Top-heavy integrated galactic stellar initial mass functions (IGIMFs) in
  starbursts
  Star formation rates (SFR) larger than 1000 Msun/ yr are observed in extreme
star bursts. This leads to the formation of star clusters with masses > 10^6
Msun in which crowding of the pre-stellar cores may lead to a change of the
stellar initial mass function (IMF). Indeed, the large mass-to-light ratios of
ultra-compact dwarf galaxies and recent results on globular clusters suggest
the IMF to become top-heavy with increasing star-forming density. We explore
the implications of top-heavy IMFs in these very massive and compact systems
for the integrated galactic initial mass function (IGIMF), which is the
galaxy-wide IMF, in dependence of the star-formation rate of galaxies. The
resulting IGIMFs can have slopes, alpha_3, for stars more massive than about 1
Msun between 1.5 and the Salpeter slope of 2.3 for an embedded cluster mass
function (ECMF) slope (beta) of 2.0, but only if the ECMF has no low-mass
clusters in galaxies with major starbursts. Alternatively, beta would have to
decrease with increasing SFR >10 Msun/ yr such that galaxies with major
starbursts have a top-heavy ECMF. The resulting IGIMFs are within the range of
observationally deduced IMF variations with redshift.
