Stellar age spreads in clusters as imprints of cluster-parent clump
  densities
  It has recently been suggested that high-density clusters have stellar age
distributions narrower than that of the Orion Nebula Cluster, indicating a
possible trend of narrower age distributions for denser clusters. We show this
effect to likely arise from star formation being faster in gas with a higher
density. We model the star formation history of molecular clumps in equilibrium
by associating a star formation efficiency (SFE) per free-fall time, \eff, to
their volume density profile. Our model predicts a steady decline of the star
formation rate (SFR), which we quantify with its half-life time, namely, the
time needed for the SFR to drop to half its initial value.
  Given the uncertainties affecting the SFE per free-fall time, we consider two
distinct values: 0.1 and 0.01. For isothermal spheres, \eff=0.1 leads to a
half-life time of order the clump free-fall time, \tff. Therefore, the age
distributions of stars formed in high-density clumps have smaller full-widths
at half-maximum than those of stars formed in low-density clumps. When
\eff=0.01, the half-life time is 10 times longer. We explore what happens if
the star formation duration is shorter than 10\tff, that is, if the half-life
time of the SFR cannot be defined. There, we build on the invariance of the
shape of the young cluster mass function to show that an anti-correlation
between clump density and star formation duration is expected. Therefore,
regardless of whether the star formation duration is longer than the SFR
half-life time, denser molecular clumps yield narrower star age distributions
in clusters.
  Published densities and stellar age spreads of young clusters actually
suggest that the time-scale for star formation is of order 1-4\tff. We conclude
that there is no need to invoke the existence of multiple cluster formation
mechanisms to explain the observed range of stellar age spreads in clusters.
