DHOST gravity in Ultra-diffuse galaxies -- Part II: NGC 1052-DF4 and Dragonfly 44

Ultra-Diffuse galaxies are gravitational systems with quite varied properties. On one hand we have cases like DF2 and DF4, both observed by the Dragonfly Array Telescope, claimed to be highly-deficient in dark matter. On the other hand, we have also observed dark matter dominated UDGs, such as DF44, estimated to be at $99\%$ dark. Such variety of behaviors might be a problem for both the standard dark matter paradigm and for alternative theories of gravity. Here we consider a modified gravity theory belonging to the family of Degenerate Higher-Order Scalar Tensor theories to study the internal kinematics of both DF4 and DF44. The peculiarity of the chosen model is the partial breaking of its Vaishtein screening mechanism for which it might have an influence not only on cosmological but also on astrophysical scales. We consider two different possibilities: one in which the model only plays the role of dark energy; and another one in which it might also mimic a sort of effective dark matter. We get conflicting results. For DF4 we confirm that the galaxy dynamics might be successfully described even only by a stellar component and that, at least at the scale which are probed, the content of dark matter is quite low. In addition to that, we show that the DHOST model is totally consistent with data and is statistically equivalent to a standard General Relativity dark matter scenario, and it might even replace dark matter. On the contrary, DF44 requires dark matter both in General Relativity and in our DHOST model. When the latter is considered only as a cosmological dark energy fluid, it is statistically fully reliable and equivalent to General Relativity. But when we try to use it to substitute dark matter, although we get good fits to the data, the constraints on the theoretical parameters are in sharp contrast with those derived from more stringent probes from the stellar scales.