Generating ultra compact boson stars with modified scalar potentials

The properties of selfinteracting boson stars with different scalar potentials going beyond the commonly used $\phi^4$ ansatz are studied. The scalar potential is extended to different values of the exponent $n$ of the form $V \propto \phi^n$. Two stability mechanism for boson stars are introduced, the first being a mass term and the second one a vacuum term. We present analytic scale-invariant expressions for these two classes of equations of state. The resulting properties of the boson star configurations differ considerably from previous calculations. We find three different categories of mass-radius relation: the first category resembles the mass-radius curve of selfbound stars, the second one those of neutron stars and the third one is the well known constant radius case from the standard $\phi^4$ potential. We demonstrate that the maximal compactness can reach extremely high values going to the limit of causality $C_\text{max} = 0.354$ asymptotically for $n\to\infty$. The maximal compactnesses exceed previously calculated values of $C_\text{max}=0.16$ for the standard $\phi^4$-theory and $C_\text{max}=0.21$ for vector-like interactions and is in line with previous results for solitonic boson stars. Hence, boson stars even described by a simple modified scalar potential in the form of $V \propto \phi^n$ can be ultra compact black hole mimickers where the photon ring is located outside the radius of the star.