Statistical Diagnostics to Identify Galactic Foregrounds in $B$-Mode Maps

Recent developments in the search for inflationary gravitational waves in the cosmic microwave background polarization motivate the search for new diagnostics to distinguish the Galactic foreground contribution to $B$ modes from the cosmic signal. We show that $B$ modes from these foregrounds should exhibit a local hexadecapolar departure in power from statistical isotropy (SI). We present a simple algorithm to search for a uniform SI violation of this sort, as may arise in a sufficiently small patch of sky. We then show how to search for these effects if the orientation of the SI violation varies across the survey region, as is more likely to occur in surveys with more sky coverage. If detected, these departures from Gaussianity would indicate some level of Galactic foreground contamination in the $B$-mode maps. Given uncertainties about foreground properties, though, caution should be exercised in attributing a null detection to an absence of foregrounds.