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The multipole moments and the tidal Love numbers of neutron stars and quarkstars satisfy certain relations which are almost insensitive to the star'sinternal structure. A natural question is whether the same relations hold fordifferent compact objects and how they possibly approach the black-hole limit.Here we consider "gravastars", which are hypothetical compact objects sustainedby their internal vacuum energy. Such solutions have been proposed as exoticalternatives to the black-hole paradigm because they can be as compact as blackholes and exist in any mass range. By constructing slowly-rotating, thin-shellgravastars to quadratic order in the spin, we compute the moment of inertia$I$, the mass quadrupole moment $Q$, and the tidal Love number $\lambda$ inexact form. The $I$-$\lambda$-$Q$ relations of a gravastar are dramaticallydifferent from those of an ordinary compact star, but the black-hole limit iscontinuous, i.e. these quantities approach their Kerr counterparts when thecompactness is maximum. Therefore, such relations can be used to discern agravastar from an ordinary compact star, but not to break the degeneracy withthe black-hole case. Based on these results, we conjecture that the fullmultipolar structure and the tidal deformability of a spinning, ultracompactgravastar are identical to those of a Kerr black hole. The approach to theblack-hole limit is nonpolynomial, thus differing from the critical behaviorrecently found for strongly anisotropic neutron stars.

I-Love-Q relations for a gravastar and the approach to the black-hole limit