Scalar stars and lumps with (A)dS core

We explore the possibility of embedding regular compact objects with (anti) de Sitter ((A)dS) core as solutions of Einstein's gravity minimally coupled to a real scalar field. We consider, among others, solutions interpolating between an inner, potential-dominated core and an outer, kinetic-term-dominated region. Owing to their analogy with slow-roll inflation, we term them gravitational vacuum inflative stars, or gravistars for short. We systematically discuss approximate solutions of the theory describing either the core or the asymptotically-flat region at spatial infinity. We extend nonexistence theorems for smooth interpolating solutions, previously proved for black holes, to compact objects without event horizons. This allows us to construct different classes of exact (either smooth or non-smooth) singularity-free solutions of the theory. We first find a smooth solution interpolating between an AdS spacetime in the core and an asymptotically-flat spacetime (a Schwarzschild solution with a subleading $1/r^2$ deformation). We proceed by constructing non-smooth solutions describing gravistars. Finally, we derive a smooth scalar lump solution interpolating between $\text{AdS}_4$ in the core and a Nariai spacetime at spatial infinity.