A Cyclic Cosmological Model Based on the f(ρ) Modified Theory of Gravity

Yaoming Shi

University of California, Berkeley, USA.

DOI: 10.4236/jmp.2013.48A004                

Abstract

We consider FLRW cosmological models for perfect fluid (with ρ as the energy density) in the frame work of the f(ρ) modified theory of gravity [V. N. Tunyak, Russ. Phys. J. 21, 1221 (1978); J. R. Ray, L. L. Smalley, Phys. Rev. D. 26, 2615 (1982)]. This theory, with total Lagrangian R-f(ρ), can be considered as a cousin of the F(R) theory of gravity with total Lagrangian F(R)-ρ. We can pick proper function forms f(ρ) to achieve, as the F(R) theory does, the following 4 specific goals, 1) producing a non-singular cosmological model (Ricci scalar and Ricci tensor curvature are bounded); 2) explaining the cosmic early inflation and late acceleration in a unified fashion; 3) passing the solar system tests; 4) unifying the dark matter with dark energy. In addition we also achieve goal number 5) unify the regular matter/energy with dark matter/energy in a seamless fashion. The mathematics is simplified because in the f(ρ) theory the leading terms in Einstein’s equations are linear in second order derivative of metric wrt coordinates but in the F(R) theory the leading terms are linear in fourth order derivative of metric wrt coordinates.

Keywords

Modified Gravity; Dark Energy; Cosmology; Nonsingular; Cosmological Constant

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. A. Starobinsky, Physics Letters B, Vol. 91, 1980, pp. 99-102. doi:10.1016/0370-2693(80)90670-X
[2]	D. Kazanas, The Astrophysical Journal, Vol. 241, 1980, pp. L59-L63. doi:10.1086/183361
[3]	A. H. Guth, Physical Review D, Vol. 23, 1981, pp. 347-356. doi:10.1103/PhysRevD.23.347
[4]	K. Sato, Monthly Notices of the Royal Astronomical Society, Vol. 195, 1981, pp. 467-479.
[5]	G. F. Smoot, C. L. Bennett, A. Kogut, E. L. Wright, J. Aymon, N. W. Boggess, E. S. Cheng, G. de Amici, S. Gulkis, M. G. Hauser, G. Hinshaw, P. D. Jackson, M. Janssen, E. Kaita, T. Kelsall, P. Keegstra, C. Lineweaver, K. Loewenstein, P. Lubin, J. Mather, S. S. Meyer, S. H. Moseley, T. Murdock, L. Rokke, R. F. Silverberg, L. Tenorio, R. Weiss and D. T. Wilkinson, The Astrophysical Journal, Vol. 396, 1992, pp. L1-L5. doi:10.1086/186504
[6]	D. H. Lyth and A. Riotto, Physics Reports, Vol. 314, 1999, pp. 1-146. doi:10.1016/S0370-1573(98)00128-8.
[7]	A. R. Liddle and D. H. Lyth, “Cosmological Inflation and Large-Scale Structure,” Cambridge University Press, Cambridge, New York, 2000. doi:10.1017/CBO9781139175180
[8]	B. A. Bassett, S. Tsujikawa and D. Wands, Reviews of Modern Physics, Vol. 78, 2006, pp. 537-589. doi:10.1103/RevModPhys.78.537
[9]	D. Huterer and M. S. Turner, Physical Review D, Vol. 60, 1999, Article ID: 081301. doi:10.1103/PhysRevD.60.081301
[10]	S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg and W. J. Couch, The Astrophysical Journal, Vol. 517, 1999, pp. 565-586. doi:10.1086/307221
[11]	A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff and J. Tonry, The Astrophysical Journal, Vol. 116, 1998, pp. 1009-1038. doi:10.1086/300499
[12]	A. G. Riess, R. P. Kirshner, B. P. Schmidt, S. Jha, P. Challis, P. M. Garnavich, A. A. Esin, C. Carpenter, R. Grashius, R. E. Schild, P. L. Berlind, J. P. Huchra, C. F. Prosser, E. E. Falco, P. J. Benson, C. Briceno, W. R. Brown, N. Caldwell, I. P. Dell’Antonio, A. V. Filippenko, A. A. Goodman, N. A. Grogin, T. Groner, J. P. Hughes, P. J. Green, R. A. Jansen, J. T. Kleyna, J. X. Luu, L. M. Macri, B. A. McLeod, K. K. McLeod, B. R. McNamara, B. McLean, A. A. E. Milone, J. J. Mohr, D. Moraru, C. Peng, J. Peters, A. H. Prestwich, K. Z. Stanek, A. Szentgyorgyi and P. Zhao, The Astrophysical Journal, Vol. 117, 1999, pp. 707-724. doi:10.1086/300738
[13]	M. Tegmark, et al. (SDSS Collaboration), Physical Review D, Vol. 69, 2004, Article ID: 103501. doi:10.1103/PhysRevD.69.103501
[14]	M. Tegmark, et al. (SDSS Collaboration), Physical Review D, Vol. 74, 2006, Article ID: 123507. doi:10.1103/PhysRevD.74.123507
[15]	D. J. Eisenstein, et al. (SDSS Collaboration), The Astrophysical Journal, Vol. 633, 2005, pp. 560-574. doi:10.1086/466512
[16]	W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol, J. A. Peacock, A. C. Pope and A. S. Szalay, Monthly Notices of the Royal Astronomical Society, Vol. 381, 2007, pp. 1053-1066. doi:10.1111/j.1365-2966.2007.12268.x
[17]	D. N. Spergel, et al. (WMAP Collaboration), The Astrophysical Journal Supplement Series, Vol. 148, 2003, pp. 175-194. doi:10.1086/377226
[18]	D. N. Spergel, et al. (WMAP Collaboration), The Astrophysical Journal Supplement Series, Vol. 170, 2007, pp. 377-408. doi:10.1086/513700
[19]	E. Komatsu, et al. (WMAP Collaboration), The Astrophysical Journal Supplement Series, Vol. 180, 2009, pp. 330-376. doi:10.1088/0067-0049/180/2/330
[20]	V. Sahni, Classical and Quantum Gravity, Vol. 19, 2002, pp. 3435-3448.
[21]	S. E. Deustua, et al., “Cosmological Parameters, Dark Energy and Large Scale Structure,” APS/DPF/DPB Summer Study on the Future of Particle Physics (Snowmass 2001) ed N Graf, eConf C010630 P342, arXiv:astro-ph/0207293v1.
[22]	M. S. Turner and D. Huterer, Journal of the Physical Society of Japan, Vol. 76, 2007, Article ID: 111015. doi:10.1143/JPSJ.76.111015
[23]	T. Padmanabhan, Current Science, Vol. 88, 2005, p. 1057.
[24]	M. Ishak, Foundations of Physics, Vol. 37, 2007, pp. 1470-1498. doi:10.1007/s10701-007-9175-z
[25]	M. Szydlowski, JCAP, Vol. 0709, 2007, p. 007.
[26]	E. V. Linder, Journal of Physics A, Vol. 40, 2007, p. 6697. doi:10.1088/1751-8113/40/25/S14
[27]	C. Clarkson, M. Cortes and B. Bassett, JCAP, Vol. 0708, 2007, p. 011.
[28]	Y.-Q. Gu, “Dynamic Behavior and Topological Structure of the Universe with Dark Energy,” arXiv:0709.2414v5 [gr-qc].
[29]	V. Sahni and A. A. Starobinsky, International Journal of Modern Physics D, Vol. 9, 2000, pp. 373-443. doi:10.1142/S0218271800000542
[30]	S. M. Carroll, Living Reviews in Relativity, Vol. 4, 2001, 1-56. http://relativity.livingreviews.org/Articles/lrr-2001-1
[31]	T. Padmanabhan, Physics Reports, Vol. 380, 2003, pp. 235-320. doi:10.1016/S0370-1573(03)00120-0
[32]	P. J. E. Peebles and B. Ratra, Reviews of Modern Physics, Vol. 75, 2003, pp. 559-606. doi:10.1103/RevModPhys.75.559
[33]	E. J. Copeland, M. Sami and S. Tsujikawa, International Journal of Modern Physics D, Vol. 15, 2006, pp. 1753-1935. doi:10.1142/S021827180600942X
[34]	L. Amendola and S. Tsujikawa, “Dark Energy: Theory and Observations,” Cambridge University Press, Cambridge, New York, 2010. doi:10.1017/CBO9780511750823
[35]	T. P. Sotiriou and V. Faraoni, Reviews of Modern Physics, Vol. 82, 2010, pp. 451-497. doi:10.1103/RevModPhys.82.451
[36]	A. De Felice and S. Shinji Tsujikawa, Living Reviews in Relativity, Vol. 13, 2010, p. 3.
[37]	S. Nojiri and S. D. Odintsov, Physics Reports, Vol. 505, 2011, pp. 59-144.
[38]	S. Nojiri and S. D. Odintsov, Academic Journal, Vol. 1241, 2010, p. 1094.
[39]	S. Capozziello, M. De Laurentis, S. Nojiri and S. D. Odintsov, Physical Review D, Vol. 79, 2009, Article ID: 124007. doi:10.1103/PhysRevD.79.124007
[40]	V. N. Tunyak, Russian Physics Journal, Vol. 21, 1978, pp. 1221-1223.
[41]	J. R. Ray and L. L. Smalley, Physical Review D, Vol. 26, 1982, pp. 2615-2618. doi:10.1103/PhysRevD.26.2615
[42]	M. Planck, Annalen der Physik, Vol. 4, 1901, pp. 553-563. doi:10.1002/andp.19013090310
[43]	L. D. Landau and E. M. Lifshitz, “Statistical Mechanics,” John Wiley, New York, 1972.
[44]	S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,” John Wiley, New York, 1972.
[45]	R. J. Nemiroff and B. Patla, American Journal of Physics, Vol. 76, 2008, pp. 265-276. doi:10.1119/1.2830536
[46]	J. M. Heinzle and P. Sandin, Communications in Mathematical Physics, Vol. 313, 2012, pp. 385-403.
[47]	S. Nojiri and S. D. Odintsov, International Journal of Geometric Methods in Modern Physics, Vol. 4, 2007, pp. 115-146.
[48]	S. Capozziello, V. F. Cardone, E. Elizalde, S. Nojiri and S. D. Odintsov, Physical Review D, Vol. 73, 2006, Article ID: 043512. doi:10.1103/PhysRevD.73.043512
[49]	S. Nojiri and S. D. Odintsov, Physical Review D, Vol. 72, 2005, Article ID: 023003. doi:10.1103/PhysRevD.72.023003
[50]	K. Lake, Physical Review D, Vol. 74, 2006, Article ID: 123505. doi:10.1103/PhysRevD.74.123505
[51]	M. Szydlowski, A. Kurek and A. Krawiec, Physics Letters B, Vol. 642, 2006, pp. 171-178. doi:10.1016/j.physletb.2006.09.052
[52]	Y. Q. Gu, International Journal of Modern Physics A, Vol. 22, 2007, pp. 4667-4678. doi:10.1142/S0217751X07037925
[53]	Y. Q. Gu, “Accelerating Expansion of the Universe with Nonlinear Spinors,” 2006.
[54]	L. Randall and R. Sundrum, Physical Review Letters, Vol. 83, 1999, pp. 3370-3373. doi:10.1103/PhysRevLett.83.3370
[55]	L. Randall and R. Sundrum, Physical Review Letters, Vol. 83, 1999, pp. 4690-4693. doi:10.1103/PhysRevLett.83.4690
[56]	Y. V. Shtanov, Reports on Progress in Physics, Vol. 67, 2004, pp. 2183-2232.
[57]	Y. Shtanov and V. Sahni, Physics Letters B, Vol. 557, 2003, pp. 1-6. doi:10.1016/S0370-2693(03)00179-5
[58]	K. Freese and M. Lewis, Physics Letters B, Vol. 540, 2002, pp. 1-8. doi:10.1016/S0370-2693(02)02122-6
[59]	J. R. Oppenheimer and H. Snyder, Physical Review, Vol. 56, 1939, pp. 455-459. doi:10.1103/PhysRev.56.455
[60]	S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,” John Wiley, New York, 1972.
[61]	P. J. Steinhardt and N. Turok, Science, Vol. 296, 2002, pp. 1436-1439. doi:10.1126/science.1070462
[62]	P. J. Steinhardt and N. Turok, New Astronomy Reviews, Vol. 49, 2005, pp. 43-57. doi:10.1016/j.newar.2005.01.003
[63]	J. Khoury, P. J. Steinhardt and N. Turok, Physical Review Letters, Vol. 92, 2004, Article ID: 031302. doi:10.1103/PhysRevLett.92.031302
[64]	S. Nojiri and S. D. Odintsov, Physical Review D, Vol. 74, 2006, Article ID: 086005. doi:10.1103/PhysRevD.74.086005
[65]	S. Nojiri and S. D. Odintsov, Journal of Physics: Conference Series, Vol. 66, 2007, Article ID: 012005.
[66]	E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, Physical Review D, Vol. 83, 2011, Article ID: 086006.
[67]	T. P. Sotiriou and V. Faraoni, Reviews of Modern Physics, Vol. 82, 2010, pp. 451-497. Eq.(10)
[68]	T. P. Sotiriou and V. Faraoni, Reviews of Modern Physics, Vol. 82, 2010, pp. 451-497. Eq.(8)
[69]	T. P. Sotiriou and V. Faraoni, Reviews of Modern Physics, Vol. 82, 2010, pp. 451-497. Eq.(20)
[70]	E. Ayon-Beato and A. Garcia, Physics Letters B, Vol. 464, 1999, pp. 25-29. doi:10.1016/S0370-2693(99)01038-2
[71]	K. A. Bronnikov, Physical Review D, Vol. 63, 2001, Article ID: 044005. doi:10.1103/PhysRevD.63.044005
[72]	J. Matyjasek, Physical Review D, Vol. 70, 2004, Article ID: 047504. doi:10.1103/PhysRevD.70.047504
[73]	M. Born and L. Infeld, Proceedings of the Royal Society of London. Series A, Vol. 144, 1934, pp. 425-451. doi:10.1098/rspa.1934.0059
[74]	M. C. Bento, O. Bertolami and A. A. Sen, Physical Review D, Vol. 66, 2002, Article ID: 043507. doi:10.1103/PhysRevD.66.043507
[75]	A. Y. Kamenshchik, U. Moschella and V. Pasquier, Physics Letters B, Vol. 511, 2001, pp. 265-268. doi:10.1016/S0370-2693(01)00571-8
[76]	M. Chevallier and D. Polarski, International Journal of Modern Physics D, Vol. 10, 2001, pp. 213-224. doi:10.1142/S0218271801000822
[77]	S. W. Hawking and G. F. R. Ellis, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,” John Wiley, New York, 1972.
[78]	Z. X. Wang and D. R. Guo, “Special Functions,” Science Press, Beijing, 1979; (English Translation, World Scientific, Singapore, 1988)