International Journal of Nonlinear Analysis and Applications
Interacting anisotropic dark energy with time dependent inhomogeneous equation of state
Document Type : Research Paper
Authors
Department of Mathematics, NIT Manipur, Imphal-795004, India
Abstract
The interaction of dark energy in the LRS Bianchi type-$I$ line element is explored on the background of $f(R,T)$ gravity, where $R$ and $T$ denotes the Ricci scalar together with the trace of energy momentum tensor of matter respectively. Here modified field equations are calculated using $f(R,T)=f_1 (R)+f_2 (T)$ together with inhomogeneous equation of state $(EoS)$, $p=\omega \rho -\Lambda (t)$, where $\omega$ is constant. The solutions of modified Einstein field equations (EFE) obtained are solved by taking a periodic time varying deceleration parameter (DP). Our model shows periodic nature with Big-Bang prevailing at time $t=0$. An investigation is done on the energy conditions and found that the conditions of null energy and strong energy are found to be violated. We analyse the geometrical and physical behaviours of these models.
Keywords
[1] K.S. Adhav, LRS Bianchi type-I cosmological model in f(R, T) theory of gravity, Astrophys. Space Sci. 339 (2012)
365–369.
[2] N. Ahmed and A. Pradhan, Bianchi Type-V Cosmology in f(R, T) Gravity with Λ(T), Int. J. Theor. Phys. 53
(2013) 289–306.
[3] F.G. Alvarenga, M.J.S. Houndjo, A.V. Monwanou and J.B.C. Orou, Testing some f(R, T) gravity models from
energy conditions, J. Mod. Phys. 4 (2013) 130–139.
[4] H. Amirhashchi, L.R.S. Bianchi type II stiff fluid cosmological model with decaying vacuum energy density Λ in
general relativity, Phys. Lett. B 697 (2011) 429–433.
[5] H. Amirhashchi, A. Pradhan and B. Saha, An interacting two-fluid scenario for dark energy in an FRW universe,
Chin. Phys. Lett. 28 (2011) 039801.
[6] K. Bamba, C. Geng, S. Nojiri and S.D. Odintsov, Equivalence of the modified gravity equation to the Clausius
relation, Europhys. Lett. 89 (2010) 50003.
[7] K. Bamba, K. Yesmakhanova, K. Yerzhanov and R. Myrzakulov, Reconstruction of the equation of state for the
cyclic universes in homogeneous and isotropic cosmology, Cent. Eur. J. Phys. 11(4) (2013) 397–411.
[8] G.R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79 (2009) 124019.
[9] M.C. Bento, O. Bertolami and A.A. Sen, Generalized Chaplygin gas, accelerated expansion, and dark energymatter unification, Phys. Rev. D. 66 (2002) 043507.
[10] O. Bertolami, C.G. Bhmer, T. Harko and F.S.N. Lobo, Extra force in f(R) modified theories of gravity, Phys.
Rev. D 75 (2007) 104016.
[11] V.K. Bhardwaj and M.K. Rana, LRS Bianchi-I transit universe with periodic varying q in f(R, T) gravity, Int.
J. Geom. Methods Mod. Phys. 16(12) (2019) 1950195.
[12] M. Bouhmadi-Lpez, L.J. Garay and P.F. Gonzlez-Daz, Quantum behavior of FRW radiation-filled universes,
Phys. Rev. D 66 (2002) 083504[13] I. Brevik, O.G. Gorbunova and A.V. Timoshkin, Accelerated expansion of the Friedmann Universe filled with
ideal liquid described by an inhomogeneous equation of state, Russ. Phys. J., 50 (2007) 8.
[14] I. Brevik, O.G. Gorbunova and A.V. Timoshkin, Dark energy fluid with time-dependent, inhomogeneous equation
of state, Eur. Phys. J. C 51 (2007) 179–183.
[15] S.M. Carroll, V. Duvvuri, M. Trodden and M.S. Turner, Is cosmic speed-up due to new gravitational physics?,
Phys. Rev. D, 70 (2004) 043528.
[16] E.J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D, 15(11) (2006) 1753–
1935.
[17] S. Dodelson, M. Kaplinghat and E. Stewart, Solving the coincidence problem: tracking oscillating energy, Phys.
Rev. Lett. 85 (2000) 5276–5279.
[18] R.Y. Donagi, J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Visible branes with negative tension in
heterotic M-theory, J. High Energy Phys. 2001(11) (2001) 041.
[19] B. Feng, X. Wang and X. Zhang, Dark energy constraints from the cosmic age and supernova, Phys. Lett. B.,
607 (2005) 35–41.
[20] P.M. Garnavich, R.P. Kirshner, P. Challis, J. Tonry, R.L. Gilliland, R.C. Smith, A. Clocchiatti, A. Diercks,
A.V. Filippenko, M. Hamuy, C.J. Hogan, B. Leibundgut, M.M. Phillips, D. Reiss, A.G. Riess, B.P. Schmidt,
J. Spyromilio, C. Stubbs, N.B. Suntzeff and L. Wells, Constraints on cosmological models from Hubble space
telescope observations of high-z supernovae, Astrophys. Phys. J. 493 (1998) 53–57.
[21] T. Harko, F.S.N. Lobo, S. Nojiri and S.D. Odintsov, f(R, T) gravity with variable deceleration parameter, Int. J.
Geom. Meth. Mod. Phys. 14 (2017) 1750097.
[22] T. Josset, A. Perez and D. Sudarsky, Dark energy from violation of energy conservation, Phys. Rev. Lett. 118
(2013) 021102.
[23] G.S. Khadekar and D. Raut, FRW viscous fluid cosmological model with time-dependent inhomogeneous equation
of state, Int. J. Geom. Methods Mod. Phys. 15(01) (2018) 1830001.
[24] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Ekpyrotic universe: Colliding branes and the origin of the
hot big bang, Phys. Rev. D 64 (2001) 123522.
[25] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, A brief comment on ”The Pyrotechnic Universe”, arXiv,
(2001).
[26] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Density perturbations in the ekpyrotic scenario, Phys. Rev.
D 66 (2002) 046005.
[27] E.V. Linder, Einsteins other gravity and the acceleration of the Universe, Phys. Rev. D 81 (2010) 127301.
[28] S. Nojiri and S.D. Odintsov, Inhomogeneous equation of state of the universe: Phantom era, future singularity,
and crossing the phantom barrier, Phys. Rev. D 2 (2005) 023003.
[29] S. Nojiri and S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, Int. J.
Geom. Methods Mod. Phys. 4 (2007) 115–145.
[30] T. Padmanabhan, Accelerated expansion of the universe driven by tachyonic matter, Phys. Rev. D., 66 (2002)
021301.
[31] T. Padmanabhan, Dark energy and gravity, Gen. Relativ. Gravit. 40 (2008) 529–564.
[32] L. Page, M.R. Nolta, C. Barnes, C.L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S.S.
Meyer, H.V. Peiris, D.N. Spergel, G.S. Tucker, E. Wollack and E.L. Wright, First-Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks,
Astrophys. J. Suppl. Ser. 148 (2003) 233–241.
[33] S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro, S. Deustua, S. Fabbro, A. Goobar,
D.E. Groom, I.M. Hook, A.G. Kim, M.Y. Kim, J.C. Lee, N.J. Nunes, R. Pain, C.R. Pennypacker, R. Quimby,
C. Lidman, R.S. Ellis, M. Irwin, R.G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B.J. Boyle, A.V.
Filippenko, T. Matheson, A.S. Fruchter, N. Panagia, H.J.M. Newberg, W.J. Couch and The Supernova Cosmology
Project, Measurements of Ω and Λ from 42 high-redshift supernovae, Astrophys. Phys. J. 517(2) (1999) 565–586.
[34] A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich, R.L. Gilliland, C.J. Hogan,
S. Jha, R.P. Kirshner, B. Leibundgut, M.M. Phillips, D. Reiss, B.P. Schmidt, R.A. Schommer, R.C. Smith, J.
Spyromilio, C. Stubbs, N.B. Suntzeff and J. Tonry, Observational evidence from supernovae for an accelerating
universe and a cosmological constant, Astrophys. Phys. J. 116 (1998) 1009–1038.
[35] M.E. Rodrigues, M.J.S. Houndjo, D. Momeni and R. Myrzakulov, A type of Levi-Civita’s solution in modified
Gauss-Bonnet gravity, Can. J. Phys. 92 (2014) 173–176.
[36] D. Saez-Gomez, Oscillating universe from an inhomogeneous equation of state and coupled dark energy, Grav.
Cosmol. 15 (2009) 134–140.
[37] V. Sahni, T.D. Saini, A.A. Starobinsky and U. Alam, Statefinder a new geometrical diagnostic of dark energy,
2180 AlamJETP Lett. 77 (2003) 201–206.
[38] P.K. Sahoo, P. Sahoo and B.K. Bishi, Anisotropic cosmological models in f(R, T) gravity with variable deceleration
parameter, Int. J. Geom. Meth. Mod. Phys. 14 (2017) 1750097.
[39] P.K. Sahoo and M. Sivakumar, LRS Bianchi type-I cosmological model in f(R, T) theory of gravity with Λ(T),
Astrophys. Space Sci. 357(1) (2015) 60.
[40] P.K. Sahoo, S.K. Tripathy and P. Sahoo, A periodic varying deceleration parameter in f(R, T) gravity, Mod.
Phys. Lett. A 3 (2018) 1850193.
[41] H. Shabani and A.H. Ziaie, Consequences of energy conservation violation: Late time solutions of Λ(T)CDM
subclass of f(R, T) gravity using dynamical system approach, Eur. Phys. J. C 77 (2017) 282.
[42] M. Sharif, S. Rani and R. Myrzakulov, Analysis of f(R, T) gravity models through energy conditions, Eur. Phys.
J. Plus 128 (2013) 123.
[43] M.S. Singh and S.S. Singh, Cosmological dynamics of anisotropic dark energy in f(R, T) gravity, New Astronom.
72 (2019) 3641.
[44] M.K. Singh, M.K. Verma and S. Ram, Two-fluid cosmological model of Bianchi type-V with negative constant
deceleration parameter, Int. J. Theor. Phys. 52 (2013) 227–232.
[45] T.P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys., 82 (2010) 451–497.
[46] D.N. Spergel, L. Verde, H.V. Peiris, E. Komatsu, M.R. Nolta, C.L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik,
A. Kogut, M. Limon, S.S. Meyer, L. Page, G.S. Tucker, J.L. Weiland, E. Wollack and E.L. Wright, Firstyear Wilkinson microwave anisotropy probe (WMAP) observations: Determination of cosmological parameters,
Astrophys. J. Suppl. Ser. 148 (2003) 175–194.
[47] P.J. Steinhardt and N. Turok, Cosmic evolution in a cyclic universe, Phys. Rev. D 65 (2002) 126003.
[48] P.J. Steinhardt and N. Turok, The cyclic universe: An informal introduction, Nucl. Phys. Proc. Suppl. 124 (2003)
38.
[49] R.K. Tiwari, D. Sofuolu and V.K. Dubey, Phase transition of LRS Bianchi type-I cosmological model in f(R, T)
gravity, Int. J. Geom. Methods Mod. Phys. 17(12) (2020) 2050187.
365–369.
[2] N. Ahmed and A. Pradhan, Bianchi Type-V Cosmology in f(R, T) Gravity with Λ(T), Int. J. Theor. Phys. 53
(2013) 289–306.
[3] F.G. Alvarenga, M.J.S. Houndjo, A.V. Monwanou and J.B.C. Orou, Testing some f(R, T) gravity models from
energy conditions, J. Mod. Phys. 4 (2013) 130–139.
[4] H. Amirhashchi, L.R.S. Bianchi type II stiff fluid cosmological model with decaying vacuum energy density Λ in
general relativity, Phys. Lett. B 697 (2011) 429–433.
[5] H. Amirhashchi, A. Pradhan and B. Saha, An interacting two-fluid scenario for dark energy in an FRW universe,
Chin. Phys. Lett. 28 (2011) 039801.
[6] K. Bamba, C. Geng, S. Nojiri and S.D. Odintsov, Equivalence of the modified gravity equation to the Clausius
relation, Europhys. Lett. 89 (2010) 50003.
[7] K. Bamba, K. Yesmakhanova, K. Yerzhanov and R. Myrzakulov, Reconstruction of the equation of state for the
cyclic universes in homogeneous and isotropic cosmology, Cent. Eur. J. Phys. 11(4) (2013) 397–411.
[8] G.R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79 (2009) 124019.
[9] M.C. Bento, O. Bertolami and A.A. Sen, Generalized Chaplygin gas, accelerated expansion, and dark energymatter unification, Phys. Rev. D. 66 (2002) 043507.
[10] O. Bertolami, C.G. Bhmer, T. Harko and F.S.N. Lobo, Extra force in f(R) modified theories of gravity, Phys.
Rev. D 75 (2007) 104016.
[11] V.K. Bhardwaj and M.K. Rana, LRS Bianchi-I transit universe with periodic varying q in f(R, T) gravity, Int.
J. Geom. Methods Mod. Phys. 16(12) (2019) 1950195.
[12] M. Bouhmadi-Lpez, L.J. Garay and P.F. Gonzlez-Daz, Quantum behavior of FRW radiation-filled universes,
Phys. Rev. D 66 (2002) 083504[13] I. Brevik, O.G. Gorbunova and A.V. Timoshkin, Accelerated expansion of the Friedmann Universe filled with
ideal liquid described by an inhomogeneous equation of state, Russ. Phys. J., 50 (2007) 8.
[14] I. Brevik, O.G. Gorbunova and A.V. Timoshkin, Dark energy fluid with time-dependent, inhomogeneous equation
of state, Eur. Phys. J. C 51 (2007) 179–183.
[15] S.M. Carroll, V. Duvvuri, M. Trodden and M.S. Turner, Is cosmic speed-up due to new gravitational physics?,
Phys. Rev. D, 70 (2004) 043528.
[16] E.J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D, 15(11) (2006) 1753–
1935.
[17] S. Dodelson, M. Kaplinghat and E. Stewart, Solving the coincidence problem: tracking oscillating energy, Phys.
Rev. Lett. 85 (2000) 5276–5279.
[18] R.Y. Donagi, J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Visible branes with negative tension in
heterotic M-theory, J. High Energy Phys. 2001(11) (2001) 041.
[19] B. Feng, X. Wang and X. Zhang, Dark energy constraints from the cosmic age and supernova, Phys. Lett. B.,
607 (2005) 35–41.
[20] P.M. Garnavich, R.P. Kirshner, P. Challis, J. Tonry, R.L. Gilliland, R.C. Smith, A. Clocchiatti, A. Diercks,
A.V. Filippenko, M. Hamuy, C.J. Hogan, B. Leibundgut, M.M. Phillips, D. Reiss, A.G. Riess, B.P. Schmidt,
J. Spyromilio, C. Stubbs, N.B. Suntzeff and L. Wells, Constraints on cosmological models from Hubble space
telescope observations of high-z supernovae, Astrophys. Phys. J. 493 (1998) 53–57.
[21] T. Harko, F.S.N. Lobo, S. Nojiri and S.D. Odintsov, f(R, T) gravity with variable deceleration parameter, Int. J.
Geom. Meth. Mod. Phys. 14 (2017) 1750097.
[22] T. Josset, A. Perez and D. Sudarsky, Dark energy from violation of energy conservation, Phys. Rev. Lett. 118
(2013) 021102.
[23] G.S. Khadekar and D. Raut, FRW viscous fluid cosmological model with time-dependent inhomogeneous equation
of state, Int. J. Geom. Methods Mod. Phys. 15(01) (2018) 1830001.
[24] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Ekpyrotic universe: Colliding branes and the origin of the
hot big bang, Phys. Rev. D 64 (2001) 123522.
[25] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, A brief comment on ”The Pyrotechnic Universe”, arXiv,
(2001).
[26] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Density perturbations in the ekpyrotic scenario, Phys. Rev.
D 66 (2002) 046005.
[27] E.V. Linder, Einsteins other gravity and the acceleration of the Universe, Phys. Rev. D 81 (2010) 127301.
[28] S. Nojiri and S.D. Odintsov, Inhomogeneous equation of state of the universe: Phantom era, future singularity,
and crossing the phantom barrier, Phys. Rev. D 2 (2005) 023003.
[29] S. Nojiri and S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, Int. J.
Geom. Methods Mod. Phys. 4 (2007) 115–145.
[30] T. Padmanabhan, Accelerated expansion of the universe driven by tachyonic matter, Phys. Rev. D., 66 (2002)
021301.
[31] T. Padmanabhan, Dark energy and gravity, Gen. Relativ. Gravit. 40 (2008) 529–564.
[32] L. Page, M.R. Nolta, C. Barnes, C.L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S.S.
Meyer, H.V. Peiris, D.N. Spergel, G.S. Tucker, E. Wollack and E.L. Wright, First-Year Wilkinson Microwave
Anisotropy Probe (WMAP) Observations: Interpretation of the TT and TE Angular Power Spectrum Peaks,
Astrophys. J. Suppl. Ser. 148 (2003) 233–241.
[33] S. Perlmutter, G. Aldering, G. Goldhaber, R.A. Knop, P. Nugent, P.G. Castro, S. Deustua, S. Fabbro, A. Goobar,
D.E. Groom, I.M. Hook, A.G. Kim, M.Y. Kim, J.C. Lee, N.J. Nunes, R. Pain, C.R. Pennypacker, R. Quimby,
C. Lidman, R.S. Ellis, M. Irwin, R.G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B.J. Boyle, A.V.
Filippenko, T. Matheson, A.S. Fruchter, N. Panagia, H.J.M. Newberg, W.J. Couch and The Supernova Cosmology
Project, Measurements of Ω and Λ from 42 high-redshift supernovae, Astrophys. Phys. J. 517(2) (1999) 565–586.
[34] A.G. Riess, A.V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich, R.L. Gilliland, C.J. Hogan,
S. Jha, R.P. Kirshner, B. Leibundgut, M.M. Phillips, D. Reiss, B.P. Schmidt, R.A. Schommer, R.C. Smith, J.
Spyromilio, C. Stubbs, N.B. Suntzeff and J. Tonry, Observational evidence from supernovae for an accelerating
universe and a cosmological constant, Astrophys. Phys. J. 116 (1998) 1009–1038.
[35] M.E. Rodrigues, M.J.S. Houndjo, D. Momeni and R. Myrzakulov, A type of Levi-Civita’s solution in modified
Gauss-Bonnet gravity, Can. J. Phys. 92 (2014) 173–176.
[36] D. Saez-Gomez, Oscillating universe from an inhomogeneous equation of state and coupled dark energy, Grav.
Cosmol. 15 (2009) 134–140.
[37] V. Sahni, T.D. Saini, A.A. Starobinsky and U. Alam, Statefinder a new geometrical diagnostic of dark energy,
2180 AlamJETP Lett. 77 (2003) 201–206.
[38] P.K. Sahoo, P. Sahoo and B.K. Bishi, Anisotropic cosmological models in f(R, T) gravity with variable deceleration
parameter, Int. J. Geom. Meth. Mod. Phys. 14 (2017) 1750097.
[39] P.K. Sahoo and M. Sivakumar, LRS Bianchi type-I cosmological model in f(R, T) theory of gravity with Λ(T),
Astrophys. Space Sci. 357(1) (2015) 60.
[40] P.K. Sahoo, S.K. Tripathy and P. Sahoo, A periodic varying deceleration parameter in f(R, T) gravity, Mod.
Phys. Lett. A 3 (2018) 1850193.
[41] H. Shabani and A.H. Ziaie, Consequences of energy conservation violation: Late time solutions of Λ(T)CDM
subclass of f(R, T) gravity using dynamical system approach, Eur. Phys. J. C 77 (2017) 282.
[42] M. Sharif, S. Rani and R. Myrzakulov, Analysis of f(R, T) gravity models through energy conditions, Eur. Phys.
J. Plus 128 (2013) 123.
[43] M.S. Singh and S.S. Singh, Cosmological dynamics of anisotropic dark energy in f(R, T) gravity, New Astronom.
72 (2019) 3641.
[44] M.K. Singh, M.K. Verma and S. Ram, Two-fluid cosmological model of Bianchi type-V with negative constant
deceleration parameter, Int. J. Theor. Phys. 52 (2013) 227–232.
[45] T.P. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys., 82 (2010) 451–497.
[46] D.N. Spergel, L. Verde, H.V. Peiris, E. Komatsu, M.R. Nolta, C.L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik,
A. Kogut, M. Limon, S.S. Meyer, L. Page, G.S. Tucker, J.L. Weiland, E. Wollack and E.L. Wright, Firstyear Wilkinson microwave anisotropy probe (WMAP) observations: Determination of cosmological parameters,
Astrophys. J. Suppl. Ser. 148 (2003) 175–194.
[47] P.J. Steinhardt and N. Turok, Cosmic evolution in a cyclic universe, Phys. Rev. D 65 (2002) 126003.
[48] P.J. Steinhardt and N. Turok, The cyclic universe: An informal introduction, Nucl. Phys. Proc. Suppl. 124 (2003)
38.
[49] R.K. Tiwari, D. Sofuolu and V.K. Dubey, Phase transition of LRS Bianchi type-I cosmological model in f(R, T)
gravity, Int. J. Geom. Methods Mod. Phys. 17(12) (2020) 2050187.
)
Volume 12, Special Issue
December 2021Pages 2167-2180
- Receive Date: 08 October 2021
- Accept Date: 06 December 2021