Domain Wall as Cosmological Oscillator

TL;DR

This work recasts cosmological domain walls as oscillators driven by a periodic background field, deriving a collective-coordinate equation for wall displacement with natural frequency $\omega_0$ and damping $\gamma$. It analyzes both $Z_2$-breaking and $Z_2$-conserving interactions, showing that resonant forcing $\omega\approx\omega_0$ can amplify wall oscillations to a universal critical deformation and trigger rupture, thereby resolving the domain-wall problem in a broad class of models. The analysis employs a perturbative expansion around the static wall, a collective-coordinate projection, and, in the $Z_2$-conserving case, a Mathieu–Floquet framework to map stability regions and resonance tongues. The results highlight the roles of Hubble damping, radiation damping, and elastic-energy balance in determining the viability and timing of resonant destruction, with implications for early-universe cosmology and potential gravitational-wave signals.

Abstract

In this study, we examine the domain wall within the framework of a cosmological harmonic oscillator. We investigate the interaction between the domain wall and a periodic background field, which can induce perturbations in the oscillatory behavior of the wall. We propose a novel mechanism for resolving the domain wall problem through the phenomenon of resonant oscillation. Resonant oscillation occurs when the frequency of the external driving force aligns with the intrinsic frequency of the domain wall. This synchrony can significantly amplify the amplitude of the oscillation. If the amplitude of oscillation exceeds a predetermined critical deformation threshold, the domain wall may be deconstructed. Furthermore, we demonstrate that this mechanism remains valid in models that preserve discrete symmetry.

Figures (7)

• Figure 1: Stage 1: Static Domain Wall. Without external forces, the domain wall is static, assumed to be at position $x=0$ and extending infinitely along the $y$-axis. Left and right of the wall are vacua $\phi = -v$ and $\phi = +v$, respectively. Here, membrane tension (analogous to the domain wall’s gradient energy) balances the system’s potential energy—this balance keeps the domain wall stable, with no holes forming. Stage 2: Stretching (Elastic Potential Energy Accumulation). A periodic external force pulls local membrane regions (representing the domain wall) left or right. This stretching raises local membrane tension, corresponding to the accumulation of elastic potential energy in the domain wall. Slight bulges or dents appear on the membrane surface, but elastic potential energy $U_{\text{elastic}}$ stays below potential barrier energy $U_{\text{barrier}}$; thus, the domain wall does not rupture. Stage 3: Rupture (Critical Strain and Crack Formation). As stretching proceeds, local membrane tension approaches its breaking strength—a critical state matching $U_{\text{elastic}} \simeq U_{\text{barrier}}$ for the domain wall. At the most stretched position, the membrane ruptures, forming a crack (denoted by //). The crack edge marks the membrane’s rupture site, mapping to the domain wall’s steep field jump region. The crack interior lacks the membrane, corresponding to the vanishing of the domain wall’s transition surface. Stage 4: Expansion and Disintegration. High tension around the crack drives its propagation, turning the crack into a hole. The hole expands outward as excess tension releases. Meanwhile, adjacent small holes from this process merge into larger ones. Eventually, the entire domain wall tears apart; the resulting fragments disintegrate and decay into radiation or gravitational waves.
• Figure 2: The upper left panel shows the double-well potential as a function of the scalar field value ($\phi$). The upper right panel presents the 1D field profile. Here, we consider three values of the field displacement: $u = 0$, $u = 0.4\delta$, and $u = \zeta_c\delta$, which correspond to the black, orange, and red lines, respectively. The lower left panel displays two key energy densities: the deformation energy density $E_{\rm def}$ (blue lines) and the energy difference $E_{\rm def} - V(\phi(x))$ (red lines). The solid, dashed, dot-dashed, and dotted lines in this panel represent results for field strains of $\zeta = 0$, $\zeta = 0.3$, $\zeta = 0.6$, and $\zeta = \zeta_c$ (critical strain), respectively. The lower right panel plots the elastic potential energy density as a function of the field strain $\zeta$. The orange line denotes the potential barrier energy ($U_{\text{barrier}}$) of the double-well potential, while the red-colored region indicates the domain wall rupture regime. All results in this figure are obtained with the parameter values set to $\lambda = 1$ and $v = 1$.
• Figure 3: The oscillating amplitude $\xi(t)$ as a function of time is illustrated in the graph. For this analysis, we set the parameters as follows: $\omega_0 = 1$, $K_0 = 0.05$, and $\omega = 2\omega_0$. The curves shown in blue, red, and green correspond to different damping coefficients: $\gamma = 0.1$, $\gamma = 0.05$, and $\gamma = 0.01$, respectively.
• Figure 4: The Strutt diagram for the damped Mathieu equation is shown in the $(a_M, q_M)$ parameter space, with damping factors set to $\beta=0.05$ (left panel) and $\beta=0.15$ (right panel). The color coding indicates the system's stability: regions where the growth rate $\alpha > 0$, $\alpha = 0$, and $\alpha < 0$ correspond to unstable, critically stable, and asymptotically stable dynamics, respectively.
• Figure 5: Tongue diagram from Floquet analysis in the $(\omega, K_0)$ plane for a fixed $\omega_0=1.0$. The left and right panels correspond to damping factors of $\beta = 0.05$ and $\beta = 0.3$, respectively, with the colorbar indicating the growth rate values.