Relativistic magnetohydrodynamics in the early Universe

Abstract: We review the conservation laws of magnetohydrodynamics (MHD) in an expanding homogeneous and isotropic Universe that can be applied to the study of early Universe physics during the epoch of radiation domination. The conservation laws for a conducting perfect fluid with relativistic bulk velocities in an expanding background are presented, extending previous results that apply in the limit of subrelativistic bulk motion. Furthermore, it is shown that the subrelativistic limit presents new corrections that have not been considered in previous work. We discuss the conformal invariance of the MHD equations for a radiation-dominated fluid and different types of scaling of the fluid variables that are relevant for other equations of state when the bulk velocity is subrelativistic. In particular, we review the super-comoving coordinates that scale the time coordinate with the Universe expansion, and present a particular choice that allows the equations to become conformally flat for any choice of the equation of state. Imperfect relativistic fluids are briefly described but their detailed study is not included in this work. We review the propagation of sound waves, Alfv\'en waves, and magnetosonic waves in the early Universe plasma. The Boris correction for relativistic Alfv\'en speeds is presented and adapted for early Universe applications. This review is an extension, including new results, of part of the lectures presented at the minicourse Simulations of Early Universe Magnetohydrodynamics'' lectured by A. Roper Pol and J. Schober at EPFL, as part of the six-week programGeneration, evolution, and observations of cosmological magnetic fields'' at the Bernoulli Center in May 2024.

• The paper extends traditional MHD by incorporating relativistic corrections using conformal transformations during the radiation-dominated epoch.
• It compares subrelativistic and relativistic approximations, highlighting the critical role of time derivatives of the Lorentz factor in energy and momentum conservation.
• The study’s findings improve simulation frameworks for primordial magnetic fields and offer insights into their potential observational signatures in cosmic microwave background and gravitational waves.

Relativistic Magnetohydrodynamics in the Early Universe: A Review

The paper “Relativistic Magnetohydrodynamics in the Early Universe” by A. Roper Pol and A. S. Midiri addresses the theoretical and practical aspects of magnetohydrodynamics (MHD) within the framework of a relativistic cosmological setting, specifically during the radiation-dominated epoch of the early Universe. This work extends existing MHD equations to the relativistic regime while maintaining conformity with the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, aligned with the cosmological principle of homogeneity and isotropy.

Theoretical Contributions

The paper sets out to derive and extend the conservation laws for a conducting perfect fluid with relativistic bulk velocities in an expanding Universe. This includes the relativistic extension of MHD equations, which are pivotal in understanding the dynamics of primordial magnetic fields and their role in early Universe phenomena. Using conformal transformations, the authors demonstrate how the equations reduce to their classical counterparts under specific conditions, such as when the equation of state is that of radiation domination, i.e., $p = \frac{1}{3}\rho$.

Subrelativistic and Relativistic Approximations

A substantial portion of the paper is devoted to comparing the subrelativistic and relativistic regimes. This includes a re-evaluation of subrelativistic limits, revealing that prior models overlook significant corrections in the energy and momentum equations due to the implicit assumption $\gamma^2 \approx 1$. The authors emphasize the necessity of including time derivatives of the Lorentz factor, $\partial_\tau \gamma^2$, which contributes non-negligibly, particularly in the subrelativistic limit with $\cs^2 \sim \mathcal{O}(1)$.

Implications of the Research

The implications of this work are twofold: theoretical and observational. Theoretically, the framework enhances the understanding of MHD processes in the relativistic regime, which is critical for examining conditions in the early universe that deviate from solely non-relativistic assumptions. Practically, these insights can inform simulations of cosmic phenomena such as the evolution of primordial magnetic fields and their potential observational signatures in the cosmic microwave background radiation and gravitational waves.

Numerical Considerations and Future Work

While the paper provides a robust theoretical basis, it also considers practical applications and numerical simulations. The limitations of the current open-source codes like the \textsc{Pencil Code} are addressed, and ongoing efforts to incorporate fully relativistic MHD into these frameworks are acknowledged. The work suggests further investigation into the relativistic effects that could lead to modifications in the understanding of the early Universe's magnetic and velocity fields.

Speculation on AI Developments

Though the paper primarily focuses on relativistic MHD, there is room for speculation on the intersection with AI developments. AI techniques, such as machine learning, could potentially be leveraged for more efficient simulations and analysis of large-scale cosmological data. AI could help in overcoming computational challenges associated with the high-dimensional parameter spaces involved in relativistic MHD simulations.

Conclusion

This paper represents a significant step in relativistic cosmology, offering detailed insights into the MHD dynamics of the early Universe with advanced theoretical frameworks. The careful mathematical extensions and considerations for relativity provide a foundation for future exploration in both theoretical and computational cosmology. Moving forward, the integration of these theoretical insights with observational data will be crucial for a comprehensive understanding of cosmological magnetism and its implications for the evolution of the Universe.