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

TL;DR

This paper explores how self-lensing of moving gravitational-wave sources by massive black holes can break degeneracies in lensing timescales, enabling direct measurement of distances and black hole masses from GW signals.

Contribution

It introduces a novel method to extract the mass and distance of black holes from self-lensed GW signals, breaking the traditional degeneracy in lensing timescales.

Findings

Self-lensing signatures can determine the distance to the black hole.

Interference patterns in GW signals reveal the black hole's redshifted mass.

The method can help constrain the environment of GW sources.

Abstract

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 Paczy\'{n}ski curve. We present a first study of the lensing signature of a massive black hole (MBH) on a frequency-dependent GW signal from a moving binary merger. 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 $t_{\rm E}$. The duration of the curve ($2 t_{\rm E}$) becomes independent on the MBH mass $M_{\rm MBH}$, and provides a direct measure of the distance $d_{\rm LS}$ to the MBH. However, $M_{\rm MBH}$ remains unknown. We show that, in GW signals, the redshifted mass $M_{{\rm MBH},z}$ can additionally be obtained from the interference pattern, by measuring the modulation period $T$, the GW frequency $f$, and $t_{\rm E}$: $M_{{\rm MBH},z}\simeq 2.5\times 10^6\,M_\odot\,(t_{\rm E}/[100\,{\rm s}])\,(f\,T)^{-1}$. If this lensing signature is not considered, it may be confused with other waveform distortions, especially in the modeling of overlapping 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 $d_{\rm LS}$ of sources, especially around low-mass MBHs at the center of star clusters and galaxies, (2) additionally estimate the mass $M_{{\rm MBH},z}$ of these MBHs, and (3) infer the orbital inclination of the binary. Simultaneously obtaining $d_{\rm LS}$ and $M_{{\rm MBH},z}$ through self-lensing can help constrain the astrophysical environments where GW signals come from.