content_image/339102.jpg content_image/339103.jpg content_image/339104.jpg content_image/339105.jpg content_image/339106.jpg
where 

Here, we set D ≡ β$_{r}$ζ$_{m}$ − β$_{m}$ζ$_{r}$ ≃ 0 . 11. 

As a result, assuming that the powers of GW from different kinks are roughly same as far as their sharpnesses are in the same order, (in other words, assuming that the quantities concerned with kinks, such as b ′′ $_{s}$, except their sharpness, are roughly same) we can estimate the total power of GWs with frequencies ∼ ω from all of the kinks in a horizon as, 

<img src='content_image/339102.jpg'>

<img src='content_image/339103.jpg'>

<img src='content_image/339104.jpg'>

<img src='content_image/339105.jpg'>

The first factor denotes the power of GWs from one kink. The second factor denotes the number of kinks which satisfy ψ ∼ ψ$_{max}$ ( ω, t ) per unit length. The third factor is length of an infinite string in a horizon. Then we finally obtain 

in the first matter era, 

in the radiation era, and 

<img src='content_image/339106.jpg'>

