How to detect gravitational waves through the cross-correlation of the galaxy distribution with the CMB polarization

Thompson scattering of cosmic microwave background (CMB) photons off of free electrons during the reionization epoch induces a correlation between the distribution of galaxies and the polarization pattern of the CMB, the magnitude of which is proportional to the quadrupole moment of radiation at the time of scattering. Since the quadrupole moment generated by gravitational waves (GWs) gives rise to a different polarization pattern than that produced by scalar modes, one can put interesting constraints on the strength of GWs on large scales by cross-correlating the small scale galaxy distribution and CMB polarization. We use this method together with Fisher analysis to predict how well future surveys can measure the tensor-to-scalar ratio $r$. We find that with a future CMB experiment with detector noise Delta_P = 2 mu K-arcmin and a beam width theta_FWHM = 2 and a future galaxy survey with limiting magnitude I<25.6 one can measure the tensor-to-scalar ratio with an error sigma_r simeq 0.09. To measure r approx 0.01, however, one needs Delta_P simeq 0.5 mu K-radian and theta_FWHM simeq 1. We also investigate a few systematic effects, none of which turn out to add any biases to our estimators, but they increase the error bars by adding to the cosmic variance. The incomplete sky coverage has the most dramatic effect on our constraints on r for large sky cuts, with a reduction in signal-to-noise smaller than one would expect from the naive estimate (S/N)^2 propto f_sky. Specifically, we find a degradation factor of f_deg=0.32 pm 0.01 for a sky cut of |b|>10^circ (f_sky=0.83) and f_deg=0.056 pm 0.004 for a sky cut of |b|>20^circ (f_sky=0.66). Nonetheless, given that our method has different systematics than the more conventional method of observing the large scale B modes directly, it may be used as an important check in the case of a detection.