Eur. Phys. J. C (2017) 77:191 DOI 10.1140/epjc/s10052-017-4760-6 Regular Article - Theoretical Physics Parameterized post-Newtonian approximation in a teleparallel model of dark energy with a boundary term H. Mohseni Sadjadia Department of Physics, University of Tehran, Tehran, Iran Received: 26 January 2017 / Accepted: 15 March 2017 / Published online: 27 March 2017 © The Author(s) 2017. This article is an open access publication Abstract We study the parameterized post-Newtonian (PPN) parameters and the theory is consistent with gravita- approximation in teleparallel model of gravity with a scalar tional tests and solar system observations. field. The scalar field is non-minimally coupled to the Recently a new coupling between the scalar field and a scalar torsion as well as to the boundary term intro- boundary term B, corresponding to the torsion divergence μ duced in Bahamonde and Wright (Phys Rev D 92:084034 B ∝∇μT , was introduced in [1], where the cosmologi- arXiv:1508.06580v4 [gr-qc], 2015). We show that, in con- cal consequences of such a coupling for some simple power trast to the case where the scalar field is only coupled to the law scalar field potential and the stability of the model were scalar torsion, the presence of the new coupling affects the discussed. There it was found that the system evolves to an parameterized post-Newtonian parameters. These parame- attractor solution, corresponding to late time acceleration, ters for different situations are obtained and discussed. without any fine tuning of the parameters. In this framework, the phantom divide line crossing is also possible. Thermo- dynamics aspects of this model were studied in [41]. This 1 Introduction model includes two important subclasses, i.e. quintessence non-minimally coupled to the Ricci scalar and quintessence In a teleparallel model of gravity, instead of the torsionless non-minimally coupled to the scalar torsion. Another impor- Levi-Civita connections, curvatureless Weitzenböck connec- tant feature of this model is its ability to describe the present tions are used [2–4]. A teleparallel equivalent of general rel- cosmic acceleration in the framework of Z2 symmetry break- ativity was first introduced in [5] as an attempt for unification ing by alleviating the coincidence problem [42]. of electromagnetism and gravity. This theory is considered In this paper, we aim to investigate whether this new as an alternative theory of usual general relativity and has boundary coupling may affect the Newtonian potential and been recently employed to study the late time acceleration of PPN parameters: γ(r) and β(r). the Universe [6–8]. This can be accomplished by considering The scheme of the paper is as follows: In the second sec- modified f (T ) models [9–24], where T is the torsion scalar, tion we introduce the model and obtain the equations of or by introducing exotic field such as quintessence. Assum- motion. In the third section, we obtain the weak field expan- ing a non-minimal coupling between the scalar field and the sion of the equations in the PPN formalism and obtain and torsion opens new windows in studying the cosmological discuss their solutions for spherically symmetric metric. We evolution [25–31], and can be viewed as a promising sce- show that the PPN parameters may show deviation from gen- nario for late time acceleration and super-acceleration [32]. eral relativity. We consider different special cases and derive A non-minimally coupled scalar field, like the scalar– explicit solutions for the PPN parameters in terms of the tensor model, may alter the Newtonian potential. So it is model parameters and confront them with observational data. necessary to check if the model can pass local gravitational We use units with h¯ = c = 1 and choose the signature tests such as solar system observations. This can be done in (−, +, +, +) for the metric. the context of the parameterized post-Newtonian formalism [35–40]. In [33,34] it was shown that when the scalar field is only coupled to the scalar torsion, there is no deviation 2 The model and the field equations from general relativity in the parameterized post-Newtonian μ In our study we use vierbeins ea = ea ∂μ, whose duals, a a a ν ν e-mail: [email protected] e μ, are defined through e μea = δμ. The metric ten- 123 191 Page 2 of 8 Eur. Phys. J. C (2017) 77 :191 μν μ ν sorisgivenbyg = ηabea eb , η = diag(−1, 1, 1, 1). ν 1 2 √ −δμ + φ e = det(eaμ) = det −g. Greek indices (indicating coordi- k2 nate bases) like the first Latin indices (indicating orthonor- −1 a λα 1 ν α ν (e e α∂λ(eSa )) − δμ∂αφ∂ φ − 2δμV (φ) mal bases) a, b, c, ... belong to {0, 1, 2, 3}, while i, j, k, ... ∈ 2 { , , } 1 2 3 . ν βα 3 ν 2 1 ν −2δμ(χ + )φSα ∂β φ − χδμφ = δμτ. (8) Our model is specified by the action [1] 2 2 T 1 μ 2 2 By combining (8) and (7) we obtain S = + (−∂μφ∂ φ + T φ + χBφ ) 2k2 2 2 2 −1 a λν ρ νβ 1 ν + 2 φ e e μ∂λ(eSa ) − T βμSρ − δμT 4 k2 4 −V (φ) + Lm ed x, (1) ν ν βν −δμV (φ) − ∂ φ∂μφ + 4(χ + )φSμ ∂β φ 2 = π where k 8 G N , and G N is the Newtonian gravitational ν 2 1 ν 2 −χ∇ ∇μφ − χδμφ constant. The torsion scalar is defined by 2 ν 1 2 −1 a λα ρ μν 1 ρ μν 1 ρ νμ ρ νμ −δμ + φ (e e α∂λeSa ) T = S μν Tρ = T μν Tρ + T μν T ρ−T μρ T ν, k2 4 2 ν βα (2) −2δμ(χ + )φSα ∂β φ ν 1 ν =−τμ + δμτ. (9) and the boundary term is [43,44] 2 τ = B = 2∂ ( μ), Note that the trace of the energy-momentum tensor is μ eT (3) μν e g τμν. In the same way, variation of the action with respect the μ = λ μ where T T λ . The Weitzenböck torsion and connection scalar field gives are given by 1 μν λ = λ − λ = λ a − ∂μeg ∂νφ − χBφ − T φ + V (φ) = 0. (10) T μν μν νμ ea T μν (4) e and Equations (9) and (10) are the main equations that we will work with in the following. λ λ a μν = ea ∂μe ν, (5) ρ respectively. S μν is defined according to 3 Post-Newtonian formalism ρ 1 ρ ρ ρ 1 ρ σ 1 ρ σ To investigate the post-Newtonian approximation [35–40]of S μν = (T μν − Tμν + Tνμ ) + δμT νσ − δν T μσ . 4 2 2 the model, the perturbation is specified by the velocity of (6) 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- =− +B Note that R T , where R is the Ricci scalar curvature. Newtonian hydrodynamics: Hence for χ =− the model reduces to a quintessence model coupled non-minimally to the scalar curvature, while τμν = (ρ + ρ + p)uμuν + pgμν, (11) for χ = 0, we recover the quintessence model coupled non- minimally to the scalar torsion. where ρ is energy density, p is the pressure and is the By variation of the action (1) with respect to the vierbeins specific internal energy. uμ is the four-vector velocity of the we obtain i vi = u fluid. The velocity of the source matter is u0 . The orders 2 2 −1 a λν ρ νβ 1 ν + φ e e μ∂λ(eS ) − T βμSρ − δ T of smallness of the energy-momentum tensor ingredients are 2 2 a μ k 4 [35–40] ν 1 α −δμ − ∂αφ∂ φ − V (φ) p 2 ρ ∼ ∼ ∼ U ∼ O(2) (12) ν βν ρ −∂ φ∂μφ + 4(χ + )φSμ ∂β φ ν 2 ν 2 ν +χ(δμφ −∇ ∇μφ ) =−τμ. (7) where U is the Newtonian gravitational potential. The com- ν ponents of the energy-momentum tensor are given by τμ is the energy-momentum tensor of matter. ν 0 2 The trace of (7), multiplied by −δμ/2, is τ0 =−ρ − ρv − ρ + O(6) 123 Eur. Phys. J. C (2017) 77 :191 Page 3 of 8 191 i i τ0 =−ρv + O(5) After these preliminaries, let us solve Eqs. (9) and (10) order τ j = ρv j v + δ j + O( ). by order in the PPN formalism. At zeroth order (9) and (10) i i p i 6 (13) imply We expand the metric around the Minkowski flat background as [33,34] V0 = V1 = 0. (22) (2) (3) (4) gμν = ημν + h μν + h μν + h μν + O(5). (14) The 0-0 component of (9)gives (1) 2 2 −1 a λ0 ρ 0β 1 Note h μν = 0[35–40]. Accordingly, the vierbeins may be + 2 φ e e ∂λ(eSa ) − T β Sρ − T k2 0 0 4 expanded as [34] 0 −V (φ) − ∂ φ∂0φ a a (2)a (3)a (4)a e μ = δ + B μ + B μ + B μ + O(5). (15) β0 0 2 1 2 μ +4(χ + )φS0 ∂β φ − χ∇ ∇0φ − χφ 2 (1)a 1 2 −1 a λα Note B μ = 0. In our analysis we need non-zero compo- − + φ (e e α∂λeS ) 2 a (2), (2), (3), (4) k nents of the metric up to order 4, i.e. hij h00 h0i h00 . = η σ δ σ a = jα 0 1 We also use the notation Bμν μσ B ν and a B ν −2(χ + )φSα ∂ j φ =−τ + τ, (23) σ 0 2 B ν.