A Bound on the Dynamical Love Number

The tidal deformability of a compact object is encoded in a single function of frequency, the retarded Green's function relating the induced multipole to the applied tide. Causality, reality, passivity, and the high-frequency conditions required for a positive-measure dispersion representation allow this response to be rescaled into a holomorphic self-map of the upper complex frequency half-plane. Using the Schwarz--Pick theorem, already employed to derive the quantum chaos bound in black hole physics, we provide a bound on the rate of the tidal response with respect to the frequency. For a neutron star the dynamical Love number is bounded in terms of the static one and of the frequency of the first internal mode, with the single-mode ($f$-mode) model saturating the bound. For a black hole, the bound gives information on the dissipative tidal-heating coefficient.