On the fraction of star formation occurring in bound stellar clusters
  We present a theoretical framework in which bound stellar clusters arise
naturally at the high-density end of the hierarchy of the interstellar medium
(ISM). Due to short free-fall times, these high-density regions achieve high
local star formation efficiencies, enabling them to form bound clusters.
Star-forming regions of lower density remain substructured and gas-rich, ending
up unbound when the residual gas is expelled. Additionally, the tidal
perturbation of star-forming regions by nearby, dense giant molecular clouds
imposes a minimum density contrast required for the collapse to a bound
cluster. The fraction of all star formation that occurs in bound stellar
clusters (the cluster formation efficiency or CFE) follows by integration of
these local clustering and survival properties over the full density spectrum
of the ISM, and hence is set by galaxy-scale physics. We derive the CFE as a
function of observable galaxy properties, and find that it increases with the
gas surface density, from ~1% in low-density galaxies to a peak value of ~70%
at densities of ~10^3 Msun pc^-2. This explains the observation that the CFE
increases with the star formation rate density in nearby dwarf, spiral, and
starburst galaxies. Indeed, comparing our model results with observed galaxies
yields excellent agreement. The model is applied further by calculating the
spatial variation of the CFE within single galaxies. We also consider the
variation of the CFE with cosmic time and show that it increases with redshift,
peaking in high-redshift, gas-rich disc galaxies. It is estimated that up to
30-35% of all stars in the Universe once formed in bound stellar clusters. We
discuss how our theory can be verified with Gaia and ALMA, and provide
implementations for future theoretical work and for simulations of galaxy
formation and evolution.
