Remembering ψ$_{max}$ dN $_{dψ}$( ψ$_{max}$ ) = ω and ψ$_{max}$ < ψ$_{l}$ , we find (B13) < (B11). Eventually, the mean square of I i + ,l is roughly estimated as (B11). In other words, in each interval around each big kink, it is sufficient to consider only the isolated big kink, while neglecting small kinks. Next, we consider the cross terms of different I i + $_{,l}$s, such as 〈 I i + $_{,l}$I i ∗ + ,m 〉 . This should vanish, and we can explicitly check this by straightforward calculation. To do so, we divide I i + ,l and I i + ,m as Eq. (B4) and evaluate the mean squares and the averages of the cross terms of the two term, using above approximations, such as Eqs. (B5) and (B12). As a result, the mean square of I i + can be evaluated by summing up Eq. (B11) for each l . Then we obtain where L denotes the integration range of I i $_{+}$. This implies that the greatest contribution to 〈 | I i + | $^{2}$〉 comes from kinks which satisfy ψ ∼ ψ$_{max}$ . Such kinks dominantly contribute to GWs with frequency ∼ ω . So far we have discussed the case where ψ$_{k}$ dN $_{dψ}$( ψ$_{k}$ ) ∼ ω has a solution. However, if ω is so large that ψ$_{k}$ dN $_{dψ}$( ψ$_{k}$ ) ≪ ω is satisfied for arbitrary values of sharpness, all kinks are thought of as “big kinks”. Therefore, the contribution from each interval of length ∼ ω around each kink becomes (B11). That from regions far from any kinks is exponentially small when ω → ∞ . Eventually, where ψ$_{max}$ denotes the value of ψ at which dN $_{dψ}$( ψ ) has a peak. This implies that kinks which satisfy ψ ∼ ψ$_{max}$ dominantly contribute to 〈 | I i + | $^{2}$〉 and GWs of frequency ∼ ω . Thus we have proved the validity of Eq. (44). 25 〈 | I i + | $^{2}$〉 ∼ ∫ 1 0 dψ dN dψ ( ψ ) ψω − 2 × L ∼ ψ$_{max}$ dN dψ ( ψ$_{max}$ ) × ψ$_{max}$ω − 2 × L, (B15) 〈 | I i + | $^{2}$〉 ∼ ∑ l ψ$_{l}$ω − 2 ∼ ∫ 1 ψ$_{max}$ dψ dN dψ ( ψ ) ψω − 2 × L ∼ ψ$_{max}$ dN dψ ( ψ$_{max}$ ) × ψ$_{max}$ω − 2 × L, (B14)