Self-lensing of moving gravitational-wave sources can break the microlensing crossing timescale degeneracy

When a moving gravitational-wave (GW) source travels behind a massive astrophysical object, its signal is gravitationally lensed, showing a waveform distortion similar to a Paczyński curve. We present a first study on the lensing signature of a massive black hole (MBH) on a frequency-dependent GW signal from a moving compact binary merger (CBC) source, focused on ground-based GW detectors. For both light and GW sources in a Keplerian circular orbit around a MBH lens, the self-lensing geometry breaks the microlensing degeneracy in the Einstein radius crossing timescale $t_{\mathrm{E}}$. The duration of the curve ($2t_{\mathrm{E}}$) becomes independent on the MBH mass $M_{\mathrm{M}\mathrm{B}\mathrm{H}}$, and provides a direct value of the orbital distance $d_{\mathrm{L}\mathrm{S}}$ of the source around the MBH. However, $M_{\mathrm{M}\mathrm{B}\mathrm{H}}$ remains unknown. In GW signals, the redshifted mass $M_{\mathrm{M}\mathrm{B}\mathrm{H},z}$ can additionally be analytically inferred from the interference pattern, once the modulation period $T$, the GW frequency $f$, and $t_{\mathrm{E}}$ are known: $M_{\mathrm{M}\mathrm{B}\mathrm{H},z}\simeq 2.5\times {10}^6{M}_{\odot }(t_{\mathrm{E}}\mathrm{/}[100\mathrm{s}])(fT{)}^{-1}$. If this lensing signature is not considered, it may be confused with other waveform distortions, especially in the modeling of overlapping CBC signals in next generation ground-based GW detectors. The observation of one of these curves and its associated parameters may help (1) constrain the orbital distance $d_{\mathrm{L}\mathrm{S}}$ of sources, especially around low-mass MBHs at the center of star clusters and galaxies, (2) additionally estimate the mass $M_{\mathrm{M}\mathrm{B}\mathrm{H},z}$ of these MBHs, and (3) infer the orbital inclination of the binary. Simultaneously obtaining $d_{\mathrm{L}\mathrm{S}}$ and $M_{\mathrm{M}\mathrm{B}\mathrm{H},z}$ through self-lensing can help constrain the astrophysical environments where GW signals come from.