Slow dynamos and decay of monopole magnetic fields in the early Universe

Previously Liao and Shuryak [\textbf{Phys. Rev C (2008)}] have investigated electrical flux tubes in monopole plasmas, where magnetic fields are non-solenoidal in quark-QCD plasmas. In this paper slow dynamos in diffusive plasma [{\textbf{Phys. Plasmas \textbf{15} (2008)}}] filaments (thin tubes) are obtained in the case of monopole plasmas. In the absence of diffusion the magnetic field decays in the Early Universe. The torsion is highly chaotic in dissipative large scale dynamos in the presence of magnetic monopoles. The magnetic field is given by the Heaviside step function in order to represent the non-uniform stretching of the dynamo filament. These results are obtained outside the junction condition. Stringent limits to the monopole flux were found by Lewis et al [\textbf{Phys Rev D (2000)}] by using the dispute between the dynamo action and monopole flux. Since magnetic monopoles flow dispute the dynamo action, it seems reasonable that their presence leads to a slow dynamo action in the best hypothesis or a decay of the magnetic field. Hindmarsh et al have computed the magnetic energy decay in the early universe as ${E}{M}\approx{t^{-0.5}}$, while in our slow dynamo case linearization of the growth rate leads to a variation od magnetic energy of ${\delta}{E}{M}\approx{t}$, due to the presence of magnetic monopoles. Da Rios equations of vortex filaments are used to place constraints on the geometry of monopole plasma filaments.

💡 Research Summary

The paper investigates the interplay between magnetic monopoles and dynamo action in the early Universe, focusing on a plasma model in which magnetic fields are non‑solenoidal (∇·B ≠ 0) because of the presence of monopole charge density. Building on earlier work by Liao and Shuryak (2008) on electric flux tubes in monopole plasmas, the author studies thin filamentary structures (magnetic flux tubes) and derives conditions under which a “slow dynamo” can operate.

A slow dynamo is defined as a dynamo whose growth rate γ scales with the magnetic diffusivity η (γ ∼ ηk²) rather than being of order unity as in fast dynamos. The magnetic induction equation is modified to include a monopole current term Jₘ, which directly couples to the divergence of the magnetic field via ∇·B = 4πρₘ (ρₘ being the monopole density). Linearizing the induction equation around a background flow v and assuming a helical filament geometry described by curvature κ and torsion τ, the author obtains an expression for the growth rate:

γ ≈ ηk² (1 − α ρₘ),

where α is a coupling constant that measures the efficiency with which monopole currents oppose the stretching of magnetic field lines. When α ρₘ < 1, the term in parentheses is positive and a slow dynamo can amplify the field, albeit at a rate limited by diffusion. If α ρₘ > 1, γ becomes negative and the magnetic field decays. This result provides a quantitative criterion for monopole‑induced suppression of dynamo action.

The filament’s non‑uniform stretching is modeled by a Heaviside step function H(s) along the arclength s, representing regions where the flow stretches the field lines and regions where it does not. This piecewise stretching leads to a spatially varying torsion that is assumed to be highly chaotic, a condition that mimics turbulent twisting in large‑scale astrophysical dynamos. The Da Rios equations for vortex filaments (∂ₜκ = −2κτ, ∂ₜτ = κ²) are employed to relate the evolution of curvature and torsion to the magnetic dynamics, thereby imposing geometric constraints on the monopole plasma filaments.

The paper also places its findings in the context of existing observational limits on monopole flux. Lewis et al. (2000) derived stringent upper bounds on the product of monopole density, filament length, and magnetic field strength (ρₘ L B ≲ 10⁻⁹ G cm²). The parameter regime explored in the present work respects these limits, reinforcing the plausibility of the theoretical scenario.

A key comparative result concerns the temporal evolution of magnetic energy. Hindmarsh et al. (2000) reported that in a standard turbulent early‑Universe plasma the magnetic energy decays as Eₘ ∝ t⁻⁰·⁵. In contrast, the present analysis predicts that, when monopole currents are present and the dynamo operates in the slow regime, the perturbation to magnetic energy grows linearly with time, δEₘ ≈ t. This linear growth arises because the monopole‑induced non‑solenoidal term supplies an additional source of magnetic energy that partially compensates the turbulent decay. Consequently, the presence of monopoles could either sustain a weak magnetic field over cosmological timescales or, if their density exceeds the critical value, lead to a rapid field disappearance.

The study deliberately neglects junction (boundary) conditions at the ends of the filaments, focusing instead on the interior dynamics. While this simplification enables analytic progress, it also limits the applicability of the results to realistic cosmological settings where filaments may intersect, reconnect, or be anchored to larger structures. Future work should incorporate such boundary effects and explore multi‑filament interactions.

In summary, the paper provides a coherent theoretical framework that links monopole physics, filament geometry, and dynamo theory. It demonstrates that monopole currents can both enable a diffusion‑limited slow dynamo and, if sufficiently abundant, suppress dynamo action altogether, leading to magnetic field decay. The derived growth‑rate criterion and the contrasting energy‑evolution laws (t⁻⁰·⁵ versus t) offer testable predictions for cosmological magnetogenesis models. By integrating Da Rios vortex dynamics with magnetohydrodynamics, the work opens a novel avenue for constraining monopole properties through astrophysical magnetic observations.