The friction-era VOS amplitude of a $\mathbb{Z}_2$ string network from nematic disclination data

Abstract

A tangle of line defects coarsens as the mean spacing $L$ between neighbouring lines grows. In a viscous medium the motion is overdamped, and the velocity-dependent one-scale (VOS) model predicts a late-time attractor $L^{2}=\mathcal{A}\,\ell_d\,t$, with $\ell_d=T/\Gamma$ the ratio of line tension to drag and $\mathcal{A}$ a dimensionless amplitude. The growth law $L\propto t^{1/2}$ holds for every value of the three model parameters---the momentum parameter $k\le1$, the sink coefficient $\tilde{c}$, and the curvature ratio $\lambda\equiv R/L$---since all three enter only through $\mathcal{A}=\kappa(\kappa+\tilde{c})$, $\kappa\equiv k/\lambda$. Thus, the exponent constrains none of them, and the amplitude is the only quantity a density history can deliver. We measure it from the disclination data of Chuang, Turok and Yurke on a nematic liquid crystal, whose companion measurement of loop collapse fixes $\ell_d$ on the same samples, canceling the 5CB material constants. Treating the unmatched per-quench $\ell_d$ as a nuisance parameter with a Gaussian prior and marginalizing it analytically, we obtain $\mathcal{A}=10.0^{+1.3}_{-1.1}$ from the three quenches in the $234~\mu$m cell; the fourth, the only one in the thinner $158~\mu$m cell, gives $\mathcal{A}=3.0^{+0.8}_{-0.6}$ and is fitted separately. Converting either into $\tilde{c}$ requires $\lambda$, which these data do not determine, so the result is a curve, $\tilde{c}(\lambda)=\mathcal{A}\lambda/k-k/\lambda$, not a number. At $\lambda=1$ with $k\le1$ they give $\tilde{c}\ge9.0$ and $\tilde{c}\ge2.0$, against $\tilde{c}=0.23$--$0.57$ from relativistic $U(1)$ simulations, values reached only at $\lambda\simeq0.33$--$0.35$ and $0.62$--$0.68$. We know of no VOS calibration for a cosmological $\mathbb{Z}_2$ network, so we cannot attribute the excess to topology, and we list what a repeat experiment must measure.