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 $rho_cpropto1/R_c^beta$ with $betaapprox1$. 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) $betagg1$, or (ii) $betall1$, 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.
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