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Observational studies showed that galaxy disks are already in place in the first few billion years of the universe. The early disks detected so far, with typical half-light radii of 3 kiloparsecs at stellar masses around 10^11 M_sun for redshift z~3, are significantly smaller than today's disks with similar masses, in agreement with expectations from current galaxy models. Here, we report observations of a giant disk at z=3.25, when the universe was only 2 billion years old, with a half-light radius of 9.6 kiloparsecs and stellar mass of 3.7^+2.6_-2.2x10^11 M_sun. This galaxy is larger than any other kinematically-confirmed disks at similar epochs and surprisingly similar to today's largest disks regarding size and mass. JWST imaging and spectroscopy reveal its spiral morphology and a rotational velocity consistent with local Tully-Fisher relation. Multi-wavelength observations show that it lies in an exceptionally dense environment, where the galaxy number density is over ten times higher than the cosmic average and mergers are frequent. The discovery of such a giant disk suggests the presence of favorable physical conditions for large-disk formation in dense environments in the early universe, which may include efficient accretion of gas carrying coherent angular momentum and non-destructive mergers between exceptionally gas-rich progenitor galaxies., Comment: 22 pages, 11 figures; submitted

Observations of $z \gtrsim 6$ quasars provide information on the early evolution of the most massive black holes (MBHs) and galaxies. Current observations, able to trace both gas and stellar properties, reveal a population of MBHs that is significantly more massive than expected from the local MBH-stellar mass relation. The population lies on, but mostly above, the relation observed in the nearby Universe. This suggests that these objects grew very rapidly. To explain their presence when the Universe was less than 1~Gyr old and to assess the physical conditions for their rapid growth, we explored whether episodes of accretion above the Eddington limit can occur across cosmic epochs. By employing state-of-the-art high-resolution cosmological zoom-in simulations of a $z\sim 7$ quasar, where different accretion regimes are included consistently, together with their associated radiative and kinetic feedback, we show that super-Eddington phases can be sustained for relatively long timescales (tens of millions of years). This allows the MBH to rapidly grow by up to three orders of magnitude, depending on the strength of the kinetic feedback. We also show by means of a semianalytic calculation that the MBH spin remains moderate and does not take on extremely high values during the super-Eddington phases. This results in a lower feedback efficiency, which may allow the rapid growth required to explain over-massive high-redshift MBHs., Comment: 8 pages, 5 figures, matches the published version