Can light dark matter solve the core-cusp problem?

Recently, there has been much interest in light dark matter, especially ultralight 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 ultralight 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 show the core density of a galaxy ${\rho }_c$ varies as a function of the core radius $R_c$ as ${\rho }_c\propto 1/R_c^{\beta }$ with $\beta \approx 1$. 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 nonrelativistic 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) $\beta \gg 1$, or (ii) $\beta \ll 1$, 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.

Figure 1

Core density ${\rho }_c$ versus core radius $R_c$ for a range of galaxies. Black dots are data taken from Ref. [4]. Orange dashed curve is the best fit power law curve ${\rho }_c\propto 1/R_c^{\beta }$ with $\beta =1.3$ for this data set, while we find $\beta \approx 1$ more generally.

Figure 2

Core density ${\rho }_c$ versus core radius $R_c$ for different exponents $\alpha$ in the potential functions $V=m^2{F}^2((1+|\mathrm{\Phi }{|}^2/F^2{)}^{\alpha }-1)/\alpha$. Here we have scaled out the constants $m$ and $G$, and chosen $F={10}^{-3}/\sqrt{G}$ ($\approx {10}^{16}\mathrm{GeV}$). The lower red curve is for $\alpha =1/2$ (attractive self-interaction), the middle blue curve is for $\alpha =1$ (no self-interaction), and the upper green curve is for $\alpha =2$ (repulsive self-interaction). The solid branches represent stable solutions (true ground states), while the dashed branches represent unstable solutions. The different branches all asymptote to power law behavior ${\rho }_c\propto 1/R_c^{\beta }$ with the value of $\beta$ labelled for each branch. The dotted black line indicates the boundary of the black hole regime, above which the scalar field would be trapped inside its own Schwarzschild radius; since the cores of galaxies are in the weak gravity regime, the important region of this plot is the part well below the dotted black line.

Figure 3

de Broglie wavelength $\lambda =h/(mv)$ versus core radius $R_c$ for a range of galaxies in the fuzzy dark matter hypothesis. Black dots correspond to the data from Ref. [4] that we used in Fig. 1. We have taken the velocity $v$ that determines the de Broglie wavelength to be the virial speed corresponding to the core density ${\rho }_c$ for that radius $R_c$, and we have taken the dark matter particle mass to be $m={10}^{-22}\mathrm{eV}$ for the sake of illustration (since $\lambda \propto 1/m$, other values of $m$ involves a simple re-scaling of the vertical axis). Dashed orange curve is the corresponding best fit power law $\lambda \propto 1/R_c^{1-\beta /2}$ from Fig. 1 with $\beta =1.3$ for this data set. The dotted cyan curve is $\lambda =2R_c$; for points that lie well above this line, the theoretical model is unphysical, and for points that lie well below this line, the core seems to require some alternate explanation.