Stellar collisions in globular clusters: constraints on the IMF of the
  first generation of stars
  Globular clusters display an anti-correlation between the fraction of the
first generation of stars ($N({\rm G1})/N({\rm tot})$) and the slope of the
present-day mass function of the clusters ($\alpha_{pd}$), which is
particularly significant for massive clusters. In the framework of the
binary-mediated collision scenario for the formation of the second generation
stars in globular clusters, we test the effect of a varying initial stellar
mass function (IMF) of the G1 stars on the $(N({\rm G1})/N({\rm
tot}))-\alpha_{pd}$ anti-correlation. We use a simple collision model which has
only two inputs, the IMF of G1 stars and the fraction of G1 stars that coalesce
to form G2 stars. We show that a variable efficiency of the collision process
is necessary in order to explain the $(N({\rm G1})/N({\rm tot}))-\alpha_{pd}$
anti-correlation, however, the scatter in the anti-correlation can only be
explained by variations in the IMF, and in particular by variations of the
slope in the mass interval $\approx$ (0.1-0.5) M$_{\odot}$. Our results
indicate that in order to explain the scatter in the $(N({\rm G1})/N({\rm
tot}))-\alpha_{pd}$ relation, it is necessary to invoke variations of the slope
in this mass range between $\approx -0.9$ and $\approx -1.9$. Interpreted in
terms of a Kroupa-like broken power law, this translates into variations of the
mean mass between $\approx 0.2$ and $0.55$ M$_{\odot}$. This level of
variations is consistent with what is observed for young stellar clusters in
the Milky Way and may reflect variations in the physical conditions of the
globular clusters progenitor clouds at the time the G1 population has formed or
to the occurrence of collisions between protostellar embryos before stars
settle on the Main Sequence.
