(S.N.).

[1] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A20 doi:10.1051/0004-6361/201525898[arXiv:1502.02114 [astro-ph.CO]].

[2] A. H. Guth, Phys. Rev. D 23 (1981) 347. doi:10.1103/PhysRevD.23.347

[3] A. A. Starobinsky, Phys. Lett. 91B (1980) 99. doi:10.1016/0370-2693(80)90670-X

[4] A. D. Linde, Phys. Lett. 129B (1983) 177. doi:10.1016/0370-2693(83)90837-7

[5] R. Brandenberger and P. Peter, Found. Phys. 47 (2017) no.6, 797 doi:10.1007/s10701-016-0057-0 [arXiv:1603.05834 [hep-th]].

[6] J. de Haro and Y. F. Cai, Gen. Rel. Grav. 47 (2015) no.8, 95 doi:10.1007/s10714-015-1936-y [arXiv:1502.03230 [gr-qc]].

[7] Y. F. Cai, Sci. China Phys. Mech. Astron. 57 (2014) 1414 doi:10.1007/s11433-014-5512-3 [arXiv:1405.1369 [hep-th]].

[8] P. A. R. Ade et al. [BICEP2 and Keck Array Collaborations], Phys. Rev. Lett. 116 (2016) 031302 doi:10.1103/PhysRevLett.116.031302 [arXiv:1510.09217 [astro-ph.CO]].

[9] S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Phys. Rept. 692 (2017) 1 doi:10.1016/j.physrep.2017.06.001[arXiv:1705.11098 [gr-qc]].

[10] S. Nojiri and S. D. Odintsov, Phys. Rept. 505 (2011) 59 doi:10.1016/j.physrep.2011.04.001 [arXiv:1011.0544 [gr-qc]].

[11] S. Nojiri and S. D. Odintsov, eConf C 0602061 (2006) 06 [Int. J. Geom. Meth. Mod. Phys. 4 (2007) 115]doi:10.1142/S0219887807001928 [hep-th/0601213].

[12] S. Capozziello and M. De Laurentis, Phys. Rept. 509 (2011) 167 doi:10.1016/j.physrep.2011.09.003 [arXiv:1108.6266 [gr-qc]].

[13] V. Faraoni and S. Capozziello, Fundam. Theor. Phys. 170 (2010). doi:10.1007/978-94-007-0165-6

[14] A. de la Cruz-Dombriz and D. Saez-Gomez, Entropy 14 (2012) 1717 doi:10.3390/e14091717 [arXiv:1207.2663 [gr-qc]].

[15] G. J. Olmo, Int. J. Mod. Phys. D 20 (2011) 413 doi:10.1142/S0218271811018925 [arXiv:1101.3864 [gr-qc]].

[16] A. Ashtekar and P. Singh, Class. Quant. Grav. 28 (2011) 213001 doi:10.1088/0264-9381/28/21/213001 [arXiv:1108.0893[gr-qc]].

[17] A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. D 74 (2006) 084003 doi:10.1103/PhysRevD.74.084003 [gr-qc/0607039].

[18] L. Areste Salo, J. Amoros and J. de Haro, Class. Quant. Grav. 34 (2017) no.23, 235001 doi:10.1088/1361-6382/aa9311[arXiv:1612.05480 [gr-qc]].

[19] H. H. Xiong, T. Qiu, Y. F. Cai and X. Zhang, Mod. Phys. Lett. A 24 (2009) 1237 doi:10.1142/S0217732309030667[arXiv:0711.4469 [hep-th]].

[20] J. Amoros, J. de Haro and S. D. Odintsov, Phys. Rev. D 89 (2014) no.10, 104010 doi:10.1103/PhysRevD.89.104010[arXiv:1402.3071 [gr-qc]].

[21] Y. F. Cai and E. Wilson-Ewing, JCAP 1403 (2014) 026 doi:10.1088/1475-7516/2014/03/026 [arXiv:1402.3009 [gr-qc]].

[22] J. de Haro and J. Amoros, JCAP 1408 (2014) 025 doi:10.1088/1475-7516/2014/08/025 [arXiv:1403.6396 [gr-qc]].

[23] K. Kleidis and V. K. Oikonomou, Int. J. Geom. Meth. Mod. Phys. 15 (2018) no.05, 1850071 doi:10.1142/S0219887818500718[arXiv:1801.02578 [gr-qc]].

[24] K. Kleidis and V. K. Oikonomou, Int. J. Geom. Meth. Mod. Phys. 15 (2017) no.04, 1850064 doi:10.1142/S0219887818500640[arXiv:1711.09270 [gr-qc]].

[25] H. Noh and J. c. Hwang, Phys. Lett. B 515 (2001) 231 doi:10.1016/S0370-2693(01)00875-9 [astro-ph/0107069].

[26] J. c. Hwang and H. Noh, Phys. Rev. D 71 (2005) 063536 doi:10.1103/PhysRevD.71.063536 [gr-qc/0412126].

[27] J. c. Hwang and H. Noh, Phys. Rev. D 66 (2002) 084009 doi:10.1103/PhysRevD.66.084009 [hep-th/0206100].

[28] D. I. Kaiser and E. I. Sfakianakis, Phys. Rev. Lett. 112 (2014) no.1, 011302 doi:10.1103/PhysRevLett.112.011302[arXiv:1304.0363 [astro-ph.CO]].

[29] A. A. Starobinsky, Phys. Lett. B 91 (1980) 99 [Phys. Lett. 91B (1980) 99] [Adv. Ser. Astrophys. Cosmol. 3 (1987) 130].doi:10.1016/0370-2693(80)90670-X

[30] S. Nojiri, S. D. Odintsov and D. Saez-Gomez, Phys. Lett. B 681 (2009) 74 doi:10.1016/j.physletb.2009.09.045[arXiv:0908.1269 [hep-th]].

[31] S. D. Odintsov and V. K. Oikonomou, Phys. Rev. D 96 (2017) no.10, 104049 doi:10.1103/PhysRevD.96.104049[arXiv:1711.02230 [gr-qc]].

[32] V. K. Oikonomou, Int. J. Mod. Phys. D 27 (2018) no.05, 1850059 doi:10.1142/S0218271818500591 [arXiv:1711.03389[gr-qc]].

[33] Work in progress.

[34] K. Kleidis and V. K. Oikonomou, arXiv:1808.04674 [gr-qc].

[35] T. Chiba, T. Okabe and M. Yamaguchi, Phys. Rev. D 62 (2000) 023511 doi:10.1103/PhysRevD.62.023511[astro-ph/9912463].

[36] C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt, Phys. Rev. Lett. 85 (2000) 4438 doi:10.1103/PhysRevLett.85.4438 [astro-ph/0004134].

[37] C. Armendariz-Picon, T. Damour and V. F. Mukhanov, Phys. Lett. B 458 (1999) 209 doi:10.1016/S0370-2693(99)00603-6[hep-th/9904075].

[38] J. Matsumoto and S. Nojiri, Phys. Lett. B 687 (2010) 236 doi:10.1016/j.physletb.2010.03.030 [arXiv:1001.0220 [hep-th]].