The decay of MHD turbulence and the primordial origin of magnetic fields in cosmic voids
Keywords: turbulence
cosmology
magnetohydrodynamics
primordial magnetism

The central result of this thesis is the solution of a decades-old problem in magnetohydrodynamic (MHD) turbulence theory: in the absence of energy injection, how do chaotic magnetic fields decay with time? We obtain novel predictions for the evolution of the magnetic energy and correlation scale by proposing that the field decays on reconnection timescales while respecting certain integral invariants that represent topological constraints. As is well known, the magnetic helicity is such an invariant for initially helical field configurations (Taylor, 1974), but does not constrain non-helical decay, where the volume-averaged magnetic-helicity density vanishes. For such a decay, we propose a new integral invariant — the “Saffman helicity invariant” — which is analogous to the Loitsyansky and Saffman invariants of hydrodynamic turbulence (Loitsyansky, 1939; Saffman, 1967), and that expresses the conservation of the random (scaling as volume^1/2) magnetic helicity contained in any sufficiently large volume. We verify that the Saffman helicity invariant is indeed well-conserved in our numerical simulations. We formulate a general principle of decay of turbulent systems subject to conservation of Saffman-like invariants, and propose how it may be applied to magnetohydrodynamic (MHD) turbulence with a strong mean magnetic field and to isotropic MHD turbulence with initial equipartition between the magnetic and kinetic energies.
We apply our theory of MHD decay to the evolution of primordial magnetic fields in the early Universe. This has become a popular problem in recent years due to the possibility that the weak magnetic field hosted by the intergalactic medium (IGM) in voids could be a relic from the early Universe [see Durrer & Neronov (2013); Subramanian (2016); Vachaspati (2021) and references therein]. If so, accurate measurement of void fields combined with a theory of turbulent decay could be used to constrain cos- mological models of the early Universe. Previous models of MHD decay have predicted that the present-day strength of fields generated at the electroweak phase transition (EWPT) should be too low to explain the observed scattering by void fields of γ-rays from TeV blazars (Neronov & Vovk, 2010; Wagstaff & Banerjee, 2016; Taylor et al., 2011). However, our new theory indicates that the accepted models greatly underpredict the present-day strength of relic fields. Our new estimates restore the consistency of the EWPT-relic hypothesis with observational constraints; moreover, we find that efficient magnetogenesis at the EWPT could produce relics with the strength that is believed sufficient to resolve the Hubble tension (Jedamzik & Pogosian, 2020; Galli et al., 2022) and explain galaxy-cluster magnetic fields without requiring dynamo amplification after structure formation (Banerjee & Jedamzik, 2003).
Further to this thesis’ core narrative on decaying MHD turbulence, we address two additional problems that are also related to the dynamics of statistically isotropic, homogeneous turbulence. The first is the role of the Saffman invariant (and its generalisations) in constraining the dynamics of forced, rather than decaying, turbulence. We show that the physical interpretation of the Saffman integral as a measure of the strength of local momentum fluctuations leads to a natural physical picture of the “large-scale thermalisation” phenomenon reported in recent numerical studies (Dallas et al., 2015; Cameron et al., 2017; Alexakis & Biferale, 2018; Alexakis & Brachet, 2019). The second is the dynamical effect that statistically isotropic “tangled” magnetic fields have on their host medium. We derive a mean-field theory of magnetoelasticity that elucidates the important role of “intermittency” in the magnetic field, and present (to the best of our knowledge) the first simulations of “magnetoelastic waves”, showing their evolution to be in good agreement with the predictions of our mean-field theory.