⇦Chinese Physics Letters, 2019, Vol. 36, No. 12, Article code 120401 An Exponential Shape Function for Wormholes in Modified Gravity * P. H. R. S. Moraes1,2, Pradyumn Kumar Sahoo3**, Shreyas Sunil Kulkarni3, Shivaank Agarwal3 Affiliations 1Università degli studi di Napoli "Federico II"–Dipartimento di Fisica, Napoli I-80126, Italy 2Instituto Tecnológico de Aeronáutica (ITA)–Departamento de Física, 12228-900, São José dos Campos, São Paulo, Brasil 3Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India Received 6 September 2019, online 25 November 2019 *Supported by the São Paulo Research Foundation (FAPESP) under Grant No 2018/20689-7.
**Corresponding author. Email: pksahoo@hyderabad.bits-pilani.ac.in Citation Text: Moraes P H R S, Sahoo P K, Kulkarni S S and Agarwal S 2019 Chin. Phys. Lett. 36 120401    Abstract We propose a new exponential shape function in wormhole geometry within modified gravity. The energy conditions and the equation-of-state parameter are obtained. The radial and tangential null energy conditions, and also the weak energy condition are validated, which indicates the absence of exotic matter due to modified gravity allied with such a new proposal. DOI:10.1088/0256-307X/36/12/120401 PACS:04.20.-q, 04.50.Kd, 04.50.-h © 2019 Chinese Physics Society Article Text Wormholes (WHs) are asymptotically flat tube-like structures. They are said to be useful for interstellar travel because they may be able to connect two different points in the same universe or two points in different universes.[1,2] WHs arise from the solutions of general relativity (GR). Schwarzschild's WH was the first WH-like solution obtained.[3] It was later found that it would collapse very quickly, preventing it being traversable.[4] This issue was delved deeper in Refs. [1,5], in which a static and spherically symmetric metric was suggested to describe WHs and the required energy constraints to make them traversable were discussed. This analysis led to the violation of the null energy condition (NEC), so that for the GRWH to be traversable, it should be filled by exotic matter (matter violating the NEC). The issue is that finding suitable contenders for exotic matter has never been carried out. Therefore, modified gravity theories (MGTs), which input some extra degrees of freedom to GR in a fundamental level, appear as a possibility to treat this issue by addressing the question of whether or not it is possible to have stable WH solutions with no need for exotic matter. In regard to MGTs, we recommend the important reviews.[6–8] MGTs have been used to address not only the exotic matter issue but also other several issues of current observational astrophysics and cosmology.[9–16] Due to the lack of WH observations so far, despite all of the efforts and proposals[17–23], some geometrical and material features of WHs, such as the shape function and equation of state (EoS), are still not precisely known. In particular, several forms for the shape function $b(r)$ have been proposed and analyzed so far, such as in Refs. [24–26]. In this Letter, we propose a new form for the WH shape function. As $b(r)$ is not arbitrary and has to obey several conditions, and therefore our proposal must be in accordance with these conditions. We also aim at the obedience of the WH energy conditions (ECs). To attain this aim, allied to the proposed shape function, we will underline our model with a particular MGT, which is named $f(R,T)$ theory.[27] The $f(R,T)$ theory starts from a gravitational action that substitutes the Ricci scalar $R$ in the usual Einstein–Hilbert action by a general function of $R$ and $T$, with $T$ being the trace of the energy momentum-tensor $T_{ij}$. The motivation to insert some material terms in the gravitational action is related to the possible existence of imperfect fluids in the universe. Since WHs' material content is described by an anisotropic fluid, their investigation in such a theory of gravity is well motivated. The $f(R,T)$ gravity authors have argued that, due to the coupling of matter and geometry, this gravity model depends on a source term, which is nothing but the variation of the matter stress-energy tensor.[27] This source term could be related to quantum effects because it could lead to a particle creation scenario.[28] Consequently, the motion of test particles in $f(R,T)$ gravity is not along geodesic path due to the presence of an extra force perpendicular to the four velocities. To obtain our WH solutions, we consider the $f(R,T)$ modified theory of gravity,[27] where the gravitational Lagrangian is given by an arbitrary function of $R$ and $T$. The gravitational action for this theory is defined as[27] $$ S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}f(R,T)+\int d^{4}x\sqrt{-g}\mathcal{L}_m.~~ \tag {1} $$ In Eq. (1), $f(R,T)$ is an arbitrary function of $R$ and $T$, $g$ denotes the determinant of the metric $g_{ij}$ and $\mathcal{L}_m$ is the matter Lagrangian. Moreover, we take natural units. By varying Eq. (1) with respect to the metric $g_{ij}$, we can obtain the field equations $$\begin{align} &f_R(R,T)R_{ij}\!-\!\frac{1}{2}f(R,T)g_{ij}\!+\!(g_{ij}\square \!-\!\nabla_i\nabla_j)f_R(R,T)\\ =&\,8\pi T_{ij}-f_T(R,T)T_{ij}-f_T(R,T){\it\Theta}_{ij}.~~ \tag {2} \end{align} $$ Here $$\begin{align} f_R(R,T)=&\,\frac{\partial f(R,T)}{\partial R},~~ f_T(R,T)=\frac{\partial f(R,T)}{\partial T},\\ T_{\mu\nu}=&\,-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g})\mathcal{L}_m}{\delta g^{\mu\nu}},~~ \tag {3} \end{align} $$ $$\begin{align} {\it\Theta}_{ij}=&\,-2T_{ij}-p g_{ij},~~ \tag {4} \end{align} $$ if we choose $\mathcal{L}_m=-p$ with $p$ being the total pressure of the fluid. $\mathcal{L}_m=-p$ represents the matter Lagrangian density of a perfect fluid, which is not uniquely defined.[27,29–31] It is quite usual to see choices such as $\mathcal{L}_m=\rho$, with $\rho$ being the matter-energy density, and $\mathcal{L}_m=-p$, but even $\mathcal{L}_m=T$ was already used.[30] It is known that geometry-matter coupling gravity theories, such as the $f(R,T)$ gravity, predict the existence of an extra force acting orthogonally to the four velocities in a (non-)geodesic motion. This extra force remarkably depends on the matter Lagrangian density and vanishes if $\mathcal{L} _m=-p$,[29,32] which is the reason why we have assumed so. We will consider $f(R,T)=R+2\lambda T$, which was assumed as the functional form for the function $f(R,T)$ in several approaches such as Refs. [33–38] among many others. The considered form is the simplest one, which reduces to general relativity for the choice of $\lambda = 0$. For this choice, one can easily correlate the obtained results with the most successful Einstein's general relativity. The $f(R,T)$ gravity field equations, for this case, read $$ G_{ij}=8\pi T_{ij}+\lambda Tg_{ij}+2\lambda(T_{ij}+pg_{ij}),~~ \tag {5} $$ with $G_{ij}$ being the Einstein tensor. To describe the geometry of WHs' spacetime, we use the modified version of the spherically symmetric spacetime metric as $$ ds^2 = -dt^2 + \Big[\frac{dr^2}{1- \frac{b(r)}{r}}+ r^2 d\theta^2 +r^2\sin^2\theta d\phi^2\Big].~~ \tag {6} $$ Here the redshift function was normalized. Constant redshift functions were assumed in several references, such as Refs. [39–41]. The radial coordinate $r$ decreases from $\infty$ to a minimum value $r_0$, called the WH throat, and then increases to $\infty$. The shape function $b(r)$ needs to satisfy the following conditions: $\bullet$ At the throat, $b(r_0)=r_0$ and for $r>r_0$, $1-\frac{b(r)}{r}>0$; $\bullet$ $b'(r_0) < 1$ (flaring-out condition); $\bullet$ $\lim_{r \to \infty} \frac{b(r)}{r} = 0$ (asymptotically flatness condition); with primes indicating radial derivatives. The Ricci scalar for the WH metric is obtained as $$ R = \frac{2b'}{r^2}.~~ \tag {7} $$ The non-zero Einstein tensor components for the WH metric are $$\begin{align} &G_{00} =\frac{b'}{r^2}, \\ &G_{11} = -\frac{b}{r^2 (r-b)}, \\ &G_{22} = \frac{b-r b'}{2 r}, \\ &G_{33} = \frac{\sin ^2\theta \left(b-r b'\right)}{2 r}.~~ \tag {8} \end{align} $$ The field Eq. (5) for the metric (6) and anisotropic energy-momentum tensor are, then, written explicitly as $$\begin{align} \frac{b'}{r^2} &= (8\pi + \lambda)\rho - \lambda(p_r + 2p_t),\\ - \frac{b}{r^3} &= (8\pi + 3\lambda)p_r + \lambda(\rho+2p_t),\\ \frac{b-rb'}{2 r^3} &= (8\pi + 4\lambda)p_t+\lambda(\rho+p_r),~~ \tag {9} \end{align} $$ with $p_r$ and $p_t$ being, respectively, the radial and transverse pressures of the WH, such that the WH total pressure is $p=(p_r+2p_t)/3$. One can obtain the values in GR by making $\lambda=0$ in this equation. In GR, the energy conditions are a set of inequalities that are required to prove various important theorems such as those related to WHs and black holes. It is well-known that static traversable GRWHs violate the energy conditions near the WH throat.[1,42] The ECs have significant theoretical applications, such as the Hawking Penrose singularity conjecture, which is based on the (strong energy condition) SEC[43] while the (dominant energy condition) DEC is applicable to proof the positive mass theorem.[44] Furthermore, the NEC is a basic requirement to derive the second law of black hole thermodynamics.[45] The cosmological terms suchlike deceleration, look back time, distance modulus and statefinder parameters are seen in terms of redshift using ECs in Ref. [46] The ECs have been studied in MGTs, such as $f(R)$ gravity, Brans-Dicke theory, $f(G)$ gravity, $f(G,\mathcal{T})$ gravity[47–50] with $\mathcal{T}$ being the torsion scalar. The generalized ECs are analyzed in MGTs considering the degrees of freedom related to scalar fields and curvature invariants.[51,52] In particular, the ECs were derived for a power law solution in $f(R,T)$ gravity and the stability of the same were established.[53] We will take into consideration the following energy conditions for a perfect fluid[2] where $i=r,t$. In the following, we will construct these energy conditions for the $f(R,T)$ WHs presented above for a new proposal for the shape function, namely an exponential shape function. In this section we propose a new exponential form for the shape function as $$ b(r)= r_0 \cdot e^{1-\frac{r}{r_0}}.~~ \tag {10} $$ In Fig. 1, we can see some features of $b(r)$ (10).
cpl-36-12-120401-fig1.png
Fig. 1. Features of the shape function (10) with $r_0=0.5$.
One can observe from Fig. 1 that the shape function satisfies all the basic requirements given in the above paragraph. Using the above shape function in the field Eq. (9), we obtain $$\begin{align} \rho &= - \frac{e^{1-\frac{r}{r_0}}}{2 (\lambda +4 \pi ) r^2},\\ p_r &=- \frac{e^{1-\frac{r}{{r_0}}} [4 (\pi -1) r+(\lambda +4 \pi ) r_0]}{2 (\lambda +4 \pi )^2 r^3},\\ p_t &=\frac{e^{1-\frac{r}{r_0}} (r+r_0)}{4 (\lambda +4 \pi ) r^3}.~~ \tag {11} \end{align} $$ We plot the radial EoS as well as the energy conditions for the present WH model with the exponential shape function. Here it is determined that $\lambda$ must remain in the range from $-80$ to $-13$ to validate NEC, WEC and DEC. In this work we have considered $\lambda=-30$. The respective behavior in GR is plotted in Fig. 4. It is quite clear that in the absence of the $f(R,T)$ gravity extra terms, most of the energy conditions are no longer respected. In fact, only one of the energy conditions is respected, even within the assumption of such a promising and fruitful proposal for the WH shape function. The $f(R,T)$ gravity, departing from many alternative gravity theories in the literature, allows one to modify the effective energy-momentum tensor worked out in GR. This is made clear in Eq. (5), in which even if one assumes the energy-momentum tensor of a perfect fluid, the effective energy-momentum tensor of the theory, namely, $$ T_{ij}^{\rm eff}=T_{ij}+\frac{\lambda}{8\pi}[Tg_{ij}+2(T_{ij}+pg_{\rm ig})],~~ \tag {12} $$ presents "imperfect" fluid terms, which may be related to viscosity or anisotropy. For such a strong modification, one can see the effective fluid permeating a particular astrophysical or cosmological system, such as in the present case the wormhole makes it possible to obtain significantly different material features and in our case. Therefore, it is possible to respect the energy conditions, as a consequence of the description of viscosity/anisotropy disguised in terms proportional to $\lambda$. WHs are tube-like structures that, as shortcuts, connect two distant regions in the universe (or even in different universes). If their geometrical structure was not singular enough, according to GR formalism, WHs are expected to be filled by exotic (negative mass) matter. The lack of observations of WHs so far means that we are unable to predict exactly some of their geometrical and material properties, such as the shape function and the EoS. In the present study we have proposed a novel functional form for the shape function, which depends only exponentially on $r$. We did not need to assume any particular form for the WH EoS, which was obtained from the model, rather than imposed to it, as happens in some cases in the literature.[54–57] Before going any further discussion of the WH EoS obtained, we should mention that for the exponential shape function presented in Eq. (10), Fig. 1 has shown that it satisfies all the requirements needed to have traversable asymptotically flat WHs. Consequently, we are allowed to obtain the EoS parameter solution $\omega_r$, as well as to construct the WH energy conditions.
cpl-36-12-120401-fig2.png
Fig. 2. Radial equation-of-state parameter $\omega_r=p_r/\rho$ as functions of $r$ and $\lambda$ with $r_0 = 0.5$.
Figure 2 shows that the WH EoS is in the phantom region; that is, $\omega_r < -1$. It is well-known that a phantom EoS parameter $ < -1$ for the universe will imply in the so-called Big Rip[58] though some alternatives to evade such a catastrophic scenario have appeared.[59–62] Phantom WHs have also appeared in the literature,[54–65] though it is important to remark that in these cases the phantom EoS was invoked rather than obtained from the model (as in the present case). Figure 3 show the energy conditions of the present WHs' scenario. They show a properly obedience of NEC and WEC, departing from standard GR solutions. The DEC is also satisfied while SEC is not. Similar approaches to WHs in $f(R,T)$ gravity can be seen in the literature, though with non-exponential shape functions.[41,57,66–73] In comparison, in our approach the important role of the exponential shape function becomes clear because none of them presents WHs fully satisfying NEC, WEC and DEC, as our model does (recall Fig. 1).
cpl-36-12-120401-fig3.png
Fig. 3. The energy conditions as a function of $r$ with $\lambda=-30$ and $r_0 = 0.5$.
cpl-36-12-120401-fig4.png
Fig. 4. The energy conditions as a function of $r$ with $\lambda=0$ and $r_0 = 0.5$.
The SEC (which is not obeyed in the present study, as well as in many others) has been discussed for some time. For instance, the SEC must be violated during the inflationary epoch and the need for this violation is the reason why inflationary models are typically driven by scalar inflation fields.[74] Furthermore, the recent observational data regarding the accelerating universe[75–77] means that the SEC is violated on cosmological scales rat the moment.[78] PKS acknowledges DST, New Delhi, India for providing facilities through DST-FIST lab, Department of Mathematics, where a part of this work was carried out. We are very much grateful to the honorable referees and the editor for the illuminating suggestions that have significantly improved our work in terms of research quality and presentation. References Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativityCausality and Multiply Connected Space-TimeWormholes, Time Machines, and the Weak Energy ConditionModified gravity theories on a nutshell: Inflation, bounce and late-time evolutionModified gravity and cosmologyExtended Theories of GravityASTROPHYSICAL TESTS OF MODIFIED GRAVITY: CONSTRAINTS FROM DISTANCE INDICATORS IN THE NEARBY UNIVERSEAstrophysical tests of modified gravity: the morphology and kinematics of dwarf galaxiesAstrophysical tests of gravity: a screening map of the nearby universeBeyond the cosmological standard modelf ( T ) teleparallel gravity and cosmologyPlanck 2015 resultsAre f ( R ) Dark Energy Models Cosmologically Viable?Cosmological constraints on f ( R ) gravity theories within the Palatini approachPossible existence of wormholes in the galactic halo regionRing-down gravitational waves and lensing observables: How far can a wormhole mimic those of a black hole?Gravitational lensing by wormholesCan the supermassive objects at the centers of galaxies be traversable wormholes? The first test of strong gravity for mm/sub-mm very long baseline interferometry facilitiesBroad K α iron line from accretion disks around traversable wormholesShadows of rotating wormholesGravitational lensing by rotating wormholesHow to tell the shape of a wormhole by its quasinormal modesA wormhole with a special shape functionTraversable wormholes and energy conditions with two different shape functions in f(R) gravity f ( R , T ) gravityThermodynamic interpretation of the generalized gravity models with geometry-matter couplingThe Dark side of gravity: Modified theories of gravityPerfect fluid Lagrangian and its cosmological implications in theories of gravity with nonminimally coupled matter fieldsThe trace of the trace of the energy–momentum tensor-dependent Einstein’s field equationsNonminimal coupling of perfect fluids to curvature$$f(R,T)=f(R)+\lambda T$$ f ( R , T ) = f ( R ) + λ T gravity models as alternatives to cosmic accelerationAnisotropic strange stars under simplest minimal matter-geometry coupling in the f ( R , T ) gravityMagnetized strange quark matter in f ( R, T ) gravity with bilinear and special form of time varying deceleration parameterStrange stars in f ( R ,𝒯) gravityEvading the non-continuity equation in the f(R, T) cosmologyExploring physical features of anisotropic strange stars beyond standard maximum mass limit in $f\left(R,\mathcal {T}\right)$ gravityWormholes supported by polytropic phantom energyCharged wormhole solutions in Einstein-Cartan gravityWormholes with ρ ( R ,   R ′ ) matter in f ( R ,   T ) gravityGeometric structure of the generic static traversable wormhole throatProof of the positive mass theorem. IIGravitational vacuum polarization. IV. Energy conditions in the Unruh vacuumEnergy conditions in f ( R ) gravityENERGY CONDITIONS IN f(R) GRAVITY AND BRANS–DICKE THEORIESEnergy conditions in modified Gauss-Bonnet gravityEnergy conditions in $$f(\mathcal {G},T)$$ f ( G , T ) gravityEnergy conditions in modified gravityGeneralized energy conditions in extended theories of gravityEnergy Conditions Constraints and Stability of Power Law Solutions in f ( R , T ) GravityPhantom energy traversable wormholesNote on linearized stability of Schwarzschild thin-shell wormholes with variable equations of stateLorentzian wormholes in Eddington-inspired Born-Infeld gravityModeling wormholes in f ( R , T ) gravityPhantom Energy: Dark Energy with w < − 1 Causes a Cosmic DoomsdayAvoidance of big rip in phantom cosmology by gravitational back reactionDoes loop quantum cosmology replace the big rip singularity by a non-singular bounce?Alternative mechanism of avoiding the big rip or little rip for a scalar phantom fieldPhantom energy-dominated universe as a transient stage in f(R) cosmologyWormholes supported by a phantom energyInstabilities of wormholes and regular black holes supported by a phantom scalar fieldStability of phantom wormholesNoncommutative inspired wormholes admitting conformal motion involving minimal couplingWormhole formation in f ( R , T ) gravity: Varying Chaplygin gas and barotropic fluidPhantom fluid supporting traversable wormholes in alternative gravity with extra material termsAnalytical general solutions for static wormholes in f ( R , T ) gravityStatic spherical wormhole models in f (R, T) gravityInfluence of modification of gravity on spherical wormhole modelsViable wormhole solutions and Noether symmetry in f ( R , T ) gravityWormhole Geometries in f(R,T) GravityENERGY CONDITIONS AND THEIR COSMOLOGICAL IMPLICATIONSObservational Evidence from Supernovae for an Accelerating Universe and a Cosmological ConstantType Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope : Evidence for Past Deceleration and Constraints on Dark Energy EvolutionCosmological Results from High‐ z SupernovaeEnergy Conditions in the Epoch of Galaxy Formation
[1] Morris M S and Thorne K S 1988 Am. J. Phys. 56 395
[2]Visser M 1995 Lorentzian Wormholes: From Einstein to Hawking (New York: AIP Press)
[3]Flamm L 1916 Phys. Z. 17 448
[4] Fuller R W and Wheeler J A 1962 Phys. Rev. 128 919
[5] Morris M S et al 1988 Phys. Rev. Lett. 61 1446
[6] Nojiri S et al 2017 Phys. Rep. 692 1
[7] Clifton T et al 2012 Phys. Rep. 513 1
[8] Capozziello S and de Laurentis M 2011 Phys. Rep. 509 167
[9] Jain B et al 2013 Astrophys. J. 779 39
[10] Vikram V et al 2013 J. Cosmol. Astropart. Phys. 2013 020
[11] Cabré A et al 2012 J. Cosmol. Astropart. Phys. 2012 034
[12] Joyce A et al 2015 Phys. Rep. 568 1
[13] Cai Y F et al 2016 Rep. Prog. Phys. 79 106901
[14] Planck Collaboration 2016 Astron. Astrophys. 594 A14
[15] Amendola L et al 2007 Phys. Rev. Lett. 98 131302
[16] Amarzguioui M et al 2006 Astron. Astrophys. 454 707
[17] Rahaman F et al 2014 Eur. Phys. J. C 74 2750
[18] Nandi K K et al 2017 Phys. Rev. D 95 104011
[19] Nandi K K et al 2006 Phys. Rev. D 74 024020
[20] Bambi C 2013 Phys. Rev. D 87 107501
[21] Bambi C 2013 Phys. Rev. D 87 084039
[22] Shaikh R 2018 Phys. Rev. D 98 024044
[23] Jusufi K and Övgün A 2018 Phys. Rev. D 97 024042
[24] Konoplya R A 2018 Phys. Lett. B 784 43
[25] Kuhfittig P K F 1999 Am. J. Phys. 67 125
[26] Godani N and Samanta G C 2019 Int. J. Mod. Phys. D 28 1950039
[27] Harko T et al 2011 Phys. Rev. D 84 024020
[28] Harko T 2014 Phys. Rev. D 90 044067
[29] Harko T and Lobo F S N 2014 Galaxies 2 410
[30] Avelino P P and Azevedo R P L 2018 Phys. Rev. D 97 064018
[31] Moraes P H R S 2019 Eur. Phys. J. C 79 674
[32] Bertolami O et al 2008 Phys. Rev. D 78 064036
[33] Sahoo P K et al 2018 Eur. Phys. J. C 78 736
[34] Deb D et al 2018 Phys. Rev. D 97 084026
[35] Sahoo P K et al 2018 New Astron. 60 80
[36] Deb D et al 2018 J. Cosmol. Astropart. Phys. 2018 044
[37] Moraes P H R S et al 2018 Eur. Phys. J. C 78 192
[38] Deb D et al 2019 Mon. Not. R. Astron. Soc. 485 5652
[39] Jamil M et al 2010 Eur. Phys. J. C 67 513
[40] Mehdizadeh M R and Ziaie A H 2019 Phys. Rev. D 99 064033
[41] Elizalde E and Khurshudyan M 2019 Phys. Rev. D 99 024051
[42] Hochberg D and Visser M 1997 Phys. Rev. D 56 4745
[43]Hawking S W and Ellis G F R 1973 The Large Scale Structure of Spacetime (Cambridge: Cambridge University Press)
[44] Schoen R and Yau S T 1981 Commun. Math. Phys. 79 231
[45]Carroll S 2004 Space-time and Geometry–An Introduction to General Relativity (San Francisco: Addison Wesley)
[46] Visser M 1997 Phys. Rev. D 56 936
[47] Santos J et al 2007 Phys. Rev. D 76 083513
[48] Atazadeh K et al 2009 Int. J. Mod. Phys. D 18 1101
[49] Garcia N M et al 2011 Phys. Rev. D 83 104032
[50] Sharif M and Ikram Ayesha 2016 Eur. Phys. J. C 76 640
[51] Capozziello S et al 2014 Phys. Lett. B 730 280
[52] Capozziello S et al 2015 Phys. Rev. D 91 124019
[53] Sharif M and Zubair M 2013 J. Phys. Soc. Jpn. 82 014002
[54] Lobbo F S N 2005 Phys. Rev. D 71 084011
[55] Varela V 2015 Phys. Rev. D 92 044002
[56] Shaikh R 2015 Phys. Rev. D 92 024015
[57] Moraes P H R S and Sahoo P K 2017 Phys. Rev. D 96 044038
[58] Caldwell R R et al 2003 Phys. Rev. Lett. 91 071301
[59] Wu P and Yu H 2005 Nucl. Phys. B 727 355
[60] de Haro J 2012 J. Cosmol. Astropart. Phys. 2012 037
[61] Xing P et al 2012 Phys. Lett. B 706 482
[62] Moraes P H R S et al 2019 Int. J. Mod. Phys. D 28 1950124
[63] Sushkov S 2005 Phys. Rev. D 71 043520
[64] Bronnikov K A et al 2012 Phys. Rev. D 86 024028
[65] Lobbo F S N 2005 Phys. Rev. D 71 124022
[66] Zubair M et al 2019 Int. J. Mod. Phys. D 28 1950067
[67] Elizalde E and Khurshudyan M 2018 Phys. Rev. D 98 123525
[68] Sahoo P K et al 2018 Int. J. Mod. Phys. D 27 1950004
[69] Moraes P H R S et al 2017 J. Cosmol. Astropart. Phys. 2017 029
[70] Yousaf et al 2017 Eur. Phys. J. Plus 132 268
[71] Yousaf Z et al 2017 Mod. Phys. Lett. A 32 1750163
[72] Sharif M and Nawazish I 2019 Ann. Phys. 400 37
[73] Azizi T 2013 Int. J. Theor. Phys. 52 3486
[74] Visser M and Barcelo C 2000 Proceedings of the Third International Workshop on Particle Physics and the Early Universe, ICTP (COSMO-99) (Trieste, Italy 27 September–2 October 1999) p 98
[75] Riess A G et al 1998 Astron. J. 116 1009
[76] Riess A G et al 2004 Astrophys. J. 607 665
[77] Tonry J L et al 2003 Astron. J. 594 1
[78] Visser M 1997 Science 276 88