Approach to scaling in axion string networks

We study the approach to scaling in axion string networks in the radiation era, through measuring the root-mean-square velocity $v$ as well as the scaled mean string separation $x$. We find good evidence for a fixed point in the phase-space analysis in the variables $(x,v)$, providing a strong indication that standard scaling is taking place. We show that the approach to scaling can be well described by a two parameter velocity-one-scale (VOS) model, and show that the values of the parameters are insensitive to the initial state of the network. The string length has also been commonly expressed in terms of a dimensionless string length density $\zeta$, proportional to the number of Hubble lengths of string per Hubble volume. In simulations with initial conditions far from the fixed point $\zeta$ is still evolving after half a light-crossing time, which has been interpreted in the literature as a long-term logarithmic growth. We show that all our simulations, even those starting far from the fixed point, are accounted for by a VOS model with an asymptote of $\zeta_*=1.20\pm0.09$ (calculated from the string length in the cosmic rest frame) and $v_* = 0.609\pm 0.014$.