  Comparing N-body calculations that include primordial residual-gas expulsion
with the observed properties of 20 Galactic globular clusters (GCs) for which
the stellar mass function (MF) has been measured, we constrain the time-scale
over which the gas of their embedded counterparts must have been removed, the
star formation efficiency the progenitor cloud must have had and the strength
of the tidal-field the clusters must have formed in. The three parameters
determine the expansion and mass-loss during residual-gas expulsion. After
applying corrections for stellar and dynamical evolution we find birth cluster
masses, sizes and densities for the GC sample and the same quantities for the
progenitor gas clouds. The pre-cluster cloud core masses were between 10^5-10^7
Msun and half-mass radii were typically below 1 pc and reach down to 0.2 pc. We
show that the low-mass present day MF slope, initial half-mass radius and
initial density of clusters correlates with cluster metallicity, unmasking
metallicity as an important parameter driving cluster formation and the gas
expulsion process. This work predicts that PD low-concentration clusters should
have a higher binary fraction than PD high-concentration clusters.
  Since the oldest GCs are early residuals from the formation of the Milky Way
(MW) and the derived initial conditions probe the environment in which the
clusters formed, we use the results as a new tool to study the formation of the
inner GC system of the Galaxy. We achieve time-resolved insight into the
evolution of the pre-MW gas cloud on short time-scales (a few hundred Myr) via
cluster metallicities. The results are shown to be consistent with a
contracting and self-gravitating cloud in which fluctuations in the pre-MW
potential grow with time. An initially relatively smooth tidal-field evolved
into a grainy potential within a dynamical time-scale of the collapsing cloud.
