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The Sagittarius dwarf spheroidal (Sgr dSph) galaxy is currently beingaccreted and disrupted by the tidal field of the Milky Way. Recent observationshave shown that the central region of the dwarf hosts at least three differentstellar populations, ranging from old and metal-poor over intermediatemetal-rich to young metal-rich. While the intermediate-age metal-richpopulation has been identified as part of the galaxy, the oldest and youngestpopulations belong to M54, the nuclear star cluster (NSC) of the Sgr dSphgalaxy. The old metal-poor component of M54 has been interpreted as at leastone decayed GC which was initially orbiting its host galaxy. The youngestpopulation formed in situ from gas accreted into M54 after its arrival at thecentre of the host. In this work, we use the observed properties of M54 toexplore the shape of the inner density profile of the Sgr dSph galaxy. To doso, we simulate the decay of M54 towards the centre of the dark matter (DM)halo of its host. We model the DM density profile using different centralslopes, and we compare the results of the simulations to the most recentobservations of the structural properties of M54. From this comparison, weconclude that a GC that decays in a DM halo with a density profile $\proptor^{-\gamma}$ and $\gamma \leq 1$ shows a rotational signal and flatteningcomparable to those observed for M54. Steeper profiles produce, instead, highlyrotating and more flattened NSCs which do not match the properties of M54.

Nuclear star clusters as probes of dark matter halos: the case of the Sagittarius Dwarf Spheroidal Galaxy