  The high-redshift Universe contains luminous Lyman Alpha (hereafter Lya)
emitting sources such as galaxies and quasars. The emitted Lya radiation is
often scattered by surrounding neutral hydrogen atoms. We show that the
scattered Lya radiation obtains a high level of polarization for a wide range
of likely environments of high-redshift galaxies. For example, the
back-scattered Lya flux observed from galaxies surrounded by a superwind-driven
outflow may reach a fractional polarization as high as ~40%. Equal levels of
polarization may be observed from neutral collapsing protogalaxies. Resonant
scattering in the diffuse intergalactic medium typically results in a lower
polarization amplitude (<7%), which depends on the flux of the ionizing
background. Spectral polarimetry can differentiate between Lya scattering off
infalling gas and outflowing gas; for an outflow the polarization should
increase towards longer wavelengths while for infall the opposite is true. Our
numerical results suggest that Lya polarimetry is feasible with existing
instruments, and may provide a new diagnostic of the distribution and
kinematics of neutral hydrogen around high-redshift galaxies. Moreover,
polarimetry may help suppress infrared lines originating in the Earth's
atmosphere, and thus improve the sensitivity of ground-based observations to
high-redshift Lya emitting galaxies outside the currently available redshift
windows.
