Charge, domain walls and dark energy

Abstract: One idea to explain the mysterious dark energy which appears to pervade the Universe is that it is due to a network of domain walls which has frozen into some kind of static configuration, akin to a soap film. Such models predict an equation of state with w=P/rho=-2/3 and can be represented in cosmological perturbation theory by an elastic medium with rigidity and a relativistic sound speed. An important question is whether such a network can be created from random initial conditions. We consider various models which allow the formation of domain walls, and present results from an extensive set of numerical investigations. The idea is to give a mechanism which prevents the natural propensity of domain walls to collapse and lose energy, almost to the point where a domain wall network freezes in. We show that when domain walls couple to a field with a conserved charge, there is a parameter range for which charge condenses onto the domain walls, providing a resistive force to the otherwise natural collapse of the walls

• The paper demonstrates that coupling a conserved U(1) charge to domain walls can freeze their evolution to mimic dark energy.
• It uses numerical simulations in models like kinky vortons and O(N)×U(1) to show how charge condensation delays collapse and frustration of the network.
• The study reveals that such charged domain walls can contribute a lasting dark energy component, offering a dynamic alternative to the cosmological constant.

Charged Domain Walls as Candidates for Dark Energy

Overview

The paper "Charge, domain walls and dark energy" (1010.3195) investigates a theoretical framework in which domain wall networks, stabilized by a conserved charge, could mimic the properties of dark energy in the present Universe. The work systematically analyzes field-theoretic models supporting domain wall formation, and demonstrates through extensive numerical simulations that domain wall networks can be partially or fully frustrated by coupling to a field with $U(1)$ symmetry, preventing their rapid collapse. This mechanism significantly alters the network’s late-time evolution and can endow the system with an effective equation of state suitable for driving cosmic acceleration.

Motivation and Background

Conventional paradigms addressing cosmic acceleration postulate a cosmological constant or other forms of dark energy. Domain wall networks offer an alternative: if their energy density dilutes sufficiently slowly, they can source a negative pressure. However, in standard $\mathbb{Z}_2$ models with disconnected vacua, the energy density in domain walls drops as $n \propto t^{-1}$ due to collapse under wall tension—precluding a sustained equation of state $w \approx -2/3$ required for dark energy. The challenge is to construct viable models where networks become "frozen" (frustrated) and maintain a high enough energy density at late times.

Domain Wall Models and Numerical Results

Standard $\mathbb{Z}_2$ Domain Walls

The baseline scenario employs a real scalar field with a spontaneously broken $\mathbb{Z}_2$ symmetry. This yields domain walls that interpolate between degenerate vacua. It is well established—analytically and through simulation—that networks in such models are unstable: the wall density scales as $n \propto t^{-1}$, and coarsening proceeds until only a negligible set of walls remains.

Kinky Vorton Model

The next extension introduces a complex scalar field $\sigma$ (with global $U(1)$ symmetry) coupled quartically to the real scalar $\phi$ supporting domain walls. This construction possesses a $\mathbb{Z}_2 \times U(1)$ symmetry, and the interaction term drives condensation of charge onto domain walls. A critical observation is that the $U(1)$ charge is strictly conserved by Noether’s theorem. Solutions called "kinky vortons" emerge—loops of domain wall carrying a current and charge. The stabilizing effect arises because the charge and current provide a resistive force counteracting tension-induced collapse.

Numerical simulations demonstrate that as the initial homogeneous charge density increases, the domain wall network’s dilution slows markedly—networks with high charge persist far longer, effectively "freezing" in the configuration. Loops forming in these charged networks exhibit properties characteristic of kinky vortons, including quantized winding number and charge-radius relationships.

Charge-Coupled Cubic Anisotropy Model

Generalization to $O(N) \times U(1)$ symmetry (with $N>1$) is realized by coupling a real vector field with discrete vacua to a complex field. Despite the richer vacuum structure admitting domain wall junctions, simulations show the uncharged case again leads to rapid wall decay. Charge condensation, however, robustly preserves walls and junctions for extended periods, with charge and current densities localized exclusively on the walls.

A key technical result is the identification of parameter regimes where phase separation is avoided and the network is maximally frustrated. These outcomes highlight the generality and robustness of the charge-induced freezing mechanism.

Theoretical and Cosmological Implications

The demonstrated mechanism of charge-induced frustration directly addresses the fundamental obstacle in domain wall dark energy scenarios: the rapid dilution and decay of the network. By coupling to a conserved current, the model achieves long-lived, glass-like configurations with an equation of state $w \approx -2/3$. Such networks can, in principle, contribute a significant and persistent dark energy component.

This approach circumvents the need for fine-tuning a cosmological constant and provides a testable, dynamical alternative. The physical realization hinges on the existence of appropriate symmetries and couplings in the early universe field content. The phenomenology of kinky vortons—charged, current-carrying wall loops—leads to signatures distinct from canonical models, possibly observable via gravitational radiation or cosmological imprints.

Open Problems and Future Directions

Several issues remain open:

• The ultimate fate of the network for ultra-long timescales and in fully 3D simulations needs exploration.
• The cosmological viability, including compatibility with CMB and large-scale structure, requires quantification of the allowed parameter space.
• The impact of explicit symmetry breaking and the role of interactions with the Standard Model sector must be analyzed.
• The possibility of direct or indirect observational signatures of kinky vortons warrants further study.

Future developments should focus on extending simulations, refining analytic approximations, and linking theoretical parameters to observable constraints.

Conclusion

The work provides compelling theoretical and numerical evidence that coupling domain wall networks to a field with conserved charge stabilizes the network, enabling a frustrated configuration compatible with dark energy phenomenology. Charge and current tightly localize on domain walls, markedly slowing their coarsening and decay. These findings motivate further investigation into charged topological defect networks as dynamical sources for cosmic acceleration and their potential observational consequences.