  The early growth of black holes (BHs) in high-redshift galaxies is likely
regulated by their feedback on the surrounding gas. While radiative feedback
has been extensively studied, the role of mechanical feedback has received
comparatively less scrutiny to date. Here we use high-resolution parsec-scale
hydrodynamical simulations to study jet propagation and its effect on BH
accretion onto 100 ${\rm M_\odot}$ BHs in the dense, low-metallicity gas
expected in early protogalaxies. As the jet propagates, it shocks the
surrounding gas and forms a jet cocoon. The cocoon consists of a
rapidly-cooling cold phase at the interface with the background gas and an
over-pressured subsonic phase of reverse shock-heated gas filling the cocoon
interior. We systematically vary the background gas density and temperature, BH
feedback efficiency, and the jet model. We found that the jet cocoon width
roughly follows a scaling derived by assuming momentum conservation in the jet
propagation direction, and energy conservation in the lateral directions.
Depending on the assumed gas and jet properties, the cocoon either stays
elongated out to a large radius or isotropizes before reaching the Bondi
radius, forming a nearly spherical bubble. Lower jet velocities and higher
background gas densities result in self-regulation to higher momentum fluxes
and elongated cocoons. In all cases, the outward momentum flux of the cocoon
balances the inward momentum flux of the inflowing gas near the Bondi radius,
which ultimately regulates BH accretion. The larger the distance the jet cocoon
reaches, the longer the variability timescale of the BH accretion rate.
Overall, the average accretion rate always remains below the Bondi rate, and
exceeds the Eddington rate only if the ambient medium is dense and cold, and/or
the jet is weak. We derive the combination of jet and ambient gas parameters
yielding super-Eddington growth.
