Relativistic magnetohydrodynamics in the early Universe

Technical Summary: Relativistic Magnetohydrodynamics in the Early Universe

Problem Statement
The study of primordial magnetic fields, fluid perturbations, and gravitational wave production in the early Universe often relies on magnetohydrodynamic (MHD) simulations. Historically, these simulations have operated under the assumption of subrelativistic bulk fluid motion ($u^2 \ll 1$) and frequently assumed the limit where the Lorentz factor $\gamma^2 \approx 1$. While this approximation holds for matter-dominated fluids, it introduces unexamined errors in radiation-dominated epochs or when the equation of state parameter $c_s^2$ is of order unity (e.g., $c_s^2 = 1/3$). Previous formulations often neglected the time derivatives of the Lorentz factor ($\partial_\tau \gamma^2$), leading to missing non-linear terms of order $U^2/\ell$ (where $U$ and $\ell$ are characteristic velocity and length scales). These omissions affect the conservation of energy and momentum, the generation of vorticity, and the propagation speeds of MHD waves, potentially leading to superluminal Alfvén speeds if not corrected.

Methodology
The authors derive the conservation laws for MHD in a homogeneous, isotropic, and expanding Friedmann–Lemaître–Robertson–Walker (FLRW) background. The work proceeds through several theoretical steps:

1. Geometric Framework: The FLRW metric is established using conformal time $\tau$ and comoving coordinates, allowing for a generic $\alpha$-time definition to explore different scaling properties of fluid variables.
2. Perfect Fluid Dynamics: The authors derive the equations of motion for a perfect fluid with a constant equation of state $p = c_s^2 \rho$. They present these equations in two forms:
   • Conservation Form: Evolving the stress-energy tensor components $\tilde{T}^{0\mu}$.
   • Non-Conservation Form: Evolving the primitive variables (comoving energy density $\tilde{\rho}$ and peculiar velocity $u$).
     Crucially, the derivation retains terms involving $\partial_\tau \gamma^2$ and $\partial_\tau u^2$, which are often discarded in the subrelativistic limit.
3. Imperfect Fluids: The framework is extended to include first-order imperfect fluid effects (viscosity and heat conduction) using the Classical Irreversible Thermodynamics (CIT) approach, providing covariant expressions for Navier-Stokes viscosity and Fourier's thermal conductivity.
4. Electromagnetism: Maxwell's equations are reviewed in the expanding background, introducing comoving electromagnetic fields ($\tilde{E}, \tilde{B}$) and a covariant generalized Ohm's law. The displacement current is analyzed to justify the ideal MHD limit in the high-conductivity regime of the early Universe.
5. MHD Coupling: The fluid equations are coupled with Maxwell's equations via the Lorentz force. The authors analyze linear perturbations to study sound, Alfvén, and magnetosonic waves.
6. Corrections: A "Boris correction" is adapted to the relativistic MHD context to ensure Alfvén speeds remain subluminal even when neglecting the displacement current.

Key Contributions and Results

• Corrected Subrelativistic Equations: The paper demonstrates that previous subrelativistic MHD equations (e.g., those used in the Pencil Code) contain errors in the coefficients of convective terms $(u \cdot \nabla) \ln \tilde{\rho}$ and velocity terms. Specifically, the time derivative of $\gamma^2$ introduces a correction factor of $1/2$ in certain terms for radiation-dominated fluids ($c_s^2 = 1/3$) and modifies coefficients for generic $c_s^2$. The corrected non-conservation form equations are presented in Eqs. (1.2) and (6.31).
• Vorticity Production: The authors show that vorticity ($\omega = \nabla \times u$) is no longer a topological invariant in the early Universe. Even in the absence of external forcing and baroclinic terms, vorticity can be generated from an initially curl-free velocity field due to relativistic corrections involving $c_s^2 \neq 0$ and the expansion of the Universe.
• Scaling and Conformal Invariance: The paper reviews various scalings of fluid variables (comoving, super-comoving) to minimize Hubble friction terms. It identifies that for subrelativistic flows with $c_s^2 \neq 1/3$, a specific scaling ($\beta = 3(1+c_s^2)$) allows the energy equation to be conformally flat, while the momentum equation retains a Hubble friction term.
• Relativistic Alfvén Speed and Boris Correction: The analysis reveals that neglecting the displacement current in ideal MHD can lead to superluminal Alfvén speeds ($v_A > 1$) when $B_0^2/\rho_0$ is large. The authors adapt the Boris correction to the relativistic MHD equations, modifying the momentum equation to ensure $v_A \to v_A \gamma_A \leq 1$, where $\gamma_A = (1+v_A^2)^{-1}$.
• Transport Coefficients: Estimates for shear viscosity, bulk viscosity, and thermal conductivity in the primordial plasma are reviewed, confirming that the plasma is effectively a perfect fluid at large scales, though dissipative effects are necessary for realistic turbulence modeling.

Significance
This work provides the first fully relativistic description of MHD equations in an expanding background that includes corrections to the subrelativistic limit previously overlooked. The authors claim that these corrections are not merely higher-order relativistic effects but are of the same order as convective derivatives in non-linear fluid dynamics. Consequently, the paper argues that future simulations of early Universe magnetohydrodynamics, particularly those involving radiation-dominated fluids or high-energy phases, must incorporate these corrected terms to accurately model non-linear dynamics, turbulence, and vorticity generation. The derived equations serve as a theoretical foundation for upcoming developments in numerical codes like Pencil Code and CosmoLattice.