Disentangling gamma-beta: the 4th-order velocity moments based on spherical Jeans analysis

Distinguishing a core and a cusp within dark matter halos is complexified by the existence of mass-anisotropy degeneracy, where various combinations of velocity anisotropy ($\beta$) and inner density slope ($\gamma$) yield similar observational signatures. We construct a dynamical model that incorporates the 4th-order velocity moments to alleviate this challenge. The inclusion of the 4th-order velocity moments enables stars' line-of-sight velocity distribution (LOSVD) to be flexible. This flexible LOSVD can cover from a thin-tailed to a heavy-tailed distribution that is inaccessible if only the 2nd-order moments are considered. We test the model on four mock galaxies having isotropic orbits, $\beta = 0$: two resembling dwarf spheroidal galaxies (dSphs) and two resembling ultra-faint dwarfs (UFDs) in terms of velocity dispersion. Each category includes one galaxy with a cuspy NFW profile and one with a cored density profile. Results show that a ratio of the global velocity dispersion to velocity error, $\sigma_{\rm los,global} / \delta v_{\rm los} \gtrsim 4$, is crucial to avoid systematic biases arising from the strong sensitivity of 4th-order moments to the LOSVD tails. In cases where this velocity ratio condition is met, our model reliably recovers $\gamma$ in dSph mock galaxies, with the true value recovered within $\sim 1\sigma$, and strongly excludes a cuspy NFW profile for the cored dSph mock galaxy. However, recovering the density profiles of UFDs remains challenging due to their intrinsically low velocity dispersions.