Parameterized post-Newtonian approximation in a teleparallel model of dark energy with a boundary term

We study the parameterized post-Newtonian approximation in teleparallel model of gravity with a scalar field. The scalar field is non-minimally coupled to the scalar torsion as well as to the boundary term introduced in Bahamonde and Wright (Phys Rev D 92:084034 arXiv:1508.06580v4 [gr-qc], 2015). We show that, in contrast to the case where the scalar field is only coupled to the scalar torsion, the presence of the new coupling affects the parameterized post-Newtonian parameters. These parameters for different situations are obtained and discussed.

A non-minimally coupled scalar field, like the scalartensor model, may alter the Newtonian potential. So it is necessary to check if the model can pass local gravitational tests such as solar system observations. This can be done in the context of the parameterized post-Newtonian formalism [35][36][37][38][39][40]. In [33,34] it was shown that when the scalar field is only coupled to the scalar torsion, there is no deviation from general relativity in the parameterized post-Newtonian a e-mail: mohsenisad@ut.ac.ir (PPN) parameters and the theory is consistent with gravitational tests and solar system observations.
Recently a new coupling between the scalar field and a boundary term B, corresponding to the torsion divergence B ∝ ∇ μ T μ , was introduced in [1], where the cosmological consequences of such a coupling for some simple power law scalar field potential and the stability of the model were discussed. There it was found that the system evolves to an attractor solution, corresponding to late time acceleration, without any fine tuning of the parameters. In this framework, the phantom divide line crossing is also possible. Thermodynamics aspects of this model were studied in [41]. This model includes two important subclasses, i.e. quintessence non-minimally coupled to the Ricci scalar and quintessence non-minimally coupled to the scalar torsion. Another important feature of this model is its ability to describe the present cosmic acceleration in the framework of Z 2 symmetry breaking by alleviating the coincidence problem [42].
In this paper, we aim to investigate whether this new boundary coupling may affect the Newtonian potential and PPN parameters: γ (r ) and β(r ).
The scheme of the paper is as follows: In the second section we introduce the model and obtain the equations of motion. In the third section, we obtain the weak field expansion of the equations in the PPN formalism and obtain and discuss their solutions for spherically symmetric metric. We show that the PPN parameters may show deviation from general relativity. We consider different special cases and derive explicit solutions for the PPN parameters in terms of the model parameters and confront them with observational data.

The model and the field equations
In our study we use vierbeins e a = e a μ ∂ μ , whose duals, e a μ , are defined through e a μ e a ν = δ ν μ . The metric ten-sor is given by g μν = η ab e a μ e b ν , η = diag(−1, 1, 1, 1 where k 2 = 8π G N , and G N is the Newtonian gravitational constant. The torsion scalar is defined by and the boundary term is [43,44] where T μ = T λ λ μ . The Weitzenböck torsion and connection are given by and λ μν = e a λ ∂ μ e a ν , respectively. S ρ μν is defined according to Note that R = −T +B, where R is the Ricci scalar curvature. Hence for χ = − the model reduces to a quintessence model coupled non-minimally to the scalar curvature, while for χ = 0, we recover the quintessence model coupled nonminimally to the scalar torsion. By variation of the action (1) with respect to the vierbeins we obtain 2 τ ν μ is the energy-momentum tensor of matter. The trace of (7), multiplied by −δ ν μ /2, is By combining (8) and (7) we obtain Note that the trace of the energy-momentum tensor is τ = g μν τ μν . In the same way, variation of the action with respect the scalar field gives Equations (9) and (10) are the main equations that we will work with in the following.

Post-Newtonian formalism
To investigate the post-Newtonian approximation [35][36][37][38][39][40] of the model, the perturbation is specified by the velocity of the source matter | v| such that e.g. O(n) ∼ | v| n . The matter source is assumed to be a perfect fluid obeying the post-Newtonian hydrodynamics: where ρ is energy density, p is the pressure and is the specific internal energy. u μ is the four-vector velocity of the fluid. The velocity of the source matter is v i = u i u 0 . The orders of smallness of the energy-momentum tensor ingredients are [35][36][37][38][39][40] where U is the Newtonian gravitational potential. The components of the energy-momentum tensor are given by We expand the metric around the Minkowski flat background as [33,34] Note h (1) μν = 0 [35][36][37][38][39][40]. Accordingly, the vierbeins may be expanded as [34] Note B (1) a μ = 0. In our analysis we need non-zero components of the metric up to order 4, i.e. h (2) i 00 . We also use the notation B μν = η μσ B σ ν and δ a σ B a ν = B σ ν . By comparing (14) and (15) we derive (like in [33,34], We introduce two functions A, and γ (which is one of the PPN parameters) through [33] B (2) The scalar field is expanded as where and φ 0 is a constant cosmological background. φ 0 is of order O(0) and may evolve in times of order of the Hubble time, so in solar system tests we assume that it is static. The time derivatives, ∂ 0 = ∂ ∂t , of the other fields are weighted with order O(1) [35][36][37][38][39][40].
The potential around the background is Defining After these preliminaries, let us solve Eqs. (9) and (10) order by order in the PPN formalism. At zeroth order (9) and (10) imply The 0-0 component of (9) gives which at order 2 reduces to resulting in where the potential is given by To obtain (25), we have used By taking the trace of the i-j component of (9), at order 2, we obtain which reduces to At the second order perturbation, the boundary term B, defined in (3), is derived: Hence from (10) the equation of motion of the scalar field becomes Equations (25), (29), and (31) are our three main equations for determining A, γ , and ψ (2) . Using these three equations, for a given U , A is derived: and ψ (2) is obtained: γ is determined by the equation where := γ −1 γ +1 . (34) is a nonhomogeneous screened Poisson equation whose solution is where Equation (32) allows us to take where G is defined through So one can define an effective G eff. through The 0-i component of (7) at the third order gives which, by using reduces to To simplify computations one may employ the gauge condition which determines B (3) 0 j in terms of second order parameters. This gauge is compatible with Eqs. (25) and (29). Using one can find that (23) at the fourth order gives Also, the scalar field equation at the fourth order is By using (47) and Equations (45) and (49) are our main results in the fourth order. These equations together with (42) and (43) in the third order, and (32)- (34), in the second order must be solved to give the post-Newtonian parameters.

Spherically symmetric metric
The source is assumed to be and the metric is given by where "Self" denotes self-energy terms of order 4, and β is the PPN parameter. The Newtonian potential is To determine γ , from (32), (33), and (35), we obtain and where in which and λ is given by (36). From h (2) 00 = 2 A = 2GU , we obtain G as (37).
For χ = 0, we may have also a situation with no deviation in the PPN parameters from general relativity, this occurs for φ 0 = 0. For example for the potentials and V 0 = V 1 = 0 (see (22)) leads to φ 0 = 0, which by using (59-61) results in γ = 1, G = 1, and β = 1. Therefore in this case too, there is no deviation from general relativity for the PPN parameters.

Range of parameters
The most precise value for γ experimentally has been obtained from Cassini [45]. The bound on this parameter is [46] |γ − 1| 2.3 × 10 −5 . (79) In this experiment the gravitational interaction, in terms of Astronomical Units takes place at r 7.44 × 10 −3 AU [47]. The parameter β is determined by lunar laser ranging experiments via the Nordtvedt effect [48]. This test indicates the bound [46] at a gravitational interaction distance r = 1AU [47]. Equations (79) and (80) restrict the parameters of our model. For V (φ) = 0, (79)and (68) give In the limiting cases (71) and (72) we find and respectively.

Conclusion
The teleparallel model of gravity with quintessence (nonminimally) coupled to the torsion and also to a boundary term (proportional to the torsion divergence) was considered (see (1)). Although the model shows some interesting aspects in cosmology and in describing the late time acceleration of the Universe, it must also pass local gravitational and solar system tests. So we studied the parameterized post-Newtonian (PPN) approximation of the model. We obtained the equations of motion (see the Sect. 2), and solve them order by order to obtain PPN parameters (see Sect. 3). Explicit expressions for the PPN parameters in a spherically symmetric metric were obtained and different possible situations were discussed. Our results show that the PPN parameters, except for some special cases, i.e. in the absence of boundary terms and also with zero scalar field background, differ from general relativity. So we conclude that coupling of the scalar field to the boundary term generally makes the model deviate from general relativity in the PPN limit.
Since T and B are not invariant under local Lorentz transformations, the teleparallel model with boundary term is not invariant under Lorentz transformations unless one takes χ = − . Despite this, in spacetimes with spherical symmetry like Schwarzschild spacetime and so on, it is possible to choose good or preferred tetrads to solve this issue [49]. In scalar-tetrad theories of gravity the effect of the preferred tetrads cannot be detected via measuring the metric components [50,51]. Similarly, in our model, the PPN parameters in the standard post-Newtonian formalism do not identify the effect of the preferred tetrads. To include these effects one must generalize the post-Newtonian approach, as was pointed out in [33].
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