A preferred-direction statistic for sky maps

Large patterns could exist on the microwave sky as a result of various non-standard possibilities for the large-scale Universe - rotation or shear, non-trivial topology, and single topological defects are specific examples. All-sky (or nearly all-sky) CMB data sets allow us, uniquely, to constrain such exotica, and it is therefore worthwhile to explore a wide range of statistical tests. We describe one such statistic here, which is based on determining gradients and is useful for assessing the level of 'preferred directionality' or 'stripiness' in the map. This method is more general than other techniques for picking out specific patterns on the sky, and it also has the advantage of being easily calculable for the mega-pixel maps which will soon be available. For the purposes of illustration, we apply this statistic to the four-year COBE DMR data. For future CMB maps, we expect this to be a useful statistical test of the large-scale structure of the Universe. In principle, the same statistic could also be applied to sky maps at other wavelengths, to CMB polarization maps, and to catalogues of discrete objects. It may also be useful as a means of checking for residual directionality (e.g. from Galactic or ecliptic signals) in maps.