Journal of Holography Applications in Physics
Journey Through Cosmic: Tsallis Holographic Dark Energy and the Deformed Starobinsky Model
Document Type : Regular article
Author
Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416--95447, Babolsar, Iran
Abstract
This study investigates the intricate dynamics of Tsallis holographic dark energy within the framework of a modified Starobinsky gravity theory. The distinctiveness of the model lies in its formulation, which involves both the Ricci scalar $R$ and an additive positive term. The primary objective is to derive the equation of state parameter, a crucial element in understanding the behavior and properties of dark energy throughout the universe. To simplify and strengthen the analysis, we adopt an exponential form for the scale factor, which is commonly used in models featuring constant expansion rates due to its analytical tractability. A comprehensive stability evaluation is also carried out, with particular attention given to the squared sound speed — a critical factor in examining how fluctuations evolve within the dark energy sector. Graphical representations are employed to highlight stable regimes and to visually interpret the viability of this holographic model under modified gravitational dynamics. The findings are presented in a detailed manner, including rigorous derivations and explicit formulations that underline the theoretical consistency and potential cosmological relevance of the model.
Keywords
Main Subjects
Article PDF
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[46] S. N. Gashti and B. Pourhassan, “Non-extensive entropy and holographic thermodynamics: topological insights”, Eur. Phys. J. C 85(4), 435 (2025), [arXiv:2412.12132 [hep-th]], DOI: 10.48550/arXiv.2412.12132
[47] S. N. Gashti, B. Pourhassan and İ. Sakallı, “Thermodynamic topology and phase space analysis of AdS black holes through non-extensive entropy perspectives”, Eur. Phys. J. C 85(3), 305 (2025), DOI: 10.1140/epjc/s10052-025-14035-x
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[50] J. Sadeghi et al., “Phase transition dynamics of black holes influenced by Kaniadakis and Barrow statistics”, Phys. Dark Univ. 47, 101780 (2025), DOI: 10.1016/j.dark.2024.101780
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[2] Z. W. Zhao et al., “Probing the Interaction between Dark Energy and Dark Matter with Future Fast Radio Burst Observations”, J. Cosmol. Astropart. Phys. 2023(04), 022 (2023), DOI: 10.1142/9789811269776_0155
[3] E. N. Saridakis et al., “Observational Constraints on Soft Dark Energy and Soft Dark Matter: Challenging ΛCDM Cosmology”, Nucl. Phys. B 986, 116042 (2023), DOI: 10.1142/9789811269776_0155
[4] F. Finster and J. M. Isidro, “A Mechanism for Dark Matter and Dark Energy in the Theory of Causal Fermion Systems”, Class. Quantum Grav. 40(7), 075017 (2023), DOI: 10.1088/1361-6382/acc0c8
[5] M. M. Ivanov et al., “Constraining Early Dark Energy with Large-Scale Structure”, Phys. Rev. D 102(10), 103502 (2020), DOI: 10.1103/PhysRevD.102.103502
[6] H. Yan, “Hyperbolic Fracton Model, Subsystem Symmetry, and Holography”, Phys. Rev. B 99(15), 155126 (2019), DOI: 10.1103/PhysRevB.99.155126
[7] J. Sadeghi, S. N. Gashti and T. Azizi, “Complex Quintessence Theory, Tsallis and Kaniadakis Holographic Dark Energy and Brans–Dicke Cosmology”, Mod. Phys. Lett. A 38(14n15), 2350076 (2023), DOI: 10.1142/S0217732323500761
[8] J. Sadeghi, S. N. Gashti and T. Azizi, “Tsallis Holographic Dark Energy under Complex Form of Quintessence Model”, Commun. Theor. Phys. 75(2), 025402 (2023), DOI: 10.1088/1572-9494/aca390
[9] C. G. Böhmer and E. Jensko, “Modified Gravity: A Unified Approach”, Phys. Rev. D 104(2), 024010 (2021), DOI: 10.1103/PhysRevD.104.024010
[10] S. I. Nojiri and S. D. Odintsov, “Introduction to Modified Gravity and Gravitational Alternative for Dark Energy”, Int. J. Geom. Meth. Mod. Phys. 4(1), 115 (2007), DOI: 10.1142/S0219887807001928
[11] X. M. Kuang et al., “Constraining a Modified Gravity Theory in Strong Gravitational Lensing and Black Hole Shadow Observations”, Phys. Rev. D 106(6), 064012 (2022), DOI: 10.1103/PhysRevD.106.064012
[12] M. B. Cantcheff et al., “Entanglement from Dissipation and Holographic Interpretation”, Eur. Phys. J. C 78, 1 (2018), DOI: 10.1140/epjc/s10052-018-5545-2
[13] J. Sadeghi et al., “Cosmic Evolution of the Logarithmic f(R) Model and the dS Swampland Conjecture”, Universe 8(12), 623 (2022), DOI: 10.3390/universe8120623
[14] M. P. Fusco, “Black Hole Entropy and the Holographic Universe”, The Phys. Metaphys. Transubstantiation, Springer Int. Pub., Cham, 2023, 221 (2023), DOI: 10.1007/978-3-031-34640-8_5
[15] R. Bousso, “The Holographic Principle for General Backgrounds”, Class. Quantum Grav. 17(5), 997 (2000), DOI: 10.1088/0264-9381/17/5/309
[16] L. A. Anchordoqui et al., “Decay of Multiple Dark Matter Particles to Dark Radiation in Different Epochs Does Not Alleviate the Hubble Tension”, Phys. Rev. D 105(10), 103512 (2022), DOI: 10.1103/PhysRevD.105.103512
[17] M. Vogelsberger et al., “Cosmological Simulations of Galaxy Formation”, Nat. Rev. Phys. 2(1), 42 (2020), DOI: 10.1038/S42254-019-0127-2
[18] T. M. C. Abbott et al., “Dark Energy Survey Year 3 Results: Constraints on Extensions to ΛCDM with Weak Lensing and Galaxy Clustering”, Phys. Rev. D 107(8), 083504 (2023), DOI: 10.1103/PhysRevD.107.083504
[19] R. F. L. Holanda et al., “Galaxy Cluster Sunyaev-Zel’dovich Effect Scaling-Relation and Type Ia Supernova Observations as a Test for the Cosmic Distance Duality Relation”, J. Cosmol. Astropart. Phys. 2019(06), 008 (2019), DOI: 10.1088/1475- 7516/2019/06/008
[20] E. Komatsu, “New Physics from the Polarized Light of the Cosmic Microwave Background”, Nat. Rev. Phys. 4(7), 452 (2022), DOI: 10.1038/s42254-022-00452-4
[21] M. Oguri, “Strong Gravitational Lensing of Explosive Transients”, Rep. Prog. Phys. 82(12), 126901 (2019), DOI: 10.1088/1361-6633/ab4fc5
[22] T. Harko et al., “Non-Minimal Geometry–Matter Couplings in Weyl–Cartan Space– Times: f(R, T, Q, Tm) Gravity”, Phys. Dark Univ. 34, 100886 (2021), DOI: 10.1016/j.dark.2021.100886
[23] N. Parbin and U. D. Goswami, “Scalarons Mimicking Dark Matter in the Hu– Sawicki Model of f(R) Gravity”, Mod. Phys. Lett. A 36(37), 2150265 (2021), DOI: 10.1142/S0217732321502655
[24] L. Granda, “Unified Inflation and Late-Time Accelerated Expansion with Exponential and R² Corrections in Modified Gravity”, Symmetry 12(5), 794 (2020), DOI: 10.3390/sym12050794
[25] A. Paliathanasis, J. L. Said and J. D. Barrow, “Stability of the Kasner Universe in f(T) Gravity”, Phys. Rev. D 97(4), 044008 (2018), DOI: 10.1103/PhysRevD.97.044008
[26] S. Casas, M. Pauly and J. Rubio, “Higgs-Dilaton Cosmology: An Inflation–Dark-Energy Connection and Forecasts for Future Galaxy Surveys”, Phys. Rev. D 97(4), 043520 (2018), DOI: 10.1103/PhysRevD.97.043520
[27] P. S. Ens and A. F. Santos, “f(R) Gravity and Tsallis Holographic Dark Energy”, Europhys. Lett. 131(4), 40007 (2020), DOI: 10.1209/0295-5075/131/40007
[28] J. Sadeghi et al., “Traversable Wormhole in Logarithmic f(R) Gravity by Various Shape and Redshift Functions”, Int. J. Mod. Phys. D 31(3), 2250019 (2022), DOI: 10.1142/S0218271822500195
[29] J. Sadeghi and S. N. Gashti, “Investigating the Logarithmic Form of f(R) Gravity Model from Brane Perspective and Swampland Criteria”, Pramana 95, 1 (2021), DOI: 10.1007/s12043-021-02234-6
[30] J. Sadeghi, E. N. Mezerji and S. N. Gashti, “Study of Some Cosmological Parameters in Logarithmic Corrected f(R) Gravitational Model with Swampland Conjectures”, Mod. Phys. Lett. A 36(05), 2150027 (2021), DOI: 10.1142/S0217732321500279
[31] J. Sadeghi, S. N. Gashti and T. Azizi, “Complex Quintessence Theory, Tsallis and Kaniadakis Holographic Dark Energy and Brans–Dicke Cosmology”, Mod. Phys. Lett. A 38, 2350076 (2023), DOI: 10.1142/S0217732323500761
[32] J. Sadeghi, S. N. Gashti and T. Azizi, “Tsallis Holographic Dark Energy under Complex Form of Quintessence Model”, Commun. Theor. Phys. 75(2), 025402 (2023), DOI: 10.1088/1572-9494/aca390
[33] J. Sadeghi et al., “Cosmic Evolution of the Logarithmic f(R) Model and the dS Swampland Conjecture”, Universe 8(12), 623 (2022), DOI: 10.3390/universe8120623
[34] J. Sadeghi et al., “Traversable Wormhole in Logarithmic f(R) Gravity by Various Shape and Redshift Functions”, Int. J. Mod. Phys. D 31(3), 2250019 (2022), DOI: 10.1142/S0218271822500195
[35] J. Sadeghi and S. N. Gashti, “Investigating the Logarithmic Form of f(R) Gravity Model from Brane Perspective and Swampland Criteria”, Pramana 95, 1 (2021), DOI: 10.1007/s12043-021-02234-6
[36] J. Sadeghi, E. N. Mezerji and S. N. Gashti, “Study of Some Cosmological Parameters in Logarithmic Corrected f(R) Gravitational Model with Swampland Conjectures”, Mod. Phys. Lett. A 36(05), 2150027 (2021), DOI: 10.1142/S0217732321500279
[37] Ph. Channuie, “Deformed Starobinsky Model in Gravity’s Rainbow”, Eur. Phys. J. C 79(6), 508 (2019), DOI: 10.1140/epjc/s10052-019-7050-7
[38] Ph. Channuie, “Refined Swampland Conjecture in Deformed Starobinsky Gravity”, Int. J. Mod. Phys. D 31(09), 2250074 (2022), DOI: 10.1142/S0218271822500742
[39] J. Sadeghi et al., “Smeared Mass Source Wormholes in Modified f(R) Gravity with the Lorentzian Density Distribution Function”, Mod. Phys. Lett. A 37(03), 2250018 (2022), DOI: 10.1142/S0217732322500183
[40] F. Baziar and J. Sadeghi, “Unveiling the Universe: Tsallis Holographic Dark Energy and Modified Logarithmic f(R) Gravity”, Iran. J. Astron. Astrophys. 11(2), 117 (2024), DOI: 10.22128/ijaa.2024.851.1188
[41] S. N. Gashti et al., “Impact of Loop Quantum Gravity on the Topological Classification of Quantum-Corrected Black Holes”, Universe 2025, DOI: 10.3390/universe11080247
[42] A. Anand and S. N. Gashti, “Universal relations with the non-extensive entropy perspective”, Phys. Dark Univ. 2025, 102015 (2025), DOI: 10.1016/j.dark.2025.102015
[43] S. N. Gashti, “Thermodynamic Topology of Einstein-Maxwell-Scalar Black Holes: Insights from Barrow entropy and Logarithmic Corrections”, J. Holography Appl. Phys. 5(2), 72 (2025), DOI: 10.22128/jhap.2025.1028.1119
[44] A. Anand and S. N. Gashti, “Universality relation and thermodynamic topology with three-parameter entropy model”, Phys. Dark Univ. 48, 101916 (2025), DOI: 10.1016/j.dark.2025.101916
[45] M. A. S. Afshar et al., “Topological insights into black hole thermodynamics: non-extensive entropy in CFT framework”, Eur. Phys. J. C 85(4), 1 (2025), [arXiv:2501.00955 [hep-th]], DOI: 10.48550/arXiv.2501.00955
[46] S. N. Gashti and B. Pourhassan, “Non-extensive entropy and holographic thermodynamics: topological insights”, Eur. Phys. J. C 85(4), 435 (2025), [arXiv:2412.12132 [hep-th]], DOI: 10.48550/arXiv.2412.12132
[47] S. N. Gashti, B. Pourhassan and İ. Sakallı, “Thermodynamic topology and phase space analysis of AdS black holes through non-extensive entropy perspectives”, Eur. Phys. J. C 85(3), 305 (2025), DOI: 10.1140/epjc/s10052-025-14035-x
[48] A. B. Brzo et al., “Thermodynamic topology of AdS black holes within noncommutative geometry and Barrow entropy”, Nucl. Phys. B 1012, 116840 (2025), DOI: 10.1016/j.nuclphysb.2025.116840
[49] S. N. Gashti et al., “Thermodynamic topology and photon spheres of dirty black holes within non-extensive entropy”, Phys. Dark Univ. 47, 101833 (2025), DOI: 10.1016/j.dark.2025.101833
[50] J. Sadeghi et al., “Phase transition dynamics of black holes influenced by Kaniadakis and Barrow statistics”, Phys. Dark Univ. 47, 101780 (2025), DOI: 10.1016/j.dark.2024.101780
[51] S. N. Gashti, “Topology of holographic thermodynamics within non-extensive entropy”, [arXiv:2412.00889 [hep-th]] (2024), DOI: 10.48550/arXiv.2412.00889
)
Volume 5, Issue 3
September 2025Pages 50-63
- Receive Date: 30 June 2025
- Revise Date: 09 August 2025
- Accept Date: 09 August 2025