Scaling density of axion strings in terasite simulations

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 $ζ_\text{w}$ and $ζ_\text{r}$, 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 $\hatζ_{\text{w},*} = 1.220(57)$ and $\hatζ_{\text{r},*} = 1.491(93)$. The two measures are related by the RMS velocity, which we estimate to be $\hat{v}_{*} = 0.5705(93)$. 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 $ζ_\text{w}$ and $ζ_\text{r}$ 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 $ζ_\text{w}$ and $ζ_\text{r}$ identified as fixed points are approached. In the case of $ζ_\text{r}$, the growth stops for simulations which started close to the fixed point length density. The difference between $ζ_\text{w}$ and $ζ_\text{r}$ can be understood to result from the continuing velocity evolution. Our results indicate that the growth of $ζ_\text{w}$ 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.