The specific frequencies of ultra-compact dwarf galaxies

We aim at quantifying the specific frequency of UCDs in a range of environments and at relating this to the frequency of globular clusters (GCs) and potential progenitor dwarf galaxies. Are the frequencies of UCDs consistent with being the bright tail of the GC luminosity function (GCLF)? We propose a definition for the specific frequency of UCDs, S_{N,UCD}=N_{UCD}*10^{0.4*(M_{V,host}-M_{V,0})}*c_{w}. The parameter M_{V,0} is the zeropoint of the definition, chosen such that the specific frequency of UCDs is the same as those of globular clusters, S_{N,GC}, if UCDs follow a simple extrapolation of the GCLF. The parameter c_{w} is a correction term for the GCLF width sigma. We apply our definition of S_{N,UCD} to results of spectroscopic UCD searches in the Fornax, Hydra and Centaurus galaxy clusters, two Hickson Compact Groups, and the Local Group. This includes a large database of 180 confirmed UCDs in Fornax. We find that the specific frequencies derived for UCDs match those of GCs very well, to within 10-50%. The ratio {S_{N,UCD}}/{S_{N,GC}} is 1.00 +- 0.44 for the four environments Fornax, Hydra, Centaurus, and Local Group, which have S_{N,GC} values. This good match also holds for individual giant galaxies in Fornax and in the Fornax intracluster-space. The error ranges of the derived UCD specific frequencies in the various environments then imply that not more than 50% of UCDs were formed from dwarf galaxies. We show that such a scenario would require >90% of primordial dwarfs in galaxy cluster centers (<100 kpc) to have been stripped of their stars. We conclude that the number counts of UCDs are fully consistent with them being the bright tail of the GC population. From a statistical point of view there is no need to invoke an additional formation channel.