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Abstract Tight relationships exist in the local Universe between the central stellar properties of galaxies and the mass of their supermassive black hole (SMBH)^1–3. These suggest that galaxies and black holes co-evolve, with the main regulation mechanism being energetic feedback from accretion onto the black hole during its quasar phase^4–6. A crucial question is how the relationship between black holes and galaxies evolves with time; a key epoch to examine this relationship is at the peaks of star formation and black hole growth 8–12 billion years ago (redshifts 1–3)⁷. Here we report a dynamical measurement of the mass of the black hole in a luminous quasar at a redshift of 2, with a look back in time of 11 billion years, by spatially resolving the broad-line region (BLR). We detect a 40-μas (0.31-pc) spatial offset between the red and blue photocentres of the Hα line that traces the velocity gradient of a rotating BLR. The flux and differential phase spectra are well reproduced by a thick, moderately inclined disk of gas clouds within the sphere of influence of a central black hole with a mass of 3.2 × 10⁸ solar masses. Molecular gas data reveal a dynamical mass for the host galaxy of 6 × 10¹¹ solar masses, which indicates an undermassive black hole accreting at a super-Eddington rate. This suggests a host galaxy that grew faster than the SMBH, indicating a delay between galaxy and black hole formation for some systems. 

We report the time-resolved spectral analysis of a bright near-infrared and moderate X-ray flare of Sgr A ⋆ . We obtained light curves in the M , K , and H bands in the mid- and near-infrared and in the 2 − 8 keV and 2 − 70 keV bands in the X-ray. The observed spectral slope in the near-infrared band is νL ν  ∝  ν 0.5 ± 0.2 ; the spectral slope observed in the X-ray band is νL ν  ∝  ν −0.7 ± 0.5 . Using a fast numerical implementation of a synchrotron sphere with a constant radius, magnetic field, and electron density (i.e., a one-zone model), we tested various synchrotron and synchrotron self-Compton scenarios. The observed near-infrared brightness and X-ray faintness, together with the observed spectral slopes, pose challenges for all models explored. We rule out a scenario in which the near-infrared emission is synchrotron emission and the X-ray emission is synchrotron self-Compton. Two realizations of the one-zone model can explain the observed flare and its temporal correlation: one-zone model in which the near-infrared and X-ray luminosity are produced by synchrotron self-Compton and a model in which the luminosity stems from a cooled synchrotron spectrum. Both models can describe the mean spectral energy distribution (SED) and temporal evolution similarly well. In order to describe the mean SED, both models require specific values of the maximum Lorentz factor γ max , which differ by roughly two orders of magnitude. The synchrotron self-Compton model suggests that electrons are accelerated to γ max  ∼ 500, while cooled synchrotron model requires acceleration up to γ max  ∼ 5 × 10 4 . The synchrotron self-Compton scenario requires electron densities of 10 10 cm −3 that are much larger than typical ambient densities in the accretion flow. Furthermore, it requires a variation of the particle density that is inconsistent with the average mass-flow rate inferred from polarization measurements and can therefore only be realized in an extraordinary accretion event. In contrast, assuming a source size of 1  R S , the cooled synchrotron scenario can be realized with densities and magnetic fields comparable with the ambient accretion flow. For both models, the temporal evolution is regulated through the maximum acceleration factor γ max , implying that sustained particle acceleration is required to explain at least a part of the temporal evolution of the flare.