On the Origin of the Salpeter Slope for the Initial Mass Function
  We suggest that the intrinsic, stellar initial mass function (IMF) follows a
power-law slope gamma=2, inherited from hierarchical fragmentation of molecular
clouds into clumps and clumps into stars. The well-known, logarithmic Salpeter
slope GAMMA=1.35 in clusters is then the aggregate slope for all the
star-forming clumps contributing to an individual cluster, and it is steeper
than the intrinsic slope within individual clumps because the smallest
star-forming clumps contributing to any given cluster are unable to form the
highest-mass stars. Our Monte Carlo simulations demonstrate that the Salpeter
power-law index is the limiting value obtained for the cluster IMF when the
lower-mass limits for allowed stellar masses and star-forming clumps are
effectively equal, m_lo = M_lo. This condition indeed is imposed for the
high-mass IMF tail by the turn-over at the characteristic value m_c ~ 1 M_sun.
IMF slopes of GAMMA ~ 2 are obtained if the stellar and clump upper-mass limits
are also equal m_up = M_up ~ 100 M_sun, and so our model explains the observed
range of IMF slopes between GAMMA ~ 1 to 2. Flatter slopes of GAMMA = 1 are
expected when M_lo > m_up, which is a plausible condition in starbursts, where
such slopes are suggested to occur. While this model is a simplistic
parameterization of the star-formation process, it seems likely to capture the
essential elements that generate the Salpeter tail of the IMF for massive
stars. These principles also likely explain the IGIMF effect seen in
low-density star-forming environments.
