Firstly, we explore the cosmological evolutions in four viable f(R) gravity models: Exponential, Hu-Sawicki, Starobinsky and Tsujikawa models. We summarize various viability conditions and explicitly demonstrate that the late-time cosmic acceleration following the matter-dominated stage can be realized in these viable models. We also study equation of state for dark energy and confirm that the crossing of the phantom divide from the phantom phase to the non-phantom (quintessence) one can occur and the future crossings of the phantom divide line wDE = −1 are the generic feature. The curvature singularities in viable f(R) gravity models are examined when the background density is dense. These singularities could be eliminated by adding the R2 term in the Lagrangian. Some of cosmological consequences, in particular the sources for the scalar mode of gravitational waves, are discussed. To illustrate the cosmological constraints on f(R), we concentrate on the exponential gravity model. We use the observational data including Supernova Cosmology Project (SCP) Union2 compilation, Two-Degree Field Galaxy Redshift Survey (2dFGRS), Sloan Digital Sky Survey Data
Release 7 (SDSS DR7) and Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP7) in our analysis.

Secondly, using the “teleparallel” equivalent of General Relativity as the gravitational sector, which is based on torsion instead of curvature, we add a canonical scalar field, allowing for a nonminimal coupling with gravity. Although the minimal case is completely equivalent to standard quintessence, the nonminimal scenario has a richer structure, exhibiting quintessence-like or phantom-like behavior, or experiencing the phantom-divide crossing. The richer structure is manifested in the absence of a conformal transformation to an equivalent minimally-coupled model. Moreover, we propose the simplest model of teleparallel dark energy with purely a non-minimal coupling
to gravity but no self-potential, leading to a single model possessing various interesting features: simplicity, self-potential-free, the guaranteed late-time cosmic acceleration driven by the non-minimal coupling to gravity, tracker behavior of the dark energy equation of state at earlier times, a crossing of the phantom divide at a late time, and the existence of a finite-time future singularity. We find the analytic solutions of the dark-energy scalar field respectively in the radiation, matter, and dark energy dominated eras, thereby revealing the above features. We further illustrate possible cosmic evolution patterns and present the observational constraint of this model obtained by numerical analysis and data fitting.

Thirdly, we examine the cosmological evolutions of the equation of state for dark energy wDE in the exponential and logarithmic as well as their combination f(T) theories. We show that the crossing of the phantom divide line of wDE = −1 can be realized in the combined f(T) theory even though it cannot be in the exponential or logarithmic f(T) theory. In particular, the crossing is from wDE > −1 to wDE < −1, in the opposite manner from f(R) gravity models. We also demonstrate that this feature is favored by
the recent observational data.

Firstly, we explore the cosmological evolutions in four viable f(R) gravity models: Exponential, Hu-Sawicki, Starobinsky and Tsujikawa models. We summarize various viability conditions and explicitly demonstrate that the late-time cosmic acceleration following the matter-dominated stage can be realized in these viable models. We also study equation of state for dark energy and confirm that the crossing of the phantom divide from the phantom phase to the non-phantom (quintessence) one can occur and the future crossings of the phantom divide line wDE = −1 are the generic feature. The curvature singularities in viable f(R) gravity models are examined when the background density is dense. These singularities could be eliminated by adding the R2 term in the Lagrangian. Some of cosmological consequences, in particular the sources for the scalar mode of gravitational waves, are discussed. To illustrate the cosmological constraints on f(R), we concentrate on the exponential gravity model. We use the observational data including Supernova Cosmology Project (SCP) Union2 compilation, Two-Degree Field Galaxy Redshift Survey (2dFGRS), Sloan Digital Sky Survey Data
Release 7 (SDSS DR7) and Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP7) in our analysis.

Secondly, using the “teleparallel” equivalent of General Relativity as the gravitational sector, which is based on torsion instead of curvature, we add a canonical scalar field, allowing for a nonminimal coupling with gravity. Although the minimal case is completely equivalent to standard quintessence, the nonminimal scenario has a richer structure, exhibiting quintessence-like or phantom-like behavior, or experiencing the phantom-divide crossing. The richer structure is manifested in the absence of a conformal transformation to an equivalent minimally-coupled model. Moreover, we propose the simplest model of teleparallel dark energy with purely a non-minimal coupling
to gravity but no self-potential, leading to a single model possessing various interesting features: simplicity, self-potential-free, the guaranteed late-time cosmic acceleration driven by the non-minimal coupling to gravity, tracker behavior of the dark energy equation of state at earlier times, a crossing of the phantom divide at a late time, and the existence of a finite-time future singularity. We find the analytic solutions of the dark-energy scalar field respectively in the radiation, matter, and dark energy dominated eras, thereby revealing the above features. We further illustrate possible cosmic evolution patterns and present the observational constraint of this model obtained by numerical analysis and data fitting.

Thirdly, we examine the cosmological evolutions of the equation of state for dark energy wDE in the exponential and logarithmic as well as their combination f(T) theories. We show that the crossing of the phantom divide line of wDE = −1 can be realized in the combined f(T) theory even though it cannot be in the exponential or logarithmic f(T) theory. In particular, the crossing is from wDE > −1 to wDE < −1, in the opposite manner from f(R) gravity models. We also demonstrate that this feature is favored by
the recent observational data.

[1] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116,1009 (1998) [astro-ph/9805201].
[2] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999) [astro-ph/9812133].
[3] D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003) [astro-ph/0302209].
[4] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arXiv:1001.4538 [astro-ph.CO]].
[5] P. A. R. Ade et al. [ Planck Collaboration], arXiv:1303.5062 [astro-ph.CO].
[6] D. J. Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005) [astro-ph/0501171].
[7] B. Jain and A. Taylor, Phys. Rev. Lett. 91, 141302 (2003) [astro-ph/0306046].
[8] S. Nojiri and S. D. Odintsov, eConf C0602061, 06 (2006) [Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)] [arXiv:hep-th/0601213].
[9] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010) [arXiv:0805.1726 [gr-qc]].
[10] A. De Felice and S. Tsujikawa, arXiv:1002.4928 [gr-qc].
[11] S. Nojiri and S. D. Odintsov, Phys. Rev. D 68, 123512 (2003); A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1 (2003); V. Faraoni, Phys. Rev. D 74, 104017 (2006); Y. S. Song, W. Hu and I. Sawicki, ibid. 75, 044004 (2007).
[12] V. Muller, H. J. Schmidt and A. A. Starobinsky, Phys. Lett. B 202, 198 (1988); [13]
[13] V. Faraoni and S. Nadeau, Phys. Rev. D 72, 124005 (2005); [14]
[14] L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, ibid. 75, 083504 (2007).
[15] T. Chiba, Phys. Lett. B 575, 1 (2003); T. Chiba, T. L. Smith and A. L. Erickcek, Phys. Rev. D 75, 124014 (2007).
[16] W. Hu and I. Sawicki, Phys. Rev. D 76, 064004 (2007).
[17] A. A. Starobinsky, JETP Lett. 86, 157 (2007).
[18] S. Tsujikawa, Phys. Rev. D 77, 023507 (2008).
[19] P. Zhang, Phys. Rev. D 73, 123504 (2006); B. Li and M. C. Chu, ibid. 74, 104010 (2006); R. Bean, D. Bernat, L. Pogosian, A. Silvestri and M. Trodden, ibid. 75, 064020 (2007); P. J. Zhang, ibid. 76, 024007 (2007); P. Zhang, M. Liguori, R. Bean and S. Dodelson, Phys. Rev. Lett. 99, 141302 (2007); G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, ibid. 77, 046009 (2008); G. Calcagni and G. Nardelli, arXiv:1004.5144 [hep-th]. [20]
[20] E. V. Linder, ibid. 80, 123528 (2009);
[21] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani and S. Zerbini, Phys. Rev. D 77, 046009 (2008).
[22] K. Bamba, C. Q. Geng and C. C. Lee, arXiv:1005.4574 [astro-ph.CO].
[23] S. Nojiri and S. D. Odintsov, Phys. Lett. B 657, 238 (2007); Phys. Rev. D 77, 026007 (2008); AIP Conf. Proc. 1241, 1094 (2010) [arXiv:0910.1464 [hep-th]].
[24] U. Alam, V. Sahni and A. A. Starobinsky, JCAP 0406, 008 (2004); 0702, 011 (2007); S. Nesseris and L. Perivolaropoulos, ibid. 0701, 018 (2007); P. U. Wu and H. W. Yu, Phys. Lett. B 643, 315 (2006); H. K. Jassal, J. S. Bagla and T. Padmanabhan, arXiv:astro-ph/0601389.
[25] R. R. Caldwell, Phys. Lett. B 545, 23 (2002); B. Feng, X. L. Wang and X. M. Zhang, ibid. 607, 35 (2005); Z. K. Guo, Y. S. Piao, X. M. Zhang and Y. Z. Zhang, ibid. 608, 177 (2005).
[26] M. Martinelli, A. Melchiorri and L. Amendola, Phys. Rev. D 79, 123516
(2009);
[27] H. Motohashi, A. A. Starobinsky and J. Yokoyama, arXiv:1002.1141 [astroph. CO];
[28] H. Motohashi, A. A. Starobinsky and J. Yokoyama, arXiv:1005.1171 [astroph. CO].
[29] M. C. B. Abdalla, S. Nojiri and S. D. Odintsov, Class. Quant. Grav. 22, L35 (2005); I. H. Brevik, Int. J. Mod. Phys. D 15, 767 (2006); Gen. Rel. Grav. 38, 1317 (2006); L. Amendola and S. Tsujikawa, Phys. Lett. B 660, 125 (2008); K. Bamba, S. Nojiri and S. D. Odintsov, JCAP 0810, 045 (2008); Y. Bisabr, arXiv:1005.5679 [gr-qc].
[30] K. Bamba, C. Q. Geng, S. Nojiri and S. D. Odintsov, Phys. Rev. D 79, 083014 (2009); Mod. Phys. Lett. A 25, 900 (2010); K. Bamba and C. Q. Geng, Phys. Lett. B 679, 282 (2009); K. Bamba, arXiv:0904.2655 [gr-qc].
[31] K. Bamba and C. Q. Geng, Prog. Theor. Phys. 122, 1267 (2009).
[32] L. Perivolaropoulos, Phys. Rev. D 71, 063503 (2005) [astro-ph/0412308].
[33] D. J. Eisenstein, W. Hu, Astrophys. J. 496, 605 (1998) [astro-ph/9709112].
[34] J. R. Bond, G. Efstathiou, M. Tegmark and , Mon. Not. Roy. Astron. Soc. 291, L33 (1997) [astro-ph/9702100].
[35] W. Hu, N. Sugiyama and , Astrophys. J. 471, 542 (1996) [astro-ph/9510117].
[36] A. V. Frolov, Phys. Rev. Lett. 101, 061103 (2008) [arXiv:0803.2500 [astroph]].
[37] T. Kobayashi and K. Maeda, Phys. Rev. D78, 064019 (2008) [arXiv:0807.2503 [astro-ph]].
[38] A. Dev, D. Jain, S. Jhingan, S. Nojiri, M. Sami and I. Thongkool, Phys. Rev. D 78, 083515 (2008) [arXiv:0807.3445 [hep-th]].
[39] S. A. Appleby and R. A. Battye, JCAP 0805, 019 (2008) [arXiv:0803.1081 [astro-ph]].
[40] E. V. Arbuzova, A. D. Dolgov and , Phys. Lett. B 700, 289 (2011) [arXiv:1012.1963 [astro-ph.CO]].
[41] T. Kobayashi and K. Maeda, Phys. Rev. D79, 024009 (2009) [arXiv:0810.5664 [astro-ph]].
[42] S. Capozziello, M. De Laurentis, S. Nojiri and S. D. Odintsov, Phys. Rev. D 79, 124007 (2009) [arXiv:0903.2753 [hep-th]].
[43] K. Bamba, S. ’i. Nojiri, S. D. Odintsov and , Phys. Lett. B 698, 451 (2011) [arXiv:1101.2820 [gr-qc]].
[44] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976).
[45] K. Hayashi and T. Shirafuji, Phys. Rev. D 19, 3524 (1979) [Addendum-ibid. D 24, 3312 (1982)].
[46] E. E. Flanagan and E. Rosenthal, Phys. Rev. D 75, 124016 (2007).
[47] J. Garecki, arXiv:1010.2654 [gr-qc].
[48] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl., 217 (1928); 401 (1930); A. Einstein, Math. Ann. 102, 685 (1930).
[49] C. Q. Geng, C. C. Lee, E. N. Saridakis and Y. P. Wu, Phys. Lett. B 704, 384 (2011) [arXiv:1109.1092 [hep-th]].
[50] A. Einstein 1928, Sitz. Preuss. Akad. Wiss. p. 217; ibid p. 224; A. Unzicker and T. Case, arXiv:physics/0503046.
[51] K. Hayashi and T. Shirafuji, Phys. Rev. D 19, 3524 (1979); Addendum-ibid. 24, 3312 (1982).
[52] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988); C. Wetterich, Nucl. Phys. B 302, 668 (1988); A. R. Liddle and R. J. Scherrer, Phys. Rev. D 59, 023509 (1999); I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999); Z. K. Guo, N. Ohta and Y. Z. Zhang, Mod. Phys. Lett. A 22, 883 (2007); S. Dutta, E. N. Saridakis and R. J. Scherrer, Phys. Rev. D 79, 103005 (2009).
[53] G. R. Bengochea and R. Ferraro, Phys. Rev. D 79, 124019 (2009).
[54] E. V. Linder, Phys. Rev. D 81, 127301 (2010) [Erratum-ibid. D 82, 109902 (2010)].
[55] R. Ferraro and F. Fiorini, Phys. Rev. D 75, 084031 (2007); ibid. 78, 124019 (2008).
[56] P. Wu and H. W. Yu, Phys. Lett. B 693, 415 (2010); R. Myrzakulov, arXiv:1006.1120 [gr-qc]; K. K. Yerzhanov, S. R. Myrzakul, I. I. Kulnazarov and R. Myrzakulov, arXiv:1006.3879 [gr-qc]; P. Wu and H. Yu, Phys. Lett. B 692, 176 (2010); R. J. Yang, arXiv:1007.3571 [gr-qc]; P. Y. Tsyba, I. I. Kulnazarov, K. K. Yerzhanov and R. Myrzakulov, arXiv:1008.0779 [astro-ph.CO]; G. R. Bengochea, arXiv:1008.3188 [astro-ph.CO]; R. Myrzakulov, arXiv:1008.4486 [astro-ph.CO]; K. Karami and A. Abdolmaleki, arXiv:1009.2459 [gr-qc]; arXiv:1009.3587 [physics.gen-ph]; R. J. Yang, arXiv:1010.1376 [gr-qc]; R. Zheng and Q. G. Huang, arXiv:1010.3512 [gr-qc].
[57] S. H. Chen, J. B. Dent, S. Dutta and E. N. Saridakis, arXiv:1008.1250 [astroph. CO].
[58] P. Wu and H. Yu, arXiv:1008.3669 [gr-qc].
[59] K. Bamba, C. Q. Geng and C. C. Lee, arXiv:1008.4036 [astro-ph.CO]; see also C. Q. Geng, talk presented at the 2nd Chuang-Xin group meeting on “Dark Energy and Dark Matter” , Weihai, China, July 19–Aug. 3, 2010.
[60] B. Li, T. P. Sotiriou and J. D. Barrow, arXiv:1010.1041 [gr-qc]; T. P. Sotiriou, B. Li and J. D. Barrow, arXiv:1012.4039 [gr-qc].
[61] J. B. Dent, S. Dutta and E. N. Saridakis, arXiv:1010.2215 [astro-ph.CO].
[62] ibid. 1011, 001 (2010) [arXiv:1007.0482 [astro-ph.CO]].
[63] L. Yang, C. C. Lee, L. W. Luo and C. Q. Geng, Phys. Rev. D 82, 103515 (2010) arXiv:1010.2058 [astro-ph.CO].
[64] A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1 (2003).
[65] V. Faraoni, Phys. Rev. D 74, 104017 (2006).
[66] Y. S. Song, W. Hu and I. Sawicki, Phys. Rev. D 75, 044004 (2007).
[67] S. Tsujikawa, R. Gannouji, B. Moraes and D. Polarski, Phys. Rev. D 80, 084044 (2009).
[68] D. F. Mota and J. D. Barrow, Phys. Lett. B 581, 141 (2004).
[69] J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104 (2004); Phys. Rev. D 69, 044026 (2004).
[70] K. i. Maeda, Phys. Rev. D 39, 3159 (1989); Y. Fujii and K. Maeda, The Scalar-Tensor Theory of Gravitation (Cambridge University Press, Cambridge,
United Kingdom, 2003).
[71] S. Capozziello and S. Tsujikawa, Phys. Rev. D 77, 107501 (2008).
[72] S. Tsujikawa, K. Uddin, S. Mizuno, R. Tavakol and J. Yokoyama, Phys. Rev.D 77, 103009 (2008).
[73] T. Tamaki and S. Tsujikawa, Phys. Rev. D 78, 084028 (2008).
[74] A. Ali, R. Gannouji, M. Sami and A. A. Sen, Phys. Rev. D 81, 104029 (2010).
[75] S. Appleby, R. Battye and A. Starobinsky, arXiv:0909.1737 [astro-ph.CO].
[76] K. Bamba, arXiv:0909.2991 [astro-ph.CO].
[77] S. Nesseris, L. Perivolaropoulos and , Phys. Rev. D 72, 123519 (2005) [astroph/ 0511040].
[78] R. Amanullah, C. Lidman, D. Rubin, G. Aldering, P. Astier, K. Barbary, M. S. Burns and A. Conley et al., Astrophys. J. 716, 712 (2010) [arXiv:1004.1711 [astro-ph.CO]].
[79] W. J. Percival et al. [SDSS Collaboration], Mon. Not. Roy. Astron. Soc. 401, 2148 (2010) [arXiv:0907.1660 [astro-ph.CO]].
[80] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330
(2009) [arXiv:0803.0547 [astro-ph]].
[81] Y. Gong, X. m. Zhu, Z. H. Zhu and , Mon. Not. Roy. Astron. Soc. 415, 1943 (2011) [arXiv:1008.5010 [astro-ph.CO]].
[82] S. A. Appleby and R. A. Battye, Phys. Lett. B 654, 7 (2007); S. Appleby, R. Battye and A. Starobinsky, JCAP 1006, 005 (2010).
[83] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
[84] A. Vilenkin, Phys. Rev. D 32, 2511 (1985).
[85] M. B. Mijic, M. S. Morris, W. M. Suen and , Phys. Rev. D 34, 2934 (1986).
[86] D. i. Hwang, B. H. Lee and D. h. Yeom, JCAP 1112, 006 (2011)
[arXiv:1110.0928 [gr-qc]].
[87] E. V. Arbuzova, A. D. Dolgov and L. Reverberi, JCAP 1202, 049 (2012)
[arXiv:1112.4995 [gr-qc]].
[88] L. Yang, C. C. Lee, C. Q. Geng and , JCAP 1108, 029 (2011) [arXiv:1106.5582 [astro-ph.CO]].
[89] Weitzenb¨ock R., Invarianten Theorie, (Nordhoff, Groningen, 1923).
[90] J. W. Maluf, J. Math. Phys. 35 (1994) 335; H. I. Arcos and J. G. Pereira, Int. J. Mod. Phys. D 13, 2193 (2004).
[91] I. Zlatev, L. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82, 896 (1999);
[92] P. J. Steinhardt, L. Wang and I. Zlatev, Phys. Rev. D 59, 123504 (1999).
[93] S. Nojiri, S. D. Odintsov and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005).
[94] C. Q. Geng, C. C. Lee and E. N. Saridakis, JCAP 1201, 002 (2012).
[95] A. de la Macorra, C. Stephan-Otto, Phys. Rev. D65, 083520 (2002); J. P. Kneller, L. E. Strigari, Phys. Rev. D68, 083517 (2003); X. Zhang, Mod. Phys. Lett. A20, 2575 (2005); E. N. Saridakis, Nucl. Phys. B819, 116 (2009); E. N. Saridakis, Nucl. Phys. B830, 374 (2010).
[96] P.G. Ferreira, M. Joyce, Phys. Rev. Lett. 79, 4740 (1997); E. J. Copeland, A. R. Liddle and D. Wands, Phys. Rev. D 57, 4686 (1998); E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006); X. m. Chen, Y. g. Gong and E. N. Saridakis, JCAP 0904, 001 (2009).
[97] V. Sahni, L. M. Wang, Phys. Rev. D62, 103517 (2000); V. Sahni, M. Sami, T. Souradeep, Phys. Rev. D65, 023518 (2002); P. Singh, M. Sami, N. Dadhich, Phys. Rev. D68, 023522 (2003).
[98] C. Xu, E. N. Saridakis, G. Leon and , JCAP 1207, 005 (2012) [arXiv:1202.3781 [gr-qc]].
[99] V. Sahni, A. A. Starobinsky and , Int. J. Mod. Phys. D 9, 373 (2000) [astroph/9904398].
[100] L. A. Urena-Lopez, T. Matos and , Phys. Rev. D 62, 081302 (2000) [astroph/ 0003364].
[101] arXiv:1011.0544 [gr-qc].
[102] P. K. S. Dunsby, E. Elizalde, R. Goswami, S. Odintsov and D. S. Gomez, Phys. Rev. D 82, 023519 (2010).
[103] B. Li, J. D. Barrow and D. F. Mota, Phys. Rev. D 76, 044027 (2007).
[104] E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, Redwood City, California, 1990).