Tidal Love numbers of gravitational atoms

Ultralight bosonic fields can form condensates, or clouds, around spinning black holes. When this system is under the influence of a secondary massive body, its tidal response can be quantified in the tidal Love numbers (TLNs). Although TLNs vanish for black holes in vacuum, it has been shown that the same is not true for black holes immersed in matter environments. In this work, we compute the gravitational TLNs of black holes surrounded by scalar clouds, in the Newtonian limit. We show that they are non-vanishing, have a strong power-law dependence on the boson's mass, and are proportional to the scalar cloud's total mass. In particular, we find that, independently of the cloud's configuration, the TLNs from axisymmetric tides scale as $\propto r_c^{2l+1}$, for $r_c$ the cloud's "radius" and $l$ the multipole order of the external tidal field. This differs by a factor $r_c$ from previous estimates based on scalar and vector tidal perturbations but is in perfect agreement with the behavior of TLNs in other matter systems. Furthermore, we show that the adiabatic tides approximation we employ is, in general, not appropriate for non-axisymmetric tidal interactions.