  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%.
