Manifestations of Horndeski theory in binary systems with pulsars.

Subclass of Horndeski gravity without Vainstein screening mechanism is tested in strong field regime of binary pulsars. Two specific cases of the Horndeski gravity are considered: the hybrid metric-Palatini f(R)-gravity and massive Brans-Dicke theory. The observational data of PSR J1738+0333, PSR J0737-3039, PSR J1012+5307 is used for obtaining constraints.


Introduction
There are two problems in modern phisics, which can not be explained by the generally recognized gravitational theory (General Relativity (GR)). The first of them is Dark Matter (DM), it's invisible matter, which fills up in galaxies and manifests itself only in the gravitational interaction [1]. The second problem is Dark Energy (DE), this phenomenon is responsible for accelerated expansion of the Universe [2]. One of the ways to solve these problems is to develope modified gravity. Adding the scalar field to GR is one of possible expansions of gravitational theory.
The Horndeski theory is the most general scalar-tensor theory with the second-order field equations which evades Ostrogradski instabilities [3]. Horndeski gravity suggests solutions for some GR's problems. For example, the scalar field can play the role of DE and explain the accelerating expansion of the Universe [4]. This theory has recently been studied extensively in the context of cosmology [5] and physics of black holes [6] and it has already been tested in cluster lensing [7], the cosmic microwave background (CMB) data [8] etc. Special attention should be paid to the recent work [9], where constraints for this theory were obtained by comparing LIGO data for event GW170817 [10] and the concomitant gamma-ray burst GRB 170817A [11]. In our work we also study the gravitational radiation in Horndeski gravity but use the binary pulsars data. In work [12] authors study the similar problem, but they restrict their consideration with massless case.
In our work we test the subclass of Horndeski gravity without Vainstein screening mechanism in the strong field regime of binary pulsars. The discovery of binary pulsar PSR 1913 + 16 in 1974 [13] has opened a new posibility for testing gravitational theories [14]. In such systems gravitational field is stronger than in Solar System. Moreover, high stability of the pulse arrival allows to extract the dynamics of the orbital motion with such accuracy at which the effects of gravitational waves emission could appear. It provides an additional opportunity to verify the predictions of theories.
In this work, we test the Horndeski gravity in mixed binary systems and neutron starneutron star binary (PSR J0737-3039) with quasi-circular orbits and impose restrictions on the parameters of Horndeski model.

Action
The action of Horndesky theory is given by [18]: where c is the speed of light, g is the determinant of the metric, and S m is the standard matter action. L i are the gravitational Lagrangian densities: where G µν is the Einstein tensor, R is Ricci scalar, φ is the scalar field, X = −1/2∇ µ φ∇ µ φ, ∇ µ is the covariant derivative, φ = g µν ∇ µ ∇ ν φ, G i (φ, X) are arbitrary functions of the scalar field φ and its kinetic term X, G iX = ∂G i ∂X . In subclass of Horndeski theory without Vinstein mechanism, where arbitrary functions take the following form: where G 2 (φ, X) contains no more than the first order in X.
The weak and strong equivalence principles are violated in some modified gravitational models. In scalar-tensor theories the inertial mass and internal structure depend on the scalar field. Eardley first showed that in the scalar-tensor gravity the influence of the scalar field on the internal structure of the body can be expressed through the assumption that the mass of the body is an arbitrary function of the scalar field [15]. So the matter action for a system of point-like masses can be written as where m a (φ) are inertial masses of particles labeled by a and τ a is the proper time of the particle a measured along its world-line x µ a .

Field equations
Pulsars have the strong surface gravitational potential Φ NS = (G N M NS )/(c 2 R NS ) = 0.2 (where G N is Newtonian gravitational constant, M NS is the mass of the neutron star, R NS is its radius), but we intend to calculate the energy flux carried away by gravitational radiation from binary pulsars at large distances from the source, and surface gravitational potential does not have a significant influence on the metric g µν and the scalar field φ. Therefore, we can expand the scalar and tensor fields in the approximation of weak field limit and small velocities (v/c 1) around cosmological background φ 0 and Minkowski background [19] η µν (post-Newtonian limit (PPN) [20]): h µν and ϕ are the small perturbations of tensor and scalar fields of order O(v 2 /c 2 ), respectively. Also four arbitrary functions G 2 (φ, X), G 3 (φ, X), G 4 (φ, X), G 5 (φ, X) can be expanded in Taylor's series around the scalar cosmological background: where G (m,n) are constants. Also we can write stress-energy tensor, its trace, and ∂T ∂ϕ in the near zone in terms of Eardly method [19] for point mass approximation (binary system case): where v a is the velocity of the object labeled in a. The quantities s a and s a are the "first and second sensitivities". These parameters firstly were introduced by Eardley [15]: If we want to consider gravitational radiation in far zone at the point of detector, we must solve a field equation in near zone, where gravitational radiation is generated. The field equations within the post-Newtonian (PN) approximation in the 1st PN order O(v/c) 2 take the form [21]: where In these equations we also use the following notations: and transverse gauge ∂ µ θ µν = 0. The Eq.(10) is analogue of inhomogeneous Klein-Gordon-Fock equation, where the parameter m ϕ is the inverse Compton wavelength of the scalar field.
In this paper we work in the CGS system thus here and further the scalar field mass m ϕ has the dimension of inverse centimeter (cm −1 ). Let's obtain the solution of these equations at leading order. For the scalar field, it takes the following form [21]: where r a = | r − r a (t)|. Then we solve Eq.(9) within the 1PN approximation in the near zone [21]: It is important to emphasize that in scalar-tensor theories weak equivalence principle (WEP) can be violated [22]. So that is why we must modify equations of motion. In Horndeski theory the changes are following: gravitational constant becomes a function of the scalar field, parameters of the theory, sensitivities and distance between objects. We obtain the explicit form for Kepler's third law using the A. Einstein, L. Infeld, and B. Hoffmann (EIH) method [23]: and the orbital binding energy of such system is here a is the semi-major axis, m = m 1 + m 2 , and µ = m 1 m 2 /m, P b is the orbital period of a binary system, G 12 is the effective gravitational coupling constant between two compact objects: In this work we use approximation where scalar field is responsible for DE effect: m ϕ r ab 1 and e −m ϕ r ab → 1.
For obtaining constraint on parameters of Horndeski theory, we selected the binary systems where the most significant dissipative effect is caused by gravitational radiation. Therefore, the energy losses can be expressed via the first derivative of the orbital period using Eq. (15) and Eq. (16): Further, we'll use this expression for obtaining constraints.
Energy flux of binary system is defined as [19] where t 0i is component of stress-energy pseudotensor. For calculation pseudotensor, we use Noether current method [24]: The energy flux consist of two part: tensor and scalar. The tensor part is where R is radius of the reduced mass µ orbit and in the case of quasi-circular orbit in Eq.(15) R = a. The quantity G 12 is the effective gravitational constant between components of binary system (17). The scalar part is where A d ,Ā d , A q , A o are functions of theory parameters, sensitivities and cosmological background. As seen from (22), the scalar part includes dipole term (term with A d ), PN correction to dipole term (term withĀ d ), quadrupole term (term with A q ), dipole-octupole term (term with A o ). In contrast to the tensor part, which includes only the quadrupole term.
We can obtain the expression for first derivative of orbital period:

Observational constrains on Horndeski gravity and special cases from binary pulsars
For obtaining constraints on the theories we use the following method: Here we compare the predicted quantityṖ th b /Ṗ GR b and the observational quantityṖ obs b /Ṗ GR b at 95% confidence level, where σ is the observational uncertainty.
For case with sensitivities we obtain the constraints from PSR J1738+0333 [26]: