V. THE STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES FROM KINKS We have found Eqs. (45),(46) and (47) as the total energy radiated per unit time in a horizon from kinks on an infinite string. Now we are in a position to calculate the density parameter of GWs defined by where ρ$_{c}$ denotes the critical energy density of the present Universe. Noting that the energy density of GWs decreases as a − 4 and the frequency redshifts as a − $^{1}$, we get where a ( t ) is the scale factor and a$_{0}$ represents its present value. Using Eqs. (45),(46) and (47), we get the spectrum as where t$_{0}$ and T$_{0}$ denote the present age and temperature of the Universe, T$_{r}$ is the reheating temperature and H ∗ is the Hubble parameter at the end of inflation. 6 This formula has 6 Here we neglected the effect of cosmological constant. The GW spectrum will be slightly modified if the cosmological constant is taken into account. Detailed estimation of this effect is beyond the scope of this paper since it needs a simulation of cosmic string network evolution in the cosmological constant dominated Universe. 14 Ω$_{gw}$( ω ) ≡ ω ρ$_{c}$ dρ dω ( ω ) , (48) Ω$_{gw}$( ω ) ∼ 1 ρ$_{c}$ ∫ t$_{0}$ t ∗ dt 1 t 3 ( ω $_{′}$dP dω ′ ) ∣ ∣ ω $^{′}$= ω × a$_{0}$/a ( t ) ( a ( t ) a$_{0}$ ) 4 , (49) Ω$_{gw}$( ω ) ∼     $^{60}$$^{π}$$^{(}$$^{Gµ}$$^{)}$$^{2}$$^{(}$$^{ωt$_{0}$}$$^{)}$$^{C$_{m}$}$ for t − 1 0 < ω < ω$_{1}$ 60 π ( Gµ ) 2 ( Ω$_{r}$ Ω$_{m}$ ) − 3 D/A$_{r}$ ( ωt$_{0}$ ) C$_{r}$ for ω$_{1}$ < ω < ω$_{2}$ 60 π ( Gµ ) 2 Ω$_{r}$ Ω$_{m}$ − 4 D/A$_{m}$ ( T$_{0}$ M$_{pl}$ ) 4 D/A$_{m}$ ( T$_{r}$ M$_{pl}$ ) − 4 D/A$_{m}$ ( ωt$_{0}$ ) C$_{m}$ for ω$_{2}$ < ω < ω$_{3}$ 60 π ( Gµ ) $^{2}$( ωt$_{0}$ ) − 2 / 3 ( Ω$_{r}$ Ω$_{m}$ ) 3 B$_{m}$/ 2 − 4 B$_{r}$ T$_{0}$ M$_{pl}$ 2 B$_{r}$ ( T$_{r}$ M$_{pl}$ ) 2 B$_{m}$ − 2 B$_{r}$ ( H ∗ M$_{pl}$ ) − B$_{m}$ for ω > ω$_{3}$ , (50) ω$_{1}$ = ω (MD) 1 ( t$_{0}$ ) = ( Ω$_{r}$ Ω$_{m}$ ) 3 A$_{m}$/ 2 t − 1 0 , (51) ω$_{3}$ = ω (MD) 3 ( t$_{0}$ ) = ( Ω$_{r}$ Ω$_{m}$ ) 3 A$_{m}$/ 2 − 2 A$_{r}$ T$_{0}$ M$_{pl}$ 2 A$_{r}$ T$_{r}$ M$_{pl}$ 2 A$_{m}$ − 2 A$_{r}$ ( H ∗ M$_{pl}$ ) − A$_{m}$ t − 1 0 (53) ω$_{2}$ = ω (MD) 2 ( t$_{0}$ ) = Ω$_{r}$ Ω$_{m}$ 3 A$_{m}$/ 2 − 2 A$_{r}$ ( T$_{0}$ M$_{pl}$ ) 2 A$_{r}$ ( T$_{r}$ M$_{pl}$ ) − 2 A$_{r}$ t − 1 0 , (52) where