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 ^ { \prime } 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 ^ { \prime } .
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 \left ( \frac { S } { N } \right ) ^ { 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 .