A Possible Solution for the M/L-[Fe/H] Relation of Globular Clusters in M3. I. A Metallicity- and Density-dependent Top-heavy IMF

The observed mass-to-light (M/L) ratios of a large sample of globular clusters (GCs) in M31 show an inverse trend with metallicity compared to what is expected from simple stellar population (SSP) models with an invariant canonical stellar initial mass function (IMF), in the sense that the observed M/L ratios decrease with increasing metallicity. We show that when incorporating the effect of dynamical evolution the SSP models with a canonical IMF cannot explain the decreasing M/L ratios with increasing metallicity for the M31 GCs. The recently derived top-heavy IMF as a function of metallicity and embedded cluster density is proposed to explain the lower-than-expected M/L ratios of metal-rich GCs. We find that the SSP models with a top-heavy IMF, retaining a metallicity- and cluster-mass-dependent fraction of the remnants within the clusters, and taking standard dynamical evolution into account, can successfully explain the observed M/L-[Fe/H] relation of M31 GCs. Thus we propose that the kinematic data of GCs can be used to constrain the top-heaviness of the IMF in GCs.