De Souza and Opher (2008) proposed that primordial magnetic fields originate in the thermal electromagnetic fluctuations, guaranteed by the fluctuation-dissipation theorem, of the hot plasma immediately after the quark-hadron phase transition; successive coalescence of the microscopic magnetic-dipole fluctuations ('bubbles') is argued to build a population of roughly 10 μ10\,\muG bubbles on comoving-parsec scales that survives to the galaxy-formation era and could seed the microgauss fields of galaxies and clusters. The paper leaves open where this bubble population resides at low redshift: it assumes, without computing, that the evolved bubbles are concentrated in cosmic-web filaments rather than spread uniformly through the intergalactic medium, and its prediction of appreciable ultra-high-energy cosmic-ray (UHECR) deflections along filaments is explicitly conditioned on that assumption. The problem is to determine, self-consistently through coupled cosmological magnetohydrodynamic structure-formation modeling, the low-redshift spatial distribution of these bubbles — filling factor, sizes, field strengths, coherence lengths, and filament/sheet concentration versus uniform intergalactic distribution — and to evaluate the resulting quantitative UHECR-deflection prediction. This low-redshift distribution question is unaddressed in the literature: the scenario has no follow-up beyond the authors' own 2010 papers, and existing cosmic-web magnetization simulations treat only volume-filling homogeneous primordial seeds.

Background

Magnetic fields of microgauss strength are observed in the Milky Way, in external galaxies (including galaxies at high redshift), and in galaxy clusters, and their origin is an open problem in cosmology. Because a mean-field galactic dynamo (the amplification of a weak seed magnetic field by turbulent plasma flow inside a galaxy) acting on a very weak seed would not have had time to build microgauss fields inside the young, already magnetized high-redshift galaxies, many primordial magnetogenesis scenarios have been proposed, operating during inflation, at electroweak or quark-hadron phase transitions, or during the formation of the first luminous objects. One candidate epoch is the quark-hadron phase transition (QHPT), when at cosmic age t∼10−4t\sim10^{-4} s and temperature T∼150T\sim150 MeV the cooling universe changed from a quark-gluon plasma to hadrons. Calculations in lattice QCD (quantum chromodynamics simulated numerically on a discrete space-time grid) indicate that in standard cosmology this change is a smooth crossover rather than a first-order transition, which disfavors mechanisms that rely on colliding nucleation bubbles (such as Cheng and Olinto 1994), but does not by itself affect the equilibrium-fluctuation mechanism described next.

De Souza and Opher (2008) proposed that primordial fields arise from the thermal electromagnetic fluctuations that the fluctuation-dissipation theorem (FDT) — the general result of statistical mechanics relating the fluctuation spectrum of an equilibrium system to its dissipation — guarantees in the hot, dense, electrically conducting plasma just after the QHPT. Following the fluctuation estimates of Tajima et al. (1992), the magnetic energy per mode at wavelength λ\lambda is ⟨B2⟩λ/8π=(T/2)(4π/3)λ−3\langle B^2\rangle_\lambda/8\pi=(T/2)(4\pi/3)\lambda^{-3}, so the fields are enormous but confined to microscopic scales of order the plasma skin depth c/ωpc/\omega_p (with ωp\omega_p the electron plasma frequency): roughly 101610^{16} G in regions of order 10−1210^{-12} cm at t∼10−4t\sim10^{-4} s. The authors model each fluctuation as a magnetic-dipole 'bubble' and argue that bubbles coalesce — dipole alignment on a flip time τflip\tau_{\rm flip} faster than tearing-mode reconnection (reconnection driven by the resistive tearing instability of current sheets) on time τtear\tau_{\rm tear} — into progressively larger, longer-lived bubbles that survive ohmic diffusion (resistive decay of magnetic fields) and recombination down to the galaxy-formation era. Their predictions at z∼10z\sim10 are: bubbles with fields ∼10 μ\sim10\,\muG over comoving ∼1\sim1 pc regions; line-of-sight averages ∼10−2 μ\sim10^{-2}\,\muG over a comoving Mpc (the quantity probed by Faraday rotation measurements, i.e., rotation of the polarization angle of radiation traversing a magnetized plasma); and volume-averaged fields capable of influencing first-star formation and reionization (the epoch when ultraviolet light from the first stars ionized the intergalactic hydrogen). These amplitudes are orders of magnitude above earlier quark-hadron-transition estimates such as Cheng and Olinto's.

What the paper explicitly does not compute is where the surviving bubbles end up at low redshift. Structure formation compresses flux-frozen plasma (in ideal magnetohydrodynamics the field lines are tied to the conducting plasma, so compression strengthens the field) into the filaments and sheets of the cosmic web (the network of matter concentrations traced by the galaxy distribution), and the authors simply assume, 'as expected', that the evolved bubbles are concentrated there rather than spread uniformly through the intergalactic medium (IGM). The assumption is load-bearing for the paper's UHECR argument: it hypothesizes that 'If our predicted magnetic fields are not spread uniformly over space but, as expected, are concentrated into the web of filaments, predicted by numerical simulations, appreciable deflections of UHECRs propagating along the filaments could occur', with small-angle deflection of ultra-high-energy cosmic rays (UHECRs, cosmic rays above ∼1018\sim10^{18} eV whose observed arrival directions are puzzlingly isotropic) over a distance DD scaling as δ∝(Dλ)1/2B/E\delta\propto(D\lambda)^{1/2}B/E for bubble coherence scale λ\lambda, field strength BB, and cosmic-ray energy EE — a prediction that depends sensitively on the bubble filling factor and spatial distribution. Synchrotron radio emission (radiation from relativistic electrons spiraling along magnetic field lines) enters the paper only as an observational-consistency remark: observed ∼0.1\sim0.1–1 μ1\,\muG fields in the low-density outskirts of cosmological collapsed objects, from diffuse radio-synchrotron clusters and a few filament cases, 'may have their origin in the primordial magnetic fields that we predict'. The generic version of the distribution question has been studied only for primordial seeds that fill space uniformly at the nanogauss level: cosmological magnetohydrodynamic (MHD) simulations of cosmic-web magnetization by Vazza et al. (2017) and UHECR propagation through those simulated fields by Hackstein et al. (2016). A small-filling-factor, pc-coherence-scale bubble population is a qualitatively different initial condition that no simulation has carried. Observationally, a magnetized cosmic-web filament has been detected in radio (Govoni et al. 2019), and observations of TeV blazars (active galactic nuclei whose relativistic jets point toward us and shine in teraelectronvolt gamma rays) impose lower limits of order 10−1510^{-15} G on line-of-sight intergalactic fields (Neronov and Vovk 2010); any predicted bubble distribution can be confronted with these constraints.

Problem Statement

Determine, self-consistently, the spatial distribution at low redshift (down to z=0z=0) of the magnetic bubbles that, in the de Souza–Opher quark-hadron-fluctuation magnetogenesis scenario, survive from the post-QHPT era: their filling factor, comoving sizes, field strengths, and coherence lengths, and specifically whether they are concentrated in the filaments and sheets of the cosmic web or distributed more uniformly through the intergalactic medium. This requires embedding the scenario's predicted bubble population (its fields and sizes at z∼10z\sim10, e.g., ∼10 μ\sim10\,\muG over comoving ∼1\sim1 pc) in a coupled cosmological magnetohydrodynamic structure-formation calculation — with stated treatments of flux freezing, ohmic diffusion, reconnection, and bubble coalescence — in place of the uniform-versus-filamentary assumption made in the source. From the computed distribution, evaluate the consequence the source explicitly conditions on it: the deflection of ultra-high-energy cosmic rays — the arrival-direction distortions produced along filaments, computed either via the small-angle scaling δ∝(Dλ)1/2B/E\delta\propto(D\lambda)^{1/2}B/E applied to the computed bubble parameters or by direct propagation through the computed fields — and whether such deflections can contribute appreciably to the observed isotropy of UHECR arrival directions. The distribution computation and the UHECR-deflection evaluation jointly serve one objective — replacing the source's uncomputed assumption with a testable prediction — and constitute the completion boundary of the problem. The paper's synchrotron remark is retained here as observational context only: any computed distribution may additionally be compared with existing radio observations of filaments (Govoni et al. 2019) and with gamma-ray-based intergalactic magnetic-field constraints (Neronov and Vovk 2010), but such comparisons are not part of the completion boundary.

References

  1. Rafael S. de Souza and Reuven Opher, 'Origin of primordial magnetic fields', Physical Review D 77, 043529 (2008). DOI:10.1103/PhysRevD.77.043529, https://arxiv.org/abs/astro-ph/0607181
  2. T. Tajima, S. Cable, K. Shibata and R. M. Kulsrud, 'On the origin of cosmological magnetic fields', The Astrophysical Journal 390, 309 (1992). DOI:10.1086/171281, https://doi.org/10.1086/171281
  3. B. Cheng and A. V. Olinto, 'Primordial magnetic fields generated in the quark-hadron transition', Physical Review D 50, 2421 (1994). DOI:10.1103/PhysRevD.50.2421, https://doi.org/10.1103/PhysRevD.50.2421
  4. Rafael S. de Souza and Reuven Opher, 'Origin of magnetic fields in galaxies', Physical Review D 81, 067301 (2010). DOI:10.1103/PhysRevD.81.067301, https://arxiv.org/abs/0910.5248
  5. Luiz Felippe S. Rodrigues, Rafael S. de Souza and Reuven Opher, 'Suppression of small baryonic structures due to a primordial magnetic field', Monthly Notices of the Royal Astronomical Society 406, 482 (2010). DOI:10.1111/j.1365-2966.2010.16677.x, https://arxiv.org/abs/1003.2829
  6. Rafael S. de Souza, Luiz Felippe S. Rodrigues and Reuven Opher, 'Random primordial magnetic fields and the gas content of dark matter haloes', Monthly Notices of the Royal Astronomical Society 410, 2149 (2010). DOI:10.1111/j.1365-2966.2010.17588.x, https://arxiv.org/abs/1005.0639
  7. Leonardo Campanelli, 'Evolution of primordial magnetic fields in mean-field approximation', The European Physical Journal C 74, 2690 (2014). DOI:10.1140/epjc/s10052-013-2690-5, https://arxiv.org/abs/1304.4044
  8. Héctor J. Hortúa, Leonardo Castañeda and J. M. Tejeiro, 'Evolution of magnetic fields through cosmological perturbation theory', Physical Review D 87, 103531 (2013). DOI:10.1103/PhysRevD.87.103531, https://arxiv.org/abs/1104.0701
  9. Ruth Durrer and Andrii Neronov, 'Cosmological magnetic fields: their generation, evolution and observation', The Astronomy and Astrophysics Review 21, 62 (2013). DOI:10.1007/s00159-013-0062-7, https://arxiv.org/abs/1303.7121
  10. Kandaswamy Subramanian, 'The origin, evolution and signatures of primordial magnetic fields', Reports on Progress in Physics 79, 076901 (2016). DOI:10.1088/0034-4885/79/7/076901, https://arxiv.org/abs/1504.02311
  11. Tanmay Vachaspati, 'Progress on cosmological magnetic fields', Reports on Progress in Physics 84, 074901 (2021). DOI:10.1088/1361-6633/ac03a9, https://arxiv.org/abs/2010.10525
  12. F. Vazza, M. Brüggen, C. Gheller, S. Hackstein, D. Wittor and P. M. Hinz, 'Simulations of extragalactic magnetic fields and of their observables', Classical and Quantum Gravity 34, 234001 (2017). DOI:10.1088/1361-6382/aa8e60, https://arxiv.org/abs/1711.02669
  13. S. Hackstein, F. Vazza, M. Brüggen, G. Sigl and A. Dundovic, 'Propagation of ultrahigh energy cosmic rays in extragalactic magnetic fields: a view from cosmological simulations', Monthly Notices of the Royal Astronomical Society 462, 3660 (2016). DOI:10.1093/mnras/stw1903, https://arxiv.org/abs/1607.08872
  14. F. Govoni, E. Orrù, A. Bonafede, M. Iacobelli, R. Paladino, F. Vazza, M. Murgia, V. Vacca, G. Giovannini, L. Feretti, F. Loi, G. Bernardi, C. Ferrari, R. F. Pizzo, C. Gheller, S. Manti, M. Brüggen, G. Brunetti, R. Cassano, F. de Gasperin, T. A. Enßlin, M. Hoeft, C. Horellou, H. Junklewitz, H. J. A. Röttgering, A. M. M. Scaife, T. W. Shimwell, R. J. van Weeren and M. Wise, 'A radio ridge connecting two galaxy clusters in a filament of the cosmic web', Science 364, 981 (2019). DOI:10.1126/science.aat7500, https://doi.org/10.1126/science.aat7500
  15. A. Neronov and I. Vovk, 'Evidence for strong extragalactic magnetic fields from Fermi observations of TeV blazars', Science 328, 73 (2010). DOI:10.1126/science.1184192, https://doi.org/10.1126/science.1184192