Smoothed one-core and core–multi-shell regular black holes

We discuss the generic properties of a general, smoothly varying, spherically symmetric mass distribution $D\left(r,\theta \right)$ , with no cosmological term ( $\theta $ is a length scale parameter). Observing these constraints, we show that (1.) the de Sitter behavior of spacetime at the origin is generic and depends only on $D\left(0,\theta \right)$ , (2.) the geometry may posses up to $2\left(k+1\right)$ horizons depending solely on the total mass M if the cumulative distribution of $D\left(r,\theta \right)$ has $2k+1$ inflection points, and (3.) no scalar invariant nor a thermodynamic entity diverges. We define new two-parameter mathematical distributions mimicking Gaussian and step-like functions and reduce to the Dirac distribution in the limit of vanishing parameter $\theta $ . We use these distributions to derive in closed forms asymptotically flat, spherically symmetric, solutions that describe and model a variety of physical and geometric entities ranging from noncommutative black holes, quantum-corrected black holes to stars and dark matter halos for various scaling values of $\theta $ . We show that the mass-to-radius ratio $\pi {c}^{2}/G$ is an upper limit for regular-black-hole formation. Core–multi-shell and multi-shell regular black holes are also derived.