The scaling density of axion strings

In the QCD axion dark matter scenario with post-inflationary Peccei-Quinn symmetry breaking, the number density of axions, and hence the dark matter density, depends on the length of string per unit volume at cosmic time $t$, by convention written $\zeta/t^2$. The expectation has been that the dimensionless parameter $\zeta$ tends to a constant $\zeta_0$, a feature of a string network known as scaling. It has recently been claimed that in larger numerical simulations $\zeta$ shows a logarithmic increase with time, while theoretical modelling suggests an inverse logarithmic correction. Either case would result in a large enhancement of the string density at the QCD transition, and a substantial revision to the axion mass required for the axion to constitute all of the dark matter. With a set of new simulations of global strings we compare the standard scaling (constant-$\zeta$) model to the logarithmic growth and inverse-logarithmic correction models. In the standard scaling model, by fitting to linear growth in the mean string separation $\xi = t/\sqrt{\zeta}$, we find $\zeta_0 = 1.19 \pm 0.20$. We conclude that the apparent corrections to $\zeta$ are artefacts of the initial conditions, rather than a property of the scaling network. The residuals from the constant-$\zeta$ (linear $\xi$) fit also show no evidence for logarithmic growth, restoring confidence that numerical simulations can be simply extrapolated from the Peccei-Quinn symmetry-breaking scale to the QCD scale. Re-analysis of previous work on the axion number density suggests that recent estimates of the axion dark matter mass in the post-inflationary symmetry-breaking scenario we study should be increased by about 50%.