Mapping the core mass function to the initial mass function
  It has been shown that fragmentation within self-gravitating, turbulent
molecular clouds ("turbulent fragmentation") can naturally explain the observed
properties of protostellar cores, including the core mass function (CMF). Here,
we extend recently-developed analytic models for turbulent fragmentation to
follow the time-dependent hierarchical fragmentation of self-gravitating cores,
until they reach effectively infinite density (and form stars). We show that
turbulent fragmentation robustly predicts two key features of the IMF. First, a
high-mass power-law scaling very close to the Salpeter slope, which is a
generic consequence of the scale-free nature of turbulence and self-gravity. We
predict the IMF slope (-2.3) is slightly steeper then the CMF slope (-2.1),
owing to the slower collapse and easier fragmentation of large cores. Second, a
turnover mass, which is set by a combination of the CMF turnover mass (a couple
solar masses, determined by the `sonic scale' of galactic turbulence, and so
weakly dependent on galaxy properties), and the equation of state (EOS). A
"soft" EOS with polytropic index $\gamma<1.0$ predicts that the IMF slope
becomes "shallow" below the sonic scale, but fails to produce the full turnover
observed. An EOS which becomes "stiff" at sufficiently low surface densities
$\Sigma_{\rm gas} \sim 5000\,M_{\odot}\,{\rm pc^{-2}}$, and/or models where
each collapsing core is able to heat and effectively stiffen the EOS of a
modest mass ($\sim 0.02\,M_{\odot}$) of surrounding gas, are able to reproduce
the observed turnover. Such features are likely a consequence of more detailed
chemistry and radiative feedback.
