situation depends on whether the reheating temperature exceeds T$_{c}$ [Eq. (14)] or not. If T$_{r}$ < T$_{c}$ , there is no period when the friction works and kinks produced in the first matter era survive. The peak of ω dP dω | tot consists of contribution from kinks produced around the turning point from the first matter era to the radiation era. Therefore, the above discussion applies and the condition (C1) is satisfied all the time. On the other hand, in the case of T$_{r}$ > T$_{c}$ , the friction becomes problematic in the early stage of the radiation dominated era. If all kinks disappear in this stage and kinks restart to emerge at the end of friction- domination, the condition (C1) is never violated as discussed above. In the opposite case, where all kinks survive the friction-dominated era, (C1) is not guaranteed. In such a case, the largest contribution to ∫ dω dP dω | tot comes from kinks produced around the end of the first matter era. The condition (C1) is satisfied if 10 Gµ ( T$_{r}$ T$_{eq}$ ) − 2 E$_{r}$/ 3 < 1 . (C5) For Gµ = 10 − $^{7}$, this leads to T$_{r}$/M$_{pl}$ ≲ 10 − $^{4}$. Therefore the gravitational backreaction might be able to be neglected unless the reheating temperature is so high. Acknowledgments K.N. would like to thank the Japan Society for the Promotion of Science for financial support. This work is supported by Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No.14102004 (M.K.) and No. 21111006(M.K. and K.N.) and also by World Premier International Research Center Initia- tive (WPI Initiative), MEXT, Japan. [1] A. Vilenkin and E. P. S. Shellard, “Cosmic Strings and Other Topological Defects,” Cambridge University Press, Cambridge, England (1994). [2] S. Sarangi and S. H. H. Tye, Phys. Lett. B 536 , 185 (2002) [arXiv:hep-th/0204074]. [3] G. Dvali and A. Vilenkin, JCAP 0403 , 010 (2004) [arXiv:hep-th/0312007]. [4] R. R. Caldwell and B. Allen, Phys. Rev. D 45 , 3447 (1992); R. R. Caldwell, R. A. Battye and E. P. S. Shellard, Phys. Rev. D 54 , 7146 (1996) [arXiv:astro-ph/9607130]. 27