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Relativistic magnetohydrodynamics in the early Universe

Alberto Roper Pol, Antonino Salvino Midiri

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, Alfvén waves, and magnetosonic waves in the early Universe plasma. The Boris correction for relativistic Alfvén 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 program ``Generation, evolution, and observations of cosmological magnetic fields'' at the Bernoulli Center in May 2024.

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, Alfvén waves, and magnetosonic waves in the early Universe plasma. The Boris correction for relativistic Alfvén 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 program ``Generation, evolution, and observations of cosmological magnetic fields'' at the Bernoulli Center in May 2024.
Paper Structure (38 sections, 286 equations, 2 figures, 1 table)

This paper contains 38 sections, 286 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Friedmann--Lemaître--Robertson--Walker background
  3. Geometry with cosmic time as $X^0$
  4. Conformal time as $X^0$
  5. Generic $\alpha$-time $\tau_\alpha$ as $X^0$
  6. Four-velocity
  7. Covariant derivative
  8. Friedmann equations
  9. Relativistic hydrodynamics in an expanding Universe
  10. Equation of state
  11. Stress-energy tensor of a perfect fluid
  12. Conservation laws of a perfect fluid
  13. Conservation form of relativistic hydrodynamics
  14. Non-conservation form of relativistic hydrodynamics
  15. Conservation laws for specific values of $c_{\rm s}^2$
  16. ...and 23 more sections

Figures (2)

  • Figure 1: Ratio $r = \sqrt{\tilde{T}^{0i} \tilde{T}^{0i}}/\tilde{T}^{00}$ of a perfect fluid as a function of the Lorentz factor $\gamma$ for different constant values of $c_{\rm s}^2 = \tilde{p}/\tilde{\rho}$.
  • Figure 2: Evolution of the sound-wave energy density perturbations $\lambda$ for an arbitrary choice of the initial conditions, $c_2 = 0$. Free-propagating waves are found when $c_{\rm s}^2 = {{1\over3}}$.