A dynamical measure of the black hole mass in a quasar 11 billion years ago.

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 × 10 8  solar masses. Molecular gas data reveal a dynamical mass for the host galaxy of 6 × 10 11  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.