On the contribution of galaxies to the magnetic field in cosmic voids

Technical Summary: On the Contribution of Galaxies to the Magnetic Field in Cosmic Voids

Problem Statement
The origin of magnetic fields in cosmic voids remains a significant open question in cosmology. While upper limits on space-filling primordial magnetic fields are constrained by the cosmic microwave background ($\lesssim 1$ nG) and ultra-high-energy cosmic ray deflections ($\lesssim 10^{-10}$ G), observational lower bounds derived from $\gamma$-ray cascade data (specifically time delays, angular broadening, and spectral changes from blazars) suggest field strengths in the range of $B \gtrsim 10^{-17}$ G. A central question is whether these observed lower bounds can be explained by purely astrophysical processes, such as the superposition of magnetic dipoles from individual galaxies, or if they necessitate a primordial origin. Recent claims suggested that the superposition of galactic dipoles could account for these fields without invoking primordial seeds, provided a magnetostatic approximation in a vacuum is used. However, this approach neglects the screening effects of the highly ionized intergalactic plasma that permeates the Universe.

Methodology
The authors provide a qualitative description of magnetic field screening by intergalactic plasmas to test the viability of the galactic dipole superposition hypothesis. The analysis proceeds through the following steps:

1. Evolution Equation: The study utilizes the evolution equation for magnetic fields in plasmas where plasma flow is neglected (a reasonable first approximation for non-relativistic intergalactic media). In Fourier space, the equation accounts for Hubble expansion, resistivity ($\eta$), and the magnetization vector-field ($M_\perp$).
2. Analytic Approximations: To solve the evolution equation, the authors introduce three key assumptions:
   • Resistivity Scaling: Resistivity follows a power law dependent on temperature ($\eta \propto T^{-n_\eta}$), which is translated to redshift dependence. The Spitzer resistivity is used as a specific example.
   • Cosmological Era: The Universe is assumed to be in a single-component dominated era (matter or dark energy), defining the Hubble parameter $H(z)$.
   • Magnetization Scaling: The magnetization is approximated to scale as $(1+z)^2$, consistent with magnetic energy density scaling.
3. Screening Length Derivation: Using these assumptions, the authors derive a critical wave-number $k_r(z)$ and the corresponding magnetic screening length $l_{r,0} = 2\pi/k_{r,0}$. This length scale determines the distance over which magnetic fields can propagate before being suppressed by plasma effects.
4. Field Calculation: The authors calculate the screened magnetic field in both Fourier space and direct space. They derive the field of a single screened dipole and then estimate the collective field strength in a void by summing the contributions of numerous galaxies modeled as dipoles.

Key Contributions and Results
The primary contribution of the paper is the demonstration that intergalactic plasma screening drastically suppresses magnetic fields generated by galactic dipoles at cosmological distances.

• Screening Scale: For realistic plasma parameters (e.g., Spitzer resistivity in a non-relativistic plasma with $T_{e,0} \approx 10$ eV in a matter-dominated universe), the screening length is calculated to be $l_{r,0} \approx 2 \times 10^{-4}$ pc. This is orders of magnitude smaller than the typical size of a cosmic void ($l_{void} \sim 1$ Mpc).
• Suppression Mechanism:
  • In Fourier space, the magnetic field $B_k$ exhibits a $(k/k_{r,0})^2$ suppression for small-$k$ modes (large distances), where $k \ll k_{r,0}$.
  • In real space, the field of a screened dipole retains its vacuum form only at distances $r \ll l_{r,0}$. For distances $r \gg l_{r,0}$, the field decays exponentially as $\sim (r/l_{r,0})^3 e^{-r^2/4l_{r,0}^2}$.
• Quantitative Estimate: Applying this to a void of radius $r_{void} \sim 20$ Mpc, the collective magnetic field from galaxies is estimated to be $B_{void} \approx (10^{-5} \text{ G}) \times e^{-r_{void}^2/4l_{r,0}^2}$. Given the tiny screening length, the resulting field is exponentially suppressed to values far below $10^{-17}$ G.
• Comparison to Observations: To reproduce the observational lower bounds of $B \sim 10^{-16}$ G via galactic dipoles, a screening length of order $l_{r,0} \approx 2$ Mpc would be required. The authors argue such a value is astrophysically implausible for intergalactic plasmas.

Significance and Claims
The paper concludes that the hypothesis of explaining cosmic void magnetic fields solely through the superposition of static galactic dipoles is untenable due to plasma screening. The authors state that:

• Contributions from the superposition of static dipoles are "highly suppressed" and "cannot explain indications for lower bounds based on observations of $\gamma$–ray cascades."
• The resulting field strength from galactic sources falls short of observed lower bounds by "many orders of magnitude."
• While the calculation is a "simplified argument," it effectively rules out the simple galactic field explanation for the observed void fields.
• The authors note that other mechanisms, such as magnetized outflows from large-scale structures or plasma instabilities (which are not the focus of this specific calculation), may still play a role, but the specific mechanism of static dipole superposition is insufficient.

The work serves as a critical counter-argument to recent claims suggesting a purely astrophysical, non-primordial origin for void magnetic fields via dipole superposition, reinforcing the need to consider primordial origins or more complex astrophysical transport mechanisms.