  The Fermi bubbles, two giant gamma-ray bubbles above and below the Galactic
center (GC), are among the most important findings of the Fermi Gamma-ray Space
Telescope. Because of the proximity, spatially resolved, multi-wavelength
observations offer excellent opportunities to learn about the physical origin
of the bubbles. One of the unique features of the observed bubbles is that
their gamma-ray spectrum, including a high-energy cutoff at ~110 GeV and the
overall shape of the spectrum, is nearly spatially uniform. The high-energy
spectral cutoff is suggestive of a leptonic origin as it could be a signature
of synchrotron and inverse-Compton (IC) cooling of cosmic-ray (CR) electrons;
however, even for a leptonic model, it is not obvious why the spectrum should
be spatially uniform. In this work, we investigate the formation of the Fermi
bubble in the leptonic jet scenario using a newly implemented CRSPEC module in
FLASH that allows us to track the evolution of CR spectrum on-the-fly during
the simulations. We show that the high-energy cutoff is caused by fast
synchrotron and IC cooling near the GC when the jets were first launched. After
the jets propagate away from the GC, the dynamical timescale of the jets become
the shortest among all relevant timescales, and therefore the spectrum is
essentially advected with only mild cooling losses. This could explain why the
bubble spectrum is nearly spatially uniform: the CRs from different parts of
the bubbles as seen today all share the same origin near the GC at early stage
of the bubble expansion. We find that the CR spatial and spectral distribution
predicted by the leptonic jet model can simultaneously match the normalization,
spectral shape, and high-energy cutoff of the observed gamma-ray spectrum and
their spatial uniformity, suggesting that past AGN jet activity is a likely
mechanism for the formation of the Fermi bubbles.
