Could compact stars in globular clusters constrain dark matter?

The dark matter content of globular clusters, highly compact gravity-bound stellar systems, is unknown. It is also generally unknow*able*, due to their mass-to-light ratios typically ranging between 1$-$3 in solar units, accommodating a dynamical mass of dark matter at best comparable to the stellar mass. That said, recent claims in the literature assume densities of dark matter around 1000 GeV/cm$^3$ to set constraints on its capture and annihilation in white dwarfs residing in the globular cluster M4, and to study a number of other effects of dark matter on compact stars. Motivated by these studies, we use measurements of stellar kinematics and luminosities in M4 to look for a dark matter component via a spherical Jeans analysis; we find no evidence for it, and set the first empirical limits on M4's dark matter distribution. Our density upper limits, a few $\times \ 10^4$ GeV/cm$^3$ at 1 parsec from the center of M4, do not negate the claims (nor confirm them), but do preclude the use of M4 for setting limits on non-annihilating dark matter kinetically heating white dwarfs, which require at least $10^5$ GeV/cm$^3$ densities. The non-robust nature of globular clusters as dynamical systems, combined with evidence showing that they may originate from molecular gas clouds in the absence of dark matter, make them unsuitable as laboratories to unveil dark matter's microscopic nature in current or planned observations.