Self-gravitating isothermal sphere in an expanding background

Spatially homogeneous thermal equilibria of self-gravitating gas, being impossible otherwise, are nevertheless allowed in an expanding background accounting for Universe's expansion. Furthermore, a fixed density at the boundary of a perturbation is a natural boundary condition keeping the mass finite inside without the need to invoke any unphysical walls. These facts allow us to develop a consistent gravitational thermodynamics of isothermal spheres inside an expanding Universe. In the canonical and grand canonical ensembles we identify an instability for both homogeneous and inhomogeneous equilibria. We discuss a potential astrophysical application. If such an instability is triggered on baryonic gas at high redshift $z > 137$ when the primary baryonic component, namely atomic hydrogen, was still thermally locked to the Cosmic Microwave Background radiation, then the corresponding destabilized gaseous clouds have baryonic mass $\geq 0.8\cdot 10^5 {\rm M}_\odot$ and radius $\geq 15{\rm pc}$.