Abstract
Recently, Verlinde proposed that gravity is an emergent phenomenon which originates from an entropic force. In this work, we extend Verlinde’s proposal to accommodate generalized uncertainty principles (GUP), which are suggested by some approaches to quantum gravity such as string theory, black hole physics and doubly special relativity (DSR). Using Verlinde’s proposal and two known models of GUPs, we obtain modifications to Newton’s law of gravitation as well as the Friedmann equation. Our modification to the Friedmann equation includes higher powers of the Hubble parameter which is used to obtain a corresponding Raychaudhuri equation. Solving this equation, we obtain a leading Planck-scale correction to Friedmann-Robertson-Walker (FRW) solutions for the p = ωp equation of state.
[1] D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 216, 41 (1989) http://dx.doi.org/10.1016/0370-2693(89)91366-X10.1016/0370-2693(89)91366-XSearch in Google Scholar
[2] M. Maggiore, Phys. Lett. B 304, 65 (1993) http://dx.doi.org/10.1016/0370-2693(93)91401-810.1016/0370-2693(93)91401-8Search in Google Scholar
[3] M. Maggiore, Phys. Rev. D 49, 5182 (1994) http://dx.doi.org/10.1103/PhysRevD.49.518210.1103/PhysRevD.49.5182Search in Google Scholar
[4] M. Maggiore, Phys. Lett. B 319, 83 (1993) http://dx.doi.org/10.1016/0370-2693(93)90785-G10.1016/0370-2693(93)90785-GSearch in Google Scholar
[5] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995) http://dx.doi.org/10.1142/S0217751X9500008510.1142/S0217751X95000085Search in Google Scholar
[6] S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer, H. Stoecker, Phys. Lett. B 575, 85 (2003) http://dx.doi.org/10.1016/j.physletb.2003.09.04010.1016/j.physletb.2003.09.040Search in Google Scholar
[7] C. Bambi, F. R. Urban, Class. Quant. Grav. 25, 095006 (2008) http://dx.doi.org/10.1088/0264-9381/25/9/09500610.1088/0264-9381/25/9/095006Search in Google Scholar
[8] F. Scardigli, Phys. Lett. B 452, 39 (1999) http://dx.doi.org/10.1016/S0370-2693(99)00167-710.1016/S0370-2693(99)00167-7Search in Google Scholar
[9] A. Kempf, G. Mangano, R. B. Mann, Phys. Rev. D52, 1108 (1995) 10.1103/PhysRevD.52.1108Search in Google Scholar PubMed
[10] A. Kempf, J. Phys. A 30, 2093 (1997) http://dx.doi.org/10.1088/0305-4470/30/6/03010.1088/0305-4470/30/6/030Search in Google Scholar
[11] F. Brau, J. Phys. A 32, 7691 (1999) http://dx.doi.org/10.1088/0305-4470/32/44/30810.1088/0305-4470/32/44/308Search in Google Scholar
[12] A. F. Ali, S. Das, E. C. Vagenas, Phys. Lett. B 678, 497 (2009) http://dx.doi.org/10.1016/j.physletb.2009.06.06110.1016/j.physletb.2009.06.061Search in Google Scholar
[13] S. Das, E. C. Vagenas, A. F. Ali, Phys. Lett. B 690, 407 (2010) http://dx.doi.org/10.1016/j.physletb.2010.05.05210.1016/j.physletb.2010.05.052Search in Google Scholar
[14] G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002) http://dx.doi.org/10.1142/S021827180200133010.1142/S0218271802001330Search in Google Scholar
[15] G. Amelino-Camelia, Phys. Lett. B 510, 255 (2001) http://dx.doi.org/10.1016/S0370-2693(01)00506-810.1016/S0370-2693(01)00506-8Search in Google Scholar
[16] J. Kowalski-Glikman, S. Nowak, Phys. Lett. B 539, 126 (2002) http://dx.doi.org/10.1016/S0370-2693(02)02063-410.1016/S0370-2693(02)02063-4Search in Google Scholar
[17] J. Magueijo, L. Smolin, Phys. Rev. D 67, 044017 (2003) http://dx.doi.org/10.1103/PhysRevD.67.04401710.1103/PhysRevD.67.044017Search in Google Scholar
[18] J. Magueijo, L. Smolin, Class. Quant. Grav. 21, 1725 (2004) http://dx.doi.org/10.1088/0264-9381/21/7/00110.1088/0264-9381/21/7/001Search in Google Scholar
[19] J. L. Cortes, J. Gamboa, Phys. Rev. D71, 065015 (2005) 10.1103/PhysRevD.71.065015Search in Google Scholar
[20] S. Das, E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008) http://dx.doi.org/10.1103/PhysRevLett.101.22130110.1103/PhysRevLett.101.221301Search in Google Scholar PubMed
[21] S. Das, E. C. Vagenas, Can. J. Phys. 87, 233 (2009) http://dx.doi.org/10.1139/P08-10510.1139/P08-105Search in Google Scholar
[22] A. F. Ali, S. Das, E. C. Vagenas, Phys. Rev. D 84, 044013 (2011) http://dx.doi.org/10.1103/PhysRevD.84.04401310.1103/PhysRevD.84.044013Search in Google Scholar
[23] W. Chemissany, S. Das, A. F. Ali, E. C. Vagenas, JCAP 1112, 017 (2011) http://dx.doi.org/10.1088/1475-7516/2011/12/01710.1088/1475-7516/2011/12/017Search in Google Scholar
[24] M. Sprenger, M. Bleicher, P. Nicolini, Class. Quant. Grav. 28, 235019 (2011) http://dx.doi.org/10.1088/0264-9381/28/23/23501910.1088/0264-9381/28/23/235019Search in Google Scholar
[25] A. F. Ali, JHEP 1209, 067 (2012) http://dx.doi.org/10.1007/JHEP09(2012)06710.1007/JHEP09(2012)067Search in Google Scholar
[26] A. F. Ali, H. Nafie, M. Shalaby, Europhys. Lett. 100, 20004 (2012) http://dx.doi.org/10.1209/0295-5075/100/2000410.1209/0295-5075/100/20004Search in Google Scholar
[27] A. Tawfik, H. Magdy, A. F. Ali, Gen. Rel. Grav. 45, 1227 (2013) http://dx.doi.org/10.1007/s10714-013-1522-010.1007/s10714-013-1522-0Search in Google Scholar
[28] A. Tawfik, H. Magdy, A. F. Ali, arXiv:1205.5998 [physics.gen-ph] Search in Google Scholar
[29] A. F. Ali, Class. Quant. Grav. 28, 065013 (2011) http://dx.doi.org/10.1088/0264-9381/28/6/06501310.1088/0264-9381/28/6/065013Search in Google Scholar
[30] B. Majumder, S. Sen, Phys. Lett. B 717, 291 (2012) http://dx.doi.org/10.1016/j.physletb.2012.09.03510.1016/j.physletb.2012.09.035Search in Google Scholar
[31] K. Nozari, S. Saghafi, JHEP 1211, 005 (2012) http://dx.doi.org/10.1007/JHEP11(2012)00510.1007/JHEP11(2012)005Search in Google Scholar
[32] B. Majumder, Phys. Lett. B 709, 133 (2012) http://dx.doi.org/10.1016/j.physletb.2012.02.02210.1016/j.physletb.2012.02.022Search in Google Scholar
[33] K. Nozari, A. Etemadi, Phys. Rev. D 85, 104029 (2012) http://dx.doi.org/10.1103/PhysRevD.85.10402910.1103/PhysRevD.85.104029Search in Google Scholar
[34] I. Pikovski et al., Nature Phys. 8, 393 (2012) http://dx.doi.org/10.1038/nphys226210.1038/nphys2262Search in Google Scholar
[35] F. Marin et al., Nature Phys. 9, 71 (2013) http://dx.doi.org/10.1038/nphys250310.1038/nphys2503Search in Google Scholar
[36] J. D. Bekenstein, Phys. Rev. D 86, 124040 (2012) http://dx.doi.org/10.1103/PhysRevD.86.12404010.1103/PhysRevD.86.124040Search in Google Scholar
[37] J. D. Bekenstein, arXiv:1301.4322 [gr-qc] Search in Google Scholar
[38] E. P. Verlinde, JHEP 1104, 029 (2011) http://dx.doi.org/10.1007/JHEP04(2011)02910.1007/JHEP04(2011)029Search in Google Scholar
[39] A. F. Ali, A. Tawfik, Adv. High Energy Phys. 2013, 126528 (2013) Search in Google Scholar
[40] J. D. Bekenstein, Phys. Rev. D 7, 2333 (1973) http://dx.doi.org/10.1103/PhysRevD.7.233310.1103/PhysRevD.7.2333Search in Google Scholar
[41] J. M. Bardeen, B. Carter, S. W. Hawking, Commun. Math. Phys. 31, 161 (1973) http://dx.doi.org/10.1007/BF0164574210.1007/BF01645742Search in Google Scholar
[42] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)] http://dx.doi.org/10.1007/BF0234502010.1007/BF02345020Search in Google Scholar
[43] R. -G. Cai, L. -M. Cao, N. Ohta, Phys. Rev. D 81, 061501 (2010) http://dx.doi.org/10.1103/PhysRevD.81.06150110.1103/PhysRevD.81.061501Search in Google Scholar
[44] G. t’Hooft, gr-qc/9310026 Search in Google Scholar
[45] P. Nicolini, Phys. Rev. D 82, 044030 (2010) http://dx.doi.org/10.1103/PhysRevD.82.04403010.1103/PhysRevD.82.044030Search in Google Scholar
[46] C. Bastos, O. Bertolami, N. C. Dias, J. N. Prata, Class. Quant. Grav. 28, 125007 (2011) http://dx.doi.org/10.1088/0264-9381/28/12/12500710.1088/0264-9381/28/12/125007Search in Google Scholar
[47] K. Nozari, S. Akhshabi, Phys. Lett. B 700, 91 (2011) http://dx.doi.org/10.1016/j.physletb.2011.04.05610.1016/j.physletb.2011.04.056Search in Google Scholar
[48] S. H. Mehdipour, A. Keshavarz, A. Keshavarz, Europhys. Lett. 91, 10002 (2012) http://dx.doi.org/10.1209/0295-5075/98/1000210.1209/0295-5075/98/10002Search in Google Scholar
[49] A. Sheykhi, Phys. Rev. D 81, 104011 (2010) http://dx.doi.org/10.1103/PhysRevD.81.10401110.1103/PhysRevD.81.104011Search in Google Scholar
[50] S. H. Hendi, A. Sheykhi, Phys. Rev. D 83, 084012 (2011) http://dx.doi.org/10.1103/PhysRevD.83.08401210.1103/PhysRevD.83.084012Search in Google Scholar
[51] B. Majumder, Adv. High Energy Phys. 2013, 296836 (2013) 10.1155/2013/296836Search in Google Scholar
[52] M. Aghaei Abchouyeh, B. Mirza, Z. Sherkatghanad, Gen. Rel. Grav. 46, 1617 (2014) http://dx.doi.org/10.1007/s10714-013-1617-710.1007/s10714-013-1617-7Search in Google Scholar
[53] A. Sheykhi, M. H. Dehghani, S. E. Hosseini, Phys. Lett. B 726, 23 (2013) http://dx.doi.org/10.1016/j.physletb.2013.08.03510.1016/j.physletb.2013.08.035Search in Google Scholar
[54] N. Komatsu, S. Kimura, Phys. Rev. D 88, 083534 (2013) http://dx.doi.org/10.1103/PhysRevD.88.08353410.1103/PhysRevD.88.083534Search in Google Scholar
[55] M. Eune, W. Kim, Phys. Rev. D 88, 067303 (2013) http://dx.doi.org/10.1103/PhysRevD.88.06730310.1103/PhysRevD.88.067303Search in Google Scholar
[56] M. Cavaglia, S. Das, R. Maartens, Class. Quant. Grav. 20, L205 (2003) http://dx.doi.org/10.1088/0264-9381/20/15/10110.1088/0264-9381/20/15/101Search in Google Scholar
[57] M. Cavaglia, S. Das, Class. Quant. Grav. 21, 4511 (2004) http://dx.doi.org/10.1088/0264-9381/21/19/00110.1088/0264-9381/21/19/001Search in Google Scholar
[58] A. J. M. Medved, E. C. Vagenas, Phys. Rev. D70, 124021 (2004) 10.1103/PhysRevD.70.124021Search in Google Scholar
[59] F. Scardigli, Nuovo Cim. B 110, 1029 (1995) http://dx.doi.org/10.1007/BF0272615210.1007/BF02726152Search in Google Scholar
[60] R. J. Adler, P. Chen, D. I. Santiago, Gen. Rel. Grav. 33, 2101 (2001) http://dx.doi.org/10.1023/A:101528143041110.1023/A:1015281430411Search in Google Scholar
[61] A. J. M. Medved, E. C. Vagenas, Phys. Rev. D 70, 124021 (2004) http://dx.doi.org/10.1103/PhysRevD.70.12402110.1103/PhysRevD.70.124021Search in Google Scholar
[62] B. Majumder, Phys. Lett. B 703, 402 (2011) http://dx.doi.org/10.1016/j.physletb.2011.08.02610.1016/j.physletb.2011.08.026Search in Google Scholar
[63] G. Amelino-Camelia, M. Arzano, A. Procaccini, Phys. Rev. D 70, 107501 (2004) http://dx.doi.org/10.1103/PhysRevD.70.10750110.1103/PhysRevD.70.107501Search in Google Scholar
[64] E. M. Lifshitz, L. P. Pitaevskii, V. B. Berestetskii, Landau-Lifshitz Course of Theoretical Physics, Volume 4: Quantum Electrodynamics, (Reed Educational and Professional Publishing, Butterworth-Heinemann in Oxford, UK, 1982) Search in Google Scholar
[65] D. Christodoulou, Phys. Lett. 25, 1596 (1970) http://dx.doi.org/10.1103/PhysRevLett.25.159610.1103/PhysRevLett.25.1596Search in Google Scholar
[66] D. Christodoulou, R. Ruffini, Phys. Rev. D 4, 3352 (1971) http://dx.doi.org/10.1103/PhysRevD.4.355210.1103/PhysRevD.4.3552Search in Google Scholar
[67] C. Adami, quant-ph/0405005 Search in Google Scholar
[68] K. A. Bronnikov, S. A. Kononogov, V. N. Melnikov, Gen. Rel. Grav. 38, 1215 (2006) http://dx.doi.org/10.1007/s10714-006-0300-710.1007/s10714-006-0300-7Search in Google Scholar
[69] R. -G. Cai, S. P. Kim, JHEP 0502, 050 (2005) http://dx.doi.org/10.1088/1126-6708/2005/02/05010.1088/1126-6708/2005/02/050Search in Google Scholar
[70] A. Awad, Phys. Rev. D 87, 103001 (2013) http://dx.doi.org/10.1103/PhysRevD.87.10300110.1103/PhysRevD.87.103001Search in Google Scholar
[71] L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999) http://dx.doi.org/10.1103/PhysRevLett.83.469010.1103/PhysRevLett.83.4690Search in Google Scholar
[72] P. Callin, F. Ravndal, Phys. Rev. D 70, 104009 (2004) http://dx.doi.org/10.1103/PhysRevD.70.10400910.1103/PhysRevD.70.104009Search in Google Scholar
[73] S. -Q. Yang et al., Phys. Rev. Lett. 108, 081101 (2012) http://dx.doi.org/10.1103/PhysRevLett.108.08110110.1103/PhysRevLett.108.081101Search in Google Scholar PubMed
[74] C. D. Hoyle et al., Phys. Rev. D 70, 042004 (2004) http://dx.doi.org/10.1103/PhysRevD.70.04200410.1103/PhysRevD.70.042004Search in Google Scholar
[75] F. Buisseret, V. Mathieu, B. Silvestre-Brac, Class. Quant. Grav. 24, 855 (2007) http://dx.doi.org/10.1088/0264-9381/24/4/00610.1088/0264-9381/24/4/006Search in Google Scholar
[76] G. L. Murphy, Phys. Rev. D 8, 4231 (1973) http://dx.doi.org/10.1103/PhysRevD.8.423110.1103/PhysRevD.8.4231Search in Google Scholar
[77] P. A. R. Ade et al., arXiv:1303.5082 [astro-ph.CO] Search in Google Scholar
[78] S. del Campo, JCAP 12, 005 (2012) http://dx.doi.org/10.1088/1475-7516/2012/12/00510.1088/1475-7516/2012/12/005Search in Google Scholar
[79] U. H. Danielsson, Phys. Rev. D 66, 023511 (2002) http://dx.doi.org/10.1103/PhysRevD.66.02351110.1103/PhysRevD.66.023511Search in Google Scholar
© 2014 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
