Jiang-Tao Li

We develop an analytic framework for the evolution of feedback-driven bubbles expanding into a hot, volume-filling circumgalactic medium (CGM), where the ambient pressure and sound speed are non-negligible and radiative cooling is often inefficient. The evolution is organized into four stages -- free expansion, Sedov--Taylor expansion, pressure-modified/transonic transition, and post-transonic relaxation -- and we derive self-consistent scalings for the characteristic radii and timescales that delimit these stages. A central result is that, in hot halos, the end of the strong-shock evolution is frequently set by pressure confinement and transonicity rather than by the onset of catastrophic cooling, implying only a modest late-time overshoot beyond the pressure-balance/transonic point. We connect the dynamics to observable outcomes by estimating bubble sizes and lifetimes, order-of-magnitude band-limited X-ray luminosities, and high-ionization ion column densities, and we provide stitched numerical trajectories that contrast our pressure-modified model appropriate for hot CGM conditions with a classical Sedov--Taylor benchmark. We then discuss physically motivated extensions beyond the single-event baseline, including continuous or episodic energy injection relevant for AGN-driven bubbles and nuclear outflows, highlighting the much higher specific energy of AGN feedback compared to supernovae and the resulting dynamical differences in how bubbles are driven. We further outline how multiphase interaction, mass loading, anisotropic dissipation, intermittency, confinement, and non-thermal channels can increase the emergent X-ray radiation efficiency without requiring changes to the intrinsic feedback energy partition at the launching site. This framework provides a transparent bridge between idealized bubble theory and feedback signatures in hot galactic halos.