frequencies ∼ ω are ones which satisfy In other words, if the interval of kinks with sharpness ∼ ψ is similar to the period of the GW under consideration, these kinks make dominant contribution to the GW. In the first matter era, using Eq. (15), Eq. (34) simplifies to 5 for ω < ( t ∗ /t ) $^{A$_{m}$}$t − $^{1}$. (Here, we set A ≡ 2 ζ − β . A$_{r}$ ≃ − 0 . 92 , A$_{m}$ ≃ − 0 . 8.) For ω > ( t ∗ /t ) $^{A$_{m}$}$t − $^{1}$, ( ψ dN dψ ) − 1 > ω − 1 is satisfied by an arbitrary value of ψ . In Appendix B it is shown that for ω > ( t ∗ /t ) $^{A$_{m}$}$t − 1 the main contribution to ω dP dω comes from the kinks corresponding to the peak of the distribution, i.e. ψ ∼ ( t ∗ /t ) 2 $^{ζ$_{m}$}$. If we denote the value of sharpness of kinks which make dominant contribution to ω dP dω as ψ$_{max}$ ( ω, t ), it is given by where In the second matter era, ψ$_{max}$ is estimated as ( ψ dN dψ ) − 1 ∼ ω − $^{1}$. (34) ψ ∼ ( ωt ) 2 ζ$_{m}$/A$_{m}$ (35) ψ$_{max}$ ( ω, t ) ∼     $^{(}$$^{ωt}$$^{)}$$^{2}$$^{ζ$_{r}$}$$^{/A$_{r}$}$ for ω < ω (RD) 1 ( t ) ( t$_{r}$ t ) 2 D/A$_{m}$ ( ωt ) 2 ζ$_{m}$/A$_{m}$ for ω (RD) 1 ( t ) < ω < ω (RD) 2 ( t ) t$_{r}$ t 2 ζ$_{r}$ ( t ∗ t$_{r}$ ) 2 ζ$_{m}$ for ω > ω (RD) 2 ( t ) , . (37) ω (RD) 1 ( t ) = ( t$_{r}$ t ) A$_{r}$ t − $^{1}$, (38) ω (RD) 2 ( t ) = t$_{r}$ t A$_{r}$ ( t ∗ t$_{r}$ ) A$_{m}$ t − $^{1}$. (39) ψ$_{max}$ ( ω, t ) ∼     $^{(}$$^{ωt}$$^{)}$$^{2}$$^{ζ$_{m}$}$$^{/A$_{m}$}$ for ω < ω (MD) 1 ( t ) ( t$_{eq}$ t ) − 2 D/A$_{r}$ ( ωt ) 2 ζ$_{r}$/A$_{r}$ for ω (MD) 1 ( t ) < ω < ω (MD) 2 ( t ) t$_{r}$ t$_{eq}$ 2 D/A$_{m}$ ( ωt ) 2 ζ$_{m}$/A$_{m}$ for ω (MD) 2 ( t ) < ω < ω (MD) 3 ( t ) ( t$_{eq}$ t ) 2 ζ$_{m}$ ( t$_{r}$ t$_{eq}$ ) 2 ζ$_{r}$ ( t ∗ t$_{r}$ ) 2 ζ$_{m}$ for ω > ω (MD) 3 ( t ) , . (40) 5 Strictly speaking, Eq. (34) has two solution for ψ , but it is sufficient to take larger one. 12