Stable Cosmic Vortons in Bosonic Field Theory

Stable ring solutions supported by the angular momentum caused by superconducting charge and current have been suggested to exist in the gauged $\mathrm{U}(1)\times \mathrm{U}(1)$ field theory. We construct potentially cosmologically relevant solutions using gradient flow for the first time and present the strongest evidence to date that they are stable to axial and, crucially, nonaxial perturbations. More importantly, we illustrate quantitative agreement with semianalytic predictions based on the thin string approximation, which validates world sheet action approaches to their formation and evolution.

Figure 1

Top: the field profiles in the $z=0$ plane, with the red surface showing $\mathrm{Re}(\varphi )$ and the yellow surface showing $\mathrm{Re}(\sigma )$. In this plane, $\mathrm{Im}(\varphi )=0$ and $\mathrm{Im}(\sigma )$ looks similar to $\mathrm{Re}(\sigma )$, acting to keep $|\sigma |$ axisymmetric. The winding of the condensate around the vorton and localization to the string core, $\varphi =0$, are clearly displayed. Bottom: the radii of vorton solutions as a function of $N$ (keeping $Q/N=31.89$ fixed) compared with the predictions from the TSA.

Figure 2

The difference between the radius of the vorton as measured by the core of the string—the measure that we use in the rest of the Letter and for the $x$ axis of this plot—and the radius measured by the peak of the condensate (the larger of the two). The curve clearly exhibits a $1/R$ shape, suggesting that it is caused by curvature effects and will be negligible in cosmological applications.

Figure 3

Simulations of perturbed vortons with 3D numerical field theory. On the top row, we show isosurfaces of the fields at $t=1000$, with $|\varphi |=\frac{3}{5}{\eta }_{\varphi }$ shown in red and $\mathrm{Re}(\sigma )=\frac{1}{10}{\eta }_{\sigma }$ shown in yellow, demonstrating the distortions to the shape of the vorton caused by the $m=2–5$ (left to right) perturbations. The central plots show the position of the string over time in the $y=0$ plane (on the $x>0$ side) and the bottom row of images displays the Fourier transforms, with the black dotted lines indicating the predicted frequencies of oscillation from the TSA. Two of the predicted frequencies are seen, and the third is expected to correspond to an almost entirely longitudinal oscillation for close to chiral vortons and therefore not be apparent in positional data.