EDGE: the puzzling ellipticity of Eridanus II's star cluster and its implications for dark matter at the heart of an ultra-faint dwarf

The Eridanus II (ErinII) 'ultra-faint' dwarf has a large (15 pc) and low-mass (4.3 x 10(3)M(circle dot)) star cluster (SC) offset from its centre by 23 +/- 3 pc in projection. Its size and offset are naturally explained if EriII has a central dark matter core, but such a core may be challenging to explain in a Lambda CDM cosmology. In this paper, we revisit the survival and evolution of EriII's SC, focusing for the first time on its puzzlingly large ellipticity (0.31(-0.06)(+0.05)). We perform a suite of 960 direct N-body simulations of SCs, orbiting within a range of spherical background potentials fit to ultra-faint dwarf (UFD) galaxy simulations. We find only two scenarios that come close to explaining EriII's SC. In the first scenario, EriII has a low-density dark matter core (of size similar to 70 pc and density less than or similar to 2 x 10(8) M-circle dot kpc(-3)). In this model, the high ellipticity of EriII's SC is set at birth, with the lack of tidal forces in the core allowing its ellipticity to remain frozen for long times. In the second scenario, EriII's SC orbits in a partial core, with its high ellipticity owing to its imminent tidal destruction. However, this latter model struggles to reproduce the large size of EriII's SC, and it predicts substantial tidal tails around EriII's SC that should have already been seen in the data. This leads us to favour the cored model. We discuss potential caveats to these findings, and the implications of the cored model for galaxy formation and the nature of dark matter.