The U-shaped distribution of globular cluster-specific frequencies in a biased globular cluster formation scenario

Using high-resolution numerical simulations, we investigate mass- and luminosity-normalized specific frequencies ( and, respectively) of globular cluster systems (GCSs) in order to understand the origin of the observed U-T S shaped relation between and the V-band magnitude (M-V) of their host galaxies. We adopt a biased GC formation S M scenario in which GC formation is truncated in galaxy halos that are virialized at a later redshift, z(trun). T-N is derived for galaxies with present- day GCs and converted into for reasonable galaxy mass-to-light ratios (M/L). We find that T-N depends on halo mass (M-h) in the sense that can be larger in more massive halos with, if Z(trun) is as high as 15. However, we find that the dependence is too weak to explain the observed S-N-M-V relation S-N and the wide range of in low- mass early- type galaxies with -20.5 < M-V < -16.0 mag for a reasonable S N constant. The dependence of for the low- mass galaxies can be well reproduced, if the mass-to-light ratio M-h/L-V proportional to M-h(alpha) where alpha is as steep as -1. Based on these results, we propose that the origin of the observed U- shaped - relation of GCSs can be understood in terms of the bimodality in the dependence of on M-h/L-V of their host galaxies. We also suggest that the observed large dispersion in in low-mass galaxies is dueNpartly to the large dispersion in T-N.