  All measurements of cosmic star formation must assume an initial distribution
of stellar masses -- the stellar initial mass function -- in order to
extrapolate from the star-formation rate measured for typically rare, massive
stars (> 8 Msun) to the total star-formation rate across the full stellar mass
spectrum. The shape of the stellar initial mass function in various galaxy
populations underpins our understanding of the formation and evolution of
galaxies across cosmic time. Classical determinations of the stellar initial
mass function in local galaxies are traditionally made at ultraviolet, optical
and near-infrared wavelengths, which cannot be probed in dust-obscured
galaxies, especially in distant starbursts, whose apparent star-formation rates
are hundreds to thousands of times higher than in our Milky Way, selected at
submillimetre (rest-frame far-infrared) wavelengths. The 13C/18O abundance
ratio in the cold molecular gas -- which can be probed via the rotational
transitions of the 13CO and C18O isotopologues -- is a very sensitive index of
the stellar initial mass function, with its determination immune to the
pernicious effects of dust. Here we report observations of 13CO and C18O
emission for a sample of four dust-enshrouded starbursts at redshifts of
approximately two to three, and find unambiguous evidence for a top-heavy
stellar initial mass function in all of them. A low 13CO/C18O ratio for all our
targets -- alongside a well-tested, detailed chemical evolution model
benchmarked on the Milky Way -- implies that there are considerably more
massive stars in starburst events than in ordinary star-forming spiral
galaxies. This can bring these extraordinary starbursts closer to the `main
sequence' of star-forming galaxies, though such main-sequence galaxies may not
be immune to changes in initial stellar mass function, depending upon their
star-formation densities.
