New strong constraints on the central behaviour of spherical galactic models -- No NFW cusp

We first stress that any spherically symmetric galactic model whose integrated mass profile $M(r)\to 0$ as $r\to 0$ is physically consistent close to the centre only provided that the circular velocity $v_c(r)\to 0$ and the gravitational field $g(r)\to 0$ as $r\to 0$. Next, we apply such a statement to a broad class of five-parameter spherical galactic models, which includes most of those used in astrophysics and cosmology. In particular, we discover that the Jaffe and Hernquist models can only be trusted for $r \gtrsim 0.2 , R_e$, while the NFW model cannot describe the central region either of regular galaxy clusters or of pure dark matter halos, thereby failing to predict any central cusp.