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epoch. This does not mean that the kinks produced at present make dominant contribution to the energy of GWs. We see above that for given frequency ω , the kinks which dominantly contribute to ω dP dω are determined by Eqs. (36),(37) or (40). Accordingly high frequency modes arise from dense, blunt and old kinks, and low frequency modes arise from thin, sharp and new ones. The first line of Eq. (50) corresponds to GWs from new kinks which were born after the matter-radiation equality, the second line corresponds to GWs from old kinks which were born in the radiation era, the third line corresponds to GWs from older kinks produced between the end of the inflation and the start of the radiation era and the last line corresponds to GWs which came from the most abundant and oldest kinks. Figure 2 sketches the shape of the spectrum. The spectrum has three inflection points at ω$_{1}$, ω$_{2}$ and ω$_{3}$ . This is due to change of the type of kinks which mainly contribute to Ω$_{gw}$. Positions of these inflection points depend on the reheating temperature T$_{r}$ . If T$_{r}$ takes the value around its lower bound, say, 10 MeV [21], the second inflection point falls in the observable region, as we will see. Even if T$_{r}$ is so low, the natural value of H ∗ makes the third inflection far above the observable region. If the reheating ends immediately and H ∗ is as small as possible, the region between the second and third inflection is so short that the third inflection enters the observable region. 

This is a crude estimation, and in order to derive the realistic spectrum we should take into account subtlety described in [7, 8] where the authors claimed that GWs from kinks are burst-like and hence GW bursts with rare event rate (“isolated” GWs ) should not be counted as constituent of the stochastic GW background. We should calculate the stochastic GW background spectrum using the following formulae [7, 8] 

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