The spectral index of polarized diffuse Galactic emission between 30 and 44 GHz

We present an estimate of the polarized spectral index between the Planck 30 and 44 GHz surveys in $3.7^\circ$ pixels across the entire sky. We use an objective reference prior that maximises the impact of the data on the posterior and multiply this by a maximum entropy prior that includes information from observations in total intensity by assuming a polarization fraction. Our parametrization of the problem allows the reference prior to be easily determined and also provides a natural method of including prior information. The spectral index map is consistent with those found by others between surveys at similar frequencies. Across the entire sky we find an average temperature spectral index of $-2.99\pm0.03(\pm1.12)$ where the first error term is the statistical uncertainty on the mean and the second error term (in parentheses) is the extra intrinsic scatter in the data. We use a clustering algorithm to identify pixels with actual detections of the spectral index. The average spectral index in these pixels is $-3.12\pm0.03(\pm0.64)$ and then when also excluding pixels within $10^\circ$ of the Galactic plane we find $-2.92(\pm0.03)$. We find a statistically significant difference between the average spectral indices in the North and South Fermi bubbles. Only including pixels identified by the clustering algorithm, the average spectral index in the southern bubble is $-3.00\pm0.05(\pm0.35)$, which is similar to the average across the whole sky. In the northern bubble we find a much harder average spectral index of $-2.36\pm0.09(\pm0.63)$. Therefore, if the bubbles are features in microwave polarization they are not symmetric about the Galactic plane.