  Recently there has been much interest in light dark matter, especially
ultra-light axions, as they may provide a solution to the core-cusp problem at
the center of galaxies. Since very light bosons can have a de Broglie
wavelength that is of astrophysical size, they can smooth out the centers of
galaxies to produce a core, as opposed to vanilla dark matter models, and so it
has been suggested that this solves the core-cusp problem. In this work, we
critically examine this claim. While an ultra-light particle will indeed lead
to a core, we examine whether the relationship between the density of the core
and its radius matches the data over a range of galaxies. We first review data
that shows the core density of a galaxy $\rho_c$ varies as a function of the
core radius $R_c$ as $\rho_c\propto1/R_c^\beta$ with $\beta\approx1$. We then
compare this to theoretical models. We examine a large class of light scalar
dark matter models, governed by some potential $V$. For simplicity, we take the
scalar to be complex with a global $U(1)$ symmetry in order to readily organize
solutions by a conserved particle number. However, we expect our central
conclusions to persist even for a real scalar, and furthermore, a complex
scalar matches the behavior of a real scalar in the non-relativistic limit,
which is the standard regime of interest. For any potential $V$, we find the
relationship between $\rho_c$ and $R_c$ for ground state solutions is always in
one of the following regimes: (i) $\beta\gg1$, or (ii) $\beta\ll1$, or (iii)
unstable, and so it never matches the data. We also find similar conclusions
for virialized dark matter, more general scalar field theories, degenerate
fermion dark matter, superfluid dark matter, and general polytropes. We
conclude that the solution to the core-cusp problem is more likely due to
either complicated baryonic effects or some other type of dark matter
interactions.
