Most stars in today’s Universe reside within spheroids, which are bulges of spiral galaxies and elliptical galaxies1,2. Their formation is still an unsolved problem3–5. Infrared/submillimetre-bright galaxies at high redshifts6 have long been suspected to be related to spheroid formation7–12. Proving this connection has been hampered so far by heavy dust obscuration when focusing on their stellar emission13–15 or by methodologies and limited signal-to-noise ratios when looking at submillimetre wavelengths16,17. Here we show that spheroids are directly generated by star formation within the cores of highly luminous starburst galaxies in the distant Universe. This follows from the ALMA submillimetre surface brightness profiles, which deviate substantially from those of exponential disks, and from the skewed-high axis-ratio distribution. Most of these galaxies are fully triaxial rather than flat disks: the ratio of the shortest to the longest of their three axes is half, on average, and increases with spatial compactness. These observations, supported by simulations, reveal a cosmologically relevant pathway for in situ spheroid formation through starbursts that is probably preferentially triggered by interactions (and mergers) acting on galaxies fed by non-coplanar gas accretion streams.
