  It has been shown recently that the dynamical V-band mass-to-light ratios of
compact stellar systems with masses from 10^6 to 10^8 Solar masses are not
consistent with the predictions from simple stellar population (SSP) models.
Top-heavy stellar initial mass functions (IMFs) in these so-called ultra
compact dwarf galaxies (UCDs) offer an attractive explanation for this finding,
the stellar remnants and retained stellar envelopes providing the unseen mass.
We therefore construct a model which quantifies by how much the IMFs of UCDs
would have to deviate in the intermediate-mass and high-mass range from the
canonical IMF in order to account for the enhanced M/L_V ratio of the UCDs. The
deduced high-mass IMF in the UCDs depends on the age of the UCDs and the number
of faint products of stellar evolution retained by them. Assuming that the IMF
in the UCDs is a three-part power-law equal to the canonical IMF in the
low-mass range and taking 20% as a plausible choice for the fraction of the
remnants of high-mass stars retained by UCDs, the model suggests the exponent
of the high-mass IMF to be approximately 1.6 if the UCDs are 13 Gyr old (i.e.
almost as old as the Universe) or approximately 1.0 if the UCDs are 7 Gyr old,
in contrast to 2.3 for the Salpeter-Massey IMF. If the IMF was as top-heavy as
suggested here, the stability of the UCDs might have been threatened by heavy
mass loss induced by the radiation and evolution of massive stars. The central
densities of UCDs must have been in the range 10^6 to 10^7 Solar masses per
cubic parsec when they formed with star formation rates of 10 to 100 Solar
masses per year.
