From the molecular-cloud- to the embedded-cluster-mass function with a
  density threshold for star formation
  The mass function of molecular clouds and clumps is shallower than the mass
function of young star clusters, gas-embedded and gas-free alike, as their
respective mass function indices are $\beta_0 \simeq 1.7$ and $\beta_\star
\simeq 2$. We demonstrate that such a difference can arise from different
mass-radius relations for the embedded-clusters and the molecular clouds
(clumps) hosting them. In particular, the formation of star clusters with a
constant mean {\it volume} density in the central regions of molecular clouds
of constant mean {\it surface} density steepens the mass function from clouds
to embedded-clusters. This model is observationally supported since the mean
surface density of molecular clouds is approximately constant, while there is a
growing body of evidence, in both Galactic and extragalactic environments, that
efficient star-formation requires a hydrogen molecule number density threshold
of $n_{th} \simeq 10^{4-5}\,cm^{-3}$. In the framework of the same model, the
radius distribution steepens from clouds (clumps) to embedded-clusters, which
contributes to explaining observed cluster radius distributions. [Abridged]
