  A viable model for the dense matter equation of state above the nuclear
saturation density is a hadron-to-quark phase transition at densities relevant
to compact objects. In this case, stable hybrid hadron-quark stars can arise.
An even more interesting scenario is one where the hadron-to-quark phase
transition results in the emergence of a third branch of stable compact objects
(in addition to white dwarfs and neutron stars). Inherent to the presence of a
third family of compact stars is the existence of twin stars -- hybrid stars
with the same mass as the corresponding neutron stars, but with smaller radii.
Interestingly, the neutron star-twin star scenario is consistent with GW170817.
If twin stars exist in nature, it raises a question about the mechanism that
leads to their formation. Here, we explore gravitational collapse as a pathway
to the formation of low-mass twin stars. We perform fully general relativistic
simulations of the collapse of a stellar iron core, modeled as a cold
degenerate gas, to investigate whether the end product is a neutron star or a
twin star. Our simulations show that even with unrealistically large
perturbations in the initial conditions, the core bounces well below the
hadron-to-quark phase transition density, if the initial total rest mass is in
the twin star range. Following cooling, these configurations produce neutron
stars. We find that twin stars can potentially form due to mass loss, e.g.,
through winds, from a slightly more massive hybrid star that was initially
produced in the collapse of a more massive core or if the maximum NS mass is
below the Chandrasekhar mass limit. The challenge in producing twin stars in
gravitational collapse, in conjunction with the fine tuning required because of
their narrow mass range, suggest the rarity of twin stars in nature.
