Loop quantum gravitational signatures via Love numbers

Loop quantum gravitational effects can resolve the central singularity of black holes, while potentially leaving tiny traces of quantization in the exterior spacetime. We show the way these residues can, in principle, be explored using tidal Love numbers (TLNs). We consider loop quantized Schwarzschild black hole, in particular the Ashtekar-Olmedo-Singh (AOS) model, and study the static response to external tidal fields of spin zero (scalar field), spin one (vector field), and spin two (axial gravitational field) types. We find that, in contrast to the classical theory, where TLNs vanish, they are nonvanishing and negative for all three responses and for all multipoles. Besides, the magnitude of TLNs decreases as the black hole mass increases, and TLNs, in response to the axial gravitational field, have the largest magnitude among these three responses. Our results show that for black holes of mass $M\ensuremath{\gtrsim}4.3\ifmmode\times\else\texttimes\fi{}{10}^{4}{M}_{\mathrm{Pl}}$, the AOS model is consistent with current and next-generation detection limits for TLNs. Our findings suggest that the quantum deformability of loop-quantized black holes, arising from the inherent fuzziness of spacetime geometry, reveals a fundamentally distinct internal structure compared to their classical counterparts. This unique feature manifests as quantum hair, which, in principle, can be detected by future observations.