Title: Unifying the micro and macro properties of AGN feeding and feedback,
Abstract: We unify the feeding and feedback of supermassive black holes with the global
properties of galaxies, groups, and clusters, by linking for the first time the
physical mechanical efficiency at the horizon and Mpc scale. The macro hot halo
is tightly constrained by the absence of overheating and overcooling as probed
by X-ray data and hydrodynamic simulations ($\varepsilon_{\rm BH} \simeq$
10$^{-3}\, T_{\rm x,7.4}$). The micro flow is shaped by general relativistic
effects tracked by state-of-the-art GR-RMHD simulations ($\varepsilon_\bullet
\simeq$ 0.03). The SMBH properties are tied to the X-ray halo temperature
$T_{\rm x}$, or related cosmic scaling relation (as $L_{\rm x}$). The model is
minimally based on first principles, as conservation of energy and mass
recycling. The inflow occurs via chaotic cold accretion (CCA), the rain of cold
clouds condensing out of the quenched cooling flow and recurrently funneled via
inelastic collisions. Within 100s gravitational radii, the accretion energy is
transformed into ultrafast 10$^4$ km s$^{-1}$ outflows (UFOs) ejecting most of
the inflowing mass. At larger radii the energy-driven outflow entrains
progressively more mass: at roughly kpc scale, the velocities of the
hot/warm/cold outflows are a few 10$^3$, 1000, 500 km s$^{-1}$, with median
mass rates ~10, 100, several 100 M$_\odot$ yr$^{-1}$, respectively. The unified
CCA model is consistent with the observations of nuclear UFOs, and ionized,
neutral, and molecular macro outflows. We provide step-by-step implementation
for subgrid simulations, (semi)analytic works, or observational interpretations
which require self-regulated AGN feedback at coarse scales, avoiding the
a-posteriori fine-tuning of efficiencies.