AI 中文总结
该研究通过向列液晶向错数据测量了线缺陷粗化的VOS模型振幅，得到不同样品池的振幅值，推导了汇系数与曲率比的关系，发现其结果与相对论U(1)模拟存在差异，为宇宙学Z₂弦网络研究提供了实验校准参考。
AI 中文摘要
线缺陷缠结会随着相邻线间平均间距$L$增大而粗化。在粘性介质中，线缺陷运动过阻尼，速度相关的单尺度（VOS）模型预言后期吸引子满足$L^2=\boldsymbol{\text{A}}\boldsymbol{\text{l}}_d t$，其中$\boldsymbol{\text{l}}_d=T/\boldsymbol{\text{\textbackslash Gamma}}$是线张力与拖拽的比值，$\boldsymbol{\text{A}}$为无量纲振幅。该模型的三个参数——动量参数$k\boldsymbol{\text{\textbackslash le}}1$、汇系数$\tilde{c}$、曲率比$\boldsymbol{\text{\textbackslash lambda}}\boldsymbol{\text{\textbackslash equiv}}R/L$——均仅通过$\boldsymbol{\text{A}}=\boldsymbol{\text{\textbackslash kappa}}(\boldsymbol{\text{\textbackslash kappa}}+\tilde{c})$（$\boldsymbol{\text{\textbackslash kappa}}\boldsymbol{\text{\textbackslash equiv}}k/\boldsymbol{\text{\textbackslash lambda}}$）参与，因此$L\boldsymbol{\text{\textbackslash prop}}t^{1/2}$的增长规律对所有参数值均成立，指数不约束任何参数，振幅是密度演化史唯一可确定的量。我们通过Chuang、Turok和Yurke向列液晶的向错数据测量该振幅，其配套的环坍缩测量在相同样品上确定了$\boldsymbol{\text{l}}_d$，抵消了5CB材料常数。将每次淬火中未匹配的$\boldsymbol{\text{l}}_d$视为带高斯先验的冗余参数并解析边缘化，得到234μm样品池的三次淬火对应的$\boldsymbol{\text{A}}=10.0^{+1.3}_{-1.1}$；更薄的158μm样品池仅有的一次淬火给出$\boldsymbol{\text{A}}=3.0^{+0.8}_{-0.6}$，单独拟合。将$\boldsymbol{\text{A}}$转换为$\tilde{c}$需要$\boldsymbol{\text{\textbackslash lambda}}$，而本数据无法确定$\boldsymbol{\text{\textbackslash lambda}}$，因此结果是曲线$\tilde{c}(\boldsymbol{\text{\textbackslash lambda}})=\boldsymbol{\text{A}}\boldsymbol{\text{\textbackslash lambda}}/k - k/\boldsymbol{\text{\textbackslash lambda}}$而非数值。在$\boldsymbol{\text{\textbackslash lambda}}=1$且$k\boldsymbol{\text{\textbackslash le}}1$时，对应$\tilde{c}\boldsymbol{\text{\textbackslash ge}}9.0$和$\tilde{c}\boldsymbol{\text{\textbackslash ge}}2.0$，而相对论$\text{U}(1)$模拟给出$\tilde{c}=0.23$--$0.57$，仅在$\boldsymbol{\text{\textbackslash lambda}}\boldsymbol{\text{\textbackslash eq}}0.33$--$0.35$和$0.62$--$0.68$时达到该范围。目前尚无宇宙学$\text{Z}_2$网络的VOS校准，无法将该差异归因于拓扑，我们列出了重复实验必须测量的量。
英文摘要
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 $\ell_d=T/Γ$ 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 $λ\equiv R/L$---since all three enter only through $\mathcal{A}=κ(κ+\tilde{c})$, $κ\equiv k/λ$. 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 $234~μ$m cell; the fourth, the only one in the thinner $158~μ$m cell, gives $\mathcal{A}=3.0^{+0.8}_{-0.6}$ and is fitted separately. Converting either into $\tilde{c}$ requires $λ$, which these data do not determine, so the result is a curve, $\tilde{c}(λ)=\mathcal{A}λ/k-k/λ$, not a number. At $λ=1$ 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 $λ\simeq0.33$--$0.35$ 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.
Comments16 pages, 6 figures, the numerical analysis files for the production of the figures can be found in the following github repository: https://github.com/Dimitrios1993/friction-era-vos-amplitude-z2-nlc
