Effects of biases in domain wall network evolution

We study the evolution of various types of biased domain wall networks in the early universe. We carry out larger numerical simulations than those currently available in the literature and provide a more detailed study of the decay of these networks, in particular by explicitly measuring velocities in the simulations. We also use the larger dynamic range of our simulations to test previously suggested decay laws for these networks, including an ad hoc phenomenological fit to earlier simulations and a decay law obtained by Hindmarsh through analytic arguments. We find the latter to be in good agreement with simulations in the case of a biased potential, but not in the case of biased initial conditions.

Figure 1

The evolution of the network density (specifically $A/V\propto \rho$) and velocity (specifically $\gamma v$) for the four cases described in the main text. The plotted quantities are the result of averaging over ten ${8192}^2$ simulations, and the fits discussed in the text were done in the last ten percent of the simulation times.

Figure 2

Box snapshots of ${1024}^2$ simulations of cases A (top), B (middle) and C (bottom), at a conformal time $\eta =20$. The three simulations are clearly distinguishable.

Figure 3

Box snapshots of ${1024}^2$ simulations of cases A (top), B (middle) and C (bottom), at a conformal time $\eta =512$. Statistically the three snapshots are now quite similar.

Figure 4

The evolution of the network density (specifically $A/V$) and velocity (specifically $\gamma v$) in the matter-dominated era for the cases $b=1.0$ (unbiased case), $b=0.8$ (weak bias) and $b=0.6$ (strong bias) described in the main text. The plotted quantities are the result of averaging over five simulations. Also plotted are the fitting functions discussed in the main text.

Figure 5

Box snapshots of a fraction of ${2048}^2$ simulations of cases $b=1$ (unbiased case, top), $b=0.8$ (middle) and $b=0.6$ (bottom), at a conformal time $\ln \eta =2.25$ when the latter network is about to disappear.

Figure 6

The evolution of the network density (specifically $A/V$) and velocity (specifically $\gamma v$) in the matter-dominated era for the cases $\mu =0$ (unbiased), $\mu =0.03$ and $\mu =0.1$ described in the main text. The plotted quantities are the result of averaging over five realizations. Expected decay time scales are shown by dashed lines, and analytic fits by dotted lines.

Figure 7

Box snapshots of a fraction of ${2048}^2$ simulations of cases $\mu =0$ (unbiased case, top), $\mu =0.03$ (middle) and $\mu =0.1$ (bottom), at a conformal time $\ln \eta =5$ when the latter network is about to disappear.