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A 3+1 Perturbative Approach to the Cosmic Dynamo Equation

Juan F. Bravo, Leonardo Castañeda, Héctor J. Hortúa

Abstract

In this work, we analyze the evolution of PMFs within a perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime using the formalisms of Numerical Relativity (NR). We apply the 3+1 decomposition to first-order cosmological perturbations to derive the cosmological dynamo equation under the kinematic-dynamo approximation. Our objective is to study the interaction between the seed magnetic field and the growing modes of scalar perturbations, whose associated velocity fields are evolved numerically using the software \texttt{Einstein Toolkit} and \texttt{FLRWSolver}. We find that these velocity fields effectively drive the amplification of the PMF, demonstrating that the extent of this growth is dependent on the electrical conductivity of the cosmic medium. Our findings provide a computational description linking primordial magnetogenesis to the evolution of magnetic seeds, ultimately explaining the ubiquity of large-scale magnetic fields in the universe

A 3+1 Perturbative Approach to the Cosmic Dynamo Equation

Abstract

In this work, we analyze the evolution of PMFs within a perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime using the formalisms of Numerical Relativity (NR). We apply the 3+1 decomposition to first-order cosmological perturbations to derive the cosmological dynamo equation under the kinematic-dynamo approximation. Our objective is to study the interaction between the seed magnetic field and the growing modes of scalar perturbations, whose associated velocity fields are evolved numerically using the software \texttt{Einstein Toolkit} and \texttt{FLRWSolver}. We find that these velocity fields effectively drive the amplification of the PMF, demonstrating that the extent of this growth is dependent on the electrical conductivity of the cosmic medium. Our findings provide a computational description linking primordial magnetogenesis to the evolution of magnetic seeds, ultimately explaining the ubiquity of large-scale magnetic fields in the universe
Paper Structure (15 sections, 78 equations, 5 figures)

This paper contains 15 sections, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Formalism
  3. 3+1 formulation of GR
  4. 1+3 formulation of GR
  5. Equivalence between formalisms
  6. FLRW Universe
  7. Background universe
  8. Perturbed universe
  9. Perturbed 3+1 formalism
  10. Perturbed 1+3 formalism
  11. Dynamo equation
  12. Obtaining the dynamo equation
  13. Computational implementation
  14. Conclusions
  15. Perturbation theory

Figures (5)

  • Figure 1: Evolution of the Hubble parameter for the matter-dominated era using the Einstein Toolkit. The red line shows the evolution using expression (\ref{['eq:fried-eq']}) where $z=-1+\left(z_{\text{CMB}}+1\right)/a$ and $z_{\text{CMB}}=1100$.
  • Figure 2: Numerical evolution of Eq. (\ref{['eq:dynamo-back']}). The decay of the magnetic field is proportional to $a^{-1}$, consistent with a frame choice that allows the field to evolve in this manner k2018magnetic.
  • Figure 3: Evolution of the magnetic field amplitude according to Eq. (\ref{['eq:dinamo1_3']}). The growth is sensitive to the conductivity $\sigma$, starting from an initial magnitude of $B\approx 10^{-22}$. Values of $\sigma$ are given in geometric units.
  • Figure 4: Limits on the magnetic field at recombination based on estimates from Neronov_2010 and constrain_pmg_lyman.
  • Figure 5: Scheme of the sub-manifold family $\mathcal{M}_\lambda$ embedded in a five-dimensional manifold $\mathcal{N}$. The comparison between manifolds is established by the flow $\phi_\lambda$.