Fundamental oscillation modes of scalar bosonic dark stars

We perform a detailed analysis of the fundamental $f$-mode frequencies and damping times of nonrotating boson stars in general relativity by solving the nonradial perturbation equations. Two parameters which govern the microscopic properties of the bosonic condensates, namely the self-coupling strength and the mass of scalar particle, are explored. These two quantities characterize oscillations of boson star. Specifically, we reexamine some empirical relations that describe the $f$-mode parameters in terms of mass and radius of the boson stars. We found it is possible to constrain the equation of state if the fundamental oscillation mode is observed.