connecting a $^{′}$s or b $^{′}$s on two different strings. Let us suppose that a ′ jumps from a ′ − to a ′ + at a kink. Then the “sharpness” of a kink is defined by Thus the norm of the difference between a ′ − and a ′ + is | Δ a $^{′}$| $_{= 2}$√$_{ψ}$$_{. The production rate}$ of kinks is given by [13] where N ( t, ψ ) dψ denotes the number of kinks with sharpness between ψ and ψ + dψ in the volume V , ¯ Δ and γ are constants related to string networks, whose values are [13], Here the subscript r ( m ) denotes the value in the radiation(matter)-dominated era. and we set g ( ψ ) = 0 for ψ < 0 , 1 < ψ . The correlation length of the cosmic strings ξ is given by ξ ≃ γt . Produced kinks are blunted by the expansion of the Universe. The blunting rate of the kink with the sharpness ψ is given by [13] where ζ is a constant which, in the radiation(matter)-dominated era, is given by ζ$_{r}$ ≃ 0 . 09 ( ζ$_{m}$ ≃ 0 . 2). On the other hand, the number of kinks on an infinite string decreases when it self- intercommutes since some kinks are taken away by the loop produced. The decrease rate of kinks due to this effect is given by [13] where η is constant which, in the radiation(matter)-dominated era, is given by η$_{r}$ ≃ 0 . 18 ( η$_{m}$ ≃ 0 . 1). Taking into account these effects, the evolution of the kink number N obeys the following equation, ψ = 1 2 (1 − a ′ + · a ′ $_{−}$) . (7) ˙ N$_{production}$ = ¯ Δ V γ $^{4}$t$_{4}$g ( ψ ) , (8) ¯ Δ$_{r}$ ≃ 0 . 20 , ¯ Δ$_{m}$ ≃ 0 . 21 , γ$_{r}$ ≃ 0 . 3 , γ$_{m}$ ≃ 0 . 55 , (9) g ( ψ ) = 35 256 √ ψ (15 − 6 ψ − ψ $^{2}$) (10) ˙ ψ ψ ∣ stretch = − 2 ζt − $^{1}$, (11) ˙ N N ∣ to loop = − η γt , (12) ˙ N = ¯ Δ V γ $^{4}$t$_{4}$g ( ψ ) + 2 ζ t ∂ ∂ψ ( ψN ) − η γt N. (13) 5