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 (T_N and S_N, respectively) of globular cluster systems (GCSs) in order to understand the origin of the observed U-shaped relation between S_N and V-band magnitude (M_V) of their host galaxies.   We adopt a biased GC formation 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 GCs today and converted into S_N for reasonable galaxy mass-to-light-ratios (M/L). We find that T_N depends on halo mass (M_h) in the sense that T_N can be larger in more massive halos with M_h > 10^9 M_sun, if z_trun is as high as 15. We however find that the dependence is too weak to explain the observed S_N-M_V relation and the wide range of S_N in low-mass early-type galaxies with -20.5 < M_V < -16.0 mag for a reasonable constant M/L. The M_V-dependence of S_N for the low-mass galaxies can be well reproduced, if the mass-to-light-ratio M_h/L_V \propto M_h^{\alpha}, where \alpha is as steep as -1. Based on these results, we propose that the origin of the observed U-shaped S_N-M_V relation of GCSs can be understood in terms of the bimodality in the dependence of M_h/L_V on M_h of their host galaxies. We also suggest that the observed large dispersionin S_N in low-mass galaxies is due partly to the large dispersion in T_N. 
Kenji Bekki, Hideki Yahagi, Duncan A. Forbes