FIG. 1: The distribution function of kinks on infinite strings produced at the end of the inflation in the second matter era. ψ (MD) 1 , ψ (MD) 2 , ψ (MD) 3 are given by Eqs. (20)-(22). IV. THE SPECTRUM OF GRAVITATIONAL WAVES FROM KINKS Given the energy-momentum tensor of the source, one can calculate the energy of the GW in the direction of ˆ k with frequency ω as [19] Thus by substituting the energy-momentum tensor (5), we find the energy of the GW after computing integral I ± in Eq. (6). It is given by 9 dE d Ω ( ˆ k ) = 2 G ∫ ∞ 0 dωω 2 ( T µν $^{∗}$( k ) T$_{µν}$ ( k ) − 1 2 | T µ $_{µ}$( k ) | 2 ) = 2 G Λ$_{ij.lm}$( ˆ k ) ∫ ∞ 0 dωω $^{2}$T ij $^{∗}$( k ) T $^{lm}$( k ) (23) Λ$_{ij,lm}$( ˆ k ) ≡ δ$_{il}$δ$_{jm}$ − 2 ˆ k$_{j}$ ˆ k$_{m}$δ$_{il}$ + 1 2 ˆ k$_{i}$ ˆ k$_{j}$ ˆ k$_{l}$ ˆ k$_{m}$ − 1 2 δ$_{ij}$δ$_{lm}$ + 1 2 δ$_{ij}$ ˆ k$_{l}$ ˆ k$_{m}$ + 1 2 δ$_{lm}$ ˆ k$_{i}$ ˆ k$_{j}$ (24) dE dωd Ω ( k ) = Gµ $^{2}$ω2 4 ( | ⃗ I$_{+}$ | $^{2}$| ⃗ I − | 2 + | ⃗ I$_{+}$ · ⃗ I − | 2 − | ⃗ I$_{+}$ · ⃗ I ∗ − | 2 − | ⃗ I$_{+}$ | $^{2}$| ˆ k · ⃗ I − | 2 − | ⃗ I − | $^{2}$| ˆ k · ⃗ I$_{+}$ | 2 + | ˆ k · ⃗ I$_{+}$ | $^{2}$| ˆ k · ⃗ I − | 2 − ( ⃗ I$_{+}$ · ⃗ I ∗ $_{−}$)( ˆ k · ⃗ I ∗ $_{+}$)( ˆ k · ⃗ I − ) − ( ⃗ I ∗ + · ⃗ I − )( ˆ k · ⃗ I$_{+}$ )( ˆ k · ⃗ I ∗ $_{−}$) + ( ⃗ I$_{+}$ · ⃗ I − )( ˆ k · ⃗ I ∗ $_{+}$)( ˆ k · ⃗ I ∗ $_{−}$) + ( ⃗ I ∗ + · ⃗ I ∗ $_{−}$)( ˆ k · ⃗ I$_{+}$ )( ˆ k · ⃗ I − ) ) (25)