Relativistic magnetohydrodynamics in the early Universe

Technical Summary: Relativistic Magnetohydrodynamics in the Early Universe

Problem and Context
The paper addresses the theoretical framework required to simulate magnetohydrodynamics (MHD) in the early Universe, specifically during the radiation-dominated epoch. Previous numerical studies, such as those using the Pencil Code, have largely relied on the subrelativistic limit where the bulk fluid velocity is small ($u^2 \ll 1$) and the Lorentz factor is approximated as $\gamma^2 \approx 1$. The authors identify a critical flaw in this approximation: neglecting the time derivative of the Lorentz factor ($\partial_\tau \gamma^2$) ignores non-linear terms of order $U^2/\ell$ (where $U$ is characteristic velocity and $\ell$ is length scale). These terms are of the same order as the convective derivative and are pivotal for describing non-linear fluid dynamics and turbulence. Consequently, previous work omitted corrections that are significant even in the subrelativistic regime when the equation of state is relativistic ($c_s^2 \sim \mathcal{O}(1)$).

Methodology
The authors derive the conservation laws for a conducting perfect fluid with relativistic bulk velocities on a Friedmann–Lemaître–Robertson–Walker (FLRW) background. The methodology involves:

1. Geometric Setup: Utilizing conformal time $\tau$ and comoving coordinates to exploit the conformal invariance of the equations for a radiation-dominated fluid ($c_s^2 = 1/3$). The metric is treated as a Weyl transformation of the Minkowski metric.
2. Fluid Dynamics: Deriving the equations of motion from the conservation of the stress-energy tensor ($T^{\mu\nu}_{;\mu} = 0$). The authors present these equations in two forms:
   • Conservation Form: Evolving the stress-energy tensor components $\tilde{T}^{0\mu}$.
   • Non-Conservation Form: Evolving primitive variables (energy density $\tilde{\rho}$ and peculiar velocity $u$). This form is presented for the first time in its fully relativistic non-conservation structure.
3. Imperfect Fluids: Extending the perfect fluid description to include first-order imperfect fluid effects (viscosity and heat conduction) using the Classical Irreversible Thermodynamics (CIT) approach, acknowledging its acausal nature but justifying its use for large-scale simulations where transport coefficients are small.
4. Electromagnetism: Coupling the fluid equations with Maxwell’s equations in an expanding background, utilizing the generalized Ohm’s law to describe the current density.
5. Wave Analysis: Analyzing linear perturbations to derive dispersion relations for sound waves, Alfvén waves, and magnetosonic waves.

Key Contributions and Results

1. Correction of Subrelativistic Equations: The paper demonstrates that the term $\partial_\tau \gamma^2$ yields non-negligible contributions in the subrelativistic limit for relativistic equations of state. This leads to specific corrections in the coefficients of the convective terms $(u \cdot \nabla) \ln \tilde{\rho}$ in both the energy and momentum equations. For a radiation-dominated fluid ($c_s^2 = 1/3$), the coefficient of this term changes from 1 to $1/2$ compared to previous literature.
2. Fully Relativistic Non-Conservation Form: The authors provide the explicit dynamical equations for $\tilde{\rho}$ and $u$ in the fully relativistic regime (Eqs. 1.2 and 6.31), which had not been previously presented in this form.
3. Vorticity Production: The paper shows that for relativistic equations of state ($c_s^2 \sim \mathcal{O}(1)$), vorticity is no longer a topological invariant. Even in the absence of external forces and baroclinic terms, the expansion of the Universe and the relativistic equation of state can generate vorticity from an initially curl-free velocity field.
4. Scaling and Coordinates: The authors review and generalize scaling choices for fluid variables. They present "super-comoving" coordinates for matter-dominated fluids and extend this concept to any constant equation of state, identifying scalings that minimize or eliminate Hubble friction terms.
5. Boris Correction for Alfvén Speeds: The paper addresses the issue of superluminal Alfvén speeds that arise when the displacement current is neglected in ideal MHD. The authors adapt the Boris correction for relativistic Alfvén speeds to the early Universe context, ensuring subluminal propagation by modifying the momentum equation with a factor $\gamma_A = (1 + v_A^2)^{-1}$.
6. Transport Coefficients: The paper provides estimates for shear viscosity, bulk viscosity, and thermal conductivity in the primordial plasma, showing that while these are small, they are non-zero and their scaling with temperature is relevant for simulations.

Significance
The authors claim that this work provides a necessary correction to the theoretical foundation of early Universe MHD simulations. By including the previously neglected $\partial_\tau \gamma^2$ terms, the derived equations correctly capture non-linear dynamics and turbulence in the subrelativistic limit for relativistic fluids. The presentation of the non-conservation form facilitates the implementation of these equations in numerical codes (such as Pencil Code and CosmoLattice) that solve for primitive variables. The review of conformal invariance and scaling choices offers a unified framework for handling different equations of state and expansion histories, while the adaptation of the Boris correction ensures physical consistency in the treatment of magnetic waves. The work is positioned as an extension of lectures given at the Bernoulli Center, aiming to provide a self-contained theoretical resource for cosmological MHD studies.