and dN dψ ( t, ψ ) ∼     $^{ψ}$$^{−}$$^{β$_{m}$}$$^{/}$$^{2}$$^{ζ$_{m}$}$$^{t}$$^{−}$$^{1}$ for ψ > ψ (MD) 1 ( t ) ( t t$_{eq}$ ) β$_{m}$ − β$_{r}$ζ$_{m}$/ζ$_{r}$ ψ − β$_{r}$/ 2 $^{ζ$_{r}$}$t − 1 for ψ (MD) 2 ( t ) < ψ < ψ (MD) 1 ( t ) t$_{eq}$ t$_{r}$ β$_{r}$ − β$_{m}$ζ$_{r}$/ζ$_{m}$ ψ − β$_{m}$/ 2 $^{ζ$_{m}$}$t − 1 for ψ (MD) 3 ( t ) < ψ < ψ (MD) 2 ( t ) ψ 1 / 2 ( t t$_{eq}$ ) β$_{m}$ + ζ$_{m}$ ( t$_{eq}$ t$_{r}$ ) β$_{r}$ + ζ$_{r}$ ( t$_{r}$ t ∗ ) β$_{m}$ + ζ$_{m}$ t − 1 for ψ < ψ (MD) 3 ( t ) (19) ψ (MD) 3 ( t ) = ( t$_{eq}$ t ) 2 ζ$_{m}$ t$_{r}$ t$_{eq}$ 2 ζ$_{r}$ ( t ∗ t$_{r}$ ) 2 ζ$_{m}$ . (22) ψ (MD) 2 ( t ) = t$_{eq}$ t 2 ζ$_{m}$ ( t$_{r}$ t$_{eq}$ ) 2 ζ$_{r}$ , (21) ψ (MD) 1 ( t ) = ( t$_{eq}$ t ) 2 ζ$_{m}$ , (20) in the second matter era, where Here β is the constant related to the string network ( β$_{r}$ ≃ 1 . 1 , β$_{m}$ ≃ 1 . 2), and t$_{eq}$ denotes the matter-radiation equality epoch. We have converted N , distribution in the volume V , to dN/dψ = N ( t, ψ ) / ( V/ξ $^{2}$), which is the distribution per unit length. The derivation of the above expression of dN/dψ is described in Appendix A. Let us consider the physical meaning of this distribution function. It is not difficult to consider how the number of kinks in the horizon changes as time goes on. When a kink is born, its sharpness ranges from 0 to 1, but the typical value is O (0 . 1). Therefore, the kink distribution has a peak at ψ ∼ O (0 . 1) at the very early stage. Then kinks are made blunt by the cosmic expansion. Thus newly produced kinks are sharp (large ψ ) and old ones are blunt (small ψ ). Therefore, the peak value of ψ is much smaller than 0.1 at the late stage. This peak consists of the oldest kinks. Fig. 1 roughly sketches the shape of the distribution function in the second matter era. It should be noted that the above distribution is derived without considering gravitational backreaction. Since the most abundant kinks are extremely blunt, they might be influenced by the gravitational backreaction and disappear. However, it is difficult and beyond the scope of this paper to take backreaction into account. We will comment on this issue in Appendix C. 8