Discovering Love numbers through resonance excitation during extreme mass ratio inspirals

General Relativity predicts that black holes do not possess an internal structure and consequently cannot be excited. This leads to a specific prediction about the waveform of gravitational waves, which they emit during a binary black hole inspiral and to the vanishing of their Love numbers. However, if astrophysical black holes do possess an internal structure, their Love numbers would no longer vanish, and they could be excited during an inspiral by the transfer of orbital energy. This would affect the orbital period and lead to an observable imprint on the emitted gravitational waves waveform. The effect is enhanced if one of the binary companions is resonantly excited. We discuss the conditions for resonant excitation of a hypothetical internal structure of black holes and calculate the phase change of the gravitational waves waveform that is induced due to such resonant excitation during intermediate- and extreme-mass-ratio inspirals. We then relate the phase change to the electric quadrupolar Love number of the larger companion, which is resonantly excited by its smaller companion. We discuss the statistical error on measuring the Love number by LISA and show that, because of this phase change, the statistical error is small even for small values of the Love number. Our results provide a strong indication that the Love number could be detected by LISA with remarkable accuracy, much higher than what can be achieved via tidal deformation effects. Our results further indicate that resonant excitation of the central black hole during an extreme- or intermediate-mass-ratio inspirals is the most promising effect for putting bounds on, or detecting, non-vanishing tidal Love numbers of black holes.