Probing the Intergalactic Medium with Ly$\mathrm{\alpha}$ and 21 cm
  Fluctuations
  We study 21cm and Ly$\mathrm{\alpha}$ fluctuations, as well as
H$\mathrm{\alpha}$, while distinguishing between Ly$\mathrm{\alpha}$ emission
of galactic, diffuse, and scattered intergalactic medium (IGM) origin.
Cross-correlation information about the state of the IGM is obtained, testing
neutral versus ionized medium cases with different tracers in a seminumerical
simulation setup. In order to pave the way toward constraints on reionization
history and modeling beyond power spectrum information, we explore parameter
dependencies of the cross-power signal between 21$\,$cm and
Ly$\mathrm{\alpha}$, which displays a characteristic morphology and a turnover
from negative to positive correlation at scales of a couple Mpc$^{-1}$. In a
proof of concept for the extraction of further information on the state of the
IGM using different tracers, we demonstrate the use of the 21$\,$cm and
H$\mathrm{\alpha}$ cross-correlation signal to determine the relative strength
of galactic and IGM emission in Ly$\mathrm{\alpha}$. We conclude by showing the
detectability of the 21$\,$cm and Ly$\mathrm{\alpha}$ cross-correlation signal
over more than one decade in scale at high signal-to-noise ratio for upcoming
probes like SKA and the proposed all-sky intensity mapping satellites SPHEREx
and CDIM, while also including the Ly$\mathrm{\alpha}$ damping tail and 21cm
foreground avoidance in the modeling.
