Biased domain walls: faster annihilation, weaker gravitational waves

TL;DR

This work analyzes biased domain-wall networks arising from a real scalar field with a slight $Z_2$-symmetry breaking via $V_{breaking}=\epsilon\chi^3$, yielding $V_{bias} \approx 2\epsilon v^3$. Through high-resolution CosmoLattice simulations, the authors find the annihilation time scales as $t_{ann} \propto V_{bias}^{-2/3}$ (or $\tau_{ann} \propto (\lambda v/\epsilon)^{2/3}$ in dimensionless form), challenging the conventional $t_{ann} \propto 1/V_{bias}$ intuition and showing that walls decay earlier than expected. This earlier decay suppresses the GW signal for a fixed bias and reshapes the GW spectrum: the IR part is reduced while the UV part gains power, and a plateau in the far UV persists, with the peak frequency and amplitude depending on $\epsilon$, $v$, $\lambda$, and $g_*(T_{ann})$. The results have direct implications for PTA interpretations of DWs and motivate higher-resolution studies to probe smaller biases and refine observational predictions. Overall, the paper demonstrates that even small symmetry-breaking biases can significantly alter DW dynamics and their GW phenomenology, affecting their viability as PTA sources.

Abstract

We study the evolution of domain wall networks and their phenomenological implications in a model of a real scalar $χ$, where a $Z_2$-symmetry is slightly broken by a potential bias $V_{bias}$. It is demonstrated that the latter triggers domain wall annihilation considerably earlier than previously thought. Namely, we observe that the scaling relation $t_{ann} \propto 1/V^{2/3}_{bias}$ for the annihilation time $t_{ann}$ fits to the simulation data better than a commonly assumed $t_{ann} \propto 1/V_{bias}$. As a result, the energy density of gravitational waves produced by the network of biased domain walls, for a given tiny $V_{bias}$, is suppressed compared to naive expectations. The spectral shape of gravitational waves is similar to that resulting from unbiased domain walls, but with more power in the close-to-maximum ultraviolet part. In the far ultraviolet region, the spectrum of gravitational waves becomes nearly flat; such a plateau has been recognised earlier in the case of unbiased walls. In our investigation we mainly focus on the symmetry breaking potential $V_{breaking} \propto χ^3$, and argue that no significant modifications of the domain walls evolution take place if one includes higher powers of $χ$.

Figures (11)

• Figure 1: Snapshots of DW evolution obtained in the case of dimensionless bias parameter $\epsilon=0.01$. DWs are shown with green, while the regions filled with true and false vacuum are shown with white and red, respectively.
• Figure 2: Histograms showing distribution of the DW network over the area $S$ for three different values of the bias parameter $\epsilon$. Simulations have been carried out on $2048^3$ lattice.
• Figure 3: Scaling parameter $\xi$ defined in Eq. \ref{['scaling']} is shown for different values of the bias parameter $\epsilon$. Simulations have been carried out with $1024^3$ lattice.
• Figure 4: Evolution of the false vacuum fraction ${\cal F}_{fv}$ is shown for different values of the dimensionless bias parameter $\epsilon$. The results have been obtained with $1024^3$ and $2048^3$ lattices and assuming vacuum initial conditions with the momentum cutoff $k_{cut}=1$ (top panel), $k_{cut}=5$ (bottom left panel), and $k_{cut}=0.3$ (bottom right panel). We have used Eq. \ref{['fvfp']} to fit numerical data.
• Figure 5: Dependence of the annihilation time $\tau_{ann}$ on the bias parameter $\epsilon$ is demonstrated for different values of the momentum cutoff $k_{cut}$ imposed on vacuum initial conditions.