Cosmological reconstruction and energy bounds in \(f(R,R_{\alpha \beta }R^{\alpha \beta },\phi )\) gravity
Eur. Phys. J. C
Cosmological reconstruction and energy bounds in f ( R, Rαβ Rαβ , φ) gravity
M. Zubair 0
Farzana Kousar 0
0 Department of Mathematics, COMSATS Institute of Information Technology Lahore , Lahore , Pakistan
We discuss the cosmological reconstruction of f (R, Rαβ Rαβ , φ) (where R, Rαβ Rαβ , and φ represent the Ricci scalar, the Ricci invariant, and the scalar field) corresponding to a power law and de Sitter evolution in the framework of the FRW universe model. We derive the energy conditions for this modified theory which seem to be more general and can be reduced to some well-known forms of these conditions in general relativity, f (R) and f (R, φ) theories. We have presented the general constraints in terms of recent values of the snap, jerk, deceleration, and Hubble parameters. The energy bounds are analyzed for reconstructed as well as known models in this theory. Finally, the free parameters are analyzed comprehensively.
1 Introduction
In current cosmic picture dark energy (DE) is introduced
as an effective characteristic which tends to accelerate the
expansion in universe. Modified theories have achieved
significant attention to explore the effect of cosmic
acceleration [1]. These models have been developed to distinguish
the source of DE as a modification to the Einstein Hilbert
action. Some modified theories of gravity are f (R)
gravity with Ricci scalar R [2], f (T ) gravity with torsion scalar
T [3], Gauss–Bonnet gravity with G invariant [4], f (R, T )
gravity with T as the trace of the stress-energy tensor [5–14],
f (R, T , Rμν )T μν [15–17] and f (R, G) gravity that contains
both R and G [18] etc. The acceleration of the expanding
universe can be explored by these theories through their
corresponding invariants.
To generalize Einstein’s theory of general relativity (GR),
there is a vast literature on relativistic theories that reduce to
GR in the proper limitations. An especially attractive class
of these generalizations are the fourth-order theories. These
theories were initially considered by Eddington in the early
1920s [19]. Whatever the inspiration to examine the
generalized fourth-order theories, it is necessary to understand
their weak-field limit, and these limits confirm the increasing
behavior of these theories in observational data.
Generally a fourth-order theory of gravity is obtained by
adding Rab Rab and Rabcd Rabcd in the standard Einstein
Hilbert action [20,21]. However, it is now established that
we can ignore the Rabcd Rabcd term if we use the Gauss–
Bonnet theorem [22]. About half a century ago, Brans and
Dicke (BD) [23] presented the scalar–tensor theory of
gravitation, which is still popular and has received great interest
in cosmological dynamics as a replacement to dark matter
and dark energy theories. The motivation behind the BD
theory was Mach’s idea [24] to present a varying gravitational
constant in general relativity. Among the theories alternative
to Einstein’s gravity, the simplest and best known is Brans–
Dicke theory. In this theory, the gravitational constant has
been taken to be inversely proportional to the scalar field φ.
The BD theory may be represented as a generalization of
f (R) theory with f (R) = F (R) = φ R [2].
In modified theories, cosmological reconstruction is one
of the important prospects in cosmology. In f (R) gravity, the
reconstruction scheme has been used in different contexts to
explain the conversion of the matter dominated era to the
DE phase. This can be examined by considering the known
cosmic evolution and the field equations are used to calculate
a particular form of the Lagrangian which can reproduce the
given evolution background. In these theories the existence
of exact power law solutions for the FRW spacetime has
been examined. In [25–27] the authors have reconstructed
f (R, T ) gravity models by employing various cosmological
scenarios. Nojiri et al. developed f (R) gravity models [28],
which were further applied to f (R, G) and modified Gauss–
Bonnet theories [29]. To reconstruct f (R) gravity models,
Carloni et al. [30] have established a new technique by using
the cosmic parameters instead of using a scale factor.
Energy conditions are necessary to study the
singularity theorems; moreover, we have the theorems related to
black hole thermodynamics. For example, the well-known
Hawking–Penrose singularity theorems [31] invoke the null
energy condition (NEC) as well as the strong energy
condition (SEC). The violation of SEC allows one to observe the
accelerating expansion, and NECs are involved in the proof
of the second law of black hole thermodynamics.
The energy conditions have been explored in different
contexts like f (T ) theory [32,33], f (R) gravity [34], and f (G)
theory [35], Brans–Dicke theory [36]. Further the energy
conditions of a very generalized second-order scalar–tensor
gravity have been discussed by Sharif and Saira [37]. Sharif
and Zubair have examined these conditions for f (R, T )
(...truncated)