Resolving the Core-Cusp and Diversity Problems with a Baryon-Correlated Dark Matter Profile

The rotation velocity profiles of galaxies (rotation curves) remain unexpectedly flat at large distances, where visible matter alone should make the rotation velocity decrease with radius. Conventionally, this requires a large amount of unseen dark matter. However, standard dark matter models face persistent small-scale challenges, such as the core-cusp and diversity problems, and struggle to explain the observed correlation between dark matter and baryons. Here, we introduce a simple empirical law for the dark matter distribution, stating that the effective dark matter energy density $\rho_{\rm DM}$ is directly correlated with the baryonic gravitational potential $\Phi_b$ with the relation of $\rho_{\rm DM} = \mu \Phi_b^2 / c^4$ in the rest frame of the galaxy. This leads to a Poisson equation for the total gravitational potential $\Phi_{\rm tot}$, $$\nabla^2\Phi_{\rm tot} = 4\pi G\,\rho_b /c^2 + 4\pi G\,\mu\,\Phi_b^2 / c^6.$$ Assuming that $\mu = K M_b^{-3/2}$ with a parameter $K$, we applied this equation to 91 galaxies from the SPARC database. This baryon-correlated dark matter profile reproduced both the inner rise and outer flat regions of the observed rotation curves, resolving the core-cusp and diversity problems using the observed baryonic mass profiles only. The fitted $K$ values for the 91 galaxies were found to be concentrated within a narrow range. In addition, our detailed analysis of the nine galaxies using specific stellar mass-to-light ratios from the THINGS survey reduced the scatter in $K$, further demonstrating the validity of this model. These results suggest the existence of new fields interacting with baryons.