Origin of Star-Forming Rings around Massive Centres in Massive Galaxies at $z\!<\!4$.

Using analytic modeling and simulations, we address the origin of an abundance of star-forming, clumpy, extended gas rings about massive central bodies in massive galaxies at $z\! \! 10^{9.5}\, M_\odot$ where merger-driven spin flips and supernova feedback are ineffective. The rings survive above this threshold mass after events of compaction to central nuggets. Ring longevity seemed to conflict with the expected inward mass transport driven by torques from violent disc instability. However, by evaluating the torques from a tightly wound spiral structure, we find that the timescale for transport per orbital time is $\propto\! \delta_{\rm d}^{-3}$, with $\delta_{\rm d}$ the cold-to-total mass ratio interior to the ring. A long-lived ring forms when the ring transport is slower than its replenishment by accretion and the interior depletion by SFR, both valid for $\delta_{\rm d} \!<\! 0.3$. The central mass that lowers $\delta_{\rm d}$ and maintains the ring is made of compaction-driven bulge and/or dark-matter, aided by the lower gas fraction at lower $z$. The ring could be Toomre unstable for clump and star formation. Mock images of simulated rings through dust indicate qualitative consistency with observed rings about bulges in massive $z\!\sim\!0.5\!-\!3$ galaxies, in $H_{\alpha}$ and in deep HST imaging. ALMA mock images indicate that $z\!\sim\!0.5\!-\!1$ rings should be detectable. We quote expected observable properties and abundances of rings and their central blue or red nuggets for comparison to observations.