Considering the cosmic fluid as a quasi-static thermodynamic system, the status of the generalized second law of thermodynamics is investigated and the valid range of the equation of state parameter is derived for a few important cosmological models. Our study shows that the satisfaction of the laws of thermodynamics in these cosmological models requires the existence of some kind of energy in our universe with . In other words, the existence of a dark energy component, or equivalently modified gravity theory, is unavoidable if the cosmological model is to approach thermal equilibrium in late times. 1. Introduction Thermodynamical nature of Einstein’s theory of general relativity was first disclosed by Jacobson [1] who showed that the hyperbolic second order partial differential field equations of gravity can be derived by applying the first law of thermodynamics on any local Rindler horizon. Generalization of this method to gravity, by introducing the entropy generation term due to nonequilibrium nature of spacetime, was investigated in [2]. More attempts to reveal the connection between thermodynamics and various theories of gravity can be found in [3–8]. An elegant example is the derivation of the Friedmann equations as a consequence of the validity of the first law of thermodynamics on the apparent horizon of the Friedmann-Robertson-Walker (FRW) universe [9]. Recently, an entropic origin for gravity was proposed by Verlinde [10]. He argued that the laws of gravity are not fundamental and in particular they emerge as an entropic force caused by the changes in the information associated with the positions of material bodies. Verlinde’s derivation of Newton’s law of gravitation at the very least offers a strong analogy with the well-understood statistical approach. Therefore, this derivation opens a new window to understanding gravity from first principles. The entropic approach to gravity has arisen a lot of enthusiasm, recently (see, e.g., [11–34] and references therein). The studies have also been generalized to higher order gravity theories such as Gauss-Bonnet and Lovelock gravity [35]. All these attempts indicate a hope to achieve a deeper connection between gravity and thermodynamics. From the second law of thermodynamics, we know that every closed system moves towards its maximum entropy state which is an equilibrium state. This leads to the conclusion that the second derivative of the entropy should be negative [36]. The assumption that the second derivative of entropy is negative comes from the fact that while the entropy is increasing as the
References
[1]
T. Jacobson, “Thermodynamics of spacetime: the Einstein equation of state,” Physical Review Letters, vol. 75, no. 7, pp. 1260–1263, 1995.
[2]
C. Eling, R. Guedens, and T. Jacobson, “Nonequilibrium thermodynamics of spacetime,” Physical Review Letters, vol. 96, no. 12, Article ID 121301, 4 pages, 2006.
[3]
T. Padmanabhan, “Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes,” Classical and Quantum Gravity, vol. 19, no. 21, pp. 5387–5408, 2002.
[4]
T. Padmanabhan, “Gravity and the thermodynamics of horizons,” Physics Reports, vol. 406, no. 2, pp. 49–125, 2005.
[5]
T. Padmanabhan, “Gravity: a new holographic perspective,” International Journal of Modern Physics D, vol. 15, no. 10, p. 1659, 2006.
[6]
M. Akbar and R. G. Cai, “Friedmann equations of FRW universe in scalar-tensor gravity, gravity and first law of thermodynamics,” Physics Letters B, vol. 635, no. 1, pp. 7–10, 2006.
[7]
R. G. Cai and L. M. Cao, “Unified first law and the thermodynamics of the apparent horizon in the FRW universe,” Physical Review D, vol. 75, no. 6, Article ID 064008, 11 pages, 2007.
[8]
M. Akbar and R. G. Cai, “Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe,” Physical Review D, vol. 75, no. 8, Article ID 084003, 9 pages, 2007.
[9]
R. G. Cai and S. P. Kim, “First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe,” Journal of High Energy Physics, vol. 2005, no. 2, article 50, 2005.
[10]
E. Verlinde, “On the origin of gravity and the laws of Newton,” Journal of High Energy Physics, vol. 2011, no. 4, article 29, 2011.
[11]
R. G. Cai, L. M. Cao, and N. Ohta, “Friedmann equations from entropic force,” Physical Review D, vol. 81, no. 6, Article ID 061501, 2010.
[12]
R. G. Cai, L. M. Cao, and N. Ohta, “Notes on entropy force in general spherically symmetric spacetimes,” Physical Review D, vol. 81, no. 8, Article ID 084012, 9 pages, 2010.
[13]
Y. S. Myung and Y. W. Kim, “Entropic force and entanglement system,” Physical Review D, vol. 81, no. 10, Article ID 105012, 7 pages, 2010.
[14]
R. Banerjee and B. R. Majhi, “Statistical origin of gravity,” Physical Review D, vol. 81, no. 12, Article ID 124006, 5 pages, 2010.
[15]
S. W. Wei, Y. X. Liu, and Y. Q. Wang, “A note on Friedmann equation of FRW universe in deformed Ho？ava-Lifshitz gravity from entropic force,” Communications in Theoretical Physics, vol. 56, no. 3, article 455, 2011.
[16]
Y. X. Liu, Y. Q. Wang, and S. W. Wei, “A note on the temperature and energy of four-dimensional black holes from an entropic force,” Classical and Quantum Gravity, vol. 27, no. 18, Article ID 185002, 2010.
[17]
R. A. Konoplya, “Entropic force, holography and thermodynamics for static space-times,” European Physical Journal C, vol. 69, no. 3-4, pp. 555–562, 2010.
[18]
H. Wei, “Cosmological constraints on the modified entropic force model,” Physics Letters B, vol. 692, no. 3, pp. 167–175, 2010.
[19]
C. M. Ho, D. Minic, and Y. J. Ng, “Cold dark matter with MOND scaling,” Physics Letters B, vol. 693, no. 5, pp. 567–570, 2010.
[20]
V. V. Kiselev and S. A. Timofeev, “The holographic screen at low temperatures,” Modern Physics Letters A, vol. 26, no. 2, p. 109, 2011.
[21]
W. Gu, M. Li, and R. X. Miao, “A new entropic force scenario and holographic thermodynamics,” Science China Physics, Mechanics and Astronomy, vol. 54, no. 11, pp. 1915–1924, 2011.
[22]
R. X. Miao, J. Meng, and M. Li, “ gravity and Maxwell equations from the holographic principle,” Science China Physics, Mechanics and Astronomy, vol. 55, no. 3, pp. 375–380, 2012.
[23]
A. Sheykhi, “Entropic corrections to Friedmann equations,” Physical Review D, vol. 81, no. 10, Article ID 104011, 4 pages, 2010.
[24]
Y. Ling and J. P. Wu, “A note on entropic force and brane cosmology,” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 8, article 17, 2010.
[25]
L. Modesto and A. Randono, “Entropic corrections to Newton's law,” http://arxiv.org/abs/1003.1998.
[26]
L. Smolin, “Newtonian gravity in loop quantum gravity,” http://arxiv.org/abs/1001.3668.
[27]
X. Li and Z. Chang, “Debye Entropic Force and Modified Newtonian Dynamics,” Communications in Theoretical Physics, vol. 55, no. 4, article 733, 2011.
[28]
Y. F. Cai, J. Liu, and H. Li, “Entropic cosmology: a unified model of inflation and late-time acceleration,” Physics Letters B, vol. 690, no. 3, pp. 213–219, 2010.
[29]
M. Li and Y. Wang, “Quantum UV/IR relations and holographic dark energy from entropic force,” Physics Letters B, vol. 687, no. 2-3, pp. 243–247, 2010.
[30]
S. H. Hendi and A. Sheykhi, “Entropic corrections to Einstein equations,” Physical Review D, vol. 83, no. 8, Article ID 084012, 5 pages, 2011.
[31]
A. Sheykhi and S. H. Hendi, “Power-law entropic corrections to Newton’s law and Friedmann equations,” Physical Review D, vol. 84, no. 4, Article ID 044023, 8 pages, 2011.
[32]
S. H. Hendi and A. Sheykhi, “Entropic Corrections to Coulomb's Law,” International Journal of Theoretical Physics, vol. 51, no. 4, pp. 1125–1136, 2012.
[33]
A. Sheykhi and Z. Teimoori, “Modified Friedmann equations from Debye entropic gravity,” General Relativity and Gravitation, vol. 44, no. 5, pp. 1129–1141, 2012.
[34]
A. Sheykhi, “Modified Friedmann equations on the brane from entropic force,” International Journal of Theoretical Physics, vol. 51, no. 1, pp. 185–192, 2012.
[35]
A. Sheykhi, H. Moradpour, and N. Riazi, “Lovelock gravity from entropic force,” General Relativity and Gravitation, vol. 45, no. 5, pp. 1033–1049, 2013.
[36]
H. B. Callen, Thermodynamics and Introduction to Thermostatics, John Wiley & Sons, New York, NY, USA, 1985.
[37]
H. B. Callen, Thermodynamics, John Wiley & Sons, New York, NY, USA, 1960.
[38]
N. Radicella and D. Pavón, “Thermodynamical analysis on a braneworld scenario with curvature corrections,” Physics Letters B, vol. 704, no. 4, pp. 260–264, 2011.
[39]
N. Radicella and D. Pavón , “A thermodynamic motivation for dark energy,” General Relativity and Gravitation, vol. 44, no. 3, pp. 685–702, 2012.
[40]
B. Wang, E. Abdalla, and T. Osada, “Behavior of shell effects with the excitation energy in atomic nuclei,” Physical Review Letters, vol. 85, no. 1, pp. 26–29, 2000.
[41]
S. Perlmutter, G. Aldering, G. Goldhaber, et al., “Measurements of ？ and Λ from 42 high-redshift supernovae,” The Astrophysical Journal, vol. 517, no. 2, article 565, 1999.
[42]
A. G. Riess, A. V. Filippenko, P. Challis, et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” The Astronomical Journal, vol. 116, no. 3, article 1009, 1998.
[43]
M. Jones, H. Robert, and J. Lambourne, An Introduction to Galaxies and Cosmology, Cambridge University Press, New York, NY, USA, 2004.
[44]
N. Ohta, “Dark energy and QCD ghost,” Physics Letters B, vol. 695, no. 1–4, pp. 41–44, 2011.
[45]
R. G. Cai, Z. L. Tuo, and H. B. Zhang, “Notes on ghost dark energy,” Physical Review D, vol. 84, no. 12, Article ID 123501, 10 pages, 2011.
[46]
R. G. Cai, Z. L. Tuo, Y. B. Wu, and Y. Y. Zhao, “More on QCD ghost dark energy,” Physical Review D, vol. 86, no. 2, Article ID 023511, 8 pages, 2012.
[47]
A. Sheykhi and A. Bagheri, “Quintessence ghost dark energy model,” Europhysics Letters, vol. 95, no. 3, Article ID 39001, 2011.
[48]
A. Sheykhi and M. S. Movahed, “Interacting ghost dark energy in non-flat universe,” General Relativity and Gravitation, vol. 44, no. 2, pp. 449–465, 2012.
[49]
E. Ebrahimi and A. Sheykhi, “Interacting ghost dark energy in Brans-Dicke theory,” Physics Letters B, vol. 705, no. 1, pp. 19–25, 2011.
[50]
E. Ebrahimi and A. Sheykhi, “Instability of QCD ghost dark energy model,” International Journal of Modern Physics D, vol. 20, no. 12, p. 2369, 2011.
[51]
A. Sheykhi, M. S. Movahed, and E. Ebrahimi, “Tachyon reconstruction of ghost dark energy,” Astrophysics and Space Science, vol. 339, no. 1, pp. 93–99, 2012.
[52]
F. R. Urban and A. R. Zhitnitsky, “The cosmological constant from the QCD Veneziano ghost,” Physics Letters B, vol. 688, no. 1, pp. 9–12, 2010.
[53]
K. Kawarabayashi and N. Ohta, “The η problem in the large-N limit: effective Lagrangian approach,” Nuclear Physics B, vol. 175, no. 3, pp. 477–492, 1980.
[54]
J. D. Bekenstein, “Black Holes and Entropy,” Physical Review D, vol. 7, no. 8, pp. 2333–2346, 1973.
[55]
G. Izquierdo and D. Pavón, “Dark energy and the generalized second law,” Physics Letters B, vol. 633, no. 4-5, pp. 420–426, 2006.
[56]
M. O. Calv？o, J. A. S. Lima, and I. Waga, “On the thermodynamics of matter creation in cosmology,” Physics Letters A, vol. 162, no. 3, pp. 223–226, 1992.
[57]
Y. Shtanov and V. Sahni, “Bouncing braneworlds,” Physics Letters B, vol. 557, no. 1-2, pp. 1–6, 2003.
[58]
A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum nature of the big bang: an analytical and numerical investigation,” Physical Review D, vol. 73, no. 12, Article ID 124038, 33 pages, 2006.
[59]
X. Zhang, “Can black holes be torn up by phantom dark energy in cyclic cosmology?” The European Physical Journal C, vol. 60, no. 4, pp. 661–667, 2009.
[60]
P. Horava, “Quantum gravity at a Lifshitz point,” Physical Review D, vol. 79, no. 8, Article ID 084008, 15 pages, 2009.
[61]
P. Horava, “Membranes at quantum criticality,” Journal of High Energy Physics, vol. 2009, no. 3, article 20, 2009.
[62]
P. Horava, “Spectral dimension of the universe in quantum gravity at a Lifshitz point,” Physical Review Letters, vol. 102, no. 16, Article ID 161301, 4 pages, 2009.
[63]
Y. S. Myung, “Thermodynamics of black holes in the deformed Ho？ava-Lifshitz gravity,” Physics Letters B, vol. 678, no. 1, pp. 127–130, 2009.
[64]
R. B. Mann, “Lifshitz topological black holes,” Journal of High Energy Physics, vol. 2009, no. 6, article 75, 2009.
[65]
R. G. Cai, L. M. Cao, and N. Ohta, “Thermodynamics of black holes in Ho？ava-Lifshitz gravity,” Physics Letters B, vol. 679, no. 5, pp. 504–509, 2009.
[66]
Y. S. Myung, “Generalized uncertainty principle, quantum gravity and Ho？ava-Lifshitz gravity,” Physics Letters B, vol. 681, no. 1, pp. 81–84, 2009.
[67]
R. G. Cai and N. Ohta, “Horizon thermodynamics and gravitational field equations in Ho？ava-Lifshitz gravity,” Physical Review D, vol. 81, no. 8, Article ID 084061, 8 pages, 2010.
[68]
M. Jamil, E. N. Saridakis, and M. R. Setare, “The generalized second law of thermodynamics in Ho？ava-Lifshitz cosmology,” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 11, article 32, 2010.
[69]
A. Sheykhi, “Thermodynamics of the apparent horizon in infrared modified Ho？ava-Lifshitz gravity,” Physical Review D, vol. 87, no. 2, Article ID 024022, 7 pages, 2013.
[70]
A. Kehagias and K. Sfetsos, “The black hole and FRW geometries of non-relativistic gravity,” Physics Letters B, vol. 678, no. 1, pp. 123–126, 2009.
[71]
A. Castillo and A. Larranaga, “Entropy for black holes in the deformed Horava-Lifshitz gravity,” Electronic Journal of Theoretical Physics, vol. 8, pp. 1–10, 2011.
[72]
Y. S. Myung, “Entropy of black holes in the deformed Ho？ava-Lifshitz gravity,” Physics Letters B, vol. 684, no. 2-3, pp. 158–161, 2010.
[73]
G. R. Dvali, G. Gabadadze, and M. Porrati, “4D gravity on a brane in 5D Minkowski space,” Physics Letters B, vol. 485, no. 1–3, pp. 208–214, 2000.
[74]
A. Sheykhi, M. H. Dehghani, and S. E. Hosseini, “Emergence of spacetime dynamics in entropy corrected and braneworld models,” Journal of Cosmology and Astroparticle Physics, vol. 2013, no. 4, article 38, 2013.
[75]
C. Deffayet, “Cosmology on a brane in Minkowski bulk,” Physics Letters B, vol. 502, no. 1–4, pp. 199–208, 2001.
[76]
C. Deffayet and G. Dvali, “Accelerated universe from gravity leaking to extra dimensions,” Physical Review D, vol. 65, no. 4, Article ID 044023, 9 pages, 2002.
[77]
L. Lombriser, W. Hu, W. Fang, and U. Seljak, “Cosmological constraints on DGP braneworld gravity with brane tension,” Physical Review D, vol. 80, no. 6, Article ID 063536, 12 pages, 2009.
[78]
A. Sheykhi, B. Wang, and R. G. Cai, “Thermodynamical properties of apparent horizon in warped DGP braneworld,” Nuclear Physics B, vol. 779, no. 1-2, pp. 1–12, 2007.
[79]
M. H. Dehghani, S.H. Hendi, A. Sheykhi, and H. R. Sedehi, “Thermodynamics of rotating black branes in Einstein-Born-Infeld-dilaton gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2007, no. 2, article 20, 2007.
[80]
A. Sheykhi and B. Wang, “Generalized second law of thermodynamics in warped DGP braneworld,” Modern Physics Letters A, vol. 25, no. 14, p. 1199, 2010.
[81]
K. Nozari, N. Behrooz, and A. Sheykhi, “Thermodynamics of viscous dark energy in DGP setup,” International Journal of Theoretical Physics, vol. 52, no. 7, pp. 2351–2362, 2013.
[82]
R. C. Myers and J. Z. Simon, “Black-hole thermodynamics in Lovelock gravity,” Physical Review D, vol. 38, no. 8, pp. 2434–2444, 1988.
[83]
R. G. Cai, “Gauss-Bonnet black holes in AdS spaces,” Physical Review D, vol. 65, no. 8, Article ID 084014, 9 pages, 2002.
[84]
R. G. Cai and Q. Guo, “Gauss-Bonnet black holes in dS spaces,” Physical Review D, vol. 69, no. 10, Article ID 104025, 9 pages, 2004.
[85]
R. G. Cai and K. S. Soh, “Topological black holes in the dimensionally continued gravity,” Physical Review D, vol. 59, no. 4, Article ID 044013, 13 pages, 1999.
[86]
T. Clunan, S. F. Ross, and D. J. Smith, “On Gauss-Bonnet black hole entropy,” Classical and Quantum Gravity, vol. 21, no. 14, article 3447, 2004.
[87]
E. V. Linder, “Uniqueness of current cosmic acceleration,” Physical Review D, vol. 82, no. 6, Article ID 063514, 6 pages, 2010.
[88]
E. V. Linder and T. L. Smith, “Dark before light: testing the cosmic expansion history through the cosmic microwave background,” Journal of Cosmology and Astroparticle Physics, vol. 2011, no. 4, article 1, 2011.
[89]
P. Serra, A. Cooray, D. E. Holz, A. Melchiorri, S. Pandolfi, and D. Sarkar, “No evidence for dark energy dynamics from a global analysis of cosmological data,” Physical Review D, vol. 80, no. 12, Article ID 121302(R), 5 pages, 2009.
