The Fermi Bubbles are giant Galactic structures observed in both gamma-rays and microwaves. Recent studies have found support for the hypothesis that the gamma-ray and microwave emission can both be understood as arising from a hard cosmic-ray electron population within the volume of the Bubbles, via inverse Compton scattering and synchrotron radiation respectively. The relative rates of these processes are set by the relative energy density of the interstellar radiation field and the magnetic field within the Bubbles; consequently, under the hypothesis of a common origin, the combination of the gamma-ray and microwave measurements can be used to estimate the magnetic field within the Bubbles. We revisit the consistency of this hypothesis on a latitude-by-latitude basis, using data from Fermi, WMAP and Planck; estimate the variation of the electron spectrum within the Bubbles; and infer bounds on the magnetic field within the Bubbles as a function of distance from the Galactic plane. We find that while the microwave and gamma-ray spectra are generally consistent with the leptonic hypothesis for few-microGauss magnetic fields, there appears to be a preference for spectral hardening in the microwaves at mid-latitudes (especially in the |b|~ 25-35 degree range) that is not mirrored in the gamma rays. This result may hint at a non-leptonic contribution to the gamma-ray spectra; however, the discrepancy can be reconciled in purely leptonic models if the cutoff energy for the electrons is lower in this latitude range and the spectrum below the cutoff is harder.
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