Coaxing cosmic 21 cm fluctuations from the polarized sky using m-mode analysis

In this paper we continue to develop the m-mode formalism, a technique for efficient and optimal analysis of wide-field transit radio telescopes, targeted at 21 cm cosmology. We extend this formalism to give an accurate treatment of the polarized sky, fully accounting for the effects of polarization leakage and cross polarization. We use the geometry of the measured set of visibilities to project down to pure temperature modes on the sky, serving as a significant compression, and an effective first filter of polarized contaminants. As in our previous work, we use the m-mode formalism with the Karhunen-Loeve transform to give a highly efficient method for foreground cleaning, and demonstrate its success in cleaning realistic polarized skies observed with an instrument suffering from substantial off axis polarization leakage. We develop an optimal quadratic estimator in the m-mode formalism which can be efficiently calculated using a Monte Carlo technique. This is used to assess the implications of foreground removal for power spectrum constraints where we find that our method can clean foregrounds well below the foreground wedge, rendering only scales k(parallel to) < 0.02 h Mpc(-1) inaccessible. As this approach assumes perfect knowledge of the telescope, we perform a conservative test of how essential this is by simulating and analyzing data sets with deviations about our assumed telescope. Assuming no other techniques to mitigate bias are applied, we find we recover unbiased power spectra when the per-feed beamwidth to be measured to 0.1%, and amplifier gains to be known to 1% within each minute. Finally, as an example application, we extend our forecasts to a wideband 400-800 MHz cosmological observation and consider the implications for probing dark energy, finding a pathfinder-scale medium-sized cylinder telescope improves the Dark Energy Task Force figure of merit by around 70% over Planck and Stage II experiments alone.