  With the axion being a prime candidate for dark matter, there has been some
recent interest in direct detection through a so called `Ferromagnetic
haloscope.' Such devices exploit the coupling between axions and electrons in
the form of collective spin excitations of magnetic materials with the readout
through a microwave cavity. Here, we present a new, general, theoretical
treatment of such experiments in a Hamiltonian formulation for strongly coupled
magnons and photons, which hybridise as cavity-magnon polaritons. Such strongly
coupled systems have an extended measurable dispersive regime. Thus, we extend
the analysis and operation of such experiments into the dispersive regime,
which allows any ferromagnetic haloscope to achieve improved bandwidth with
respect to the axion mass parameter space. This experiment was implemented in a
cryogenic setup, and initial search results are presented setting laboratory
limits on the axion-electron coupling strength of $g_{aee}>3.7\times10^{-9}$ in
the range $33.79~\mu$eV$< m_a<33.94~\mu$eV with $95\%$ confidence. The
potential bandwidth of the Ferromagnetic haloscope was calculated to be in two
bands, the first of about $1$GHz around $8.24$GHz (or $4.1~\mu$eV mass range
around $34.1~\mu$eV) and the second of about $1.6$GHz around $10$GHz
($6.6~\mu$eV mass range around $41.4~\mu$eV). Frequency tuning may also be
easily achieved via an external magnetic field which changes the ferromagnetic
resonant frequency with respect to the cavity frequency. The requirements
necessary for future improvements to reach the DFSZ axion model band are
discussed in the paper.
