the most general scalar-tensor theories with second-order equations of motion resulted from a spherically symmetric space-time with a matter source. They applied their general results to a number of concrete models such as the covariant/extended Galileons and the Dirac-Born-Infeld Galileons with Gauss-Bonnet and other terms. They found that in these theories the fifth force can be suppressed and be consistent to solar system constraints, provided that non-linear field kinetic terms coupled to the Einstein tensor do not dominate over other nonlinear field self-interactions.
The aim of this work is to constrain the parameters of the subclass of the Horndeski theory with a scalar field coupled kinetically to Einstein tensor using the solar system observations. We will work with a well studied black hole solution of the Horndeski theory with the derivative coupling [12]. In this solution the coupling function of the kinetic scalar term to Einstein tensor is constant independent of the scalar field itself, therefor we do not expect any nonlinearities to appear in our model [50]. In this model because of the shift symmetry the scalar field appears only through its derivative and then ψ = φ ′ appears as an extra degree of freedom, expressed by the real quantity ψ 2 . Following mainly the work in [51,52] we will study the effects of the solar system tests by considering the perihelion precession, the gravitational red-shift, the deflection of light and the gravitational time delay.
The paper is organized as follows. In Section II we give a brief review of the four-dimensional Horndeski black hole of [12] that we will consider as background. In Section III we study the motion of massless and massive particles, and we perform some classical tests such as the perihelion precession, the deflection of light and the gravitational time delay. Finally, in Section IV we conclude.

II. FOUR-DIMENSIONAL HORNDESKI BLACK HOLE
In this section we will present briefly the black hole solution with the non-minimal derivative coupling in the Horndeski theory [12]. Consider the Lagrangian where m p is the Planck mass, z a real number, G µν the Einstein tensor, ϕ a scalar field, and g µν is the metric. The absence of scalar potential allows for the shift symmetry ϕ → ϕ+const, which is the relevant Galileon symmetry that survives in curved space. Consider the metric ansatz Setting ρ = r the equations of motion are [12] r where K is an integration constant and ψ ≡ ϕ ′ . We can see that if ψ = 0 implies K = 0 and the resulting metric turns out to be the Schwarzschild one. When K = 0 and z = 0, analytical exact solutions of the system were found which they depend on the sign of z and to avoid nonphysical modes for the scalar field z > 0 was considered where we defined l 2 = 12z/m 2 p in the above expressions not appear l 2 and M is a constant of integration that will play the role of a mass. As it was discussed in [12] z is a non-perturbative parameter when we regard the Lagrangian (1) as a theory of modified gravity. Indeed, the deviation from GR vanishes when z diverges and the scalar field is strongly coupled. Also, the parameter z clearly interpolates between the flat black hole solution and the Schwarzschild AdS one as 1/z essentially plays the role of an effective negative cosmological constant.

III. SOLAR TEST FOR THE HORNDESKI BLACK HOLE
In order to find the effects of the solar system to the Horndeski Black hole we have to study the geodesics of the space-time described by (2). For this we will solve the Euler-Lagrange equations for the variational problem associated with this metric. The Lagrangian associated to the metric (2) is given by whereq = dq/dτ , and τ is an affine parameter along the geodesic. Since the Lagrangian (9) is independent of the cyclic coordinates (t, φ), then their conjugate momenta (Π t , Π φ ) are conserved and the equations of motion reaḋ where Π q = ∂L/∂q is the conjugate momenta of the coordinate q. Using (9), the above equation yieldṡ and the conjugate momenta are given by Now, without loss of generality, we consider that the motion develops in the invariant plane θ = π/2 andθ = 0, which is characteristic of central fields. With this choice, Eqs. (12) and (13) become where E and L are integration constants associated to each of them. So, inserting equations (14) into equation (9) we obtain where V (r) is the effective potential given by where m is the test mass, and by normalization we shall consider that m = 1 for massive particles and m = 0 for photons. Finally, using (14) and (15) we obtain the following equations dr dφ In the following we will consider the regime: r < √ z/m p . Thus, for 0 < m p r/ √ z < 1 arctan m p r/ √ z ≈ ∞ n=0 (−1) n 2n+1 (m p r/ √ z) 2n+1 . Therefore, the lapsus function Eq. (6) can be written as: Now, by considering the first three terms of the summation, we obtain Note that, the lapsus function approximates to Schwarzschild when z → ∞, and m p = 1. With this approximation, the event horizon corresponds to the real solution of F (r) = 0, given by is the generalized hypergeometric function.

A. Time-like geodesics
In order to observe the possible orbits, we plot the effective potential for massive particles (16) which is shown in Fig. 1. In the following, we describe the radial motion and the angular motion.

Radial motion
In this case L = 0. The particles always fall into the horizon from an upper distance R 0 . Note that the proper time (τ ) depends on the energy of the test particle, while that the coordinate time (t) does not depend on the energy of the test particle. In Fig. 2 we plot the proper time (τ ) and the coordinate time (t) as a function of r for a particle falling from a finite distance with zero initial velocity, we can see that the particle falls towards the horizon in a finite proper time. The situation is very different if we consider the trajectory in the coordinate time, where t goes to infinity. This physical result is consistent with the Schwarzschild black hole.

Angular motion
For the angular motion we consider L > 0. The allowed orbits depend on the value of the constant E.
• If E = E s ≈ 1.204 the particle can orbit in a stable circular orbit at r s = 14.230, see Fig. 3. • If E = E u ≈ 1.331 the particle can orbit in an unstable circular orbit at r u = 7.126. Also, there are two critical orbits that approximate asymptotically to the unstable circular orbit. For the first kind orbit the particle starts from rest and at a finite distance greater than the unstable radius. For the second kind orbit the particle starts from a finite distance greater than the horizon, but smaller than the unstable radius.
• The planetary orbits are constrained to oscillate between an apoastro and a periastro. We plot in Fig. 4 the planetary orbit for E = 1.259. We can observe that the particle completes an oscillation in an angle greater than 2π which is similar to the Schwarzschild black hole [53].
It is possible to calculate the periods of the circular orbits (r c.o. ), which can be stable (r s ) or unstable (r u ) orbits using the constants of motion √ E and L, given by (14), which yields and where T τ is the period of the orbit with respect to the proper time and T t is the period of the orbit with respect to the coordinate time. It is worth to mention that the periods depend on the lapsus function and their derivative that contain the M and z parameter. On the other hand, for the stable circular orbits is possible to find the epicycle frequency, given by κ 2 = V ′′ (r s )/2, which yields

Perihelion precession
The following treatment, performed by Cornbleet [54], allows us to derive the formula for the advance of the perihelia of planetary orbits. The starting point is to consider the line element in unperturbed Lorentz coordinates together with line element (2), where F (r) and G(r) are given by (20). So, considering only the radial and time coordinates in the binomial approximation, the transformation gives We will consider two elliptical orbits: the classical Kepler orbit in (r, t) space and a Horndeski orbit in an (r,t) space. Then, in the Lorentz space dA = R 0 rdrdφ = R 2 dφ/2, and hence which corresponds to Kepler's second law. For the Horndeski case we have where dr is given by Eq. (27). So, we can write (29) as Therefore, applying the binomial approximation wherever necessary, we obtain So, using this increase to improve the elemental angle from dφ to dφ, then for a single orbit where we have neglected products of M and z. The polar form of an ellipse is given by where ǫ is the eccentricity and l is the semi-latus rectum. In this way, plugging Eq. (33) into Eq. (32), we obtain which at first order yields Note that if we consider the limit M m 2 p → M ⊙ and z → ∞, we recovered the classical result for the Schwarzschild spacetime. Therefore, the perihelion advance has the standard value of GR plus the correction term coming from the Horndeski's theory. It is worth to mention that there is a (negative) discrepancy between the observational value of the precession of perihelion for Mercury, ∆φ Obs. = 5599.74 (arcsec/Julian − century) and the total ∆φ T otal = 5603.24 (arcsec/Julian − century), the discrepancy is ∆φ Dis. = −3.50 (arcsec/Julian − century), see [52], which is possible to be attributed as a correction coming from a scalar-tensor theory, in particular coming from the parameter z of the Horndeski's theory (∆φ Dis. = − πm 4 p l 4 15z 2 ), giving the constrain √ z/m p = 2.6×10 9 (Km).

B. Null geodesics
In the next analysis, we consider two kinds of motion: radial motion (L = 0) and angular motion (L > 0) of the photons (m = 0).

Radial motion
In this case, the master equation (15) can be written as where (+) stands for outgoing photons and (−) stands for ingoing photons. The solution of the above equation yields where r 0 is an integration constant that corresponds to the initial position of the photon, as in the Schwarzschild case. The photons always fall into the horizon from an upper distance. In Fig. 5 we plot the affine (τ ) and coordinate (t) time as a function of r for a photon falling from a finite distance (r 0 = 12), we can see that photons fall towards the horizon in a finite affine time. The situation is very different if we consider the trajectory in the coordinate time, where t goes to infinity.

FIG. 5:
The behavior of the affine (τ ) and the coordinate (t) time as a function of r, with z = 100 and E = 1.

Angular motion
In this case, the allowed orbits for photons depend on the value of the impact parameter b ≡ L/ √ E. Next, based on the impact parameter values shown in Fig. 6, where E u is the energy of the unstable circular orbit and 12z , we give a brief qualitative description of the allowed angular motions for photons, described in the following • Capture zone: If 0 < b < b u , photons fall inexorably to the horizon, and their cross section, σ, in this geometry is [55] σ = π b 2 u .
• Critical trajectories: If b = b u (E u ≈ 0.009), photons can stay in one of the unstable inner circular orbit of radius r u (r u ≈ 6.03). Therefore, photons that arrive from the initial distance r i (r + < r i < r u , or r u < r i < ∞) can fall asymptotically into a circle of radius r u . The period respect to the affine parameter (τ ) for the unstable circular orbit is Also, the coordinate period is given by • Deflection zone. If b u < b < b 0 ≡ L/ √ E ∞ , photons can fall from infinity to a minimum distance r D and return to infinity. This photons are deflected, see Fig. 7. Also, we can observe a zone where the deflection is attractive and other one repulsive. The other allowed orbits correspond to photons moving into the other side of the potential barrier, which plunges into the singularity.

Gravitational redshift
Since Horndeski black hole is a stationary spacetime there is a time-like Killing vector so that in coordinates adapted to the symmetry the ratio of the measured frequency of a light ray crossing different positions is given by for M/(m 2 p r) << 1 and rm p / √ z << 1, we obtain where we neglected products of z and M . Obviously, if we consider the limit M m 2 p → M ⊕ and z → ∞, we recover the classical result for the Schwarzschild spacetime. The clock can be compared with an accuracy of 10 −15 , the H-maser in the GP-A redshift experiment [56] reached an accuracy of 10 −14 . Therefore, by considering that all observations are well described within Einstein's theory, we conclude that the extra terms of Horndeski must be < 10 −14 . Thus, √ z m p > 2.7 × 10 7 Km , where we assume a clock comparison between Earth and a satellite at 15,000 km height, as in Ref. [57].

Deflection of light
The deflection of light is important because the deflection of light by the Sun is one of the most important test of general relativity, and the deflection of light by galaxies is the mechanism behind gravitational lenses. The Eq. (18) for photons can be written as where b is the impact parameter, and we have used Eq. (20). Now, by performing the change of variables r = 1/u, the above equation yields Notice that for z → ∞, and m p = 1, the above equation is reduced to the classical equation of Schwarzschild for the motion of photons given by du dφ So, the derivative of Eq. (45) with respect to φ yields where ′ denotes the derivative with respect to φ. So, following the procedure established in Ref. [58], we obtain In the limit u → 0, φ approaches φ ∞ , with Therefore, for the Horndeski black holes the deflection of lightα is equal to 2 |−φ ∞ | and yieldŝ Note that if we consider the limit M m 2 p → M ⊙ and z → ∞, we recovered the classical result of GR; that is, α GR = 4M ⊙ /b. If the impact parameter is equal to the radius of the sun, the value obtained isα GR = 4M ⊙ /R ⊙ = 1.75092 ′′ . The parameterized post-Newtonian (PPN) formalism introduces the phenomenological parameter γ, which characterizes the contribution of space curvature to gravitational deflection. In this formalism the deflection angle isα = 0.5(1 + γ)1.7426, and currently γ = 0.9998 ± 0.0004 [59]. So,α = 1.74277 ′′ for γ = 0.9998 + 0.0004 and α = 1.74208 ′′ for γ = 0.9998 − 0.0004. The observational values, compared to the classic result, are smaller, and the contribution of the Horndeski term to the deflection angle is positive, therefore, there is no observable effect. Thus, if the Horndeski term contributes it does so thatα Horndeski < 0.00001 ′′ , or √ z/m p > 1.6 × 10 8 (Km).

Gravitational time delay
An interesting relativistic effect in the propagation of light rays is the apparent delay in the time of propagation for a light signal passing near the Sun, which is a relevant correction for astronomic observations, and is called the Shapiro time delay. The time delay of Radar Echoes corresponds to the determination of the time delay of radar signals which are transmitted from the Earth through a region near the Sun to another planet or spacecraft and then reflected back to the Earth. The time interval between emission and return of a pulse as measured by a clock on the Earth is t 12 = 2 t(r 1 , ρ 0 ) + 2 t(r 2 , ρ 0 ) , where ρ 0 as closest approach to the Sun. Now, in order to calculate the time delay we use (17), (20) and the coordinate timeṙ so, (15) can be written as By considering ρ 0 as closest approach to the Sun, dr/dt vanishes, so that Now, by inserting (54) in (53) Therefore, for the circuit from point 1 to point 2 and back the delay in the coordinate time is ∆t := 2 t(r 1 , ρ 0 ) + t(r 2 , ρ 0 ) − r 2 1 − ρ 2 0 − r 2 2 − ρ 2 0 = ∆t M + ∆t z , where ∆t M = 2M m 2 p 2 ln (r 1 + r 2 1 − ρ 2 0 )(r 2 + r 2 2 − ρ 2 0 )