#### Higher-dimensional inhomogeneous perfect fluid collapse in f(R) gravity

Eur. Phys. J. C
Higher-dimensional inhomogeneous perfect fluid collapse in f ( R) gravity
G. Abbas 2
M. S. Khan 1
Zahid Ahmad 1
M. Zubair 0
0 Department of Mathematics, COMSATS Institute of Information Technology , Lahore , Pakistan
1 Department of Mathematics, COMSATS Institute of Information Technology , Abbottabad, KPK , Pakistan
2 Department of Mathematics, The Islamia University of Bahawalpur , Bahawalpur , Pakistan
This paper is about the n + 2-dimensional gravitational contraction of an inhomogeneous fluid without heat flux in the framework of a f (R) metric theory of gravity. Matching conditions for two regions of a star are derived by using the Darmois junction conditions. For the analytic solution of the equations of motion in modified f (R) theory of gravity, we have taken the scalar curvature constant. Hence the final result of gravitational collapse in this framework is the existence of black hole and cosmological horizons, and both of these form earlier than the singularity. It is shown that a constant curvature term f (R0) (R0 is the constant scalar curvature) slows down the collapsing process.
1 Introduction
Recently, the modified theories of gravity have attracted the
attention of many researchers in theoretical and observational
cosmology and astrophysics. One of the most active research
directions is the exploration of many astrophysical problems
in a modified f (R) metric theory of gravity. It is most
reliable to consider this theory due to its simplicity as regards the
derivation as it is obtained by taking a function f (R) of the
Ricci scalar, R, for the action as in the Einstein–Hilbert
gravitational action. All the modifications of general relativity
(GR) explore the problem of dark energy in a more scientific
way [
1–3
]. Some major modifications of GR are f (R) gravity
[
2
], f (R, T ) gravity (T is the trace of the energy-momentum
tensor Tαβ ) [
4
], f (R, T , Q) gravity (where Q = Rαβ T αβ )
[
5
], Gauss–Bonnet gravity [
6
], teleparallel modified theories
[
7,8
], and scalar–tensor theories [9]. It has been pointed out
by many researchers [
3–8
] that the f (R) theory confirms
that when major interactions are unified this leads to actions
which involve curvature invariants of nonlinear order.
During the last decades many renowned researchers have
investigated the gravitational collapse in modified theories. It
has been shown that as one goes beyond the general relativity,
one has more chances of admitting an uncovered singularity.
A lot of work has been done in GR as regards the
gravitational collapse [
10–24
]. The nonlinear electrodynamics static
black holes (BH) solutions have been formulated within the
frame work of f (R) [
25
]. In this connection Borisov et al.
[
26
] have investigated the spherical gravitational collapse
in f (R) gravity by performing one-dimensional numerical
simulations. In this study, the nonlinear self-interaction
coupling of the scalar field has been included in the dynamical
equations and a relation scheme has been used to follow the
gravitationally contracting solutions. During the scalar-field
collapse in f (R) gravity, the density increases rapidly near
the virial radius, which may provide an observable test of
gravity. Schmidt [
27
] has used large scale simulations for
spherical collapse in f (R) gravity to estimate the halo mass
function. Capozziello et al. [
28
] have investigated the
hydrostatic equilibrium and stellar structure in f (R) gravity.
Cembranos et al. [
29
], have explored the gravitational
collapse of matter with uniform density in f (R) gravity. This
analysis provides information as regards the structure
formation in the early universe. It has been remarked that, for
some particular models of f (R) gravity, the gravitational
collapse process would help to constrain the models that exhibit
the late-time cosmological acceleration. The time of collapse
in this frame work has been observed to be much smaller as
compared to the age of the universe while it is much longer to
form the matter clustering. All of the previous investigations
of gravitational collapse in f (R) gravity imply that gravity
is a highly attractive force—this is in the agreement with the
observed consequences of f (R) gravity. It is an admitted fact
that a scalar force would reduce the time for gravitational
collapse because of its attractive behavior. Ghosh and Maharaj
[
30
] have explored the exact models of null dust collapse in
metric f (R) theory with the constant scalar curvature
condition. Further, in this situation the null dust collapse leads to
the formation of naked singularities, hence violating the
cosmic censorship conjecture (CCC) in f (R) gravity. Goswami
et al. [
31
] have proved that a gravitational collapse of heat
conducting, shearing and anisotropic fluid in f (R) seems
to be unstable with respect to matter perturbations. There
exist no apparent horizons and hence there occur naked
singularities. It is important to note that investigating CCC in
modified gravity may be more complicated than in GR. It has
been well established that inhomogeneity is closely related
to spacetime shear and Weyl curvature of the collapsing star,
which produces the naked singularities.
The Oppenheimer–Snyder–Datt model [
10,32
], which is
a widely acceptable model for BH formation via dynamical
collapse is no longer a viable model in modified f (R)
gravity. Hence in order to establish the existence of BH solutions
via stellar collapse in modified f (R) theories, we have to
find some new physically reasonable solutions in modified
f (R) gravity that may predict BHs. Hwang et al. [33] have
investigated the collapse of a charged BH in f (R) gravity
using the double null formalism and constant scalar
curvature assumption. In such charged BH solutions there appears
a new type of singularity due to higher curvature corrections,
the so-called f (R) induced singularity. Pun et al. [
34
] have
confirmed the existence of a Schwarzschild-like BH
solution in f (R) gravity. The modified f (R) theories of gravity
provide toy models for the existence and stability of neutron
stars [
35
]. The stars which satisfy the baro-tropic equation of
state (ρ = ωp) in f (R) gravity, when they undergo
gravitational collapse, have an end state of the collapse that would
be a naked singularity violating CCC [
36
].
Sharif and Nasir [
37
] have discussed the stability of
expansion-free axially symmetric fluids in f (R) gravity. The
gravitating source preserves its axial symmetry due to the
f (R) extra degree of freedom. Also, the axially
symmetric solutions have been formulated in f (R) gravity by using
the Newman–Janis method [
38
]. The rotating black string
solutions have been investigated in f (R)-Maxwell gravity
[
39
] using the constant scalar curvature assumption. Sharif
et al. [
40–44
] have studied the dynamical stabilities of many
gravitating stellar systems in f (R) theories with the general
form of f (R) models. The instability and anti-evaporation of
Reissner–Nordstro¨m BHs have been explored in the modified
f (R) gravity [
45
]. The inhomogeneous dust as well perfect
fluid collapse with the geodesic flow condition in 4D have
been explored in [
46,47
]. It has been remarked that f (R)
with constant scalar curvature would appear as an alternative
to the cosmological constant.
Recent advancements in string theory and other field
theories indicate that gravity is a higher-dimensional interaction.
It would be interesting to determine an analytic model of
stellar contraction and singularity formation in more than 4D.
The most general forms of a Vaidya solution for a null fluid
in Lovelock theory of gravity have been explored by many
authors [
48–50
] and they arrived at the conclusion that the
uncovered singularities are feasible for an odd dimension for
several values of the parameters and, due to the gravitational
collapse, a BH is formed for any value of the parameters.
Banerjee et al. [
51
] studied the uncovered singularities in
higher-dimensional gravitational collapse and concluded that
there is a great chance of an uncovered singularity. Feinstein
[
52
] investigated the formation of a black string for
gravitational collapse in a higher-dimensional vacuum. In this
paper, the work done by Sharif and Kausar [
47
] is extended
for n + 2-dimensional spacetime. The scheme of the paper
is as follows. In Sect. 2, the field equations are given.
Section 3 is devoted to solutions of the field equations. In Sect.
4, trapped surfaces and apparent horizons are discussed in
detail. Finally, the results are summarized in Sect. 5.
2 LTB model and equations of motion in f ( R) gravity
For the interior region we take the n + 2-dimensional
nonstatic spherically symmetric LTB metric given by
(2.1)
(2.3)
(2.4)
ds−2 = dt 2 − A2dr 2 − Y 2d 2,
where A = A(r, t ) and Y = Y (r, t ). We have
d 2 =
n
k−1
sin2θl dθk2 = dθ12 + sin2θ1dθ22
k=1 l=1
+ sin2θ1sin2θ2dθ32
+ · · · + sin2θ1sin2θ2sin2θ3 · · · sin2θn−1dθn2.
(2.2)
In f (R) gravity the equations of motion are [
1–3
]
F (R)Rπχ − 21 f (R)gπχ − ∇π ∇χ F (R) + gπχ ∇σ ∇σ F (R)
= κ Tπχ .
Here F (R) = d f (R)/d R, ∇π is the covariant derivative,
Tπχ is the standard energy-momentum tensor and κ is the
coupling constant. The perfect fluid source is
Tπχ = (ρ + p)uπ uχ − pgπχ ,
where ρ = ρ(r, t ) is the fluid matter density, p is the fluid
pressure and uπ is the n + 2-dimensional velocity vector
defined by uπ = δπ0 . For the metric (2.1), we get the following
set of independent partial differential equations:
,
Here · and are for the partial derivatives with respect to t
and r , respectively.
Using the scalar perturbation constraints, Cooney et al.
[
53
] have explored the formation of compact objects like
a neutron star in f (R) gravity. The Schwarzschild metric
cosmological constant has been considered in the external
region, which has been matched smoothly with the interior
fluid solution using Darmois junction conditions in a very
similar way to GR. According to the authors of [
54,55
] the
Schwarzschild solution is the most suitable solution for the
exterior geometry of the star. In the same way in f (R) gravity
many researchers [
40–44,56,57
] have examined the
matching conditions for gravitational collapse.
For the exterior region, we consider the n + 2-dimensional
Schwarzschild metric
ds+2 = 1 −
2RM dt 2 − 1 −12RM dr 2 − R2d 2.
The matching conditions require that:
1. The first fundamental form of the metrics must be
continuous over , i.e.,
(ds2 )
+
= (ds2 )
−
= (ds2) .
(2.9)
(2.10)
−
2. Also, the extrinsic curvature must be continuous over
i.e.,
Now from the continuity of the extrinsic curvature (2.11), it
follows that
[Kcd ] = Kc+d − Kc−d = 0, (i, j = 0, 2, 3 . . . n + 1), (2.11)
K0+0 = 0,
K2+2 = K2−2.
where Kcd is the extrinsic curvature tensor defined as
(2.5)
Kc±d = −nω±
∂2x±ω ω ∂ x ±γ∂ x ±δ
∂ξ c∂ξ d + γ δ ∂ξ c∂ξ d
,
(ω, γ , δ = 0, 1, 2, 3 . . . n + 1).
F
Here nω±, x ω and ξ c denotes the outward coordinates on V ±,
±
, respectively. The equations of hypersurfaces for the inner
and outer metrics are given by
Now from the junction condition equation (2.10), we get
R
= Y (r , T ),
1
Z (R) − Z (R)
d R
dT
where Z (R) = 1 − 2RM . The possible components of Kc±d are
(2.22)
(2.24)
= 0,
.
Equations (2.18), (2.19), (2.25) and (2.26) are the required
conditions for the matching of two regions.
3 Solution
We need the explicit value of A, for the solution of the set of
fields Eqs. (2.5)–(2.9). It follows from Eq. (2.9) that
A =
nY˙ F + F˙ Y
dt.
nY F + F Y
To solve the above equation, we assume R = R0, and
F (R0) = const ant and this assumption provides us with
p = p0 and ρ = ρ0; here the quantities with subscript 0 are
constant quantities. In modified gravitation theories the
stability of models is tested through the Dolgov and Kawasaki
[
58
] stability criterion,
F (R) = f R (R) > 0, f R R (R) > 0,
R ≥ R0.
Using the above assumptions, Eqs. (2.5)–(2.9) yield
Y A Y
Y − AY
1 f
= F 2 − κ p , (3.4)
A˙Y˙ 1 Y
− AY + A2 Y
2
Now using Eqs. (2.20–2.24) along with (2.18) and (2.19), the
junction takes the form
1
= − F (R0)
where c(t ) is an arbitrary function of t . The function m(r )
must be positive. Using the second junction condition from
Eqs. (3.5) and (3.9), we get
(n − 1)Y n+1
M = (n − 1)m(r ) − 2(n + 1)F (R0)
×
(3.14)
Now we assume that
1
F (R0)
f (R0)
2
> 0,
(3.15)
and the condition W (r ) = 1 to find the solution, so Eq. (3.9)
implies that
The above equation yields
m(r ) Y 2
(Y˙ )2 = W 2 − 1 + 2 Y n−1 − (n + 1)F (R0)
8π f (R0)
× n ((n − 1) p0 − ρ0) − 2
here m = m(r ) and its value is given by
8π
m = n F (R0) ((n − 1) p0 + ρ0)Y Y n.
Also, the above equation leads to
8π
m(r ) = n F (R0) ( p0(n −1)+ρ0)
(Y Y n)dr +c(t ), (3.11)
,
(3.9)
(3.10)
(3.12)
(3.13)
Y =
A =
2(n + 1)m F(R0)
8nπ ((n − 1) p0 − ρ0) − 21 f (R0)
2(n + 1)m F(R0)
8nπ ((n − 1) p0 − ρ0) − 21 f (R0)
We would like to mention that in the limit f (R0) →
8π( p0−ρ0) , we have the Tolman–Bondi solution [
59
]
n
Y =
A =
1
(n + 1)2m(r ) (ts − t )2 n+1 ,
2
m (ts − t ) + 2mts
[2(n + 1)n−1mn(ts − t )n−1] n +11 .
From Eqs. (3.2) and (3.15), we get F (R0) > 0 and f (R0) <
16nπ [(n −1) p0 −ρ0]. The Dolgov and Kawasaki [
58
] stability
criterion does not restrict the sign of f (R0). Therefore, for
ρ0 > 0, we must have (n − 1) p0 < ρ0; it holds for all n ≥ 1,
and finally we get f (R0) < 0. Hence these are the viability
conditions for f (R) gravity that must hold throughout the
discussion.
4 Apparent horizons
For spacetime (2.1) the boundary of a trapped n-sphere is
given by
(n + 1) 8nπ ((n − 1) p0 − ρ0) − 21 f (R0)
4F(R0)
[ts (r ) − t].
(3.18)
gπχ Y,π Y,χ =
A2Y˙ 2 − Y
A2
= 0.
Using Eq. (3.9), the above equation yields
1
F (R0)
×Y n+1 − (n + 1)Y n−1 + 2(n + 1)m = 0.
The values of Y give the boundaries of trapped surfaces which
are the apparent horizons. For f (R0) = 2( 8nπ ((n − 1) p0 −
1
ρ0))), one gets Y = (2m) n−1 , which is the Schwarzschild
n-radius. It gives a de Sitter horizon, when m = 0, i.e.,
Y =
(n + 1)F (R0)
8nπ ((n − 1) p0 − ρ0) − f (2R0) .
The approximate solution of Eq. (4.2) up to first order in m
and F(1R0) [ 8nπ ((n − 1) p0 − ρ0) − f (2R0) )], respectively, are
given by
(Y )ch and (Y )bh are called the cosmological and black hole,
respectively. The existence of (Y )ch is mainly due to the
appearance of the f (R) term. Now from Eqs. (3.18) and
(4.2), the time for the trapped surfaces’ formation is
(n + 1)F (R0)
8nπ ((n − 1) p0 − ρ0) − f (2R0)
n F (R0)
8π((n − 1) p0 − ρ0) − 21 f (R0)
(n + 1)
8π((n − 1) p0 − ρ0) − 21 f (R0)
n F (R0)
n
2
In the limiting case when f (R0) → 2 8nπ (ρ0 − (n − 1) p0) ,
the result coincides with the Tolman–Bondi solution [
59
],
tn = ts −
1
(2nm) n−1
n + 1
.
It is clear from Eq. (4.6) that the formation of trapped
surfaces takes less time as compared to the time of formation of
a singularity, t = ts . This implies that horizons form earlier
than the singularity, hence the singularity is covered by the
event horizons, and the end state of gravitational collapse is
a BH. The present solutions in f (R) are in agreement with
Oppenheimer–Snyder–Datt models. Hence the end state of
gravitational collapse is a BH. From Eq. (4.7) it is to be
noted that the time for forming trapped surfaces in
higherdimensional Tolman–Bondi spacetime is a special case of
our present investigation. Further, the f (R0) term affects the
time lag between the formation of trapped surfaces and the
singularity. The Misner–Sharp mass [
11
] has been modified
by f (R0). Also, the exterior trapped surface, the so-called
cosmological horizon is due to the presence of the f (R0)
term. We explored the physical aspects of the solutions and
found a suitable counterterm in the analytic solutions which
(4.5)
(4.6)
(4.7)
avoids the occurrence of a naked singularity during
gravitational collapse.
The Dolgov and Kawasaki [
58
] stability criterion F (R) =
f R (R) > 0, R≥R0, explains that f (R) theory must avoid a
ghost state, while the second condition, f R R (R) > 0, R≥R0
is introduced to avoid a negative mass squared of a scalar-field
degree of freedom. Hence the present solutions are ghost free
and free of any exotic matter instability caused by the external
perturbation [
41
]. This means the final state of gravitational
collapse in the present case is not a two phase transition. In
other words one may not have any condition to convert a BH
into naked singularity. This shows that, in f (R) gravity, the
instability of a gravitating system decreases rapidly and the
system tends to a stable state naturally. This is the important
consequence of what we expect as the f (R) theory modifies
the interaction of gravity by the inclusion of a new scalar
field. We have investigated that f (R0) plays a dominant role
in trapping the collapsing fields and contributes to the black
hole formation. Due to the repulsive nature of the scalar force,
the f (R0) term slows down the collapse rate.
5 Conclusion
It is particularly interesting to establish the predictions of
f (R) theories concerning the gravitational collapse, and
particularly the collapse time, for several astrophysical objects.
The outcomes of the analysis of gravitational collapse in
f (R) theory may provide constraints for the validity of
models and be helpful to discard the models which appear to
contradict experimental investigations. Here, we have examined
the gravitational contraction of an inhomogeneous perfect
fluid in f (R) gravity by considering the metric approach.
We have assumed an n + 2-dimensional spacetime with an
inhomogeneous and isotropic perfect fluid as the
gravitating source. The n-dimensional fluid sphere is taken as the
interior and Schwarzschild spacetimes as exterior region,
respectively. The general conditions for the smooth
matching of two regions have been formulated. For the solution of
the field equations, the assumption of constant curvature is
used, which implies that pressure and density are constants
in this case. Two physical apparent horizons, the black hole
horizon and the cosmological horizon, have been found. We
have shown that trapped surfaces are formed earlier than the
singular point of the collapsing sphere, hence a singularity
is covered by the black hole horizon. This favors the cosmic
censorship conjecture.
From Eq. (3.9), the rate at which fluid sphere collapse
occurs is given by
Y¨ = −
×
(n − 1)m
Y n
Y
+ (n + 1)F (R0)
f (R0)
2
(5.1)
For the collapsing process, the acceleration should be
negative, which is possible when
Y <
(n − 1)(n + 1)m F (R0)
− 8nπ ((n − 1) p0 − ρ0) − f (2R0)
1
n+1
It is evident from Eq. (5.1) that the f (R0) term slows
down the collapsing process (as mentioned in [
26
]) when
F(1R0) ( 8nπ ((n − 1) p0 − ρ0) − f (2R0) ) < 0 is satisfied.
Further, due to the f (R0) term there exist two physical horizons,
namely the black hole horizon and the cosmological horizon.
We would like to point out that, for n = 2, our results match
the results of Sharif and Kausar [
47
].
As mentioned earlier (in the introduction), there are two
types of solutions concerning the gravitational collapse in
f (R) gravity: one predicts a naked singularity and the other
a BH. In [
30,31,35
], the final state of gravitational collapse
in f (R) gravity is a naked singularity, while in [
33,34,36,
37,45
] a BH has been found as a final outcome of collapse
in f (R) gravity. Thus our results favor the investigations
of [
33,34,36,37,45
] and may be considered as one example
of Oppenheimer–Snyder–Datt models in f (R) gravity. The
f (R0) term slows down the process of gravitational collapse,
and this favors the finding of Ref. [
26
]. Finally, we would like
to mention that the results of this paper can be extended in the
frame work of other modified theories of gravity, like f (T ),
f (G), f (R, G) and f (R, T ).
Acknowledgements The constructive comments and suggestions of
the anonymous referee are highly acknowledged.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. T.P. Sotiriou , V. Faraoni , Rev. Mod. Phys . 82 , 451 - 497 ( 2010 )
2. T.P. Sotiriou , Modified action for gravity: theory and phenomenolgy . PhD thesis (SISSA, Trieste) ( 2007 ). arXiv: 0710 .4438 [gr-qc]
3. T.P. Sotiriou , Class. Quantum Grav . 26 , 152001 ( 2009 )
4. S. Nojiri , S.D. Odintsov , Int. J. Geom. Methods Mod. Phys. 4 , 115 ( 2007 )
5. S. Capozziello , M. Francaviglia , Gen. Relativ. Grav. 40 , 357 ( 2008 )
6. S. Capozziello , V. Faraoni , Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics (Springer, Netherlands, 2011 )
7. S. Nojiri , S.D. Odintsov , Phys. Rept . 505 , 59 ( 2011 )
8. S. Nojiri , S.D. Odintsov, in Problems of Modern Theoretical Physics, A Volume in Honour of Prof. Buchbinder, I. L in the Occasion of his 60th Birthday (TSPU Publishing , Tomsk, 2008 ), pp. 266 - 285 . arXiv: 0807 . 0685
9. M. Sharif , Z. Yousaf , Int. J. Theor Phys . 55 , 470 - 480 ( 2016 )
10. J.R. Oppenheimer , H. Snyder, Phys. Rev . 56 , 455 ( 1939 )
11. C.W. Misner , D. Sharp , Phys. Rev. 136B , 571 ( 1964 )
12. L. Herrera , N.O. Santos , Phys. Rep . 286 , 53 ( 1997 )
13. L. Di Herrera , A. Prisco , J.R. Hernandez , N.O. Santos , Phys. Lett. A 237 , 113 ( 1998 )
14. L. Herrera , N.O. Santos , Phys. Rev. D 70 , 084004 ( 2004 )
15. L. Herrera , A. Di Prisco , J. Ospino , Gen. Relativ. Gravit. 44 , 2645 ( 2012 )
16. L. Herrera , Int. J. Mod. Phys. D 15 , 2197 ( 2006 )
17. L. Herrera , A. Di Prisco , W. Barreto, Phys. Rev. D 73 , 024008 ( 2006 )
18. L. Herrera , N.O. Santos , A. Wang , Phys. Rev. D 78 , 084024 ( 2008 )
19. G. Abbas, Sci. China. Phys. Mech. Astron . 57 , 604 ( 2014 )
20. S.M. Shah , G. Abbas Eur , Phys. J. C 77 , 251 ( 2017 )
21. G. Abbas, Astrophys. Space Sci . 350 , 307 ( 2014 )
22. G. Abbas, Adv. High Energy Phys . 2014 , 306256 ( 2014 )
23. G. Abbas, Astrophys. Space Sci . 352 , 955 ( 2014 )
24. G. Abbas, U. Sabiullah, Astrophys. Space Sci . 352 , 769 ( 2014 )
25. S.H. Mazharimousavi , M. Halilsoy , T. Tahamtan, Eur. Phys. J. C 72 , 1851 ( 2012 )
26. A. Borisvo , B. Jain , P. Zhang, Phys. Rev. D 85 , 063578 ( 2004 )
27. H. - J.A. Schmidth , Int. J. Geom. Methods Phys . 4 , 209 ( 2007 )
28. S. Capozziello , M. De Laurentis , S.D. Odintsov , A. Stabile , Phys. Rev. D 83 , 064004 ( 2011 )
29. J.A.R. Cembranos , A. de la Cruz-Dombriz, B.M. Nu ´n˜ez , JCAP 04 , 021 ( 2012 )
30. S.G. Ghosh , S.D. Maharaj , Phys. Rev. D 85 , 124064 ( 2012 )
31. R. Goswami , A.M. Nzioki , S.D. Maharaj , S.G. Ghosh , Phys. Rev. D 90 , 084011 ( 2014 )
32. B. Datt , Z. Phys . 108 , 314 ( 1938 ) (reprinted as Golden Oldie Datt , B. , Gen . Relativ. Gravit. 31 , 1619 ( 1999 ))
33. D. Hwang , B.H. Lee , D. Yeom , JCAP 12 , 006 ( 2011 )
34. C.S.J. Pun , Z. Kovacs , T. Harko, Phys. Rev. D 78 , 084015 ( 2008 )
35. E. Santos, Astrophys. Space Sci . 341 , 411 ( 2012 )
36. A.H. Ziaie , K. Atazadeh , S.M.M. Rasouli , Gen. Relativ. Gravit. 43 , 2943 ( 2011 )
37. M. Sharif , Z. Nasir , Astrophys. Space Sci . 357 , 89 ( 2015 )
38. S. Capozziello et al., Class. Quantum Gravity 27 , 165008 ( 2010 )
39. A. Sheykhi , S. Salarpour , Y. Bahrampour , Phys. Scr . 87 , 045004 ( 2013 )
40. M. Sharif , Z. Yousaf , Phys. Rev. D 88 , 024020 ( 2013 )
41. M. Sharif , Z. Yousaf , MNRAS 440 , 3479 ( 2014 )
42. M. Sharif , H.R. Kausar , Phys. Lett. B 697 , 1 ( 2011 )
43. M. Sharif , H.R. Kausar , J. Cosmol . Astropart. Phys. 07 , 022 ( 2012 )
44. M. Sharif , Z. Yousaf , Astropart. Phys. 56 , 19 ( 2014 )
45. S. Nojiri , S.D. Odintsov , Phys. Lett. B 735 , 376 ( 2014 )
46. M. Farasat Shamir , Z. Ahmad , Z. Raza , Int. J. Theor. Phys . 54 , 1450 ( 2015 )
47. M. Sharif , H.R. Kausar , Astrophys. Space Sci . 331 , 281 ( 2011 )
48. P. Rudra , R. Biswas , U. Debnath, Astrophys. Space Sci . 335 , 505 ( 2011 )
49. S.G. Ghosh , D.W. Deshkar , Int. J. Mod. Phys. D 12 , 913 ( 2003 )
50. M. Sharif , Z. Ahmad , Acta Phys. Polon. B 39 , 1337 ( 2008 )
51. A. Banerjee , U. Debnath , S. Chakraborty , Int. J. Mod. Phys. D 12 , 1255 ( 2003 )
52. A. Feinstein, Phys. Lett. A 372 , 4337 ( 2008 )
53. A. Cooney et al., Phys. Rev. D 83 , 064033 ( 2010 )
54. R. Goswami et al., Phys. Rev. D 90 , 084011 ( 2014 )
55. A. Ganguly et al., Phys. Rev. D 89 , 064019 ( 2014 )
56. N. Ifra , M. Zubair , Eur. Phys. J. C 75 , 62 ( 2015 )
57. N. Ifra et al., JCAP 1502 , 033 ( 2015 )
58. A.D. Dolgov , M. Kawasaki , Phys. Lett. B 573 , 1 ( 2003 )
59. D.M. Eardley , L. Smarr, Phys. Rev. D 19 , 2239 ( 1979 )