Dynamics of Biased Domain Walls: The Rocket Effect

Technical Summary: Dynamics of Biased Domain Walls: The Rocket Effect

Problem Statement
Domain walls, topological defects arising from the spontaneous breaking of discrete symmetries, pose a significant cosmological problem. Due to their $(2+1)$-dimensional sheet-like structure, their energy density decreases more slowly than radiation or matter during cosmic expansion. Unless suppressed by instability or specific mechanisms, domain wall networks would eventually dominate the Universe's energy density, contradicting observational constraints from the cosmic microwave background and standard cosmology.

While mechanisms such as vacuum population bias and explicit symmetry breaking (introducing vacuum energy differences, $\Delta V$) are known to drive network decay, the origin of bias in theories with exactly degenerate vacua ($\Delta V = 0$) remains a subject of investigation. Previous studies suggested that asymmetries in the shape of the scalar potential near its local maximum (parameterized by the third derivative, $d^3V$) could induce biased evolution. This paper investigates whether the vacuum dependence of the scalar field mass ($\Delta m^2 \neq 0$), rather than potential barrier asymmetry, is the primary driver of such bias.

Methodology
The authors employ a combination of analytical arguments and numerical simulations across three distinct regimes:

1. Model Setup: Four distinct scalar field potential models are constructed to systematically disentangle two types of asymmetry:

   • Model A: Symmetric potential ($d^3V = 0, \Delta m^2 = 0$).
   • Model B: Asymmetric barrier, degenerate masses ($d^3V \neq 0, \Delta m^2 = 0$).
   • Model C: Asymmetric barrier and mass splitting ($d^3V \neq 0, \Delta m^2 \neq 0$).
   • Model D: Symmetric barrier, mass splitting ($d^3V = 0, \Delta m^2 \neq 0$).
     All models are normalized to have identical domain wall tensions ($\sigma = 4/3$).

2. Analytical Framework: In Minkowski spacetime, the authors analyze the kinematic constraints on scalar radiation emitted by an oscillating domain wall. They derive that modes with frequencies $\omega$ satisfying $m_R < \omega < m_L$ (where $m_L$ and $m_R$ are the masses in the left and right vacua) can propagate in the lower-mass vacuum but remain evanescent in the higher-mass vacuum. This creates an anisotropic momentum flux.

3. Numerical Simulations:

   • 1+1 Dimensions: Simulations of perturbed planar walls track radiation spectra, emission bias, and wall recoil velocity using a staggered leapfrog scheme with absorbing boundary conditions.
   • 2+1 Dimensions: Simulations of perturbed planar walls with transverse undulations (wiggles) to test the robustness of the recoil mechanism against geometric degrees of freedom.
   • FLRW Cosmology: Simulations of full domain wall networks in radiation- and matter-dominated universes using the Press–Ryden–Spergel (PRS) algorithm to maintain constant comoving wall thickness. Initial conditions are generated via Gaussian white noise with antithetic pairing to minimize statistical biases.

Key Contributions and Results

• The Rocket Effect Mechanism: The paper demonstrates that a non-zero mass splitting ($\Delta m^2 \neq 0$) acts as a kinematic filter. Scalar radiation emitted by an oscillating wall is preferentially transmitted into the vacuum with the smaller scalar field mass. This anisotropic emission generates a net recoil force (the "rocket effect") that drives the domain wall toward the vacuum with the larger scalar field mass.
• Dominance of Mass Splitting over Barrier Asymmetry:
  • In Model B (asymmetric barrier, $\Delta m^2 = 0$), the recoil velocity is negligible ($v \approx 0.0036$ in 1+1D), indicating that potential barrier asymmetry alone produces only a weak bias.
  • In Model D (symmetric barrier, $\Delta m^2 \neq 0$), the recoil velocity is significant ($v \approx 0.096$ in 1+1D), comparable to Model C.
  • This establishes that the vacuum dependence of the scalar field mass is the primary driver of the bias, not the asymmetry of the potential barrier near the local maximum.
• Radiation Spectra: In mass-asymmetric models (C and D), the fundamental shape mode or its harmonics often fall within the frequency window $m_R < \omega < mL$. This results in strong suppression of radiation propagation into the high-mass vacuum, leading to predominantly unidirectional emission and substantial wall acceleration.
• Cosmological Implications:
  • In FLRW simulations, models with $\Delta m^2 \neq 0$ (C and D) exhibit a systematic drift where the volume fraction of the lower-mass vacuum increases over time, leading to network decay.
  • The symmetric model (A) remains stable at $f_- = 0.5$, and the barrier-asymmetric model (B) shows only a weak drift.
  • The rocket effect acts as a complementary source of dynamical bias in theories with non-degenerate vacua. Depending on its direction relative to the conventional volume pressure (driven by $\Delta V$), it can either hasten or delay network decay.

Significance and Claims
The authors claim that this work reinterprets the origin of biased domain wall evolution in theories with asymmetric potentials. They argue that previously observed biases, often attributed to potential barrier asymmetries ($d^3V$), are instead primarily controlled by the vacuum dependence of the scalar field mass ($\Delta m^2$).

The paper posits that the "rocket effect" constitutes a generic mechanism for resolving the cosmological domain wall problem, even in scenarios with exactly degenerate vacuum energy densities. Furthermore, it suggests that in models with non-degenerate vacua, the recoil mechanism provides an additional dynamical contribution that modifies the dependence of the network decay time on the vacuum energy difference, offering a potential explanation for non-trivial decay behaviors observed in recent high-resolution simulations. The study concludes that whenever the scalar excitation spectrum differs between vacua, the rocket effect is a potentially significant ingredient in the dynamics of cosmological domain wall networks.