Small-scale structure on a cosmic-string network

We construct an analytic model of the accumulation of small-scale structure on a cosmic-string network in an expanding universe. The structure is composed of kinks which form when the string segments cross and intercommute, and then decay from stretching and gravitational radiation back reaction. We calculate the linear density of kinks, $K\left(t\right)\mathrm{}$, for a general power-law expansion $a\left(t\right)={\left(\frac{t}{t_0}\right)}^p$; after a period of rapid initial growth, the linear density approaches scaling $K\left(t\right)\mathrm{}\propto t^{-1\mathrm{}}$. We also examine the transition from the radiation- to matter-dominated eras; after the transition between eras, $K\left(t\right)\mathrm{}$ drops before settling once more into scaling. Because of the slow decay of kinks, we find that the kink density is many orders of magnitude larger than one might expect. The predictions of our model are in good quantitative agreement with the kink density observed in the numerical simulations. Our model may explain the observed lack of scaling behavior in the small-scale structure and in the formation of loops observed in numerical simulations.