Cracking the relation between mass and 1P-star fraction of globular clusters: III. Initial distributions of in-situ and ex-situ clusters

Galactic globular clusters consist of two main stellar populations, the pristine (1P) and polluted (2P) stars. The fraction of 1P stars in clusters, $F_{1P}$, is a decreasing function of the cluster present-day mass, $m_{prst}$. The information about cluster formation it contains has yet to be unlocked. Paper I demonstrated that the observed distribution $(m_{prst},F_{1P})$ of Galactic globular clusters can result from a pristine-star fraction that is inversely proportional to their birth mass, $m_{ecl}$. This relation was then calibrated with a fixed stellar mass threshold for 2P-star formation, $m_{th}$, i.e., $F_{1P}=m_{th}\mathrm{/}{m}_{ecl}$. We now estimate the masses $m_{init}$ of Galactic globular clusters as they start their long-term gas-free evolution in the Galaxy and we map their behavior in the $(m_{init},F_{1P})$ space. Several dissolution time-scales are tested (with and without primordial mass segregation), each yielding its own initial cluster distribution $(m_{init},F_{1P})$. The $(m_{init},F_{1P})$ distributions are mapped according to cluster origin, with the emphasis on the Disk, Low-Energy and Gaia-Enceladus cluster groups of Massari et al. (2019). All three initial distributions $(m_{init},F_{1P})$ are more compact than their present-day counterparts since dynamical evolution scatters clusters in the $F_{1P}$ versus cluster-mass space. The Disk initial distribution is the tightest one and potential reasons for this are discussed. Its power-law representation allows us to generalize the initial mass threshold of Paper I and prompts us to represent the cluster $(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},F_{1P})$ distribution in a log-log space. No evidence is found suggesting that, initially, the pristine-star fraction of globular clusters depends on their metallicity on top of their mass.