Dynamics of $Z_N$ domain walls with bias directions

TL;DR

This work investigates how bias directions in $Z_N$ domain-wall networks influence their nonlinear dynamics and annihilation times in a radiation-dominated universe. Using CosmoLattice with the Press-Ryden-Spergel (PRS) approach, the authors analyze a $Z_3$ system with a biased potential and quantify how annihilation times depend on the relative potential differences encoded by $\zeta$, in addition to the overall bias. They derive semi-analytic fits for the annihilation times of the three wall types: $01$-DWs follow a generalized inverse-power law, $12$-DWs scale roughly as $\sigma/\Delta V$, and $02$-DWs exhibit a more intricate, $\zeta$-dependent behavior with no universal analytic form yet; these dynamics imply possible multi-peak gravitational-wave spectra from stepwise annihilation. The results sharpen predictions for GW signatures from domain walls and motivate extending the analysis to larger $N$, where richer multi-vacuum structures may yield distinct observational imprints.

Abstract

The spontaneous breaking of a discrete symmetry can lead to the formation of domain walls in the early Universe. In this work, we explore the impact of bias directions on the dynamics of $Z_N$ domain walls, mainly focusing on the $N = 3$ model with a biased potential. Utilizing the Press-Ryden-Spergel method, we numerically investigate the dynamics of domain walls with lattice simulations. We find notable differences in the dynamics of domain walls due to bias directions. Our results indicate that the annihilation time depends not only on the vacuum energy difference $δV$ but also on bias directions described by the relative potential difference $ ζ$.

Figures (13)

• Figure 1: (Left) The potential landscape and (right) energy level diagram for the three vacua structure.
• Figure 2: Evolution of the area parameters for each type of domain wall, as well as the total area parameter, with respect to the program time $\tilde{\tau}$, in the absence of bias, $\Xi_1 = \Xi_2 = 0$. Each data point with error bars corresponds to the average over 8 independent realizations.
• Figure 3: Phase distributions of the field at different moments, with model parameters are $\Xi_2=0.02, \Xi_1=-0.01$, are shown in slices. Arranged from top left to bottom right, these correspond to program times $\tilde{\tau}$ of 5, 13, 37, and 256, respectively. The phase values are mapped continuously within the range of $-\pi$ to $\pi$, represented by a gradient of colors.
• Figure 4: Evolution of the area parameters for each type of domain wall, as well as the total area parameter, with respect to the program time $\tilde{\tau}$, when parameters are set to $\Xi_1 = -0.01$ and $\Xi_2 = 0.02$. The plot shows the transition from the scaling regime to the annihilation stage, with the duration of the scaling regime varying across different domain wall types. Each data point with error bars corresponds to the average over 8 independent realizations.
• Figure 5: Dependence of the annihilation time (in program units) of various domain walls on the parameter $\zeta$. The plot shows raw simulation results with error bars. Each data point with error bars corresponds to the average over 8 independent realizations.