The Inflationary with Inverse Power-Law Potential in Tsallis Entropy

Year 2024, , 37 - 42, 01.06.2024

M. Faruk Karabat

https://doi.org/10.5541/ijot.1356867

Abstract

In this article, we focus on the inflation dynamics of the early Universe using an inverse power law potential scalar field (V_((ϕ))=V_0 ϕ^(-n)) within the framework of Tsallis entropy. First, we derive the modified Friedmann equations from the non-additive Tsallis entropy by applying the first law of thermodynamics to the apparent horizon of the Friedmann–Robertson–Walker (FRW) Universe. We assume that the inflationary era of the Universe consists of two phases; the slow roll inflation phase and the kinetic inflation phase. We obtained the scalar spectral index n_s and tensor-to-scalar ratio r and compared our results with the latest Planck data for these phases. By choosing the appropriate values for the Tsallis parameters, which bounded by β<2, and the inverse power-term of the potential n, we determined that the inflation era of the Universe in Tsallis entropy can only occur in the second phase (kinetic inflation), while the slow-roll inflation phase is incompatible with the Planck data.

Keywords

Tsallis entropy, inflationary, inverse power law potential

References

• A. G. Riess et al., ‘‘Observational evidence from supernovae for an accelerating Universe and a cosmological constant,’’ Astron. J. 116, 1009, 1998.
• S. Perlmutter et al., ‘‘Measurements of Omega and Lambda from 42 High-Redshift supernovae,’’ Astrophys. J. 517, 565, 1998.
• P. A. R. Ade et al., ‘‘Joint Analysis of BICEP2/Keck Array and Planck Data’’ Astron. Astrophys. 571, A16, 2014.
• R. G. Vishwakarma, ‘‘Mon. Not. Roy,’’ Astron. Soc. 331, 776, 2002.
• S. L. Briddle, O. Lahav, J. P. Ostriker, P.J. Steinhardt, “Precision cosmology? Not just yet,’’ Science, vol.299, pp.1532–1533, 2003.
• P. J. E. Peebles, B. Ratra, “Cosmic Discordance: Planck and luminosity distance data exclude LCDM’’ Rev. Mod. Phys. 75, 559, 2003.
• Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia., “Holographic dark energy through Tsallis entropy,’’ Phys. Rept. 493, 1, 2010.
• P. J. E. Peebles, B. Ratra, ‘‘The cosmological constant and dark energy,’’ Rev. Mod. Phys. 75, 559, 2003, arXiv:astro-ph/0207347.
• Y. F. Cai, E. N. Saridakis, M. R. Setare, J. Q. Xia, ‘‘Quintom cosmology: theoretical implications and observations,’’ Phys. Rept. 493, 1, 2010, arXiv:0909.2776 [hep-th].
• N. Bartolo, E. Komatsu, S. Mattarrese and A. Riotto., ‘‘Non-Gaussianity from Inflation: Theory and Observations,’’ Phys. Rept. 402, 103, 2004.
• S. Nojiri, S. D. Odintsov, ‘‘Modified f(R) gravity consistent with realistic cosmology: from matter dominated epoch to dark energy Universe,’’ Phys. Rev. D 74, 086005, 2006.
• S. Nojiri, S. D. Odintsov, ‘‘Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models,’’ Phys. Rept. 505, 59, 2011.
• S. Nojiri, S. D. Odintsov, ‘‘Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration,’’ Phys. Rev. D 68, 123512, 2003, arXiv:hep-th/0307288.
• S. Nojiri, S. D. Odintsov, ‘‘Introduction to modified gravity and gravitational alternative for dark energy,’’ Int. J. Geom. Meth. Mod. Phys. 4, 115, 2007.
• S. Capozziello, M. D. Laurentis, ‘‘Introduction to modified gravity and gravitational alternative for dark energy,’’ Phys. Rept. 509, 167, 2011, arXiv:1108.6266 [gr-qc].
• Y. F. Cai, S. Capozziello, M. D. Laurentis, E.N. Saridakis, ‘‘f (T) teleparallel gravity and cosmology,’’ Rept. Prog. Phys. 79, 106901, 2016, arXiv:1511.07586 [gr-qc].
• S. Nojiri, S. D. Odintsov, ‘‘Modified Gauss–Bonnet theory as gravitational alternative for dark energy,’’ Phys. Lett. B 631, 1, 2005, arXiv:hep-th/0508049.
• S. Nojiri, S. D. Odintsov, Petr V. Tretyakov. ‘‘From inflation to dark energy in the non-minimal modified gravity,’’ Progress of Theoretical Physics Supplement 172, 81-89, 2008,
• C. Deffayet, G. Esposito-Farese, A. Vikman, ‘‘Covariant galileon,’’ Phys. Rev. D 79, 084003, 2009, arXiv:0901.1314 [hep-th].
• G. Leon, E.N. Saridakis, ‘‘Dynamical analysis of generalized Galileon cosmology,’’ JCAP 1303, 025, 2013, arXiv:1211.3088 [astro-ph.CO].
• P. D. Mannheim, D. Kazanas, ‘‘Exact vacuum solution to conformal Weyl gravity and galactic rotation curves,’’ Astrophys. J. 342, 635, 1989.
• E. E. Flanagan, ‘‘Fourth order Weyl gravity,’’ Phys. Rev. D 74, 023002, 2006, arXiv:astro-ph/0605504.
• T. Padmanabhan, ‘‘Gravity and the thermodynamics of horizons,’’ Phys. Rep. 406, 49, 2005.
• E. Verlinde, ‘‘On the origin of gravity and the laws of Newton,’’ J. High Energy Phys. 1104, 029, 2011.
• J. M. Bardeen, B. Carter, S. Hawking, ‘‘The four laws of black hole mechanics,’’ Commun. Math. Phys. 31, 161, 1973.
• J. D. Bekenstein, ‘’Black holes and entropy,’’ Phys. Rev. D 7, 2333, 1973.
• J. B. Hartle, S. W. Hawking. ‘‘Path-integral derivation of black-hole radiance,’’ Physical Review D 13.8, 2188, 1976.
• L. Buoninfante, G. G. Luciano, L. Petruzziello, F. Scardigli, ‘‘Bekenstein bound and uncertainty relations,’’ Phys. Lett. B 824, 136818, 2022.
• G. T. Hooft, ‘‘Dimensional reduction in quantum gravity,’’ arXiv preprint gr-qc/9310026, 1993.
• L. Susskind, ‘‘The world as a hologram,’’ J. Math. Phys. 36, 6377, 1995.
• G. Wilk, Z. Wlodarczyk, ‘‘Interpretation of the none-xtensivity parameter q in some applications of Tsallis statistics and Lévy distributions,’’ Physical Review Letters 84.13, 2770, 2000.
• M. L. Lyra, C. Tsallis, ‘‘Non-extensivity and multifractality in low-dimensional dissipative systems,’’ Phys. Rev. Lett. 80, 53, 1998.
• C. Tsallis, R. S. Mendes, A. R. Plastino, ‘‘The role of constraints within generalized non-extensive statistics,’’ Physica A: Statistical Mechanics and its Applications 261.3-4, 534-554, 1998.
• C. Tsallis, L. J. L. Cirto, ‘‘Black hole thermodynamical entropy,’’ Eur. Phys. J. C 73, 2487, 2013.
• Y. Akrami et al., Planck. ‘‘Planck 2018 results-X. Constraints on inflation,’’ Astron. Astrophys. 641, A10, 2020.
• A. Sheykhi, ‘‘Modified Friedmann equations from Tsallis entropy,’’ Physics Letters B 785, 118, 2018.
• K. Huang, Statistical Mechanics, John Wiley & Sons, 2008.
• T. Padmanabhan, ‘‘Thermodynamical aspects of gravity: new insights,’’ Reports on Progress in Physics 73.4, 046901, 2010.
• M. Akbar, R. G. Cai, ‘‘Thermodynamic behavior of Friedmann equation at apparent horizon of FRW Universe,’’ Phys. Rev. D 75, 08400, 2007, arXiv:0609128 [hep-th].
• A. I. Keskin, ‘’The inflationary era of the universe via Tsallis cosmology,’’ Int. J. Geom. Methods Mod. Phys. 19, 2250005, 2022.
• R. Brandenberger, P. Peter, ‘‘Bouncing cosmologies: progress and problems,’’ Found. Phys. 47, 797, 2017, https:// doi.org/10.1007/s10701-016-0057-0.
• L. Randall, M. Soljacic, ‘‘Supernatural inflation: Inflation from supersymmetry with no (very) small parameters,’’ A.H. Guth, Nucl. Phys. B 472, 377, 1996.
• V. Barger, H. S. Lee, D. Marfatia, ‘‘WMAP and inflation,’’ Phys. Lett. B 565, 33, 2003.
• J. D. Barrow, ‘‘Graduated inflationary universes,’’ Physics Letters B 235, 40, 1990.
• J. D. Barrow, A. R. Liddle, ‘‘Perturbation spectra from intermediate inflation,’’ Physical Review D 47, R5219, 1993.
• J. D. Barrow, A. R. Liddle, C. Pahud, ‘‘Intermediate inflation in light of the three-year WMAP observations,’’ Physical Review D 74, 127305, 2006.
• J. D. Barrow, N. J. Nunes, ‘‘Dynamics of “logamediate” inflation,’’ Physical Review D 76, 043501, 2007.
• P. A. R. Ade, et al., ‘‘Planck 2015 results-XX. Constraints on inflation,’’ Astronomy & Astrophysics 594, A20, 2016.