  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.
