Generating ultra compact boson stars with modified scalar potentials

The properties of selfinteracting boson stars with different scalarpotentials going beyond the commonly used $\phi^4$ ansatz are studied. Thescalar potential is extended to different values of the exponent $n$ of theform $V \propto \phi^n$. Two stability mechanism for boson stars areintroduced, the first being a mass term and the second one a vacuum term. Wepresent analytic scale-invariant expressions for these two classes of equationsof state. The resulting properties of the boson star configurations differconsiderably from previous calculations. We find three different categories ofmass-radius relation: the first category resembles the mass-radius curve ofselfbound stars, the second one those of neutron stars and the third one is thewell known constant radius case from the standard $\phi^4$ potential. Wedemonstrate that the maximal compactness can reach extremely high values goingto 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 solitonicboson stars. Hence, boson stars even described by a simple modified scalarpotential in the form of $V \propto \phi^n$ can be ultra compact black holemimickers where the photon ring is located outside the radius of the star.