Reconciling primordial magnetic fields with observations via turbulent decay

The paper addresses whether non-helical primordial magnetic fields (PMFs), plausibly generated during the electroweak phase transition (EWPT), can evolve to present-day strengths and coherence scales that satisfy observational constraints from blazar-induced gamma-ray cascades (Equation (1)). Classical expectations based on selective decay (invariance of the magnetic Loitsyansky integral I_μ) and Alfvénic decay timescales predicted relics inconsistent with observations. The study revisits PMF decay by incorporating reconnection-controlled decay and constraints from magnetic-helicity conservation (including effects of local helicity fluctuations even when global helicity vanishes), to determine if EWPT-origin non-helical PMFs can be consistent with current extragalactic magnetic-field (EGMF) bounds and potentially address cosmological issues such as the Hubble tension.

Prior work assumed selective decay preserving the small-k asymptotic of the magnetic-energy spectrum, implying invariance of the magnetic Loitsyansky integral I_μ (Equations (4)–(6)). Earlier analyses often posited Alfvénic decay timescales, leading to B–λ_g relations that conflicted with observational bounds when extrapolated from EWPT initial conditions. Subsequent numerical discoveries showed inverse transfer of magnetic energy even in non-helical MHD turbulence due to conserved mean-square fluctuations of locally nonzero helicity. Additional results established that magnetic energy decay occurs on reconnection timescales associated with plasmoid-unstable current sheets rather than simple Alfvénic turnover times. The paper builds on these advances, integrating reconnection physics, potential viscosity suppression at high magnetization, and radiative drag, to re-evaluate non-helical PMF evolution.

• Cosmological-MHD framework: Use conformal rescaling to map expanding-Universe MHD to Minkowski spacetime for analysis, then transform back via scaling relations (Equation (3)). Post-recombination evolution uses a different rescaling appropriate to matter domination (Equation (21)).
• Decay invariants and scaling: Assume selective decay preserving I_μ and correlation/integral scale Λ_g, leading to B^2Λ_g ≈ const (Equation (6)). For late-time decay, regardless of early-time microphysics, the decay timescale scales with cosmic time τ ∼ t (Equation (8)), enabling late-time predictions insensitive to early details.
• Timescale models: Critically, decay proceeds on the reconnection-controlled timescale. In resistive MHD with plasmoid instability, the global reconnection timescale is set by the critical sheet, T_rec = (1+Pm)^{1/2} min(S^{5/2}λ_s/v_A, S^{1/2}λ_s/v_A) with S = v_A λ_s/[(1+Pm)^{1/2} η] and S_c ~ 10^6 (Equations (14)–(15)). If the critical sheet becomes kinetic (δ_c < r_i or d_i), the reconnection rate saturates at τ_rec ∼ 0.1 λ_B/v_A (Equation (17)). Radiative (Thomson) drag on electrons introduces a large-scale inflow limitation with τ_a = λ_B/α (Equations (18)–(19), (54)). The operative decay timescale is τ = max(τ_rec, τ_a) (Equation (19)).
• Recombination benchmark: Using Friedmann and thermodynamic relations, obtain t_recomb ≈ 10^16 s at T ≈ 0.3 eV (Equations (26)–(28)).
• Regime delineation at recombination: Compute Spitzer collisional viscosity and resistivity to estimate Pm_SP and η (Equations (32)–(37)), Lundquist number S (Equation (38)), and evaluate B–λ_g relations at t_recomb for multiple regimes: (i) collisional Pm = Pm_SP (Equation (40)), (ii) viscosity suppression for B > B_iso via Pm ≈ (B_iso/B)^2 Pm_SP (Equation (45)), (iii) microinstability-limited Pm ≤ 1 (Equations (48)–(50)), and (iv) radiation-drag-limited decay (Equation (55)). Kinetic-scale thresholds for the critical sheet (δ_c vs r_i, d_i) are assessed (Equations (41)–(47)).
• Observational comparison: Intersections of evolutionary tracks (lines of constant I_H ∼ B λ_B) with τ ∼ t constraints at recombination define allowed present-day B and λ_g (Fig. 3 lines (i)–(iv)).
• Stability checks: Verify that pressure-anisotropy-driven firehose instability remains unexcited during decay (Equations (56)–(60)).
• Numerical support: Incompressible MHD simulations (Snoopy code) with hyper-dissipation provide qualitative visualization (Equation (61)), though primary results are analytic/scaling.

• Alfvénic-timescale assumption alone (τ ≈ λ_g/v_A) yields B(t_recomb) ≈ 10^(-8) G (λ_g/1 Mpc) (Equation (10)), which conflicts with observational constraint (1) when extrapolated from plausible EWPT initial conditions (e.g., B(t_) ~ 10^(-5) G, λ_g(t_) ~ 10^(-10) Mpc).
• Incorporating reconnection-controlled decay reconciles non-helical EWPT PMFs with observations. The decay locus in B–λ_g space is governed by τ = max(τ_rec, τ_a): • Line (i): For I_H ≤ 10^(-29) I_H,max with collisional Spitzer Pm, B ≈ 10^(-8) G (λ_g/1 Mpc) (Equation (40)). • Line (ii): For 10^(-29) I_H,max ≤ I_H ≤ 10^(-27) I_H,max and B > B_iso ≈ 10^(-13) G, viscosity suppression reduces Pm, modifying the τ_rec constraint (Equation (45)). • Line (iii): If microinstabilities reduce Pm ≤ 1 for B > B_iso, τ_rec leads to B ≈ 10^(-6) G (λ_g/1 Mpc)^(1/2) at recombination (Equation (50)), with the evolution continuing along B ≈ B_iso into line (iv). • Line (iv): Radiation drag limits inflow for sufficiently strong/large-scale fields, giving B scaling set by Equation (55).
• For I_H ≥ 10^(-23) I_H,max, reconnection-controlled decays yield present-day states consistent with observational constraint (1). This corresponds to EWPT initial conditions with (B(t_)/10^(-10) G) ~ (λ_g(t_)/10^(-10) Mpc) ~ 2×10^(-23) (Equation (20)).
• Stronger initial conditions (e.g., ρ_g(t_) ~ ρ_r(t_), λ_g(t_) ≥ 10^2 r_H(t_)) can evolve to B ~ 10^(-11) G at recombination, producing viable seeds for cluster fields requiring minimal subsequent turbulent-dynamo amplification.
• Fields of order 10^(-10)–10^(-9) G are compatible with resolving the Hubble tension via modified recombination; the framework allows non-helical EWPT PMFs to reach such comoving strengths.
• Kinetic thresholds: For many relevant decays, fluid theory is valid at recombination (δ_c > r_i, d_i) when B < B_iso (Equations (41)–(44)). Higher-I_H decays that transiently enter kinetic regimes ultimately terminate when B decreases to ~10^(-11) G (Equation (46)).
• Microinstability impacts: Even with extreme Pm ≤ 1 for B > B_iso, the predicted present-day locus (red–gold line) remains consistent with constraints for I_H ≥ 10^(-20) I_H,max.
• Firehose instability remains unexcited for parameter ranges consistent with observations (Equation (60)).
• Post-recombination decay in comoving variables is only logarithmic, so recombination-era predictions provide robust estimates of present-day relic strengths.

By replacing the naive Alfvénic decay assumption with reconnection-controlled decay, and accounting for helicity-related constraints and radiative drag, the study demonstrates that non-helical PMFs generated at the EWPT can evolve into present-day EGMFs consistent with blazar constraints. The analysis identifies clear parameter regimes—set by collisionality (Pm), kinetic thresholds (δ_c vs r_i, d_i), and photon drag—determining the terminal B–λ_g states at recombination. This resolves earlier inconsistencies where selective decay with Alfvénic times led to over-strong relics. The results suggest that viable non-helical EWPT PMFs could: (i) seed cluster magnetic fields at levels (~10^(-11) G at recombination) requiring only modest subsequent amplification, and (ii) reach comoving strengths around 10^(-10)–10^(-9) G potentially relevant to the Hubble tension. Even under uncertain microinstability physics that may suppress effective viscosity (Pm ≤ 1), the allowed parameter space remains consistent with observational bounds, indicating robustness of the EWPT-magnetogenesis scenario when reconnection physics is included.

The paper provides a reconnection-controlled decay framework for non-helical PMFs in the expanding Universe, showing that EWPT-generated fields can be consistent with current EGMF observational constraints. Key contributions include: (1) demonstrating τ ∼ t late-time decay insensitivity to early microphysics; (2) deriving recombination-era B–λ_g relations across collisional, kinetic, and radiation-drag-limited regimes; (3) establishing compatibility with observations for a broad range of initial helicity measures (I_H ≥ 10^(-23) I_H,max), with relics potentially reaching ~10^(-11) G at recombination and up to ~10^(-10)–10^(-9) G comoving strengths relevant to cosmology; and (4) showing that firehose instability does not invalidate the decay pathways. Future work should quantify effective viscosity and resistivity in weakly collisional early-Universe plasmas (role of microinstabilities), refine the transition criteria between fluid and kinetic reconnection, explore helical analogues in detail, and couple these decay laws to structure-formation contexts to assess memory of primordial seeds in observed cluster fields.

• Title and abstract are not provided in the excerpt; results are synthesized from main-text and Methods fragments.
• Analysis relies on order-of-magnitude scaling arguments (∼ equalities) rather than full cosmological MHD simulations including radiation and baryon-photon coupling.
• Effective viscosity and resistivity may be modified by plasma microinstabilities for B > B_iso; their quantitative impact on reconnection rate remains uncertain.
• Applicability of fluid (resistive-MHD) reconnection vs kinetic reconnection depends on δ_c relative to r_i and d_i; while argued to be valid for many cases at recombination, transitions introduce uncertainty.
• Scale-invariant (inflationary) magnetic spectra are excluded; conclusions apply to causal (phase-transition) fields with peaked spectra.
• Early-time processes (e.g., neutrino viscosity, high-temperature drag) are argued to be subdominant to late-time τ ∼ t behavior, but detailed early-Universe microphysics is not modeled explicitly.