A periodic varying deceleration parameter in gravity
The phenomenon of accelerated expansion of the present universe and a cosmic transit aspect is explored in the framework of a modified gravity theory known as gravity (where is the Ricci scalar and is the trace of the energy momentum tensor of the matter content). The cosmic transit behaviour signifies a signature flipping behaviour of the deceleration parameter. We employ a periodic varying deceleration parameter and obtained the exact solution of field equations. The dynamical features of the model including the oscillatory behaviour of the EOS parameter are studied. We have also explored the obvious violation of energy momentum conservation in gravity. The periodic behaviour of energy conditions for the model are also discussed with a wide range of the free parameters.
Keywords: FRW metric; gravity; periodic varying deceleration parameter.
The phenomenon of accelerated expansion of the present universe becomes a centre of attention for all astrophysicists, cosmologists, and astronomers. Observations from Supernovae Ia, CMB, BAO and other astrophysical measurements have confirmed this fact. The late time behavior of the universe is one of the major challenges in modern cosmology. In order to get a proper theoretical explanation of this phenomenon, various concepts and theories have been proposed in recent times by many researchers. Einstein’s General theory of Relativity (GR) fails to explain this late time behaviour of the universe. Modified gravity theories emerge as alternatives to the conventional cosmology and increasingly become popular to describe the late time cosmic speed up process. Geometrically modified theories are the generalizations of GR, in which the Einstein-Hilbert action is modified by replacing the Ricci scalar by a more general function, may be of Ricci scalar or any other function with matter-geometry coupling. In particular, gravity ( is Ricci scalar) carroll04 (); nojiri07 (); bertolami07 (), gravity ( is torsion scalar) bengocheu09 (); linder10 (), gravity ( is Gauss-Bonnet scalar ) bamba10a (); bamba10b (); rodrigues14 () and gravity ( is Ricci scalar and is trace of stress energy momentum tensor) Harko2011 () are some widely used alternative modified theories. Among these geometrically modified theories, theory has attracted a lot of attention for the astrophysicists in recent times because of its ability to explain several issues in cosmology and astrophysics Zubair16 (); Singh016 (); Sahoo16 (); Sahoo16a (); Sahu17 (); Mishra16 (); Moraes17 (); Yousaf17 (). The big rip singularity of LRS Bianchi type-I space time has been obtained in this theory Sahoo/2015 () . Also, it has an elegant geometrical structure and can be reduced to the GR and theory under suitable functional forms of the functional . Moreover, the dependence of the geometrical action in theory may be due to the existence of some imperfect fluids and intrinsically may have some quantum effects like particle production Harko14 ().
The matter energy coupling in gravity plays a significant role to provide a complete theoretical description for the late time acceleration of the universe, without resorting to the existence of dark energy. In literature, there have been a lot of investigations on different aspects of gravity such as scalar perturbation Alvarenga13 (), energy conditions Sharif13 (); Kiani14 (); Alvarenga013 (), thermodynamics Sharif12 (); Jamil012 (), wormhole solutions Zubair/2016 (); Moraes17a (); Moraes/2017 (), and solutions in higher dimensions Yousaf16 (); Moraes15 (). In this theory, the interactions of matter with space time curvature become a well motivation to consider cosmological consequence with different matter components Sahoo17 (); Sahoo017 (). The role of violation of energy momentum conservation in modified theories have yet not been studied properly in literature. Josset et al. Josset17 ()have shown that a violation of energy momentum conservation leads to accelerated expansion in modified gravity models. It will be more interesting to consider this phenomena in the context of gravity.
On the basis of the isotropic and spatially homogeneous universe, the Friedmann Robertson Walker (FRW) metric is adequate for describing the present state of the universe. Due to this FRW models are globally acceptable with the perfect fluid matter on account of flat space-time. According to the literature, an FRW cosmological model has been investigated in gravity along with perfect fluid matter and linearly varying deceleration parameter Ramesh16 (), and with magnetized strange quark matter and Aygun16 (). After that, a time periodic varying deceleration parameter (PVDP) is introduced by Shen and Zhao Shen2014 () in the account of an oscillating cosmological model with quintom matter. Oscillating cosmological models have been studied in literature Dodelson00 (); Nojiri06 (). These models are very natural to resolve the coincidence problem due to periods of acceleration.
We have organized the paper as follows. In Section-II, we formulate the gravitational field equations of for a flat universe. In Section-III, solutions to the field equations are obtained by using the PVDP ansatz. Section-IV contains a discussion on the dynamical properties of the model based on PVDP. We discuss briefly on the non-conservation of energy-momentum at back drop of our present model in Section-V. The energy conditions are discussed in Section-VI. We discuss the stability of the model though linear homogeneous perturbation in Section-VII. The results of the present work are summarized in the last section.
Ii Basic Field equations
In gravity theory, we have a geometrically modified action
which can be varied with respect to the metric tensor to obtain the gravitational field equation for gravity as
where, , , ; is the co-variant derivative. , where and are the Newtonian Gravitational constant and speed of light in vacuum respectively. The energy-momentum tensor for a perfect fluid distribution of the universe, and are derived from the matter Lagrangian . Following Harko et al. Harko2011 (), we choose the matter Lagrangian as which yields . Here, and are the energy density and pressure respectively. is the four-velocity of the fluid satisfying in comoving coordinates.
The functional can be chosen in many ways corresponding to viable models. In the present work, we have consider the functional as where is an arbitrary function of the trace of the energy momentum tensor. The corresponding field equations become
where denotes the partial derivative of with respect to . Assuming , being a constant, the field equations for a flat FriedmannRobertsonWalker (FRW) metric
are obtained as
In the above equations, is the scale factor of the universe and the Hubble parameter is defined through . We have chosen the unit system such that .
Iii Periodically varying Deceleration parameter
Deceleration parameter is one of the geometrical parameters through which the dynamics of the universe can be quantified. It is defined as where the overhead dots denote time derivatives. In the context of the late time cosmic speed up phenomena with a cosmic transit from a phase of deceleration to acceleration at some redshift , one can speculate a signature flipping of the deceleration parameter. Obviously, at a decelerated phase, is positive and at the accelerating phase, it becomes negative. General dynamical behaviour can also be assessed through the values of the deceleration parameter in the negative domain. While de Sitter expansion occurs for , accelerating power-law expansion can be achieved for . A super-exponential expansion of the universe occurs for . Even though, there is uncertainty in the determination of the deceleration parameter from observational data, most of the studies in recent times constrain this parameter in the range . Keeping in view the signature flipping nature of , in the present work, we assume a periodic time varying deceleration parameter Shen2014 ()
where and are positive constants. decides the periodicity of the PVDP and can be considered as a cosmic frequency parameter. is an enhancement factor that enhances the peak of the PVDP. This model simulates a positive deceleration parameter (for ) at an initial epoch and evolves into a negative peak of . After the negative peak, it again increases and come back to the initial states. The evolutionary behaviour of is periodically repeated. In other words, the universe in the model, starts with a decelerating phase and evolves into a phase of super-exponential expansion in a cyclic history.
Integration of Eq. (7) yields
where is a constant of integration. Here we have used the definitions and so that . Without loss of generality, we may consider and the Hubble function becomes
The scale factor is obtained by integrating the Hubble function in Eq. (9) as
where is the scale factor at the present epoch and can be taken as 1. Inverting Eq. (10), we obtain
In the above equation, redshift is defined through the relation . An equivalent present epoch can be derived from Eq.(11) as corresponding to , where is a positive integer including zero. It can be straightforward to express the deceleration parameter in Eq. (7) in terms of redshift using Eq.(11). In Fig.2, we have shown the evolutionary aspect of the deceleration parameter as a function of cosmic time for three different domain of the parameter namely and . The periodic nature of the PVDP is clearly depicted in the figure. In Fig. 2, the evolutionary aspect of the deceleration parameter as a function of redshift is shown. The evolutionary behaviour of the PVDP is greatly affected by the choice of the parameter . In general, the deceleration parameter oscillates in between and . For , deceleration parameter becomes a constant quantity with a value of and can lead to a de Sitter kind of expansion. For , it varies periodically in the negative domain and provide accelerated models. However, for , evolves from a positive region to a negative region showing a signature flipping at some redshift . It is worth to mention here that, the transition redshift depends on the choice of the parameter . This parameter can be constrained from the cosmic transit behaviour and transit redshift . We have adjusted the values of so as to get a compatible with that extracted from observations Capozziello14 (); Capozziello15 (); Farooq17 (); Moraes16 (). In Fig.2, the signature flipping of the deceleration parameter is shown to occur at for . In the event of non availability of any observational data regarding cosmic oscillation and corresponding frequency, we consider as a free parameter. In the present work, we are interested for a time varying deceleration parameter that oscillates in between the decelerating and accelerating phase to simulate the cosmic transit phenomenon. In order to assess the dynamical features of the model through numerical plots, we assume a small value for , say .
The scale factor and the Hubble parameter derived from the PVDP are shown in Fig. 3 and Fig. 4 respectively for some specific time frame. Within the time frame, the scale factor increases with cosmic time whereas the Hubble parameter decreases with time. However, the evolutionary behaviour of the scale factor is governed by a function and that of the Hubble parameter is governed by a function and therefore both can either be positive or negative at some epoch.
Iv Dynamical properties of the model
For a PVDP as defined Eq. (7), the above expressions reduce to
Equation (14) sets up a condition for the parameters and to get a positive energy density,
A signature flipping behaviour of the deceleration parameter fixes to be greater than 1 (refer Fig.2). We have constrained from the cosmic transit redshift to be . From Eq. (16), it is certain that, this value of allows any positive values for . In view of this, one may take as a free parameter with positive values only. In the present work, we have considered three moderate values, and for numerical calculation of the dynamical parameters. In Fig. 6, the periodic variation of the energy density is shown for these values of the parameter . It is evident that, energy density has periodic singularities at the cosmic times , is an integer. The periodic variation clearly depends on the choice of . Since we have taken , the cosmic singularity occurs corresponding to the time period, . The interesting feature is that, in a given cosmic cycle, it starts from a very large value at an initial time () and decreases to a minimum, , and then again increases with the growth of time. The minimum in energy density occurs at a time given by . The evolutionary trend of the energy density is not changed by a variation of , rather an increase in simply decreases the value of at a given time. In other words, with an increase in , there occurs a decrement in .
The evolutionary behaviour of pressure is shown in Fig. 6. Pressure is also found to have a periodic variation with singularities at or at . Within a given cycle, pressure decreases from large positive values at the beginning to large negative values and then reverses the trend. However, pressure is a negative quantity at the present epoch in a given cycle. The choice of the parameter has some effects on the evolutionary trend. In a given cycle, in general, lower value of results in a pressure curve that lies to the left side in the figure. The crossing over time from positive domain to negative domain is decided by the value of .
In Fig. 8, we have plotted the evolution of EOS parameter, , as a function of cosmic time. The EOS parameter exhibits an oscillatory behaviour. In the first half of the cosmic cycle, decreases from a positive value close to to negative values after crossing the phantom divide at . After attending a minimum it again increases to positive value at the end of the cycle. One interesting feature of the equation of state parameter is that, unlike the energy density and pressure, it does not acquire any singular values during the cosmic cycle. This fact is due to the cancellation of the factor from the pressure by the same factor of energy density and depends only the value of the PVDP. Since the PVDP does not a have singularity, the same thing also occurs in the EOS parameter. The oscillatory behaviour comes only from the factor appearing both in the numerator and denominator of . The evolutionary trend is affected by the choice of . Curves of with low values of remain on top before the phantom divide whereas after the phantom divide, they remain in the bottom of all the curves. At an equivalent present epoch in any given cycle, the EOS parameter remains within the quintessence region with a value close to . One can decipher the evolution of the EOS parameter, in the purview of usual Friedman model, as an evolution from a radiation dominated phase () to a matter dominated phase () and then to a dark energy driven accelerated phase (). At the present epoch, our model predicts an EOS that behaves more like a cosmological constant and the model is somewhat close to that of CDM model. This aspect of the EOS parameter is clearly visible in Fig. 8, where we have shown the evolution of as a function of redshift. It is clear from Fig. 8 that, at , . It is interesting to note that, the overlapping of the present model with CDM at equivalent present epochs is independent of the choice of the parameter . In this model from Fig. 8, it can be observed that the evolution of as a function of redshift shows a transitional behavior from at low redshift to at higher redshift. This transitional behavior of of this model fits with SNL3 data Feng2005 (). In particular, the use of SNL3 data suggests that BAO data is also partly responsible for this. Some recent reconstruction of the EOS from different observational data sets including the high redshift Lyman- forest (LyFB) measurement favours a non-constant dynamical dark energy. In these reconstructed models, the EOS evolves with time and crosses the phantom divide Zhao12 (); Zhao17 (). The behavior of EoS in this model is consistent with quintom model which allows to cross . As inferred in Ref.Zhao17 (), the departure from is more evident in the reconstruction history of the dynamical dark energy with more recent data sets including the LyFB measurement.
For the present model with a PVDP, we obtain, the density parameter as
which can be expressed as a function of redshift as
The density parameter as a function of redshift is plotted in Fig. 9 for three representative values of . remains almost unaltered in the range of redshift greater than 1 for all the three values of considered in the work. However, with the cosmic evolution, decreases with cosmic time after . The density parameter, at a given redshift, is observed to have lower value for higher values of .
V Violation of Energy-Momentum Conservation
Friedman models in GR ensure the energy conservation through the continuity equation
which implies . Here , the volume scale factor of universe and the quantity gives an account of the total energy. As the universe expands the amount of dark energy in an expanding volume increases in proportion to the volume. If that space time is standing completely still, the total energy is constant; if it’s evolving, the energy changes in a completely unambiguous way. It decreases or increases in time. However in modified gravity theories, one may get a different picture. Taking a covariant derivative of Eq. (2), one can obtain Harko14 (); alva13 (); bar14 (); das17 ()
With the substitution of and , Eq. (21) reduces to
One should note here that, for , one would get . However for , the conservation of energy-momentum is violated. Recently some researchers have investigated the consequence of the violation of energy-momentum conservation (i.e. ) in modified gravity theories. The non-conservation of energy-momentum may arise due to non unitary modifications of quantum mechanics and in phenomenological models motivated by quantum gravity theories with spacetime discreteness at the Planck scale Josset17 (). In the context of unimodular gravity, Josset et al. have shown that a non-conservation of energy-momentum leads to an effective cosmological constant which increases or decreases with the creation or annihilation of energy during the cosmic expansion and can be reduced to a constant when matter density diminishes Josset17 (). Shabani and Ziaie Shabani17 () have studied the consequence of the non-conservation of energy-momentum in some classes of gravity with pressure-less cosmic fluid and showed a violation of energy-momentum conservation in modified theories of gravity can provide accelerated expansion. Also, the non-conservation of the energy-momentum tensor implies in non-geodesic motions for test particles in gravitational fields as it was deeply investigated Baffou17 (). In Shabani18 (); Moraes18 () the authors have constructed a formalism in which an effective fluid is conserved in gravity, rather than the usual energy-momentum tensor non-conservation. We have investigated the non-conservation of energy-momentum for the class of models in gravity with the suggested PVDP. We quantify the violation of energy-momentum conservation through a deviation factor , defined as
In case of the model satisfying energy-momentum conservation, we have , otherwise, we get a non zero value for this quantity. can be positive or negative depending on whether the energy flows away from or into the matter field. In Fig. 10, we have shown the non-conservation of energy-momentum for a periodic cosmic cycle. It is clear that, except for a very limited period, the conservation is violated along with the cosmic evolution. However, the nature of energy flow changes periodically. This behaviour is obtained for all the values of considered in the work. At an equivalent present epoch in a given cosmic cycle, at least within the purview of the present model, there is a signal of non-conservation.
Vi Energy Conditions
The energy conditions (ECs) of GR are a variety of different ways to show positive energy density more precisely. The ECs take the form of various linear combinations of the stress-energy tensor components (at any specified point in spacetime) should be positive, or at least non-negative Visser/2000 (). The ECs are the fundamental tools for the study of wormholes (WHs) and black holes in different physical scenario. Particularly ECs are the outcomes of Raychaudhuri’s equation for the expansion nature Carroll/2004 (). The study of singularities in the spacetime was based on ECs. The main ECs in GR such as null, weak, strong, and dominant (NEC, WEC, SEC, and DEC) respectively for the energy-momentum tensor are expressed as
One can observe from the above Figures 12-14, all the ECs are behaved periodic for fixed values of and with accepted range for in this model. The free parameters are considered based on the positivity of energy density as shown in Figure-12. In the present model DEC is satisfied and all other ECs are violated.
Vii Stability Analysis
In this section we wish to analyse the stability of our model under linear homogeneous perturbations in the FRW background. We consider linear perturbations for the Hubble parameter and the energy density as Sharif2013 ()
where and are the perturbation parameters. In the above, we have assumed a general solution which satisfies the background FRW equations. The matter energy density can be expressed in terms of as
The Friedman equation and the trace equation for the modified gravity model with a functional can be obtained as
Here, is the expansion scalar. For a standard matter field, we can have the first order perturbation equation
Integration of Eq.(35) leads to
where is a non zero positive constant. Consequently, the evolution of the perturbation becomes
the factor is evaluated as
The growth and decay of the perturbation depend on the factors and periodically. We found that the considered values for and in the physical parameters , and are compatible with the decay of perturbation.
In this work, we have studied the background cosmology of an isotropic flat universe in the framework of gravity. According to the observations and associated analysis, the universe undergoes an accelerated expansion in the present epoch. The universe might have transitioned from a decelerated phase at some past epoch to an accelerated phase. This behaviour clearly hints for a time varying deceleration parameter which should evolve from a positive value in past to negative values at late phase of cosmic time. In other words, the evolving deceleration parameter displays a signature flipping behaviour. Keeping in view the signature flipping nature of the deceleration parameter, in the present work, we assume a periodically varying deceleration parameter to reconstruct the cosmic history. The PVDP has two adjustable parameters one of which can be constrained from the cosmic transit behaviour. The assumed deceleration parameter oscillates in between two limits usually set by the transit redshift. Consequently, the universe in this model, starts with a decelerating phase and evolves into a phase of super-exponential expansion with a periodic repetition of the phenomenon.
The PVDP obviously generates periodically varying dynamical properties of the universe. The energy density and pressure vary cyclically within a given cosmic period decided by the cosmic frequency parameter of the model. At some finite time, the magnitude of these physical parameters become infinitely large. This behaviour leads to a future type I singularity as classified by Nojiri et al Noj2005 (). There appears to have a big Rip at certain finite time during the cosmic repetition of the phenomenon since and . Since the parameters repeats their behaviour after a time period , the big Rip also occurs periodically after a time gap of .
The equation of state parameter for the PVDP has a cyclic behaviour that repeats with time. The EOS parameter evolves from a radiation dominated phase () to a matter dominated phase () and then to a dark energy driven accelerated phase (). It may cross the phantom divide for some cosmic time range. How far the well of the EOS parameter will go beyond the phantom divide is decided by the coupling constant . The EOS acquires a deeper well for a larger value of coupling constant. At the present epoch, our model predicts an EOS that behaves more like a cosmological constant.
In the present work, we have investigated an important aspect of gravity theory, the violation of energy-momentum conservation. It is well known that, in modified theories like gravity, the energy-momentum conservation is violated and this non-conservation can lead to a sort of accelerated expansion of the universe. theories of gravity, as originally proposed, predict a non-conservation of the usual energy-momentum tensor of matter Harko2011 (). As a result, although the form of the field equations remain same but now the test particles move in a geodesics and the choice of the Lagrangian function is not totally arbitrary. We have shown that, a PVDP leads to a kind of universe model, where within a given cycle, the energy momentum conservation is continuously violated except for a small period of cosmic time. As pointed out by Josset and Perez Josset17 (), modified gravity models can explain the accelerated expansion at the cost of energy momentum non conservation.
We have analyzed the energy conditions for the modified gravity theory in a general way. For the perfect fluid source of matter in Sect. VI we have shown the violation/validity of the energy conditions both analytically as well as graphically. The energy conditions in modified theories of gravity have a well defined physical motivation, i.e., Raychaudhuri’s equation along with attractive nature of gravity. For the considered model of gravity, the NEC and SEC are derived from the Raychaudhuri equation together with the condition that gravity is attractive. It is shown that these conditions behaved periodic from those derived in the context of GR. Finally, we have discussed the stability of the solutions under linear homogeneous perturbations. The stability depends on the values of the parameters and as they behaves periodically.
Acknowledgements.PKS and PS acknowledges DST, New Delhi, India for providing facilities through DST-FIST lab, Department of Mathematics, where a part of this work was done.
- (1) T. Harko, F. S. N. Lobo, S. Nojiri, S. D. Odintsov, Phys. Rev. D 84(2) (2011) 024020.
- (2) S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D 70 (2004) 043528.
- (3) S. Nojiri, S. D. Odintsov, Int. J. Geom. Methods Mod. Phys. 04 (2007) 115.
- (4) O. Bertolami, et al., Phys. Rev. D 75 (2007) 104016.
- (5) G. R. Bengochea, R. Ferraro, Phys. Rev. D 79 (2009) 124019.
- (6) E. V. Linder, Phys. Rev. D 81 (2010) 127301.
- (7) K. Bamba, C. Q. Geng, S. Nojiri, S. D. Odintsov, Euro. Phys. Lett. 89 (2010) 50003.
- (8) K. Bamba, S. D. Odintsov, L. Sebastiani, S. Zerbini, Eur. Phys. J. C. 67 (2010) 295.
- (9) M. E. Rodrigues, M. J. S. Houndjo, D. Mommeni, R. Myrzakulov, Can J. Phys. 92 (2014) 173.
- (10) M. Zubair, S. M. Ali Hassan, Astrophys Space Sci., 361 (2016) 149.
- (11) G. P. Singh, B. K. Bishi, P. K. Sahoo; Int. J. Geom. Methods Mod. Phys., 13 (2016) 1038.
- (12) P. K. Sahoo, B. Mishra, P. Sahoo, S. K. J. Pacif, Eur. Phys. J. Plus 131 (2016) 333.
- (13) P. K. Sahoo, B. Mishra, S K Tripathy, Ind. J. Phys. 90 (2016) 485.
- (14) S. K. Sahu, S. K. Tripathy, P. K. Sahoo, A. Nath, Chin. J. Phys. 55 (2017) 862.
- (15) B. Mishra, S. Tarai, S. K. Tripathy, Adv. High. Energy. Phys. 2016 (2016) 8543560.
- (16) P. H. R. S. Moraes, P. K. Sahoo, Eur. Phys. J . C 77 (2017) 480.
- (17) Z. Yousaf, M. Ilyas, M. Z. Bhatti, Eur. Phys. J. Plus, 132 (2017) 268.
- (18) P. K. Sahoo, M. Sivakumar, Astrophys Space Sci. 357 (2015) 60.
- (19) T. Harko, Phys. Rev. D. 90 (2014) 044067.
- (20) F. G. Alvarenga, A. de la Cruz-Dombriz, M. J. S. Houndjo, M. E. Rodrigues, D. Saez-Gaomez, Phys. Rev. D 87 (2013) 103526.
- (21) M. Sharif, M. Zubair, J. High Energ. Phys. 12 (2013) 079.
- (22) F. Kiani, K. Nozari,. Phys. Lett. B 728 (2014) 554.
- (23) F. G. Alvarenga, M. J. S. Houndjo, A. V. Monwanou, J.B. Chabi Orou, J. Mod. Phys. 04 (2013) 130.
- (24) M. Sharif, M. Zubair, JCAP 03 (2012) 028.
- (25) M. Jamil, D. Momeni, M. Ratbay, Chinese Phys. Lett. 29 (2012) 109801.
- (26) M. Zubair, Saira Waheed, Yasir Ahmad, Eur. Phys. J. C 76 (2016) 444.
- (27) P. H. R. S. Moraes, P. K. Sahoo, Phys. Rev. D. 96 (2017) 044038.
- (28) P. H. R. S. Moraes, R.A.C. Correa, R.V. Lobato, JCAP 07 (2017) 029.
- (29) Z. Yousaf, K. Bamba, M. Z. H. Bhatti, Phys. Rev. D 93 (2016) 124048.
- (30) P. H. R. S. Moraes, Eur. Phys. J. C 75 (2015) 168.
- (31) P. K. Sahoo, P. Sahoo, B. K. Bishi, Int. J. Geom. Methods Mod. Phys. 14 (2017) 1750097.
- (32) P. K. Sahoo, P. Sahoo, B. K. Bishi, S. Aygun, Mod. Phys. Lett. A, 32 (2017) 1750105.
- (33) T. Josset, A. Perez, Phys. Rev. Lett. 118 (2017) 021102.
- (34) G. Ramesh, S. Umadevi, Astrophys Space Sci. 361 (2016) 2.
- (35) S. Aygun, C. Aktas, I. Yilmaz, Astrophys Space Sci. 361 (2016) 380.
- (36) M. Shen, L. Zhao, Chin. Phys. Lett. 31 (2014) 010401.
- (37) S. Dodelson, M. Kaplinghat, E.Stewart, Phys. Rev. Lett. 85 (2000) 5276.
- (38) S. Nojiri, S. D. Odintsov; Phys. Lett. B 637 (2006) 139.
- (39) S. Capozziello, O. Farooq, O. Luongo, B. Ratra, Phys. Rev. D, 90 (2014) 044016.
- (40) S. Capozziello, O. Luongo, E. N. Saridakis, Phys. Rev. D, 91 (2015) 124037.
- (41) O. Farooq, F. Madiyar, S. Crandall, B. Ratra, Astrophys. J., 835 (2017) 26.
- (42) P. H. R. S. Moraes, G. Ribeiro, R. A. C. Correa, Astrophys Space Sci., 361 (2016) 227.
- (43) B. Feng, X. L. Wang, and X. M. Zhang, Phys. Lett. B 607 (2005) 35.
- (44) G.-B. Zhao, R. G. Crittenden, L. Pogosian, X. Zhang, Phys. Rev. Lett. 109 (2012) 171301.
- (45) G.-B. Zhao et al., Nature Astronomy, 1 (2017) 627, arXiv:1701.08165.
- (46) F. G. Alvarenga, A de la Cruz-Dombriz, M. J. S. Houndjo, M.E. Rodrigues, D. Saez Gomez, Phys. Rev. D, 87 (2013) 103526.
- (47) O. J. Barrientos, G. F. Rubilar, Phys. Rev. D, 90 (2014) 028501.
- (48) A. Das, S. Ghosh, B. K. Guha, Phys. Rev. D, 95 (2017) 124511.
- (49) H. Shabani, A. H. Ziaie, Eur. Phys. J. C 77 (2017) 282.
- (50) E. H. Baffou et al., Chin. J. Phys. 55 (2017) 467.
- (51) H. Shabani, A. H. Ziaie, Int. J. Mod. Phys. A 33 (2018) 1850050.
- (52) P. H. R. S. Moraes, R. A. C. Correa, G. Ribeiro, Eur. Phys. J. C 78 (2018) 192.
- (53) M. Visser, C. Barcelo, COSMO-99 (1999) 98, arXiv:gr-qc/0001099
- (54) S. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison Wesley, 2004).
- (55) F. G. Alvarenga, M. J. S. Houndjoa, A. V. Monwanoub, Jean B. Chabi Oroub, arXiv:1205.4678
- (56) M. Sharif, S. Rani, R. Myrzakulov, Eur. Phys. J. Plus 128 (2013) 123.
- (57) S. Nojiri, S. D. Odintsov, S. Tsujikawa, Phys. Rev. D, 71 (2005) 063004.
- (58) M. Sharif and M. Zubair, J. Phys. Soc. of Japan, 82 (2013)014002.