# Tsallis Holographic Dark Energy in f(G,T) Gravity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reconstruction of THDE $\mathit{f}(\mathit{G},\mathit{T})$ Models

- $f(G,T)={f}_{1}\left(G\right)+{f}_{2}\left(T\right)$,
- $f(G,T)=F\left(G\right)+\eta T$,

#### 2.1. Conserved EMT Based Reconstruction

#### 2.2. Non-Conserved EMT Based Reconstruction

## 3. Cosmological Analysis

#### 3.1. Cosmic Diagnostics Parameters for Conserved EMT

#### 3.2. Cosmic Diagnostics Parameters for Non-Conserved EMT

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Capozziello, S.; De Laurentis, M.; Luongo, O.; Ruggeri, A.C. Cosmographic constraints and cosmic fluids. Galaxies
**2013**, 1, 216–260. [Google Scholar] [CrossRef] - Susskind, L. The world as a hologram. J. Math. Phys.
**1995**, 36, 6377–6396. [Google Scholar] [CrossRef] [Green Version] - Cohen, A.G.; Kaplan, D.B.; Nelson, A.E. Effective field theory, black holes, and the cosmological constant. Phys. Rev. Lett.
**1999**, 82, 4971. [Google Scholar] [CrossRef] - Li, M. A model of holographic dark energy. Phys. Lett. B
**2004**, 603, 1–5. [Google Scholar] [CrossRef] [Green Version] - Karami, K.; Khaledian, M.S. Reconstructing f(R) modified gravity from ordinary and entropy-corrected versions of the holographic and new agegraphic dark energy models. J. High Energy Phys.
**2011**, 3, 86. [Google Scholar] [CrossRef] - Houndjo, M.J.S.; Piattella, O.F. Reconstructing f(R,T) gravity from holographic dark energy. Int. J. Mod. Phys. D
**2012**, 21, 1250024. [Google Scholar] [CrossRef] - Daouda, M.H.; Rodrigues, M.E.; Houndjo, M.J.S. Static anisotropic solutions in f(T) theory. Eur. Phys. J. C
**2012**, 72, 1890. [Google Scholar] [CrossRef] - Jawad, A.; Pasqua, A.; Chattopadhyay, S. Holographic reconstruction of f(G) gravity for scale factors pertaining to emergent, logamediate and intermediate scenarios. Eur. Phys. J. Plus
**2013**, 128, 156. [Google Scholar] [CrossRef] - Sharif, M.; Zubair, M. Cosmology of holographic and new agegraphic f(R,T) models. J. Phys. Soc. Jpn.
**2013**, 82, 064001. [Google Scholar] [CrossRef] - Fayaz, V.; Hossienkhani, H.; Amirabadi, M.; Azimi, N. Anisotropic cosmological models in f(R,T) gravity according to holographic and new agegraphic dark energy. Astrophys. Space Sci.
**2014**, 353, 301–309. [Google Scholar] [CrossRef] - Horava, P.; Minic, D. Probable values of the cosmological constant in a holographic theory. Phys. Rev. Lett.
**2000**, 85, 1610. [Google Scholar] [CrossRef] [PubMed] - Thomas, S. Holography stabilizes the vacuum energy. Phys. Rev. Lett.
**2002**, 89, 081301. [Google Scholar] [CrossRef] [PubMed] - Hsu, S.D.H. Entropy bounds and dark energy. Phys. Lett. B
**2004**, 594, 13–16. [Google Scholar] [CrossRef] [Green Version] - Guberina, B.; Horvat, H.; Nikolić, H. Non-saturated holographic dark energy. J. Cosmol. Astropart. Phys.
**2007**, 1, 012. [Google Scholar] [CrossRef] - Wang, B.; Abdalla, E.; Atrio-Barandela, F.; Pavon, D. Dark matter and dark energy interactions: Theoretical challenges, cosmological implications and observational signatures. Rep. Prog. Phys.
**2016**, 79, 096901. [Google Scholar] [CrossRef] [PubMed] - Wang, S.; Wang, Y.; Li, M. Holographic dark energy. Phys. Rep.
**2017**, 699, 1–57. [Google Scholar] [CrossRef] - Moradpour, H. Implications, consequences and interpretations of generalized entropy in the cosmological setups. Int. J. Theor. Phys.
**2016**, 55, 4176–4184. [Google Scholar] [CrossRef] - Wen, W.Y. Thermodynamic metric of deformed Schwarzschild black holes. Int. J. Mod. Phys. D
**2017**, 26, 1750106. [Google Scholar] [CrossRef] [Green Version] - Moradpour, H.; Bonilla, A.; Abreu, E.M.C.; Neto, A.J. Accelerated cosmos in a nonextensive setup. Phys. Rev. D
**2017**, 96, 123504. [Google Scholar] [CrossRef] [Green Version] - Jahmori, A.S.; Moosavi, S.A.; Moradpour, H.; Graca, J.M.; Lobo, I.P.; Salako, I.G.; Jawad, A. Generalized entropy formalism and a new holographic dark energy model. Phys. Lett. B
**2018**, 780, 21–24. [Google Scholar] [CrossRef] - Moradpour, H.; Moosavi, S.A.; Lobo, I.P.; Graca, J.P.; Jawad, A.; Salako, I.G. Thermodynamic approach to holographic dark energy and the Rényi entropy. Eur. Phys. J. C
**2018**, 78, 829. [Google Scholar] [CrossRef] - Biró, T.S.; Ván, P. Zeroth law compatibility of nonadditive thermodynamics. Phys. Rev. E
**2011**, 83, 061187. [Google Scholar] [CrossRef] [PubMed] - Majhi, A. Non-extensive statistical mechanics and black hole entropy from quantum geometry. Phys. Lett. B
**2017**, 775, 32–36. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Tsallis, C.; Citro, L.J.L. Black hole thermodynamical entropy. Eur. Phys. J. C
**2013**, 73, 2487. [Google Scholar] [CrossRef] [Green Version] - Czinner, V.G.; Iguchi, H. A zeroth law compatible model to Kerr black hole thermodynamics. Universe
**2017**, 3, 14. [Google Scholar] [CrossRef] - Komatsu, N. Cosmological model from the holographic equipartition law with a modified Rényi entropy. Eur. Phys. J. C
**2017**, 77, 229. [Google Scholar] [CrossRef] - Tavayef, M.; Sheykhi, A.; Bamba, K.; Moradpour, H. Tsallis holographic dark energy. Phys. Lett. B
**2018**, 781, 195–200. [Google Scholar] [CrossRef] - Harko, T.; Lobo, F.S.; Nojiri, S.I.; Odintsov, S.D. f(R,T) gravity. Phys. Rev. D
**2011**, 84, 024020. [Google Scholar] [CrossRef] - Sharif, M.; Ikram, A. Energy Conditions in f($\mathcal{G}$,T) Gravity. Eur. Phys. J. C
**2016**, 76, 640. [Google Scholar] [CrossRef] - Sharif, M.; Ikram, A. Stability analysis of some reconstructed cosmological models in f($\mathcal{G}$,T) gravity. Phys. Dark Universe
**2017**, 17, 11–19. [Google Scholar] [CrossRef] - Bhatti, M.Z.U.; Sharif, M.; Yousaf, Z.; Ilyas, M. Role of f(G,T) gravity on the evolution of relativistic stars. Int. J. Mod. Phys. D
**2017**, 27, 1850044. [Google Scholar] [CrossRef] - Shamir, M.F.; Ahmad, M. Noether symmetry approach in f($\mathcal{G}$,T) gravity. Eur. Phys. J. C
**2017**, 77, 55. [Google Scholar] [CrossRef] - Jamil, M.; Saridakis, E.N. New agegraphic dark energy in Horava-Lifshitz cosmology. J. Cosmol. Astropart. Phys.
**2010**, 1007, 028. [Google Scholar] [CrossRef] - Jawad, A.; Chattopadhyay, S. Cosmological analysis of F($\tilde{R}$) models via pilgrim dark energy. Astrophys. Space Sci.
**2015**, 357, 37. [Google Scholar] [CrossRef] - Sharif, M.; Nazir, K. Cosmological evolution of generalized ghost pilgrim dark energy in f(T) gravity. Astrophys. Space Sci.
**2015**, 360, 57. [Google Scholar] [CrossRef] - Sharif, M.; Nazir, K. Cosmological analysis of reconstructed $\mathcal{F}$(T,T
_{$\mathcal{G}$}) models. Eur. Phys. J. C**2018**, 78, 77. [Google Scholar] [CrossRef] - Kleidis, K.; Oikonomou, V.K. Unification of late-and early-time acceleration, with that of the intermediate eras, by scalar fields. Astrophys. Space Sci.
**2017**, 362, 74. [Google Scholar] [CrossRef] - Ghaffari, S.; Moradpour, H.; Lobo, I.P.; Graca, J.M.; Bezerra, V.B. Tsallis holographic dark energy in the Brans-Dicke cosmology. Eur. Phys. J. C
**2018**, 78, 706. [Google Scholar] [CrossRef] - Ghaffari, S.; Dehghani, M.H.; Sheykhi, A. Holographic dark energy in the DGP braneworld with Granda-Oliveros cutoff. Phys. Rev. D
**2014**, 89, 123009. [Google Scholar] [CrossRef] - Sharif, M.; Ikram, A. Stability analysis of Einstein universe in f($\mathcal{G}$,T) gravity. Int. J. Mod. Phys. D
**2017**, 26, 1750084. [Google Scholar] [CrossRef] - Aviles, A.; Bravetti, A.; Capozziello, S.; Luongo, O. Cosmographic reconstruction of f(T) cosmology. Phys. Rev. D
**2013**, 87, 064025. [Google Scholar] [CrossRef] - Capozziello, S.; Luongo, O.; Saridakis, E.N. Transition redshift in f(T) cosmology and observational constraints. Phys. Rev. D
**2015**, 91, 124037. [Google Scholar] [CrossRef] - Capozziello, S.; Luongo, O.; Pincak, R.; Ravanpak, A. Cosmic acceleration in non-flat f(T) cosmology. Gen. Relativ. Gravit.
**2018**, 50, 53. [Google Scholar] [CrossRef] - Caldwell, R.; Linder, E.V. Limits of quintessence. Phys. Rev. Lett.
**2005**, 95, 141301. [Google Scholar] [CrossRef] [PubMed] - Sahni, V.; Saini, T.D.; Starobinsky, A.A.; Alam, U. Statefinder-a new geometrical diagnostic of dark energy. J. Exp. Theor. Phys. Lett.
**2003**, 77, 201–206. [Google Scholar] [CrossRef] - Aghanim, N.; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Basak, S.; et al. Planck 2018 results. VI. Cosmological parameters. arXiv, 2018; arXiv:1807.06209. [Google Scholar]
- Aviles, A.; Gruber, C.; Luongo, O.; Quevedo, H. Cosmography and constraints on the equation of state of the Universe in various parametrizations. Phys. Rev. D
**2012**, 86, 123516. [Google Scholar] [CrossRef] - Dunsby, P.K.S.; Luongo, O. On the theory and applications of modern cosmography. Int. J. Geom. Meth. Mod. Phys.
**2016**, 13, 1630002. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Plot of THDE $f(G,T)$ model against z for ${m}_{1}=1.7$ (red), ${m}_{2}=1.8$ (green) and ${m}_{3}=1.9$ (blue) for conserved EMT. The model shows gradual increase with convergence to zero from both sides, indicates realistic one.

**Figure 2.**Plot of THDE $f(G,T)$ model against z for ${m}_{1}=1.8$ (red), ${m}_{2}=2.1$ (green) and ${m}_{3}=2.4$ (blue) for non-conserved EMT. In this case, the developed model shows reverse behavior as compared to conserved one which coincides with it for late-time phase of redshift parameter.

**Figure 3.**Plot of EoS parameter against z for ${m}_{1}=1.7$ (red), ${m}_{2}=1.8$ (green) and ${m}_{3}=1.9$ (blue) for conserved EMT shows phantom regime of the universe.

**Figure 4.**Plot of deceleration parameter versus z for ${m}_{1}=1.7$ (red), ${m}_{2}=1.8$ (green) and ${m}_{3}=1.9$ (blue) predicts accelerating phase for conserved EMT.

**Figure 5.**Plot of ${\nu}_{s}^{2}$ against z for conserved EMT for ${m}_{1}=1.7$ (red), ${m}_{2}=1.8$ (green) and ${m}_{3}=1.9$ (blue) illustrates instability of the model.

**Figure 6.**Trajectories of ${\omega}_{GT}-{\omega}_{GT}^{\prime}$ against z for conserved EMT for ${m}_{1}=1.7$ (red), ${m}_{2}=1.8$ (green) and ${m}_{3}=1.9$ (blue) shows freezing region which is more accelerating era of cosmos.

**Figure 7.**Trajectories of $r-s$ plan against z for ${m}_{1}=1.7$ (red), ${m}_{2}=1.8$ (green) and ${m}_{3}=1.9$ (blue) shows phantom and quintessence phase of the universe for conserved EMT.

**Figure 8.**Plot of ${\omega}_{GT}$ parameter against z for ${m}_{1}=1.8$ (red), ${m}_{2}=2.1$ (green) and ${m}_{3}=2.4$ (blue) shows phantom expansion for entire cosmic evolutionary regime in case of non-conserved EMT.

**Figure 9.**Plot of deceleration parameter against z for ${m}_{1}=1.8$ (red), ${m}_{2}=2.1$ (green) and ${m}_{3}=2.4$ (blue) indicates accelerating cosmic era for non-conserved matter distributions.

**Figure 10.**Squared speed of sound parameter against z for ${m}_{1}=1.8$ (red), ${m}_{2}=2.1$ (green) and ${m}_{3}=2.4$ (blue) shows stability of the model for current cosmic evolution while becomes unphysical for late-time cosmic regime in case of non-conserved EMT.

**Figure 11.**Trajectories of ${\omega}_{GT}-{\omega}_{GT}^{\prime}$ for ${m}_{1}=1.8$ (red), ${m}_{2}=2.1$ (green) and ${m}_{3}=2.4$ (blue) indicates more accelerating (freezing) phase as compared with thawing for non-conserved EMT.

**Figure 12.**Trajectories of $r-s$ phase plane for ${m}_{1}=1.8$ (red), ${m}_{2}=2.1$ (green) and ${m}_{3}=2.4$ (blue) performs Chaplygin gas model regime for non-conserved EMT.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sharif, M.; Saba, S.
Tsallis Holographic Dark Energy in *f*(*G*,*T*) Gravity. *Symmetry* **2019**, *11*, 92.
https://doi.org/10.3390/sym11010092

**AMA Style**

Sharif M, Saba S.
Tsallis Holographic Dark Energy in *f*(*G*,*T*) Gravity. *Symmetry*. 2019; 11(1):92.
https://doi.org/10.3390/sym11010092

**Chicago/Turabian Style**

Sharif, Muhammad, and Saadia Saba.
2019. "Tsallis Holographic Dark Energy in *f*(*G*,*T*) Gravity" *Symmetry* 11, no. 1: 92.
https://doi.org/10.3390/sym11010092