The distribution of old stars around the Milky Way's central black hole I: Star counts

(abridged) In this paper we revisit the problem of inferring the innermost structure of the Milky Way's nuclear star cluster via star counts, to clarify whether it displays a core or a cusp around the central black hole. Through image stacking and improved PSF fitting we push the completeness limit about one magnitude deeper than in previous, comparable work. Contrary to previous work, we analyse the stellar density in well-defined magnitude ranges in order to be able to constrain stellar masses and ages. The RC and brighter giant stars display a core-like surface density profile within a projected radius R<0.3 pc of the central black hole, in agreement with previous studies, but show a cusp-like surface density distribution at larger R. The surface density of the fainter stars can be described well by a single power-law at R<2 pc. The cusp-like profile of the faint stars persists even if we take into account the possible contamination of stars in this brightness range by young pre-main sequence stars. The data are inconsistent with a core-profile for the faint stars.Finally, we show that a 3D Nuker law provides a very good description of the cluster structure. We conclude that the observed stellar density at the Galactic Centre, as it can be inferred with current instruments, is consistent with the existence of a stellar cusp around the Milky Way's central black hole, Sgr A*. This cusp is well developed inside the influence radius of about 3 pc of Sgr A* and can be described by a single three-dimensional power-law with an exponent gamma=1.23+-0.05. The apparent lack of RC stars and brighter giants at projected distances of R < 0.3 pc (R<8") of the massive black hole may indicate that some mechanism has altered their distribution or intrinsic luminosity.