Exploring the nature of dark matter with the extreme galaxy AGC 114905

AGC 114905 is a dwarf gas-rich ultra-diffuse galaxy seemingly in tension with the cold dark matter (CDM) model. Specifically, the galaxy appears to have an extremely low-density halo and a high baryon fraction, while CDM predicts dwarfs to have very dense and dominant dark haloes. The alleged tension relies on the galaxy's rotation curve decomposition, which depends heavily on its inclination. This inclination, estimated from the gas (neutral atomic hydrogen, H I) morphology, remains somewhat uncertain. We present unmatched ultra-deep optical imaging of AGC 114905 reaching surface brightness limits mu(r, lim) approximate to 32 mag/arcsec(2) (3 sigma; 10 arcsec x 10 arcsec) obtained with the 10.4 m Gran Telescopio Canarias. With the new imaging, we characterise the galaxy's optical morphology, surface brightness, colours, and stellar mass profiles in great detail. The stellar disc has a similar extent to the H I disc, presents spiral arms-like features, and shows a well-defined edge. Stars and gas have a similar morphology, and crucially, we find an inclination of 31 +/- 2 degrees, in agreement with the previous determinations. We revisit the rotation curve decomposition of the galaxy, and we explore different mass models in the context of CDM, self-interacting dark matter (SIDM), fuzzy dark matter (FDM) or Modified Newtonian dynamics (MOND). We find that the last does not fit the circular speed of the galaxy, while CDM only does so with dark halo parameters rarely seen in cosmological simulations. Within the uncertainties, SIDM and FDM remain feasible candidates to explain the observed kinematics of AGC 114905.