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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<jats:sup>4\u20136<\/jats:sup>. 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\u201312\u2009billion years ago (redshifts 1\u20133)<jats:sup>7<\/jats:sup>. 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\u2009billion years, by spatially resolving the broad-line region (BLR). We detect a 40-\u03bcas (0.31-pc) spatial offset between the red and blue photocentres of the H\u03b1 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\u2009\u00d7\u200910<jats:sup>8<\/jats:sup>\u2009solar masses. Molecular gas data reveal a dynamical mass for the host galaxy of 6\u2009\u00d7\u200910<jats:sup>11<\/jats:sup>\u2009solar 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.<\/jats:p>","DOI":"10.1038\/s41586-024-07053-4","type":"journal-article","created":{"date-parts":[[2024,1,29]],"date-time":"2024-01-29T17:03:35Z","timestamp":1706547815000},"page":"281-285","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":59,"title":["A dynamical measure of the black hole mass in a quasar 11 billion years ago"],"prefix":"10.1038","volume":"627","author":[{"given":"R.","family":"Abuter","sequence":"first","affiliation":[]},{"given":"F.","family":"Allouche","sequence":"additional","affiliation":[]},{"given":"A.","family":"Amorim","sequence":"additional","affiliation":[]},{"given":"C.","family":"Bailet","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4863-6211","authenticated-orcid":false,"given":"A.","family":"Berdeu","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5025-0428","authenticated-orcid":false,"given":"J.-P.","family":"Berger","sequence":"additional","affiliation":[]},{"given":"P.","family":"Berio","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bigioli","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1390-758X","authenticated-orcid":false,"given":"O.","family":"Boebion","sequence":"additional","affiliation":[]},{"given":"M.-L.","family":"Bolzer","sequence":"additional","affiliation":[]},{"given":"H.","family":"Bonnet","sequence":"additional","affiliation":[]},{"given":"G.","family":"Bourdarot","sequence":"additional","affiliation":[]},{"given":"P.","family":"Bourget","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1939-6351","authenticated-orcid":false,"given":"W.","family":"Brandner","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5301-1326","authenticated-orcid":false,"given":"Y.","family":"Cao","sequence":"additional","affiliation":[]},{"given":"R.","family":"Conzelmann","sequence":"additional","affiliation":[]},{"given":"M.","family":"Comin","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8382-2020","authenticated-orcid":false,"given":"Y.","family":"Cl\u00e9net","sequence":"additional","affiliation":[]},{"given":"B.","family":"Courtney-Barrer","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4949-7217","authenticated-orcid":false,"given":"R.","family":"Davies","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3499-2506","authenticated-orcid":false,"given":"D.","family":"Defr\u00e8re","sequence":"additional","affiliation":[]},{"given":"A.","family":"Delboulb\u00e9","sequence":"additional","affiliation":[]},{"given":"F.","family":"Delplancke-Str\u00f6bele","sequence":"additional","affiliation":[]},{"given":"R.","family":"Dembet","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3903-0373","authenticated-orcid":false,"given":"J.","family":"Dexter","sequence":"additional","affiliation":[]},{"given":"P. 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