Fast Dynamical Modelling of Milky Way Globular Clusters -- I. Implications for Initial Cluster Densities

We infer the initial conditions of Milky Way (MW) globular clusters (GCs) from present-day observations, through the coupling of recently updated rapid cluster evolution models with multimass equilibrium models. This novel method is validated by fitting to simulated observations of a large grid of star-by-star Monte Carlo models, demonstrating that we are able to recover cluster properties like the total mass, half-mass radius/density and black hole (BH) mass fraction, both initially and at the present day, across a large region of parameter space. We apply this framework to a sample of 40 MW GCs, fitting to a suite of observed radial profiles of number densities, proper motions, line-of-sight velocities and stellar mass functions. From these fits we infer a distribution of initial half-mass densities with a median and $1\sigma$ width, across our sample, of $\rho_{h,0} = 10^{6.4\pm0.9}\,{M_\odot pc^{-3}}$, higher than what is found for young massive clusters in the Local Universe and in line with young clusters at high redshift. We also find stellar initial mass functions that are bottom-light in comparison to canonical prescriptions, and relatively small present-day BH mass fractions ($\lesssim 1.5\%$). We discuss the implications of these initial cluster densities for observations of high-redshift proto-GCs, binary BH merger rates and intermediate-mass BHs (IMBHs) in GCs. Finally, we quantify how these densities may depend on assumptions typically made surrounding BH formation and natal kicks.