The Stellar IMF, Core Mass Function, & The Last-Crossing Distribution
  Hennebelle & Chabrier 2008 (HC08) attempted to derive the stellar IMF as a
consequence of turbulent density fluctuations, using an argument similar to
Press & Schechter 1974 for Gaussian random fields. Like that example, however,
this solution does not resolve the 'cloud in cloud' problem; it also does not
extend to large scales that dominate the velocity/density fluctuations. In
principle, these can change the results at the order-of-magnitude level. Here,
we use the results from Hopkins 2011 (H11) to generalize the excursion set
formalism and derive the exact solution in this regime. We argue that the
stellar IMF and core mass function (CMF) should be associated with the
last-crossing distribution, i.e. the mass spectrum of bound objects defined on
the smallest scale on which they are self-gravitating. This differs from the
first-crossing distribution (mass function on the largest self-gravitating
scale) which is defined cosmologically and which H11 show corresponds to the
GMC mass function in disks. We derive an analytic equation for the
last-crossing distribution that can be used for an arbitrary collapse threshold
in ISM and cosmological studies. With this, we show that the same model that
predicts the GMC mass function and large-scale structure of galaxy disks also
predicts the CMF (and by extrapolation IMF) in good agreement with
observations. The only adjustable parameter in the model is the turbulent
velocity power spectrum, which in the range p~5/3-2 gives similar results. We
also use this to justify why the approximate solution in HC08 is reasonable (up
to a normalization) over the CMF/IMF mass range; however there are significant
corrections at intermediate and high masses. We discuss how the exact solutions
here can be embedded into time-dependent models that follow density
fluctuations, fragmentation, successive generations of star formation.
