The Conditional CMB Visibility of High-Redshift X-Ray Cavities in Galaxy clusters

At high redshift, the cosmic microwave background (CMB) both cools relativistic leptons and provides seed photons for inverse-Compton (IC) X-ray emission. These two effects link the particle content of a bubble directly to the soft X-ray depression used to identify a cavity. We derive a physical criterion connecting lepton cooling, bubble volume, enthalpy evolution, and 0.5--2 keV X-ray visibility, while separating intrinsic filling of the X-ray depression from detector-dependent cavity recovery. For uncooled electron populations described by a power law with spectral index $s=2.5$, minimum Lorentz factor $100\leqγ_{\min}\leq1000$, and maximum Lorentz factor $γ_{\max}=10^5$, the measured atmosphere of the testbed high-redshift cluster SPT-CL J2215$-$3537 requires 0.6--1.2\% of the pressure in radiating leptons to fill the soft-band deficit. Across the full tested ranges of $s$, $γ_{\min}$, and atmospheric properties, the required fraction is 0.2--3.7\%. Complete loss of this component reduces the enthalpy by less than 3\% relative to an otherwise identical adiabatic bubble at the same pressure. A low-cutoff control with $γ_{\min}=10$ and $s=3$ instead requires 18.9\%, identifying the poorly constrained low-energy spectrum as the main physical uncertainty. For a fixed atmosphere and spectrum, the required fraction scales as $(1+z)^{-4}$. The declining-pressure tracks show that the X-ray deficit can cross the adopted detection threshold while the enthalpy deficit remains below 0.4\%; the time of this transition depends on the assumed pressure history. Thus, high-redshift cavity visibility depends on both the surrounding atmosphere and the pressure-carrying particle spectrum. This criterion provides the physical basis for an instrument-specific recovery analysis.