The stellar initial mass function, core mass function and the last-crossing distribution

Hennebelle & Chabrier attempted to derive the stellar initial mass function (IMF) as a consequence of lognormal density fluctuations in a turbulent medium, using an argument similar to Press & Schechter for Gaussian random fields. Like that example, however, the solution there does not resolve the 'cloud-in-cloud' problem; it also does not extend to the large scales that dominate the velocity and density fluctuations. In principle, these can change the results at the order-of-magnitude level or more. In this paper, we use the results from Hopkins 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 in cosmological applications and which, Hopkins shows, corresponds to the giant molecular cloud (GMC) mass function in discs. We derive an analytic equation for the last-crossing distribution that can be applied for an arbitrary collapse threshold shape in interstellar medium and cosmological studies. With this, we show that the same model that predicts the GMC mass function and large-scale structure of galaxy discs also predicts the CMF - and by extrapolation stellar IMF - in good agreement with observations. The only adjustable parameter in the model is the turbulent velocity power spectrum, which in the range gives similar results. We also use this to formally justify why the approximate solution in Hennebelle & Chabrier is reasonable (up to a normalization constant) over the mass range of the CMF/IMF; however, there are significant corrections at intermediate and high masses. We discuss how the exact solutions here can be used to predict additional quantities such as the clustering of stars, and embedded into time-dependent models that follow density fluctuations, fragmentation, mergers and successive generations of star formation.