  Context: The shape of the mass density profiles of cosmological halos informs
us of the nature of DM and DM-baryons interactions. Previous estimates of the
inner slope of the mass density profiles of clusters of galaxies are in
opposition to predictions derived from numerical simulations of CDM. Aims: We
determine the inner slope of the DM density profile of a massive cluster of
galaxies, Abell S1063 (RXC J2248.7$-$4431) at $z=0.35$, with a dynamical
analysis based on an extensive spectroscopic campaign carried out with the
VIMOS and MUSE spectrographs at the ESO VLT. This new data set provides an
unprecedented sample of 1234 spectroscopic members, 104 of which are located in
the cluster core (R < 200 kpc), extracted from the MUSE integral field
spectroscopy. The latter also allows the stellar velocity dispersion profile of
the brightest cluster galaxy (BCG) to be measured out to 40 kpc. Methods: We
used an upgraded version of the MAMPOSSt technique to perform a joint maximum
likelihood fit to the velocity dispersion profile of the BCG and to the
velocity distribution of cluster member galaxies over a radial range from 1 kpc
to the virial radius (~ 2.7 Mpc). Results: We find a value of $\gamma_{DM}
=0.99 \pm 0.04$ for the inner logarithmic slope of the DM density profile after
marginalizing over all the other parameters. The newly determined dynamical
mass profile is found to be in excellent agreement with the mass density
profiles obtained from the independent X-ray hydrostatic analysis based on deep
Chandra data, as well as the strong and weak lensing analyses. Our value of
\gamma_{DM} is in very good agreement with predictions from cosmological CDM
simulations. We will extend our analysis to more clusters in future works. If
confirmed on a larger cluster sample, our result makes this DM model more
appealing than alternative models.
