Advances in Astronomy
Volume 2019 (2019), Article ID 8574798, 8 pages
https://doi.org/10.1155/2019/8574798

Research Article

A Cosmological Scenario from the Starobinsky Model within the Formalism

P. H. R. S. Moraes,1,2 P. K. Sahoo,³ G. Ribeiro,⁴ and R. A. C. Correa2,5

¹Universitá degli Studi di Napoli Federico II (UNINA), Dipartimento di Fisica, Napoli 80126, Italy
²Instituto Tecnológico de Aeronáutica (ITA), Departamento de Física, 12228-900 São José dos Campos, SP, Brazil
³Department of Mathematics, Birla Institute of Technology and Science, Pilani, Hyderabad Campus, Hyderabad 500078, India
⁴Universidade Estadual Paulista “Júlio de Mesquita Filho” (UNESP), Departamento de Física e Química, 12516-410 Guaratinguetá, SP, Brazil
⁵Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Correspondence should be addressed to R. A. C. Correa; fis04132@gmail.com

Received 19 November 2018; Accepted 27 March 2019; Published 2 June 2019

Academic Editor: Charalampos C. Moustakidis

Copyright © 2019 P. H. R. S. Moraes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper we derive a novel cosmological model from the theory of gravitation, for which is the Ricci scalar and is the trace of the energy-momentum tensor. We consider the functional form , with being the Starobinsky Model, named , and , with and being constants. We show that a hybrid expansion law form for the scale factor is a solution for the derived Friedmann-like equations. In this way, the model is able to predict both the decelerated and the accelerated regimes of expansion of the universe, with the transition redshift between these stages being in accordance with recent observations. We also apply the energy conditions to our material content solutions. Such an application makes us able to obtain the range of acceptability for the free parameters of the model, named and .

1. Introduction

The theories of gravity [1, 2] are an optimistic alternative to the shortcomings General Relativity (GR) faces as the underlying gravitational theory, such as those discussed by Padmanabhan [3]; Antoniadis et al. [4]; Demorest et al. [5]; Bull et al. [6]. They can account for the cosmic acceleration [7, 8] with no need for a cosmological constant, providing good match between theory and cosmological observations [9–11].

Particularly, in standard cosmology, derived from GR, the dark energy and dark matter should compose of the universe [12], but their nature is still dubious [13, 14].

Another crucial trouble surrounding GR is the difficulty in quantizing it. Attempts to do so have been proposed [15–17] and can, in future, provide us a robust and trustworthy model of gravity-quantum mechanics unification.

Meanwhile it is worthwhile to attempt to consider the presence of quantum effects in gravitational theories. Those effects can rise from the consideration of terms proportional to the trace of the energy-momentum tensor in the gravitational part of the action, yielding the gravity theories [18]. Those theories were also motivated by the fact that although gravity is well behaved in cosmological scales, the solar system regime seems to rule out most of the models proposed so far [19–22]. Furthermore, rotation curves of spiral galaxies were constructed in gravity, but the results did not favour the theory, as it can be checked in Chiba [23]; Dolgov & Kawasaki [24]; Olmo [25]. The structure and cosmological properties of the modified gravity starting from theory to power-counting renormalizable covariant gravity were presented by Nojiri & Odintsov [26]. The review in Nojiri et al. [27] describes the cosmological developments regarding inflation, bounce, and late-time evolution in , , and modified theories of gravity, with and being the Gauss-Bonnet and torsion scalars.

Despite its recent elaboration, gravity has already been applied to a number of areas, such as cosmology ([28–34] and astrophysics [35–42]). Particularly, solar system tests have been applied to gravity [43] and the dark matter issue was analysed by Zaregonbadi et al. [44]. The late-time behaviour of cosmic fluids consisting of collisional self-interacting dark matter and radiation was discussed by Zubair et al. [45]. Moreover, considering the metric and the affine connection as independent field variables, the Palatini formulation of the gravity can be seen in [46, 47].

By investigating the features of an or model, one realizes the strong relation they have with the functional form of the chosen functions for and , as well as with their free parameters values. In fact, a reliable method to constraint those “free” parameters to values that yield realistic models can be seen in Correa & Moraes [48] and Correa et al. [49] for these theories, respectively.

In the gravity, a reliable and reputed functional form was proposed by A.A. Starobinsky as [50, 51]

which is known as Starobinsky Model (SM), with a constant. It predicts a quadratic correction of the Ricci scalar to be inserted in the gravitational part of the Einstein-Hilbert action.

SM has been deeply applied to the cosmological and astrophysical contexts in the literature. Starobinsky showed that a cosmological model obtained from (1) can satisfy cosmological observational tests [51]. On the other hand, the model seems to predict an overproduction of scalarons in the very early universe. In an astrophysical context, SM is also of great importance. In Sharif & Siddiqa [52], the authors have explored the source of a gravitational radiation in SM by considering axially symmetric dissipative dust under geodesic condition. In Resco et al. [53], it has been shown that in SM it is possible to find neutron stars with , which raises as an important alternative to some of the GR shortcomings mentioned above [4, 5]. In Astashenok et al. [54], the macroscopical features of quark stars were obtained.

Our proposal in this paper is to construct a cosmological scenario from an functional form whose dependence is the same as in the SM, i.e., with a quadratic extra contribution of , as in (1). The dependence will be considered to be linear, as , with being a constant. Therefore, we will take

As far as the present authors know, the SM has not been considered for the dependence within models for cosmological purposes so far, only in the study of astrophysical compact objects [55–57] and wormholes [58, 59]. We believe this is due to the expected high nonlinearity of the resulting differential equation for the scale factor. Anyhow, the consideration of linear material corrections together with quadratic geometrical terms can imply interesting outcomes in a cosmological perspective as it does in the astrophysical level.

2. The Gravity

Following the steps in Harko et al. [18], we can write the gravity total action as

in which is the determinant of the metric , is the matter Lagrangian, and we are working with natural units.

By taking , with being the pressure of the universe, the variational principle applied in (3) yields the following field equations:

In (4), is the Einstein tensor, is the Ricci tensor, , is the matter-energy density of the universe, and . Still in (4), it can be straightforwardly seen that the limit recovers the GR field equations.

Also, the above choice for the matter Lagrangian is usually assumed in the literature as it can be checked in Harko et al. [18]; Moraes et al. [35, 38], among many others.

3. The Cosmology

Let us assume a flat Friedmann-Robertson-Walker metric in the field equations above. Such a substitution yields the following Friedmann-like equations:

where we are using the following definitions:

In the equations above, is the scale factor and dots represent time derivatives. Once again, the limit retrieves the standard formalism.

In terms of the Hubble parameter , the values of and from (5)-(6) are

with the following definitions:

For (9)-(10), we considered . Moreover, and , with , are functions of and its time derivatives, expressed by the following:

We can consider, as a solution for (9)-(10), the scale factor in the hybrid expansion law form [60]:

with and being constants. It can be seen that such a form for the scale factor consists of a product of power law and exponential law functions. Equation (23) mimics the power law and de-Sitter cosmologies as particular cases and can predict the transition from a decelerated to an accelerated regime of the universe expansion. It has been applied to Brans-Dicke models in Ozgur et al. [60], yielding observational constraints to and .

From (23), the Hubble and deceleration parameters are

such that the deceleration parameter is defined in such a way that its negative values describe an accelerated expansion of the universe.

We know that the universe not always has accelerated its expansion [7, 8]. The accelerated regime of expansion is considered to be a late-time phenomenon.

In this way, one can choose the constants and in such a way that the power law dominates over exponential law in the early universe and the exponential law dominates over power law at late times. Therefore, the decelerated and accelerated regimes of the universe expansion can be, respectively, well described, as well as the transition between these regimes.

From (25) it is clear that there is a transition from deceleration to acceleration phases of the universe expansion at , with . Since the negativity of the second term leads to a negative time, which indicates an nonphysical situation, we conclude that the cosmic transition may have occurred at .

From , with being the present value of the scale factor and being the redshift, we obtain the following time-redshift relation:

where denotes the Lambert function (also known as “product logarithm”).

Using (26), we can plot the deceleration parameter with respect to the redshift, which can be appreciated in Figure 1, in which the values chosen for and are in agreement with the observational constraints found in Ozgur et al. [60].

Figure 1: Deceleration parameter evolution in redshift for different values of and .

Plotting as a redshift function has the advantage of checking the reliability of the model, through the redshift value in which the decelerated-accelerated expansion of the universe transition occurs. We will call the transition redshift as and in our model it can be seen that it depends directly on the parameter . From Figure 1, the transition occurs at , corresponding to , respectively. The values of the transition redshift for our model are in accordance with the observational data, as one can check in Capozziello et al. [61, 62]; Farooq et al. [63].

Now, let us write the solutions for the material content of our model, named and . Using (24) in (9)-(10), we have

where we are using the definitions

with and , with , being functions of only, defined as

The evolution of the energy density, pressure, and corresponding equation of state (EoS) parameter with are shown in Figures 2–4, in which the time units are Gyr.

Figure 2: Evolution of with time.

Figure 3: Evolution of with time.

Figure 4: Evolution of with time.

4. Energy Conditions

Energy conditions (ECs), in the context of a wide class of covariant theories including GR, are relations one demands for the energy-momentum tensor of matter to satisfy in order to try to capture the idea that “energy should be positive”. By imposing so, one can obtain constraints to the free parameters of the concerned model.

The standard point-wise ECs are [64–66]

 Null energy condition (NEC): . Weak energy condition (WEC): , . Strong energy condition (SEC): . Dominant energy condition (DEC): .

Generally the above ECs are formulated from the Raychaudhuri equation, which describes the behaviour of space-like, time-like, or light-like curves in gravity.

In the present model the energy-momentum tensor has the form of a perfect fluid. So, we will use the above relations for analysing the ECs in theory. The behaviour of the ECs with , and is given in Figures 5–8.

Figure 5: Evolution of .

Figure 6: Validation of NEC.

Figure 7: Violation of SEC.

Figure 8: Validation of DEC.

5. Discussion and Conclusions

In this article we have proposed an cosmological model. For the functional form of the function , we investigated a quadratic correction to , as in the well-known SM, together with a linear term on . The substitution of such an function in the gravitational action generates extra terms in the field equations (4) and, consequently, in the Friedmann-like equations (5) and (6). Those extra terms bring some new information regarding the dynamics of the universe, as we discuss below.

Our solution for the scale factor is a hybrid expansion law, well described in Ozgur et al. [60]. From (23), we have obtained the cosmological parameters, namely, Hubble factor and deceleration parameter. Specifically, for the deceleration parameter, we could separate it in two phases: one describing the decelerated and the other describing the accelerated regime of the universe expansion. This can be well-checked in Figure 1, in which the transition redshift between these two stages agrees with observational data. Moreover, one can also note that, remarkably, the present () values for , named for and for , are also in agreement with observational data [12].

We have also obtained solutions for the material content of the universe, named and (see (27)-(41)). In Figures 2–4 we plot the evolution of , , and , the EoS parameter. The values chosen for and respect the energy conditions outcomes presented in Section 4. In Figure 3 we see that the pressure of the universe starts in positive values and then assumes negative values. In standard model of cosmology, a negative pressure fluid is exactly the mechanism responsible for accelerating the universe expansion. In the present model, such a behaviour for the pressure was naturally obtained.

It is also worth stressing that the EoS parameter shows a transition from a decelerated to an accelerated regime of the expansion of the universe. This can well be seen in Figure 4, by recalling that, from standard cosmology, the latter regime may happen if [67]. Moreover, it can be seen from such a figure that as time passes by, , in accordance with recent observational data on the cosmic microwave background temperature fluctuations [12].

Furthermore, in Figures 5–8 we plotted the ECs from the material solutions of our cosmological model. Those figures were plotted in terms of , for fixed . They show the validation of WEC, NEC, and DEC, with a wide range of acceptable values for .

On the other hand, Figure 7 shows violation of SEC. However, as it has been deeply discussed by Barcelo & Visser [68] an early and late-time accelerating universe must violate SEC.

As a further work, we can look for the gravity universe at very early times, particularly investigating the production of scalarons in this model. Moraes & Santos [30] have shown that the trace of the energy-momentum tensor contribution in the theory is higher for the early universe when compared to the late-time contribution. According to Starobinsky [51], some mechanism should work in the early universe to prohibit the scalaron overproduction within SM. The high contribution of the terms proportional to in the early universe may well be this mechanism.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

P. H. R. S. Moraes thanks São Paulo Research Foundation (FAPESP), grant 2015/08476-0, for financial support. P. K. Sahoo acknowledges DST, New Delhi, India, for providing facilities through DST-FIST Lab, Department of Mathematics, where a part of this work was done. R. A. C. Correa is partially supported by FAPESP (Foundation for Support to Research of the State of São Paulo) under grants numbers 2016/03276-5 and 2017/26646-5.

References

T. P. Sotiriou and V. Faraoni, “ theories of gravity,” Reviews of Modern Physics, vol. 82, no. 1, pp. 451–497, 2010.

A. De Felice and S. J. Tsujikawa, “f(R) theories,” Living Reviews in Relativity, vol. 13, no. 3, pp. 1–161, 2010.

T. Padmanabhan, “Cosmological constant—the weight of the vacuum,” Physics Reports, vol. 380, no. 5-6, pp. 235–320, 2003.

J. Antoniadis, P. C. C. Freire, N. Wex et al., “A massive pulsar in a compact relativistic binary,” Science, vol. 340, p. 448, 2013.

P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels, “A two-solar-mass neutron star measured using Shapiro delay,” Nature, vol. 467, no. 7319, pp. 1081–1083, 2010.

P. Bull, Y. Akrami, J. Adamek et al., “Beyond ΛCDM: problems, solutions, and the road ahead,” Physics of the Dark Universe, vol. 12, p. 56, 2016.

A. G. Riess, A. V. Filippenko, P. Challis et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” The Astronomical Journal, vol. 116, no. 3, pp. 1009–1038, 1998.

S. Perlmutter, G. Aldering, and G. Goldhaber, “Measurements of and from 42 high-redshift supernovae,” The Astrophysical Journal, vol. 517, no. 2, pp. 565–586, 1999.

S. Tsujikawa, “Observational signatures of f(R) dark energy models that satisfy cosmological and local gravity constraints,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 77, no. 2, Article ID 023507, 2008.

S. Capozziello, V. F. Cardone, and A. Troisi, “Reconciling dark energy models with f(R) theories,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 71, no. 4, 2005.

S. Nojiri and S. D. Odintsov, “Modified f(R) gravity consistent with realistic cosmology: from matter dominated epoch to dark energy universe,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 74, Article ID 086005, 2006.

G. Hinshaw, D. Larson, and E. Komatsu, “Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological parameter results,” The Astrophysical Journal Supplement Series, vol. 208, no. 2, p. 19, 2013.

J. Evslin, “Model-independent dark energy equation of state from unanchored baryon acoustic oscillations,” Physics of the Dark Universe, vol. 13, pp. 126–131, 2016.

Y. Zhang, “Self-interacting dark matter without direct detection constraints,” Physics of the Dark Universe, vol. 15, pp. 82–89, 2017.

E. S. Fradkin and A. A. Tseytlin, “Quantum string theory effective action,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 261, no. 1, pp. 1–27, 1985.

E. Witten, “Noncommutative geometry and string field theory,” Nuclear Physics B, vol. 268, no. 2, pp. 253–294, 1986.

D. Friedan, E. Martinec, and S. Shenker, “Conformal invariance, supersymmetry and string theory,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 271, no. 1, pp. 93–165, 1986.

T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, “F(R, T) gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 84, no. 2, Article ID 024020, 2011.

A. L. Erickcek, T. L. Smith, and M. Kamionkowski, “Solar system tests do rule out 1/R gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 74, no. 12, Article ID 121501, 2006.

T. Chiba, T. L. Smith, and A. L. Erickcek, “Solar System constraints to generalf(R)gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 75, no. 12, Article ID 124014, 2007.

S. Capozziello, A. Stabile, and A. Troisi, “Newtonian limit of gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 76, no. 10, Article ID 104019, 2007.

G. J. Olmo, “Limit to general relativity in theories of gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 75, no. 2, Article ID 023511, 2007.

T. Chiba, “1/R gravity and scalar-tensor gravity,” Physics Letters B, vol. 575, no. 1-2, pp. 1–3, 2003.

A. D. Dolgov and M. Kawasaki, “Can modified gravity explain accelerated cosmic expansion?” Physics Letters B, vol. 573, pp. 1–4, 2003.

G. J. Olmo, “Post-Newtonian constraints onf(R) cosmologies in metric and Palatini formalism,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 72, no. 8, Article ID 083505, 2005.

S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R)F(R) theory to Lorentz non-invariant models,” Physics Reports, vol. 505, pp. 59–144, 2011.

S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, “Modified gravity theories on a nutshell: inflation, bounce and late-time evolution,” Physics Reports, vol. 692, pp. 1–104, 2017.

P. H. R. S. Moraes, “Cosmological solutions from induced matter model applied to 5Df(R, T) gravity and the shrinking of the extra coordinate,” European Physical Journal C, vol. 75, p. 168, 2015.

P. H. Moraes, G. Ribeiro, and R. A. Correa, “A transition from a decelerated to an accelerated phase of the universe expansion from the simplest non-trivial polynomial function of in the formalism,” Astrophysics and Space Science. An International Journal of Astronomy, Astrophysics and Space Science, vol. 361, no. 7, p. 227, 2016.

P. H. R. S. Moraes and J. R. L. Santos, “A complete cosmological scenario from f(R, Tϕ) gravity theory,” European Physical Journal C, vol. 76, p. 60, 2016.

P. H. R. S. Moraes and P. K. Sahoo, “The simplest non-minimal mattergeometry coupling in thef(R, T)cosmology,” The European Physical Journal C, vol. 77, p. 480, 2017.

R. Myrzakulov, “FRW cosmology in F (R,T) gravity,” The European Physical Journal C, vol. 72, article 2203, 2012.

D. R. K. Reddy, R. Santikumar, and R. L. Naidu, “Bianchi type-III cosmological model in f(R,T) theory of gravity,” Astrophysics and Space Science, vol. 342, no. 1, pp. 249–252, 2012.

V. Singh and C. P. Singh, “Friedmann cosmology with matter creation in modified f(R,T) gravity,” International Journal of Theoretical Physics, vol. 55, no. 2, pp. 1257–1273, 2016.

P. H. Moraes, J. D. Arbañil, and M. Malheiro, “Stellar equilibrium configurations of compact stars in theory of gravity,” Journal of Cosmology and Astroparticle Physics, no. 6, p. 005, 2016.

M. E. Alves, P. H. Moraes, J. C. de Araujo, and M. Malheiro, “Gravitational waves in f(R,Tφ) and f(R,T) theories of gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 2, Article ID 024032, 2016.

P. H. Moraes and P. K. Sahoo, “Modeling wormholes in gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 96, no. 4, Article ID 044038, 2017.

P. H. Moraes, R. A. Correa, and R. V. Lobato, “Analytical general solutions for static wormholes in gravity,” Journal of Cosmology and Astroparticle Physics, vol. 07, 029 pages, 2017.

A. Das, F. Rahaman, B. K. Guha, and S. Ray, “Compact stars inf(R, T)gravity,” European Physical Journal C, vol. 76, p. 654, 2016.

M. Sharif and Z. Yousaf, “Dynamical analysis of self-gravitating stars in f(R,T) gravity,” Astrophysics and Space Science, vol. 354, no. 2, pp. 471–479, 2014.

Z. Yousaf, M. Ilyas, and M. Z. Bhatti, “Influence of modification of gravity on spherical wormhole models,” Modern Physics Letters A, vol. 32, no. 30, Article ID 1750163, 2017.

E. H. Baffou, M. J. Houndjo, and I. G. Salako, “Viscous generalized Chaplygin gas interacting with f(R, T) gravity,” International Journal of Geometric Methods in Modern Physics, vol. 14, no. 4, Article ID 1750051, p. 14, 2017.

H. Shabani and M. Farhoudi, “Cosmological and solar system consequences of f(R,T) gravity models,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, Article ID 044031, 2014.

R. Zaregonbadi, M. Farhoudi, and N. Riazi, “Dark matter from f(R,T) gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 8, p. 084052, 2016.

M. Zubair, “Impact of collisional matter on the late-time dynamics of f(R, T) gravity,” Symmetry, vol. 10, p. 463, 2018.

J. Wu, G. Li, T. Harko, and S. Liang, “Palatini formulation of f(R, T) gravity theory, and its cosmological implications,” The European Physical Journal C, vol. 78, no. 5, p. 430, 2018.

E. Barrientos, F. S. Lobo, S. Mendoza, G. J. Olmo, and D. Rubiera-Garcia, “Metric-affine theories of gravity and their applications,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 97, no. 10, Article ID 104041, 2018.

R. A. C. Correa and P. H. R. S. Moraes, “Configurational entropy inf(R, T)brane models,” European Physical Journal C, vol. 76, p. 100, 2016.

R. A. Correa, P. H. Moraes, A. de Souza Dutra, and R. da Rocha, “Information content in brane models with nonconstant curvature,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 92, no. 12, Article ID 126005, 2015.

A. A. Starobinsky, “A new type of isotropic cosmological models without singularity,” Physics Letters B, vol. 91, no. 1, pp. 99–102, 1980.

A. A. Starobinsky, “Disappearing cosmological constant in f(R) gravity,” JETP Letters, vol. 86, no. 3, pp. 157–163, 2007.

M. Sharif and A. Siddiqa, “Axial dissipative dust as a source of gravitational radiation in f(R) gravity,” Physics of the Dark Universe, vol. 15, pp. 105–113, 2017.

M. Aparicio Resco, Á. de la Cruz-Dombriz, F. J. Llanes Estrada, and V. Zapatero Castrillo, “On neutron stars in f(R) theories: Small radii, large masses and large energy emitted in a merger,” Physics of the Dark Universe, vol. 13, pp. 147–161, 2016.

A. V. Astashenok, S. Capozziello, and S. D. Odintsov, “Nonperturbative models of quark stars in f(R) gravity,” Physics Letters B, vol. 742, pp. 160–166, 2015.

M. Zubair and I. Noureen, “Evolution of axially symmetric anisotropic sources in f(R,T) gravity,” European Physical Journal C, vol. 75, p. 265, 2015.

I. Noureen and M. Zubair, “Dynamical instability and expansion-free condition inf(R, T)gravity,” European Physical Journal C, vol. 75, p. 62, 2015.

I. Noureen, M. Zubair, A. A. Bhatti, and G. Abbas, “Shear-free condition and dynamical instability inf(R, T) gravity,” European Physical Journal C, vol. 75, p. 323, 2015.

M. Zubair, S. Waheed, and Y. Ahmad, “Static spherically symmetric wormholes in f(R, T) gravity,” The European Physical Journal C, vol. 76, no. 8, p. 444, 2016.

Z. Yousaf, M. Ilyas, and M. Zaeem-ul-Haq Bhatti, “Static spherical wormhole models in f (R, T) gravity,” The European Physical Journal Plus, vol. 132, p. 268, 2017.

Ö. Akarsu, S. Kumar, R. Myrzakulov, M. Sami, and L. Xu, “Cosmology with hybrid expansion law: scalar field reconstruction of cosmic history and observational constraints,” Journal of Cosmology and Astroparticle Physics, vol. 01, p. 022, 2014.

S. Capozziello, O. Luongo, and N. E. Saridakis, “Cosmographic bounds on the cosmological deceleration-acceleration transition redshift inf(R)gravity,” Physical Review D, vol. 90, Article ID 044016, 2014.

S. Capozziello, O. Luongo, and E. N. Saridakis, “Transition redshift in cosmology and observational constraints,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 91, no. 12, Article ID 124037, 2015.

O. Farooq, F. R. Madiyar, S. Crandall, and B. Ratra, “Hubble parameter measurement constraints on the redshift of the deceleration-acceleration transition, dynamical dark energy, and space curvature,” The Astrophysical Journal, vol. 835, no. 1, 2017.

S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, UK, 1973.

R. M. Wald, General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984.

M. Visser, Lorentzian wormholes, AIP Series in Computational and Applied Mathematical Physics, American Institute of Physics, New York, NY, USA, 1995.

B. Ryden, Introduction to Cosmology, Addison Wesley, San Francisco, Mass, USA, 2003.

C. Barcelo and M. Visser, “Twilight for the energy conditions?” International Journal of Modern Physics D: Gravitation, Astrophysics, Cosmology, vol. 11, no. 10, pp. 1553–1560, 2002.