Fermi bubble edges: spectrum and diffusion function
  Current measurements of the $\gamma$-ray Fermi bubbles (FB) are based on
model-dependent tracers, carry substantial systematic uncertainties, and are at
some tension with each other. We show that gradient filters pick out the FB
edges, which are found to smoothly connect to the bipolar X-ray structure
emanating from the Galactic center, thus supporting the interpretation of the
FBs as a Galactic-scale phenomenon. The sharp edges facilitate a direct,
model-free measurement of the peripheral FB spectrum. The result is strikingly
similar to the full FB-integrated spectrum, softened by a power law of index
$\eta\simeq (0.2\mbox{--}0.3)$. This is naturally explained, in both hadronic
and leptonic models, if cosmic rays are injected at the edge, and diffuse away
preferentially at higher energies $E$. The inferred, averaged diffusion
function in the (more plausible) leptonic model, $D(E)\simeq
10^{29.5}(E/10\mbox{ GeV})^{0.48\pm0.02}\mbox{ cm}^2\mbox{ s}^{-1}$, is
consistent with estimates for Kraichnan-like turbulence. Our results, in
particular the minute spatial variations in $\eta$, indicate that the FB edge
is a strong, Mach $\gtrsim5$, forward shock.
