Energy conditions in \(f(\mathcal {G},T)\) gravity
Eur. Phys. J. C
Energy conditions in f (G , T ) gravity
M. Sharif 0
Ayesha Ikram 0
0 Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus , Lahore 54590 , Pakistan
The aim of this paper is to introduce a new modified gravity theory named f (G, T ) gravity (G and T are the Gauss-Bonnet invariant and trace of the energy-momentum tensor, respectively) and investigate energy conditions for two reconstructed models in the context of FRW universe. We formulate general field equations, divergence of energymomentum tensor, equation of motion for test particles as well as corresponding energy conditions. The massive test particles follow non-geodesic lines of geometry due to the presence of an extra force. We express the energy conditions in terms of cosmological parameters like the deceleration, jerk, and snap parameters. The reconstruction technique is applied to this theory using de Sitter and power-law cosmological solutions. We analyze the energy bounds and obtain feasible constraints on the free parameters.
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Current cosmic accelerated expansion has been affirmed
from a diverse set of observational data coming from
several pieces of astronomical evidence, including supernova
type Ia, large scale structure, cosmic microwave background
radiation etc. [1–4]. This expanding paradigm is considered
as a consequence of mysterious force dubbed dark energy
(DE), which possesses a large negative pressure. Modified
theories of gravity are considered as the favorite candidates
to unveil the enigmatic nature of this energy. These modified
theories are usually developed by including scalar invariants
and their corresponding generic functions in the Einstein–
Hilbert action.
A remarkably interesting gravity theory is the modified
Gauss–Bonnet (GB) theory. A linear combination of the form
G = Rαβξη Rαβξη − 4Rαβ Rαβ + R2,
a e-mail:
b e-mail:
where Rαβξη, Rαβ and R represent the Riemann tensor, the
Ricci tensor, and the Ricci scalar, respectively, is called a
Gauss–Bonnet invariant (G). It is a second order Lovelock
scalar invariant and thus free from spin-2 ghosts instabilities
[5–7]. The Gauss–Bonnet combination is a four-dimensional
topological invariant which does not involve the field
equations. However, it provides interesting results in the same
dimensions when either coupled with a scalar field or when
an arbitrary function f (G) is added to the Einstein–Hilbert
action [8–10]. The latter approach is introduced by Nojiri and
Odintsov; it is known as the f (G) theory of gravity [11]. Like
other modified theories, this theory is an alternative to study
DE and is consistent with solar system constraints [12]. In
this context, there is a possibility to discuss a transition from
decelerated to accelerated as well as from non-phantom to
phantom phases and also to explain the unification of early
and late times accelerated expansion of the universe [13,14].
The fascinating problem of cosmic accelerated expansion
has successfully been discussed by taking into account
modified theories of gravity with curvature–matter coupling. The
motion of test particles is studied in f (R) and f (G) gravity
theories non-minimally coupled with the matter Lagrangian
density (Lm ). Consequently, the extra force experienced by
test particles is found to be orthogonal to their four velocities
and the motion becomes non-geodesic [15–17]. It is found
that, for certain choices of Lm , the presence of the extra force
vanishes in a non-minimal f (R) model, while it remains
preserved in a non-minimal f (G) model. The geodesic deviation
is weaker in f (G) gravity for small curvatures as compared to
non-minimal f (R) gravity. Nojiri et al. [18] studied the
nonminimally coupling of f (R) and f (G) theories with Lm and
found that such a coupling naturally unifies the inflationary
era with current cosmic accelerated expansion.
In order to describe some realistic matter distribution,
certain conditions must be imposed on the energy-momentum
tensor (Tαβ ) known as energy conditions. These conditions
originate from the Raychaudhuri equations with the
requirement that not only gravity is attractive but also the energy
density is positive. The null (NEC), weak (WEC), dominant
(DEC), and strong (SEC) energy conditions are the four
fundamental conditions. They play a key role to study the
theorems related to singularity and black hole thermodynamics.
The null energy condition is important to discuss the second
law of black hole thermodynamics while its violation leads
to a Big-Rip singularity of the universe [19]. The proof of
the positive mass theorem is based on DEC [20], while SEC
is useful to study the Hawking–Penrose singularity theorem
[21].
The energy conditions have been investigated in
different modified theories of gravity like f (R) gravity, Brans–
Dicke theory, f (G) gravity, and generalized teleparallel
theory [22–25]. Banijamali et al. [26] investigated the energy
conditions for non-minimally coupling f (G) theory with
Lm (...truncated)