Fermi-bubble bulk and edge analysis reveals dust, cooling breaks, and cosmic-ray diffusion, facilitating a self-consistent model

The full, radio to $\gamma$-ray spectrum of the Fermi bubbles is shown to be consistent with standard strong-shock electron acceleration at the bubble edge, without the unnatural energy cutoffs and unrealistic electron cooling of previous studies, if the ambient interstellar radiation is strong; the $\gamma$-ray cooling break should then have a microwave counterpart, undetected until now. Indeed, a broadband bubble-edge analysis uncovers a pronounced downstream dust component, which masked the anticipated $\sim 35$ GHz spectral break, and the same overall radio softening consistent with Kraichnan diffusion previously reported in $\gamma$-rays.