  Feeding and feedback of active galactic nuclei (AGN) are critical for
understanding the dynamics and thermodynamics of the intracluster medium (ICM)
within the cores of galaxy clusters. While radio bubbles inflated by AGN jets
could be dynamically supported by cosmic rays (CRs), the impact of CR-dominated
jets are not well understood. In this work, we perform three-dimensional
simulations of CR-jet feedback in an isolated cluster atmosphere; we find that
CR jets impact the multiphase gas differently than jets dominated by kinetic
energy. In particular, CR bubbles can more efficiently uplift the cluster gas
and cause an outward expansion of the hot ICM. Due to adiabatic cooling from
the expansion and less efficient heating from CR bubbles by direct mixing, the
ICM is more prone to local thermal instabilities, which will later enhance
chaotic cold accretion onto the AGN. The amount of cold gas formed during the
bubble formation and its late-time evolution sensitively depend on whether CR
transport processes are included or not. We also find that low-level, subsonic
driving of turbulence by AGN jets holds for both kinetic and CR jets;
nevertheless, the kinematics is consistent with the Hitomi measurements.
Finally, we carefully discuss the key observable signatures of each bubble
model, focusing on gamma-ray emission (and related comparison with Fermi), as
well as thermal Sunyaev-Zel'dovich constraints.
