Parameterized post-Newtonian approximation in a teleparallel model of dark energy with a boundary term
Eur. Phys. J. C
Parameterized post-Newtonian approximation in a teleparallel model of dark energy with a boundary term
H. Mohseni Sadjadi 0
0 Department of Physics, University of Tehran , Tehran , Iran
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.
-
In a teleparallel model of gravity, instead of the torsionless
Levi-Civita connections, curvatureless Weitzenböck
connections are used [2–4]. A teleparallel equivalent of general
relativity was first introduced in [5] as an attempt for unification
of electromagnetism and gravity. This theory is considered
as an alternative theory of usual general relativity and has
been recently employed to study the late time acceleration of
the Universe [6–8]. This can be accomplished by considering
modified f (T ) models [9–24], where T is the torsion scalar,
or by introducing exotic field such as quintessence.
Assuming a non-minimal coupling between the scalar field and the
torsion opens new windows in studying the cosmological
evolution [25–31], and can be viewed as a promising
scenario for late time acceleration and super-acceleration [32].
A non-minimally coupled scalar field, like the scalar–
tensor 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
[35–40]. 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
(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 ∝ ∇μT 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 Z2 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.
We use units with h¯ = c = 1 and choose the signature
(−, +, +, +) for the metric.
2 The model and the field equations
In our study we use vierbeins ea = ea μ∂μ, whose duals,
ea μ, are defined through ea μea ν = δμν. The metric
tensor is given by gμν = ηabea μebν , η = diag(−1, 1, 1, 1).
e = det(ea μ) = det √−g. Greek indices (indicating
coordinate bases) like the first Latin indices (indicating
orthonormal bases) a, b, c, ... belong to {0, 1, 2, 3}, while i, j, k, ... ∈
{1, 2, 3}.
Our model is specified by the action [1]
S =
where k2 = 8π G N , and G N is the Newtonian gravitational
constant. The torsion scalar is defined by
T = Sρ μν Tρ μν = 41 T ρ μν Tρ μν
+ 21 T ρ μν T νμρ −T ρ μρ T νμν ,
3 Post-Newtonian formalism
To investigate the post-Newtonian approximation [35–40] 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
postNewtonian hydrodynamics:
where ρ is energy density, p is the pressure and is the
specific internal energy. uμ is the four-vector velocity of the
fluid. The velocity of the source matter is vi = uu0i . The orders
of smallness of the energ (...truncated)