Time-Dependent Cosmic Ray Halos from Bursty Star Formation and Active Galactic Nuclei: Semi-Analytic Formalism and Galaxy Formation Implications

Cosmic ray (CR) feedback in galaxy evolution has seen a theoretical resurgence in the past decade, but significant uncertainties remain in CR transport through the interstellar and circum-galactic media (ISM and CGM). While several works indicate CR effects may be notable in both star-forming and quenched massive galaxies, modeling the vast CR transport parameter space currently allowed by observations is computationally restrictive to survey. Analytic treatments of CR feedback have provided useful insights to potential ramifications in different regimes, but have relied on time-steady assumptions which may not well characterize CR effects at different cosmic epochs and galaxy mass scales. We present semi-analytic approximations and numerical solutions describing the time-dependent evolution of CR pressure in the CGM under simplified assumptions, which allow for quick evaluation of the vast allowable CR transport parameter space. We demonstrate that time-dependent injection from bursty star formation and/or episodic black hole accretion can substantially alter CR pressure profiles, particularly in the outer halos of massive galaxies ($\gtrsim R_{vir}$). Finally, we benchmark the approximate solutions from our semi-analytic formalism against a cosmic ray-magnetohydrodynamic (CR-MHD) cosmological zoom-in galaxy simulation directly modeling the CR scattering rate and emergent transport in full generality, highlighting the validity of our approach. We conclude by motivating careful consideration of time-dependent ``softening" effects in sub-grid routines for CR feedback, particularly for use in large cosmological volumes.