Where is Love? Tidal deformability in the black hole compactness limit

Abstract: One of the macroscopically measurable effects of gravity is the tidal deformability of astrophysical objects, which can be quantified by their tidal Love numbers. For planets and stars, these numbers measure the resistance of their material against the tidal forces, and the resulting contribution to their gravitational multipole moments. According to general relativity, nonrotating deformed black holes, instead, show no addition to their gravitational multipole moments, and all of their Love numbers are zero. In this paper we explore different configurations of nonrotating compact and ultracompact stars to bridge the compactness gap between black holes and neutron stars and calculate their Love number $k_2$. We calculate $k_2$ for the first time for uniform density ultracompact stars with mass $M$ and radius $R$ beyond the Buchdahl limit (compactness $M/R > 4/9$), and we find that $k_2 \to 0^+$ as $M/R \to 1/2$, i.e., the Schwarzschild black hole limit. Our results provide insight on the zero tidal deformability limit and we use current constraints on the binary tidal deformability $\tilde{\Lambda}$ from GW170817 (and future upper limits from binary black hole mergers) to propose tests of alternative models.