turnaround of each cycle for this implementing is shown. In this design, the causal diagram of cyclic multiverse likes that of inflationary multiverse. Thus the measure for the multiverse might be discussed similarly. Dark en- ergy is an important issue of current cosmology, which is being intensively explored. In some sense, this work might provide an alternative motivation for the existence of dark energy. Acknowledgments We thank R. Brandenberger, Y.F. Cai, J. Zhang for helpful discussions. This work is supported in part by NSFC under Grant No:10775180, in part by the Scientific Research Fund of GUCAS(NO:055101BM03), in part by National Ba- sic Research Program of China, No:2010CB832805 6 [1] R.C. Tolman, Relativity, Thermodynamics and Cosmol- ogy, (Oxford U. Press, Clarendon Press, 1934). [2] P.J. Steinhardt, N. Turok, Science 296 , (2002) 1436; Phys. Rev. D65 126003 (2002). [3] J. Barrow, M. P. Dabrowski, Mon. Not. R. Astr. Soc. 275 , 850 (1995). [4] N. Kanekar, V. Sahni, Y. Shtanov, Phys. Rev. D63 , 083520 (2001). [5] Y.S. Piao, Phys. Rev. D70 , 101302 (2004); Y.S. Piao, Y.Z. Zhang, Nucl. Phys. B725 , 265 (2005). [6] J.E. Lidsey, D.J. Mulryne, N.J. Nunes, R. Tavakol, Phys. Rev. D70 , 063521 (2004). [7] T. Clifton, J.D. Barrow, Phys. Rev. D75 , 043515 (2007). [8] H.H. Xiong, Y.F. Cai, T. Qiu, Y.S. Piao, X.M. Zhang, Phys. Lett. B666 , 212 (2008). [9] X. Zhang, Eur. Phys. J. C60 , 661 (2009). [10] J.L. Lehners, P.J. Steinhardt, Phys. Rev. D79 , 063503 (2009). [11] T. Biswas, S. Alexander, Phys. Rev. D80 , 043511 (2009). [12] T. Biswas, A. Mazumdar, Phys. Rev. D80 , 023519 (2009). [13] M. Novello, S.E.P. Bergliaffa, Phys. Rept. 463 , 127 (2008). [14] Y.S. Piao, Phys. Lett. B677 , 1 (2009). [15] B. Feng, M.Z. Li, Y.S. Piao, X.M. Zhang, Phys. Lett. B634, 101 (2006). [16] C. Ilie, T. Biswas, K. Freese. arXiv:0908.0991. [17] Y.S. Piao, E Zhou, Phys. Rev. D68 , 083515 (2003). [18] Y.S. Piao, Y.Z. Zhang, Phys. Rev. D70 , 063513 (2004). [19] T. Biswas, T. Koivisto, A. Mazumdar, arXiv:1005.0590. [20] Y.F. Cai, T. Qiu, Y.S. Piao, M.Z. Li, X.M. Zhang, JHEP 0710 , 071 (2007). [21] I.Y. Arefeva, L.V. Joukovskaya, A.S. Koshelev, JHEP 0309 , 012 (2003). [22] I.Y. Arefeva, A.S. Koshelev, S.Y. Vernov, Phys. Rev. D72 , 064017 (2005). [23] V.F. Mukhanov, JETP lett. 41, 493 (1985); Sov. Phys. JETP. 68, 1297 (1988). [24] H. Kodama, M. Sasaki, Prog. Theor. Phys. Suppl. 78 1 (1984). [25] D. Wands, Phys. Rev. D60 , 023507 (1999). [26] R. Brandenberger, F. Finelli, JHEP 0111 (2001) 056; F. Finelli, R. Brandenberger, Phys. Rev. D65 , 103522 (2002). [27] P. Peter, N. Pinto-Neto, Phys. Rev. D78 , 063506 (2008). [28] D. Wands, arXiv:0809.4556. [29] Y.F.Cai, T. Qiu, R. Brandenberger, X.M. Zhang, Phys. Rev. D80 , 023511 (2009). [30] A.A. Starobinsky, JETP Lett. 30 , 682 (1979). [31] B. Carr, S.W. Hawking, Mon. Not. R. Astro. Soc. 168 , 399 (1974). [32] R. Brandenberger, Phys. Rev. D80 , 023535 (2009). [33] Y.S. Piao, B. Feng, X.M. Zhang, Phys. Rev. D69 , 103520 (2004). [34] Y.S. Piao, S. Tsujikawa, X.M. Zhang, Class. Quant. Grav. 21 , 4455 (2004). [35] J. Mielczarek, JCAP 0811 , 011 (2008); arXiv:0908.4329. [36] Y.F. Cai, X.M. Zhang, JCAP 0906 , 003 (2009). [37] S. Winitzki, Lect. Notes Phys. 738 , 157 (2008). [38] J.R. Gott, Nature 295 , 304 (1982). [39] P.J. Steinhardt, in “The Very Early Universe” ed. by G.W. Gibbons, S.W. Hawking and S. T. C. Siklos (Cam- bridge University Press, Cambridge, 1983). [40] A. Vilenkin, Phys. Rev. D27 , 2848 (1983). [41] A. Linde, Phys. Lett. B175 , 395 (1986). [42] S. Li, Y. Liu, Y.S. Piao, Phys. Rev. D80 , 123535 (2009). [43] J.K. Erickson, S. Gratton, P.J. Steinhardt, N. Turok, Phys. Rev. D75 , 123507 (2007). [44] D.H. Lyth, Phys. Lett. B524, 1 (2002); Phys. Lett. B 526, 173 (2002). [45] R. Durrer, F. Vernizzi, Phys. Rev. D66, 083503 (2002). [46] S. Tsujikawa, R. Brandenberger, F. Finelli, Phys. Rev. D66, 083513 (2002). [47] S. Gratton, J. Khoury, P.J. Steinhardt, N. Turok, Phys. Rev. D69, 103505 (2004). [48] Y.S. Piao, Y.Z. Zhang, Phys. Rev. D70 043516 (2004). [49] V. Bozza, G. Veneziano, JCAP 009, 005 (2005); V. Bozza, JCAP 0602, 009 (2006). [50] S. Alexander, T. Biswas, R.H. Brandenberger, arXiv:0707.4679. [51] Y.F. Cai, T. Qiu, R. Brandenberger, Y.S. Piao, X.M. Zhang, JCAP 0803, 013 (2008). [52] L.E. Allen, D. Wands, Phys. Rev. D70, 063515 (2004). [53] K. Freese, M.G. Brown, W.H. Kinney, arXiv:0802.2583. [54] L. Baum and P.H. Frampton, Phys. Rev. Lett. 98 , 071301 (2007). [55] P.F. Gonzalez-Diaz, arXiv:0908.3244. [56] P.F. Gonzalez-Diaz, Phys. Rev. Lett. 93 , 071301 (2004); P.F. Gonzalez-Diaz, J.A. Jimenez-Madrid, Phys. Lett. B596 , 16 (2004).