Hubble-Lema\^itre fragmentation and the path to equilibrium of
  merger-driven cluster formation
  This paper discusses a new method to generate self-coherent initial
conditions for young substructured stellar cluster. The expansion of a uniform
system allows stellar sub-structures (clumps) to grow from fragmentation modes
by adiabatic cooling. We treat the system mass elements as stars, chosen
according to a Salpeter mass function, and the time-evolution is performed with
a collisional N-body integrator. This procedure allows to create a
fully-coherent relation between the clumps' spatial distribution and the
underlying velocity field. The cooling is driven by the gravitational field, as
in a cosmological Hubble-Lema\^itre flow. The fragmented configuration has a
`fractal'-like geometry but with a self-grown velocity field and mass profile.
We compare the characteristics of the stellar population in clumps with that
obtained from hydrodynamical simulations and find a remarkable correspondence
between the two in terms of the stellar content and the degree of spatial
mass-segregation. In the fragmented configuration, the IMF power index is ~0.3
lower in clumps in comparison to the field stellar population, in agreement
with observations in the Milky Way. We follow in time the dynamical evolution
of fully fragmented and sub-virial configurations, and find a soft collapse,
leading rapidly to equilibrium (timescale of 1 Myr for a ~ 10^4 Msun system).
The low-concentration equilibrium implies that the dynamical evolution
including massive stars is less likely to induce direct collisions and the
formation of exotic objects. Low-mass stars already ejected from merging clumps
are depleted in the end-result stellar clusters, which harbour a top-heavy
stellar mass function.
