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 ${\zeta }_{*}=1.20\pm 0.09$ (calculated from the string length in the cosmic rest frame) and $v_{*}=0.609\pm 0.014$.

Figure 1

Comparison of the mean string separation defined from the winding length estimator ${\xi }_w$ (solid black) and the rest-frame estimator ${\xi }_r$ (solid blue) as presented after Eq. (29), from a single simulations with correlation length $l_{\varphi }\eta =5$ and $s=1$ (upper panel) and $s=0$ (lower panel). The dashed black line corresponds to the winding estimator ${\xi }_w$ modified via Eq. (27).

Figure 2

Ratios of the winding length estimator ${\ell }_w$ to rest-frame estimator ${\ell }_r$ (solid line) for different correlation lengths. The lines correspond to a single simulations. In dashed, we plot $(1-v_s^2{)}^{1/2}$.

Figure 3

Mean string separation ${\xi }_r$ from simulations with $s=1$ (top panel) and $s=0$ (bottom panel) with initial field correlation lengths $l_{\varphi }\eta =5$ (black), $l_{\varphi }\eta =10$ (red), $l_{\varphi }\eta =20$ (blue) and $l_{\varphi }\eta =40$ (green—only for $s=1$). The solid line represents the mean over realizations of $\xi$ at each time, with the shaded regions showing the $1\text{−}\sigma$ variation. The vertical green line corresponds to the end of the core growth period, after which the system is evolved with the physical equations of motion in the $s=1$ case.

Figure 4

Comparison of velocity estimators presented in Sec. 2b for a simulation with correlation length $l_{\varphi }\eta =5$ and $s=1$ (upper panel) and $s=0$ (lower panel). The values of the scalar field velocity estimator $v_s$ (26), the equation of state velocity estimator $v_{\omega }$ (24) and the Lagrangian-derived velocity estimator $v_L$ (22) are shown in black, red, and blue, respectively.

Figure 5

Velocities from scalar field estimator $v_s$ (26) from simulations with $s=1$ (top panel) and $s=0$ (bottom panel) with initial field correlation lengths $l_{\varphi }\eta =5$ (black), $l_{\varphi }\eta =10$ (red), $l_{\varphi }\eta =20$ (blue) and $l_{\varphi }\eta =40$ (green—only for $s=1$). The solid line represents the mean over realizations at each time, with the shaded regions showing the $1\text{−}\sigma$ variation. The vertical green line corresponds to the end of the core growth period.

Figure 6

Phase space plot for $v_s$ and $x_{\mathrm{r}}$ for $s=1$ (top panel) and $s=0$ (bottom panel), and the same color scheme correspond to different initial correlation lengths, as the previous figure. Larger dots are plotted every 20 cosmic time units, starting at cosmic time $t\eta =4$. As time increases, all curves spiral in toward an apparent fixed point, analysed in Sec. 5.

Figure 7

Phase plane for $s=1$ (top), $s=0$ (bottom), with stream lines of the best-fit values of the VOS model parameters shown in Table 3, with $t_{\mathrm{fit}}\eta =50$. The color scheme is the same as in previous figures. The larger markers correspond to the same points as in Fig. 6, with empty circles denoting points with $t_{\mathrm{fit}}\eta <50$.

Figure 8

Relative difference of the VOS prediction and the simulation data of the dimensionless string separation $x$ and rms velocity $v$. The shaded bands represent errors propagated from the simulations’ energy estimators.

Figure 9

String network evolution expressed in terms of the rest-frame length density parameter $\zeta$ and RMS velocity $v$, plotted against $\log (t\eta )$, with VOS models (dotted line) that correspond to best fit values for each $l_{\varphi }$ shown in Table 2. The prediction of the VOS model is plotted from $t_{\mathrm{fit}}$ on. The horizontal dashed black line and grey band in the top panel show the mean and uncertainly obtained from our previous analysis [40], translated from the universe-frame length used in that paper by multiplying by $(1-v_{*}^2{)}^{-1/2}$. The solid purple line and shaded purple bands are the mean and uncertainly obtained from the analysis in this work. The vertical green dashed line marks the end of the core growth phase, and the fit start time $t_{\mathrm{fit}}$ is indicated by the vertical grey dashed line.