In-situ formation of star clusters at z > 7 via galactic disk fragmentation; shedding light on ultra-compact clusters and overmassive black holes seen by JWST

We investigate the nature of star formation in gas-rich galaxies at $z > 7$ forming in a markedly overdense region, in the whereabouts of a massive virialized halo already exceeding $10^{12}$ M$_{\odot}$. We find that not only the primary galaxy, but also the lower-mass companion galaxies rapidly develop massive self-gravitating compact gas disks, less than 500~pc in size, which undergo fragmentation by gravitational instability into very massive bound clumps. Star formation proceeds fast in the clumps, which quickly turn into compact star clusters with masses in the range $10^5$-$10^8$ M$_{\odot}$ and typical half-mass radii of a few pc, reaching characteristic densities above $10^5$ M$_{\odot}$ pc$^{-2}$. The properties of the clusters in the lowest-mass galaxy bear a striking resemblance to those recently discovered by the James Webb Space Telescope (JWST) in the lensed Cosmic Gems arc system at $z = 10.2$. We argue that, due to their extremely high stellar densities, intermediate-mass black holes (IMBHs) would form rapidly inside the clusters, which would then swiftly sink and merge on their way to the galactic nucleus, easily growing a $10^7$~M$_{\odot}$ supermassive black hole (SMBH). Due to the high fractional mass contribution of clusters to the stellar mass of the galaxies, in the range $20$-$40\%$, the central SMBH would comprise more than $10\%$ of the mass of its host galaxy, naturally explaining the overmassive SMBHs discovered by JWST at $z > 6$.