Observing primordial magnetic fields through Dark Matter

Primordial magnetic fields are often thought to be the early Universe seeds that have bloomed into what we observe today as galactic and extra-galactic magnetic fields. Owing to their minuscule strength, primordial magnetic fields are very hard to detect in cosmological and astrophysical observations. We show how this changes if a part of neutral Dark Matter has a magnetic susceptibility. In this way, by studying Dark Matter one can obtain information about the properties of primordial magnetic fields, even if the latter have a comoving amplitude $B_0 \lesssim0.01~\mbox{nG}$. In our model Dark Matter is a stable singlet scalar $\chi$, which interacts with electromagnetism through the Rayleigh operator as $\chi^2 F_{\mu \nu} F^{\mu \nu}/\Lambda^2$. For primordial magnetic fields present in the early Universe this operator forces the $Z_2$-symmetry of the model to be spontaneously broken. Later, when the primordial magnetic field redshifts below a critical value, the symmetry is restored through an "inverse phase transition". At that point the field $\chi$ begins to oscillate and acts as a "magnetomorphic" Dark Matter component, inheriting the properties of the primordial magnetic field space distribution. In particular, for a nearly flat spectrum of magnetic field fluctuations, the scalar $\chi$ carries a statistically anisotropic isocurvature mode. We discuss the parameter space of the model and consider the possibility that the bulk of the Dark Matter is composed of the same particles $\chi$ produced via the freeze-in mechanism.
Comment: 29 pages and 4 figures, v2 with updated references