Scaling density of axion strings in terasite simulations

Abstract

We report on a study of axion string networks using fixed-grid simulations of up to $16384$ points per side. The length of string can be characterised in terms of standard dimensionless parameters ${\zeta }_{\text{w}}$ and ${\zeta }_{\text{r}}$, the length density measured in the cosmic rest frame and the string rest frame, scaled with the cosmic time. The motion of the string can be characterised by the root-mean-square (RMS) velocity of the string. Starting from a range of initial length densities and velocities, we analyse the string network in the standard scaling framework and find evolution towards a fixed point with estimated values ${\hat{\zeta }}_{\text{w},\ast }=1.220(57)$ and ${\hat{\zeta }}_{\text{r},\ast }=1.491(93)$. The two measures are related by the RMS velocity, which we estimate to be ${\hat{v}}_{\ast }=0.5705(93)$. The length density is consistent with previous measurements, while the velocity is about 5% lower. For simulations starting from low enough density, the length density parameters ${\zeta }_{\text{w}}$ and ${\zeta }_{\text{r}}$ remain below their fixed point values throughout, while growing slowly, giving rise to an impression of approximately logarithmic increase with time. This has been proposed as the true long-term behaviour. We find that the growth tends to slow down as the values of ${\zeta }_{\text{w}}$ and ${\zeta }_{\text{r}}$ identified as fixed points are approached. In the case of ${\zeta }_{\text{r}}$, the growth stops for simulations which started close to the fixed point length density. The difference between ${\zeta }_{\text{w}}$ and ${\zeta }_{\text{r}}$ can be understood to result from the continuing velocity evolution. Our results indicate that the growth of ${\zeta }_{\text{w}}$ is a transient appearing at low densities and while the velocity is converging. This highlights the importance of studying the string density and the velocity together, and the preparation of initial conditions.

Comments: 19 pp, 18 figures, typos corrected and link to data added