content_image/368446.jpg content_image/368443.jpg content_image/368444.jpg content_image/368445.jpg
## Appendix C 

Here we discuss a subtlety related to validity to use the distribution function of kinks [Eqs. (15), (16) and (19)]. These formulae are derived without considering gravitational backreaction. The distribution may be altered if such an effect is taken into account. It may be necessary to define the residual lifetime for blunt kinks and set lower cutoff of sharpness. It is difficult to clarify how we should take into account this effect at this moment. However, at least we can find a crude condition which must be satisfied regardless of the detail of backreaction; the energy of GWs emitted from strings must be less than the string energy. This condition is expressed as 

<img src='content_image/368446.jpg'>

LHS represents the energy emitted from kinks on one infinite string in a Hubble horizon per Hubble time, and RHS denotes the energy of one infinite string in a Hubble horizon. (Note that this is only a necessary condition that the backreaction does not affect kink distribution.) 

First, let us assume that strings emerged in the radiation era. In the radiation era, the condition (C1) is satisfied for 

<img src='content_image/368443.jpg'>

( E ≡ 8 ζ − β, E$_{r}$ ≃ − 0 . 38 , E$_{m}$ ≃ − 0 . 4.) For Eq. (C2) to be satisfied in the whole radiation era, 

<img src='content_image/368444.jpg'>

If we take Gµ ∼ 10 − $^{7}$, this becomes T ∗ ≲ 10 14 GeV. In the matter era, the condition (C1) is written as 

<img src='content_image/368445.jpg'>

The condition that the backreaction is not problematic in the matter era also leads to (C3). Eventually, if we assume that kinks had not appeared until friction domination ended or strings emerged at low temperature at which friction can be neglected, the gravitational backreaction is not important. 

Next, let us assume that strings were born at the end of inflation. It is easy to see that (C1) is satisfied in the first matter era, using Eq. (45). After the first matter era, the 

