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Large-scale (i.e., $\gtrsim {\rm kpc}$) and micro-Gauss scale magnetic fieldshave been observed throughout the Milky Way and nearby galaxies. These fieldsdepend on the geometry and matter-energy composition, can display complicatedbehavior such as direction reversals, and are intimately related to theevolution of the source galaxy. Simultaneously, gravitational-wave astronomyoffers a new probe into astrophysical systems, for example the LaserInterferometer Space Antenna (LISA) will observe the mergers of massive (i.e.,$M ~> 10^6$ M$_{\odot}$) black-hole binaries and provide extraordinaryconstraints on the evolution of their galactic hosts. In this work, we show howgalactic, large-scale magnetic fields and their electromagnetic signatures areconnected with LISA gravitational-wave observations via their common dependenceon the massive black-hole binary formation scenario of hierarchical galaxymergers. Combining existing codes, we astrophysically evolve a population ofmassive binaries from formation to merger and find that they are detectable byLISA with signal-to-noise ratio $\sim 10^3$ which is correlated with quantitiesfrom the progenitors' phase of circumbinary disk migration such as the maximummagnetic field magnitude $|\mathbf{B}| \approx 7 \,\mu$G, polarized intensity,and Faraday rotation measure. Interesting correlations result between theseobservables arising from their dependencies on the black-hole binary totalmass, suggesting a need for further analyses of the full parameter space. Weconclude with a discussion on this new multi-messenger window into galacticmagnetic fields.

A multi-messenger window into galactic magnetic fields and black hole mergers with LISA