On the Evolution of Abelian-Higgs String Networks

TL;DR

This study investigates the scale-invariant evolution of Abelian-Higgs string networks using large-scale field-theory simulations in flat and expanding backgrounds, extracting the true invariant string length and rms velocity to test the velocity-dependent one-scale (VOS) model. The authors develop a robust diagnostic toolkit, including a novel fat-string algorithm, and show that loop production, supplemented by limited massive radiation, provides a consistent description of network decay across backgrounds. Analyses of flat-space, radiation-era, matter-era, and global-string systems reveal quantitative fits with VOS parameters (notably the loop chopping efficiency ${\tilde c}$ and radiation coefficients $\Sigma$ and $L_d$) that reproduce the observed evolution of the correlation length and velocity, while highlighting the role of proto-loops in energy losses. The results support the standard cosmological picture where loop production and gravitational radiation (and, for global strings, Goldstone boson radiation) govern scaling, with important implications for predicting cosmic-ray and gravitational signatures and for informing extrapolations from simulations to cosmological scales.

Abstract

We study the evolution of Abelian-Higgs string networks in large-scale numerical simulations in both a static and expanding background. We measure the properties of the network by tracing the motion of the string cores, for the first time estimating the rms velocity of the strings and the invariant string length, that is, the true network energy density. These results are compared with a velocity-dependent one scale model for cosmic string network evolution. This incorporates the contributions of loop production, massive radiation and friction to the energy loss processes that are required for scaling evolution. We use this analysis as a basis for discussing the relative importance of these mechanisms for the evolution of the network. We find that the loop distribution statistics in the simulations are consistent with the long-time scaling of the network being dominated by loop production. Making justifiable extrapolations to cosmological scales, these results appear to be consistent with the standard picture of local string network evolution in which loop production and gravitational radiation are the dominant decay mechanisms.

Figures (18)

• Figure 1: String network formation in the Abelian-Higgs model using gradient flow (diffusive) evolution. Given random initial phases, a symmetry-breaking phase transition occurs, strings form and then begin to evolve in a scale-invariant manner (the correlation length is $L\propto t^{1/2}$). This dissipative evolution is used to create the initial configuration for a string network with a specified $L$ for subsequent relativistic evolution.
• Figure 2: String network evolution during the radiation era. Positions for the string cores were found using phase information and the trajectories connected appropriately. Note that lattice discretisation effects have been reduced by smoothing. The respective times for the advancing evolution are the conformal times $t= 24, 32, 40, 48$, by which time the horizon is comparable to the box size.
• Figure 3: Correction to the measured velocity required for the velocity-position diagnostic to obtain the true velocity (solid blue line). This data represents the velocity inferred from the growth in the kinetic energy for an initially static network (see Fig. \ref{['figv_normalization']}). Tests of the diagnostic for a distribution of straight string segments (circles) showed almost direct proportionality, whereas idealized small and large rings (red triangles) also underestimated the true velocity.
• Figure 4: The string velocity (solid blue line) inferred from the growth in kinetic energy of a network field theory simulation in flat space (shown as red '$+$' data points). The actual proportion of kinetic energy relative to the total energy is the red dashed line. The raw velocity diagnostic before normalisation (circular data points) underestimates the true rms network velocity by about 15%.
• Figure 5: Flat spacetime simulation results for a series of simulations with a grid size of $250^3$ or above. The top panel shows the spatial correlation length $L$ as a function of time. The dotted lines give the static correlation length $L_{\rm s}$ inferred from the static string density, rather than the total invariant string length used for $L$. The lower panel shows the corresponding average rms velocity $v$ as a function of time (with the same colour coding). The dotted lines give the raw velocity before applying the velocity corrections plotted in Fig. \ref{['figv_correction']}.