Parameterized Post-Newtonian Approximation in a Teleparallel Model of Dark Energy with a Boundary Term

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 © 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. By comparing (14) and (15) we derive (like in [33,34], (2) which at order 2 reduces to Bij is assumed to be diagonal) (2) (2) 1 2 j0 1 2 1 2 ji h = 2B + φ ∂ j S − V (φ) − χφ − + φ ∂ j Si ij ij k2 0 0 2 k2 ( ) ( ) h 2 = 2B 2 ρ 00 00 = , (24) (3) = (3) 2 h0i 2B0i ( ) ( ) ( ) resulting in h 4 = 2B 4 − (B 2 )2. (16) 00 00 00 We introduce two functions A, and γ (which is one of the 1 2 2 2 (2) 1 2 − + φ ∇ A − χφ0∇ ψ =− ∇ U, (25) PPN parameters) through [33] k2 0 k2 (2) = B00 A where the potential is given by (2) B = γ Aδij. (17) ij k2 ∇2U =− ρ. (26) The scalar ﬁeld is expanded as 2

To obtain (25), we have used φ = φ0 + ψ, (18) (2)0 (2) j S j0 =−∂ j (γ A), S ij = ∂i ((1 − γ)A) , where (2)i S 0 j = 0 ( ) ( ) ψ = ψ 2 + ψ 4 + O(6), (19) (2) (2)0 ∂μe = ∂μ ((3γ − 1)A) , T i0 =−∂i A, (2)0 S 0i = ∂i (γ A). (27) and φ0 is a constant cosmological background. φ0 is of order O(0) and may evolve in times of order of the Hubble time, By taking the trace of the i-j component of (9), at order 2, so in solar system tests we assume that it is static. The time we obtain ∂ = ∂ derivatives, 0 ∂t , of the other ﬁelds are weighted with order O(1) [35–40]. − 1 + φ2 ∂ j0 − 1 + φ2 ∂ ji − χφ ∇2ψ(2) 3 0 j S0 0 j Si 5 0 The potential around the background is k2 k2 3 =− ρ, (28) (φ ) V 0 2 2 V (φ) = V (φ0) + V (φ0)ψ + ψ + O(6). (20) 2 which reduces to

(n) V (φ0) Deﬁning V (φ0) = V0, = Vn we ﬁnd 1 ( ) 3 n! + φ2 ∇2 ((4γ − 1)A) − 5χφ ∇2ψ 2 = ∇2U. k2 0 0 k2 2 V = V1 + 2V2ψ + 3V3ψ + O(6). (21) (29) 123 191 Page 4 of 8 Eur. Phys. J. C (2017) 77:191

At the second order perturbation, the boundary term B, The 0-i component of (7) at the third order gives deﬁned in (3), is derived: 2 2 μi (3) i i (2) 2 + 2 φ ∂μ S =−τ = ρv , (40) B = 2∇ ((1 − 2γ)A). (30) k2 0 0 0

Hence from (10) the equation of motion of the scalar ﬁeld which, by using becomes (3)0 (3)0 (3)0 T ij = ∂i B j − ∂ j B i −∇2ψ(2) + ψ(2) = χ( − γ) φ . 2V2 2 1 2 A 0 (31) (3)i = ∂ (3)i − δi ∂ (γ ) T j0 j B 0 j 0 A (3)i (3)i Equations (25), (29), and (31) are our three main equations T i0 =−3∂0(γ A) + 3∂i B 0, (41) for determining A, γ , and ψ(2). Using these three equations, for a given U, A is derived: reduces to 2 2 1 2 (3)0 1 j (3)0 2 + 2 φ ∂0∂i (γ A) − ∇ B i + ∂ ∂i B j A = U, (32) k2 0 2 2 (1 + φ2k2)(1 + γ) 0 = ρvi . (42) ( ) and ψ 2 is obtained: To simplify computations one may employ the gauge con- dition γ − (2) 1 ψ = U. (33) μ χ 2φ 2χφ (γ + ) j (2)i 1 i (2) k 0 i (2) k 0 1 −∂ B + ∂ B μ = ∂ ψ j 2 + φ2 2 k 0 γ is determined by the equation 2 ( ) j 1 ( ) j χk φ ( ) −∂ B 3 + ∂ B 2 = 0 ∂ ψ 2 , (43) j 0 0 j 2 + φ2 0 2 k 0 6k2χ 2φ2 k4χ 2φ2 1 − 0 ∇2(U) − 2V (U) =− 0 ρ, + 2φ2 2 + 2φ2 (3) j 1 k 0 1 k 0 which determines B 0 in terms of second order parame- (34) ters. This gauge is compatible with Eqs. (25) and (29). Using γ − where := 1 .(34) is a nonhomogeneous screened Pois- γ +1 (4)i = γ ∂ (γ ) + ∂ − ∂ (4)0 + ∂ (3)0 son equation whose solution is S ji A j A A j A j B 0 0 B j (4)0 = γ ∂ (γ ) S j0 A j A 4 2 2 exp −λ r − r ( ) i 3 k χ φ 3 =− ∂ (γ ), U = 0 ρ(x , t)d3x , S i0 0 A (44) + 2φ2 − 2χ 2φ2 2 1 k 0 6k 0 4π r − r one can ﬁnd that (23) at the fourth order gives (35) 1 0 + φ2 ∇2 B(4) +∇2(γ A)2 where 2 0 0 k

− ∇(γ ) ·∇ − ∇2 2 2 3 A A A A 2V2(1 + k φ ) λ = 0 . (36) 2 2 2 2 2 (2) 2 (2) 1 + k φ − 6k χ φ −4 φ0ψ ∇ A − 2(χ + )φ0∇ψ 0 0 χ ·∇ A − ∇2(ψ(2))2 − V (ψ(2))2 Equation (32) allows us to take 2 2 2 −χφ ∇2ψ(4) + χφ ∇((γ − ) ) ·∇ψ(2) + ∂ ψ(2) 2 0 3 0 1 A 0 = , G 2 (37) (1 + k2 φ )(γ + 1) ( ) 1 ( )i 0 +3χφ ∂2ψ 2 + + φ2 ∂ 3∂ (γ A) − ∂ B 3 0 0 k2 0 0 0 i 0 where G is defined through 1 0 1 0 − + φ2 ∂ j ∂ B(3) = τ (4) − τ (4) . 2 0 0 j 0 (45) (2) = = . k 2 h00 2A 2GU (38) Also, the scalar field equation at the fourth order is So one can define an effective G . through eff −∇2ψ(4) + ψ(4) + ∂2ψ(2) = χφ (4) 2V2 0 0 B (2) (2) (4) (2) 2 Geff = GGN . (39) +ψ B + φ0T − 3V3(ψ ) . (46) 123 Eur. Phys. J. C (2017) 77:191 Page 5 of 8 191 2 2 By using = 2χφ0 − 4γ A + 7∇.(γ A∇ A)

B(4) =− ∇2(γ 2 2) + ∇·(γ ∇ ) + ( − γ) ∇2 8 A 14 A A 2 1 5 A A +(1 − 5γ)A∇2 A + 6γ A∇2(γ A) + γ ∇2(γ ) −∇2 (4)0 + ∂2(γ ) 12 A A B 0 6 0 A (2) 2 +2ψ ∇ ((1 − 2γ)A) + 2 φ0∇(γ A) (3)i −2∂i ∂0 B 0 (47) (2) 2 ·∇((2 − γ)A) − 3V3(ψ ) and − χφ ∇2 (4)0 + ∇( − γ) ·∇ψ(2), 2 0 B 0 3 1 A (53) ( ) T 4 = 2∇(γ A) ·∇((2 − γ)A), (48) respectively. To obtain the post-Newtonian parameters we must obtain A, ψ(2), and γ(r). By inserting them in (52) and (4)0 (46) becomes (53), we obtain solutions for B 0. To do so we consider a −∇2ψ(4) + ψ(4) + ∂2ψ(2) spherically symmetric metric with a point source. 2V2 0 ( )i = 6χφ ∂2(γ A) − 2χφ ∂ ∂ B 3 0 0 0 i 0 0 3.1 Spherically symmetric metric 2 2 2χφ0 −4γ A + 7∇.(γ A∇ A) The source is assumed to be + (1 − 5γ)A∇2 A + 6γ A∇2(γ A) ( ) ρ = Mδ(r), = 0, p = 0,v= 0, (54) +2ψ 2 ∇2((1 − 2γ)A) i + φ ∇(γ ) ·∇(( − γ) ) − (ψ(2))2 2 0 A 2 A 3V3 and the metric is given by ( )0 ( ) −2χφ ∇2 B 4 + 3∇(1 − γ)A ·∇ψ 2 . (49) 0 0 =− + − 2 β 2 + + O( ) g00 1 2GeffU 2Geff U Self 6 Equations (45) and (49) are our main results in the fourth gij = O(5) order. These equations together with (42) and (43) in the third g = (1 + 2G γ U) δ + O(4), (55) order, and (32)–(34), in the second order must be solved to ij eff ij give the post-Newtonian parameters. where “Self” denotes self-energy terms of order 4, and β is To solve these complicated equations, we consider solu- the PPN parameter. The Newtonian potential is tions speciﬁed by U = U(r), which results in 2 ( ) ( ) k M A = A(r), γ = γ(r), ψ 2 = ψ 2 (r). (50) U = . (56) 8πr

j (3)0 The gauge (43) implies ∂ B j = 0. Therefore (42) To determine γ ,from(32), (33), and (35), we obtain reduces to χφ ψ(2) = 2 0 (−λ ) 1 ( )0 exp r (57) − + φ2 ∇2 B 3 = ρv . (51) 1 + k2φ2 − 6χ 2k2φ2 k2 0 i i 0 0 and i (3)0 For v = 0, (51)givesB i = 0 (by the assumption that perturbation terms vanish at large distance). In this situation k2 M A = , (58) Eqs. (45) and (49) become 4π(1 + k2φ2)(1 + γ)r 0 1 0 + φ2 ∇2 B(4) +∇2(γ A)2 2 0 0 where k −3∇(γ A) ·∇A − A∇2 A 1 + α exp(−λr) γ = , (59) (2) 2 1 − α exp(−λr) −4 φ0ψ ∇ A − 2(χ + ) ( ) χ ( ) ( ) ×φ ∇ψ 2 ·∇A − ∇2(ψ 2 )2 − V (ψ 2 )2 in which 0 2 2 −χφ ∇2ψ(4) + χφ ∇((γ − ) ) ·∇ψ(2) 0 3 0 1 A 2k2χ 2φ2 α = 0 , 1 ( ) ( )0 (60) = τ 4 − τ 4 (52) 1 + k2 φ2 − 6k2χ 2φ2 2 0 0 0 and λ (2) = = and is given by (36). From h00 2A 2GU, we obtain 2 (4) (4) −∇ ψ + 2V2ψ G as (37). 123 191 Page 6 of 8 Eur. Phys. J. C (2017) 77:191

(4)0 (φ) = To obtain B 0, one must insert (57)–(59)in(52) and 3.2.1 V 0 0 (53), and solve them together. From B(4) we determine the 0 λ = other PPN parameter, β,as If we ignore the scalar field potential, we obtain 0 (see (36)), and γ becomes a constant ( )0 2B 4 + A2 = 2G2β(r)U 2(r). (61) 0 1 − (4χ 2 − )k2φ2 γ = 0 . (68) − ( χ 2 − ) 2φ2 To determine the PPN parameters γ and β, we will con- 1 8 k 0 sider different situations. (4)0 By solving the system of Eqs. (52) and (53)forB 0 and 3.1.1 χ = 0 by considering Eqs. (57)–(61), after some computations we find For χ = 0, from (59) and (60), we find γ = 1, hence P β = , (69) 2 ( + ( χ 2 + ) 2φ2)( − ( χ 2 − ) 2φ2)2 k M 1 1 2 k 0 1 8 k 0 A = , G = . (62) 8π(1 + k2φ2)r 1 + k2 φ2 0 0 where ψ(2) = 3 3 3 Equation (57)gives 0. So we write (52)as P = 1 + 160 χ 6 + χ5 + χ4 − 2χ 3 10 40 16 2 (4)0 1 2 2 2 1 3 1 ∇ B 0 − ∇ (A) + 2A∇ A = 0, (63) − 2χ 2 + 3χ + 3 k6φ6 2 0 10 160 160 3 3 where ∇ A ·∇A = 1 ∇2 A2 − A∇2 A has been used. Putting +2 χ 3 − 8χ 2 + χ + k2φ2 2 2 2 0 (62)in(63), and ignoring the gravitational self-energy, we 7 8 1 1 obtain +12 χ 4 − χ3 − χ2 + χ 2 + 2 k4φ4. (70) 3 3 2 4 0 2 4 2 ( )0 A k M B 4 =− + . (64) Let us consider some limiting values: for small χ, χ 1we 0 π 2( + 2φ2)2 2 2 64 1 k 0 r have 2 2 2 2 β( ) = χ = 3 k φ 2k φ Therefore (61) yields r 1. So for 0 we find β = 1 + 0 χ − 0 χ 2 + O(χ 3) + 2φ2 + 2φ2 1 k 0 1 k 0 β(χ = 0) = γ(χ = 0) = 1. (65) 4k2φ2 γ = 1 + 0 χ 2 + O(χ 4), (71) + 2φ2 1 k 0 Therefore there is no deviation from general relativity for the φ φ PPN parameters. This is in complete agreement with [33,34]. and for small k 0, k 0 1wehave β = 1 + χ(2χ 2 − 2χ + 3 )k2φ2 + O(k4φ4) φ = 0 0 3.2 0 0 γ = + χ 2 2φ2 + O( 4φ4). 1 4 k 0 k 0 (72) For χ = 0, we may have also a situation with no deviation λ in the PPN parameters from general relativity, this occurs for 3.2.2 r 1 φ0 = 0. For example for the potentials In this limit from (57) and (59)wehaveψ(2) = 0 and γ = 1, 1 respectively. The solution of (52) is obtained, thus: V (φ) =− μ2φ2 + φ4,>0, (66) 2 4 + (4)0 1 2 1 (4) B 0 = A + ψ , (73) and 2 2χφ0

n 2χ2k2φ2 V (φ) = φ ,>0, n > 1, (67) =− + 0 where 1 + 2φ2 . The equation of motion of the 1 k 0 scalar field (53) becomes V0 = V1 = 0(see(22)) leads to φ0 = 0, which by using (59–61) results in γ = 1, G = 1, and β = 1. Therefore in ∇2ψ(4)+ ψ(4) = ( − χ)φ ∇2 2+( χ− )φ ∇2 , this case too, there is no deviation from general relativity for 2V2 2 0 A 8 0 A A the PPN parameters. (74) 123 Eur. Phys. J. C (2017) 77:191 Page 7 of 8 191 V r 2 2 whose solution, in the limit 2 1, is 3 k φ − 0 χ 2 2.3 × 10 4 (82) 1 + k2φ2 0 4 2φ ( − χ) ψ(4) = k M 0 2 1 . and 64π 2(1 + k2φ2)(2χ 2k2φ2 − k2φ2 − 1) r 2 0 0 0 χ 2 2φ2 . × −5 (75) 4 k 0 2 3 10 χ( χ 2 − χ + ) 2φ2 . × −4, 2 2 3 k 0 2 3 10 (83) From (75), (73), and (61), we find respectively. (χ − 1)k2φ2 − 1 For λr 1, we have β = 0 . (76) ( χ 2 − ) 2φ2 − 2 k 0 1 2 2 2V2(1 + k φ ) − 2φ2 2φ2 0 (1AU) 1, (84) For small k 0 , k 0 1 this gives + 2 φ2 − 2χ 2φ2 1 k 0 6k 0 β = 1 + (2χ 2 − χ )k2φ2 + O(k4φ4), (77) 0 0 and (80) restricts our parameters: and for small χ, χ 1gives 2 2 χ(2χ − )k φ − 0 2.3 × 10 4. (85) k2φ2 k2φ2 (2χ 2 − )k2φ2 − 1 β = 1 − 0 χ + 2 0 χ 2 + O(χ 3). (78) 0 + 2φ2 + 2φ2 1 k 0 1 k 0 λ λ Finally, let us note that, for small r, r 1, we take 4 Conclusion exp(−λr) 1. In this case γ and β take the same form as (68) and (69), respectively. The teleparallel model of gravity with quintessence (non- minimally) coupled to the torsion and also to a boundary 3.3 Range of parameters term (proportional to the torsion divergence) was considered (see (1)). Although the model shows some interesting γ The most precise value for experimentally has been aspects in cosmology and in describing the late time accel- obtained from Cassini [45]. The bound on this parameter eration of the Universe, it must also pass local gravitational is [46] and solar system tests. So we studied the parameterized post- Newtonian (PPN) approximation of the model. We obtained |γ − | . × −5. 1 2 3 10 (79) the equations of motion (see the Sect. 2), and solve them order by order to obtain PPN parameters (see Sect. 3). Explicit In this experiment the gravitational interaction, in terms of expressions for the PPN parameters in a spherically sym- −3 Astronomical Units takes place at r 7.44 × 10 AU [47]. metric metric were obtained and different possible situations The parameter β is determined by lunar laser ranging were discussed. Our results show that the PPN parameters, experiments via the Nordtvedt effect [48]. This test indicates except for some special cases, i.e. in the absence of boundary the bound [46] terms and also with zero scalar field background, differ from general relativity. So we conclude that coupling of the scalar −4 |β − 1| 2.3 × 10 , (80) field to the boundary term generally makes the model deviate from general relativity in the PPN limit. at a gravitational interaction distance r = 1AU [47]. Equa- Since T and B are not invariant under local Lorentz trans- tions (79) and (80) restrict the parameters of our model. formations, the teleparallel model with boundary term is For V (φ) = 0, (79)and (68)give not invariant under Lorentz transformations unless one takes χ =− . Despite this, in spacetimes with spherical symme- 2 2 2 4k χ φ − try like Schwarzschild spacetime and so on, it is possible to 0 2.3 × 10 5. (81) − ( χ 2 − ) 2φ2 choose good or preferred tetrads to solve this issue [49]. In 1 8 k 0 scalar-tetrad theories of gravity the effect of the preferred In the limiting cases (71) and (72) we find tetrads cannot be detected via measuring the metric compo- nents [50,51]. Similarly, in our model, the PPN parameters 2 2 4k φ − 0 χ 2 2.3 × 10 5 in the standard post-Newtonian formalism do not identify + 2φ2 1 k 0 the effect of the preferred tetrads. To include these effects 123 191 Page 8 of 8 Eur. Phys. J. C (2017) 77:191 one must generalize the post-Newtonian approach, as was 21. M. Sharif, S. Rani, Mod. Phys. Lett. A 28, 1350118 (2013) pointed out in [33]. 22. K. Bamba, S.D. Odintsov, E.N. Saridakis, arXiv:1605.02461 [gr- qc] 23. K. Bamba, S. Nojiri, S.D. Odintsov, Phys. Lett. B 725, 368 (2013) 24. R.C. Nunes, S. Pan, E.N. Saridakis, JCAP 08, 011 (2016). Open Access This article is distributed under the terms of the Creative arXiv:1606.04359 [gr-qc] Commons Attribution 4.0 International License (http://creativecomm 25. L. Jarv, A. Toporensky, Phys. Rev. D 93, 024051 (2016) ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 26. M. Skugoreva, A. Toporensky, Eur. Phys. J. C 76, 340 (2016). and reproduction in any medium, provided you give appropriate credit arXiv:1605.01989 [gr-qc] to the original author(s) and the source, provide a link to the Creative 27. B. Fazlpour, Gen. Rel. Grav. 48, 159 (2016). arXiv:1604.03080 Commons license, and indicate if changes were made. [gr-qc] Funded by SCOAP3. 28. C.Q. Geng, C.C. Lee, E.N. Saridakis, Y.P. Wu, Phys. Lett. B 704, 384 (2011) 29. G. Otalora, JCAP 07, 044 (2013) References 30. J.A. Gu, C.C. Lee, C.Q. Geng, Phys. Lett. B 718, 722 (2013). arXiv:1204.4048v2 [astro-ph.CO] 31. K. Bamba, S.D. Odintsov, D. Saez-Gomez, Phys. Rev. D 88, 1. S. Bahamonde, M. Wright, Phys. Rev. D 92, 084034 (2015). 084042 (2013) arXiv:1508.06580v4 [gr-qc] 32. H.M. Sadjadi, Phys. Rev. D 87, 064028 (2013) 2. A. Unzicker, T. Case, arXiv:physics/0503046 [physics.hist-ph] 33. Z.C. Chen, Y. Wu, H. Wei, Nucl. Phys. B 894, 422 (2015) 3. K. Hayashi, T. Nakano, Prog. Theor. Phys. 38, 491 (1967) 34. J.T. Li, Y.P. Wu, C.Q. Geng, Phys. Rev. D 89, 044040 (2014) 4. C. Pellegrini, J. Plebanski, Mat. Fys. Skr. Dan. Vid. Selsk. 2(4), 35. C.M. Will, Theory and Experiment in Gravitational Physics (Cam- 1–39 (1963) bridge University Press, Cambridge, 1993) 5. A. Einstein, Sitzber. Preuss. Akad. Wiss. 217, 221 (1928) 36. C.M. Will, Liv. Rev. Rel. 9, 3 (2006) 6. E.V. Linder, Phys. Rev. D 81, 127301 (2010) 37. L.L. Smalley, Phys. Rev. D 21, 328 (1980) 7. H. Wei, Phys. Lett. B 712, 430 (2012) 38. M. Hohmann, Phys. Rev. D 92, 064019 (2015) 8. H.M. Sadjadi, Phys. Rev. D 92, 123538 (2015) 39. Z. Yi, Y. Gong, Int. J. Mod. Phys. D 26, 1750005 (2017). 9. Y.F. Cai, S. Capozziello, M. De Laurentis, E.N. Saridakis, Rept. arXiv:1512.05555 [gr-qc] Prog. Phys. 79, 106901 (2016). arXiv:1511.07586 [gr-qc] 40. Kh Saaidi, A. Mohammadi, H. Sheikhahmadi, Phys. Rev. D 88, 10. H.M. Sadjadi, Phys. Lett. B 718, 270 (2012) 084054 (2013) 11. S. Carloni, F.S.N. Lobo, G. Otalora, E.N. Saridakis, Phys. Rev. D 41. M. Zubair, S. Bahamonde, arXiv:1604.02996 [gr-qc] 93, 024034 (2016) 42. H.M. Sadjadi, JCAP 01, 031 (2017). arXiv:1609.04292 [gr-qc] 12. Y. Zhang, H. Li, Y. Gong, Z.H. Zhu, JCAP 07, 015 (2011) 43. S. Bahamonde, C.G. Boehmer, M. Wright, Phys. Rev. D 92, 104042 13. K. Karami, S. Asadzadeh, A. Abdolmaleki, Z. Safari, Phys. Rev. (2015). arXiv:1508.05120v2 [gr-qc] D 88, 084034 (2013) 44. M. Wright, Phys. Rev. D 93, 103002 (2016) 14. M.E. Rodrigues, M.J.S. Houndjo, D. Momeni, R. Myrzakulov, 45. B. Bertotti, L. Iess, P. Tortora, Nature 425, 374 (2003) Int. J. Mod. Phys. D 22, 1350043 (2013). arXiv:1302.4372v2 46. C.M. Will, Liv. Rev. Rel. 9, 3 (2006). arXiv:gr-qc/0510072 (Table [physics.gen-ph] 4) 15. F. Darabi, M. Mousavi, K. Atazadeh, Phys. Rev. D 91, 084023 47. M. Hohmann, L. Jarv, P.Kuusk, E. Randla, Phys. Rev. D 88, 084054 (2015) (2013) 16. G.G.L. Nashed, Gen. Relativ. Gravit. 47, 75 (2015). 48. F. Hofmann, J. Müller, L. Biskupek, Astron. Astrophys. 522,L5 arXiv:1506.08695 [gr-qc] (2010) 17. X. Fu, P. Wu, H. Yu, Int. J. Mod. Phys. D 21(09), 1250074 (2012) 49. N. Tamanini, C.G. Boehmer, Phys. Rev. D 86, 044009 (2012) 18. K. Myrzakulov, P. Tsyba, R. Myrzakulov, arXiv:1601.07357 50. J. Hayward, Phys. Rev. D 20, 3039 (1979) [physics.gen-ph] 51. J. Hayward, Gen. Rel. Grav. 13, 43 (1981) 19. E.L.B. Junior, M.E. Rodrigues, I.G. Salako, M.J.S. Houndjo, Class. Quantum Grav. 33, 125006 (2016) 20. A. Behboodi, S. Akhshabi, K. Nozari, Phys. Lett. B 718, 30 (2012)

123