Whether the weak magnetic fields inferred in cosmic voids are relics of magnetogenesis at the electroweak phase transition (EWPT) was restored to consistency by Hosking & Schekochihin (2023), whose reconnection-controlled decay theory — conserving the Hosking integral, the mean-square fluctuation level of magnetic helicity — predicts relic fields orders of magnitude stronger than earlier selective-decay treatments. The source work itself flags five uncontrolled links that must be closed before the EWPT-relic hypothesis can be tested quantitatively against the gamma-ray lower bound on void fields: (i) possible modification of that bound by plasma-instability cooling of cascade electrons; (ii) microinstability-set effective transport coefficients and magnetic Prandtl number in the primordial plasma; (iii) the reconnection rate of the critical current sheet once its width approaches kinetic scales, where only a phenomenological proxy is available; (iv) the asymptotic scaling of the generated Hosking integral with numerical resolution and Prandtl number; and (v) whether early-Universe mechanisms such as chiral magnetohydrodynamics can generate enough net helicity to make primordial fields maximally helical. The problem is open.

Background

Cosmic voids — the vast, nearly empty regions of the intergalactic medium (IGM) that occupy most of the Universe's volume — host magnetic fields whose existence is inferred from a gamma-ray effect: TeV photons emitted by distant blazars (active galactic nuclei with relativistic jets pointed toward us) collide with the extragalactic background light, producing electron–positron pairs that inverse-Compton-scatter cosmic microwave background photons up to GeV energies. Weak magnetic fields threading the void IGM deflect these pairs, delaying and smearing the secondary GeV emission; the observed absence of such halos around TeV blazars lower-bounds the field strength BB and coherence scale λB\lambda_B (Neronov & Vovk 2010): B≳10−17 G (λB/1 Mpc)−1/2B \gtrsim 10^{-17}\,\mathrm{G}\,(\lambda_B/1\,\mathrm{Mpc})^{-1/2}, with the normalisation rising to ∼10−15 G\sim 10^{-15}\,\mathrm{G} depending on modelling assumptions such as time delays. A leading explanation is that these are primordial magnetic fields (PMFs) generated in the early Universe, prominently at the electroweak phase transition (EWPT, the symmetry-breaking event at temperature ∼100\sim 100 GeV at which the electromagnetic and weak forces separated). Predicting the present-day strength of such a relic requires modelling magnetohydrodynamic (MHD) decay across cosmic history. The central physical quantity is magnetic helicity, H=∫A⋅B d3xH=\int \mathbf{A}\cdot\mathbf{B}\,d^3x, a topological measure of the twist and linkage of field lines that is conserved even as field energy decays; a field saturating the bound ∣H∣≤λBB2|H| \le \lambda_B B^2 is called maximally helical. Under the traditional 'selective decay' theory, non-helical PMFs from the EWPT were predicted to decay too fast to meet the gamma-ray bound (Wagstaff & Banerjee 2016), and generating the required helicity exceeded what the baryon asymmetry allows. Hosking & Schekochihin (2021, 2023) overturned this picture: decaying non-helical turbulence exhibits 'inverse transfer' (magnetic energy migrating to larger scales) because local fluctuations of magnetic helicity exist even when the net helicity vanishes, and their mean-square level per unit volume — the Hosking integral IHI_H, a Saffman-type invariant analogous to Saffman's conserved momentum-fluctuation integral in hydrodynamic turbulence — is conserved; moreover, the decay proceeds on the timescale of magnetic reconnection (the breaking and rejoining of field lines in thin current sheets), not on a turbulent-cascade timescale. With IH∼B~4λB5∼constI_H \sim \tilde{B}^4 \lambda_B^5 \sim \mathrm{const} (tildes denote comoving, rescaled variables), decay effectively terminates at recombination — in the matter-dominated era, power-law decay becomes only logarithmic — leaving relic fields several orders of magnitude stronger than previously thought. This restores consistency of the non-helical EWPT-relic hypothesis with the gamma-ray bound, and efficient EWPT magnetogenesis could even yield ∼10−11 G\sim 10^{-11}\,\mathrm{G} comoving fields, the strength invoked in magnetic-field solutions of the Hubble tension and sufficient to seed galaxy-cluster fields. The quantitative case, however, rests on five links that the source work explicitly leaves open. First, the gamma-ray bound itself: if the pair beams are cooled by plasma instabilities (collective electromagnetic modes excited by the beam) rather than by inverse-Compton losses, the inferred lower bound on BB could change — the source calls this effect 'poorly understood'. Second, in the primordial plasma the effective transport coefficients (viscosity ν~\tilde{\nu}, resistivity η~\tilde{\eta}) may be set by plasma microinstabilities rather than by particle collisions — e.g. firehose or mirror modes, instabilities of a magnetized plasma driven by pressure anisotropy, in which field lines buckle or the plasma pinches into flux tubes, altering the magnetic Prandtl number Pm=ν~/η~\mathrm{Pm}=\tilde{\nu}/\tilde{\eta} that enters the reconnection rate. Third, the reconnection timescale used, τrec∼10 λB/v~A\tau_\mathrm{rec} \sim 10\,\lambda_B/\tilde{v}_A (with v~A\tilde{v}_A the Alfvén speed), is a phenomenological proxy for the regime in which the critical current-sheet width δc\delta_c becomes comparable to kinetic scales — the proton Larmor radius rLr_L (gyro-radius of protons in the mean field) or the ion inertial length did_i (the scale below which ions decouple from electrons) — and its universality and parametric dependence at recombination-era plasma parameters is unsettled. Fourth, the size of the generated IHI_H: simulations show that configurations starting with IH=0I_H = 0 develop IH≠0I_H \ne 0 and then conserve it, but the asymptotic scaling with numerical resolution and Pm\mathrm{Pm}, and the role of the early small-scale dynamo (exponential amplification of magnetic field by turbulence), remain open. Fifth, whether any early-Universe mechanism — e.g. chiral MHD, in which an imbalance between left- and right-handed fermions (a chiral chemical potential) drives the chiral magnetic effect, an electric current parallel to B\mathbf{B}, and the chiral plasma instability that can generate helical field — can produce enough net helicity for PMFs to become maximally helical during their evolution is an open question of cosmological magnetogenesis. Recent work has made major but partial progress on these links — laboratory evidence that pair-beam instabilities are suppressed (Arrowsmith et al. 2025), first kinetic simulations of helicity-constrained decay (Li 2026), dual-invariant decay phenomenology (Brandenburg & Banerjee 2025), relativistic chiral-MHD frameworks (Garg & Schober 2026), and new EWPT helicity-generation models (Domcke, Ema & Patel 2025) — while each channel retains a well-defined open core.

Problem Statement

Make the connection between magnetogenesis at the electroweak phase transition and present-day observations of magnetic fields in cosmic voids fully quantitative: replace each of the five uncontrolled links identified by Hosking & Schekochihin (2023) with a controlled calculation, simulation, or measurement, so that the present-day field strength BB and coherence scale λB\lambda_B can be predicted — or the hypothesis excluded — from specified EWPT-era initial conditions (B~(t∗),λB(t∗))(\tilde{B}(t_*), \lambda_B(t_*)) (the comoving field strength and coherence scale at the magnetogenesis epoch t∗t_*) with an uncertainty budget comparable to that of the observational bound. Concretely, a complete answer must determine: (1) the gamma-ray cascade lower bound on void magnetic fields including the physics of plasma-instability cooling of cascade electrons, or demonstrate that bound's robustness quantitatively; (2) the effective viscosity, resistivity, and magnetic Prandtl number of the primordial plasma when microinstabilities rather than collisions set the transport, and the resulting reconnection rates; (3) the reconnection rate of the critical current sheet once its width δc\delta_c approaches kinetic scales (δc≲rL,di\delta_c \lesssim r_L, d_i) at plasma parameters relevant around recombination, replacing the phenomenological proxy τrec∼10 λB/v~A\tau_\mathrm{rec} \sim 10\,\lambda_B/\tilde{v}_A; (4) the asymptotic value and scaling of the Hosking integral IHI_H generated from initially IH=0I_H = 0 fields, including its dependence on numerical resolution and magnetic Prandtl number and the interplay of early dynamo action with helicity-fluctuation generation; and (5) whether early-Universe mechanisms (chiral MHD or other parity-violating processes) can generate sufficient net helicity for PMFs to become maximally helical before or during their decay. These deliverables are jointly required: they are the uncertainty channels of one end-to-end inference chain (EWPT initial conditions → reconnection-controlled, IHI_H-conserving decay → recombination-era relic → comparison with the gamma-ray bound), and each feeds the same quantitative verdict on the electroweak-relic hypothesis. The problem is solved when that verdict — consistency region or exclusion — is robust to the uncertainties in all five channels.

References

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