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 initial mass function (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 than the CMF slope (−2.1), owing to the slower collapse and easier fragmentation of large cores. Secondly, 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 γ < 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 Σgas ∼ 5000 M_⊙ pc^(−2), and/or models, where each collapsing core is able to heat and effectively stiffen the EOS of a modest mass (∼0.02 M_⊙) of surrounding gas, are able to reproduce the observed turnover. Such features are likely a consequence of more detailed chemistry and radiative feedback