Enigmatic Velocity Dispersions of Ultra-Diffuse Galaxies in Light of Modified Gravity Theories and Radial Acceleration Relation

Recent observations of anomalous line-of-sight velocity dispersions of two ultra-diffuse galaxies (UDGs) provide a stringent test for modified gravity theories. While NGC 1052-DF2 exhibits an extremely low dispersion value ($\sigma \sim 7.8_{-2.2}^{+5.6}$ km/s), the reported dispersion value for NGC 1052-DF44 is quite high ($\sigma \sim 41.0 \pm 8$ km/s). For DF2, the dynamical mass is almost equal to the luminous mass suggesting the galaxy have little to no `dark matter' in $\Lambda$CDM whereas DF4 requires a dynamical mass-to-light ratio of $\sim 30$ making it to be almost entirely consists of dark matter. It has been claimed that both these galaxies, marking the extreme points in terms of the estimated dynamical mass-to-light ratio among known galaxies, would be difficult to explain in modified gravity scenarios. Extending the analysis presented in \cite{islam2019modified}, we explore the dynamics of DF2 and DF44 within the context of three popular alternative theories of gravity [Modified Newtonian Dynamics (MOND), Weyl Conformal gravity and Modified gravity (MOG)] and examine their viability against the dispersion data of DF2 and DF44. We further show that the galactic `Radial Acceleration Relation' (RAR) is consistent with DF44 dispersion data but not with DF2.