A Possible Solution for the M/L-[Fe/H] Relation of Globular Clusters in M31: A metallicity and density dependent top-heavy IMF

Abstract: The observed mass-to-light ($M/L$) ratios of a large sample of 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 IMF, in the sense that the observed $M/L$ ratios decrease with increasing metallicity. We show that incorporating the effect of dynamical evolution the SSP models with a canonical IMF can not 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 kinematical data of GCs can be used to constrain the top-heaviness of the IMF in GCs.