Physics Letters B 728 (2014) 554–561
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Physics Letters B
www.elsevier.com/locate/physletb
Energy conditions in F (T ,Θ) gravity and compatibility with a stable
de Sitter solution
Faeze Kiani, Kourosh Nozari
∗
Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
article info abstract
Article history:
Received 8 September 2013
Received in revised form 10 December 2013
Accepted 10 December 2013
Available online 17 December 2013
Editor: J. Hisano
Keywords:
Modified gravity
Teleparallel gravity
Energy conditions
We study a new type of the modified teleparallel gravity of the form F (T ,Θ) in which T ,thetorsion
scalar, is coupled with
Θ, the trace of the stress–energy tensor. In a perturbational approach, we study
the stability of the solutions and as a special case we find a condition for stability of the de Sitter
phase. Then we adopt a suitable form for F
(T ,Θ) that realizes a stable de Sitter solution so that the
stability condition creates a specific constraint on the parametric space of the model. Finally, the energy
conditions in the framework of F
(T ,Θ) gravity is investigated.
© 2013 The Authors. Published by Elsevier B.V.
1. Introduction
Einstein’s general relativity is a completely geometrical theory
so that gravitation is described not as a force, but as a geometric
deformation of flat Minkowski space–time. In this point of view,
the gravitational field creates a curvature in space time so that
its action on the particles is determined by allowing them to fol-
low the geodesics of the space time. In this approach, trajectories
are described by the geodesic equation not the force equation [1].
On the other hand, in 1928, Einstein in an attempt to build a uni-
fied gauge theory of gravitation and electrodynamics presented the
other theory of gravity, the so-called teleparallel gravity [2].Inthis
theory torsion, the antisymmetric part of connection, is non-zero
and torsion instead of curvature describes the gravitational inter-
action. In teleparallel gravity, tetrad (or vierbein) fields form the
(pseudo) orthogonal bases for the tangent space at each point of
flat space time. Similarly to the metric tensor in general relativity,
here tetrad plays the role of the dynamical variables. Teleparal-
lel gravity also uses the curvature-free Weitzenböck connection
instead of Levi-Civita connection of general relativity to define co-
variant derivatives [3]. In spite of such fundamental conceptual
differences between teleparallel theory and general relativity, it has
been shown that teleparallel Lagrangian density only differs from
Ricci scalar by a total divergence [4,5].
*
Corresponding author.
E-mail addresses: fkiani@umz.ac.ir (F. Kiani), knozari@umz.ac.ir (K. Nozari).
In general relativity, the dark energy puzzle can be addressed
by introducing additional geometrical degree of freedom into the
theory, that is called F
(R) modified gravity. In F (R) gravity the
late time acceleration of the universe is catched by dark geometry
instead of dark energy [6]. The modification of gravity in teleparal-
lel gravity is accomplished by supplementing an additional torsion
term into Einstein–Hilbert Lagrangian [7].TheF
(T ) gravity has in-
teresting properties that the field equations are of second order,
unlike F
(R) gravity which is of fourth order in the metric ap-
proach. In this context, F
(T ) models have been extensively used
in cosmology to explain the late time cosmic speed-up expansion
without the need of dark energy [7,8].
In this Letter we construct a generalization of F
(T ) modified
gravity by considering coupling between torsion scalar T and trace
of the stress–energy tensor
Θ via a general function as F (T ,Θ).
Then we investigate stability of the de Sitter solution (when sub-
jected to homogeneous perturbations) in this framework. In this
sense, we obtain a stability condition for the de Sitter phase in
the general F
(T ,Θ) theories. Then we propose a specific F (T ,Θ)
model and show that the stability condition can be expressed as
a constraint equation between the parameters of the model. We
also consider the constraints imposed by the energy conditions
and investigate whether the parameters ranges of the proposed
model are consistent with the stability conditions. We note that
since homogeneous and isotropic perturbations can be considered
as the route to determine the stability of different modified gravity
theories (see for instance [9]), the full anisotropic analysis of the
cosmological perturbations is not considered here.
The Letter is organized as follows: in Section 2 w
e explore the
general features of the F
(T ,Θ) gravity theories by writing the
0370-2693 © 2013 The Authors. Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.physletb.2013.12.036
Open access under CC BY license.
Open access under CC BY license.
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