  The recent detection of the sky-averaged 21-cm cosmological signal indicates
a stronger absorption than the maximum allowed value based on the standard
model. One explanation for the required colder primordial gas is the energy
transfer between the baryon and dark matter fluids due to non-gravitational
scattering. Here, we explore the thermal evolution of primordial gas,
collapsing to form Population III (Pop III) stars, when this energy transfer is
included. Performing a series of one-zone calculations, we find that the
evolution results in stars more massive than in the standard model, provided
that the dark matter is described by the best-fit parameters inferred from the
21-cm observation. On the other hand, a significant part of the dark matter
parameter space can be excluded by the requirement to form massive Pop III
stars sufficiently early in cosmic history. Otherwise, the radiation background
needed to bring about the strong Wouthuysen-Field coupling at z >~ 17, inferred
to explain the 21-cm absorption feature, could not be builtup. Intriguingly,
the independent constraint from the physics of first star formation at high
densities points to a similarly narrow range in dark matter properties. This
exploratory study has to be followed-up with self-consistent three-dimensional
simulations for a more rigorous derivation.
