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

TL;DR

This paper shows that self-lensing of moving gravitational-wave sources by a massive black hole yields a Paczyński-like envelope whose width $2t_E$ directly measures the orbital distance $d_{LS}$, independent of MBH mass. For GW signals, wave-optics interference between the two lens images introduces a modulation with period $T$, enabling a measurement of the redshifted MBH mass $M_{\rm MBH,z}$ when combined with $t_E$ and the GW frequency $f$, via $M_{\rm MBH,z}\simeq 2.5\times 10^6\,M_\odot\,(t_E/100\,\mathrm{s})/(fT)$. The study outlines conditions under which both $d_{LS}$ and $M_{\rm MBH,z}$ can be jointly inferred, discusses observability in AGN-disk environments, and shows that self-lensing can further constrain the binary’s orbital inclination. These results offer a new avenue to probe MBH environments and the astrophysical contexts of GW sources, potentially improving distance estimates and characterizing host environments for next-generation GW detectors. All relations are expressed with explicit $t_E$, $d_{LS}$, $f$, $T$, and $M_{\rm MBH,z}$ dependencies, enabling direct application to data.

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 Paczyński 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.

Figures (6)

• Figure 1: Schema of the lensing configuration considered in this study. Top: The GW source (S) is a compact binary orbiting a MBH at a distance $d_{\rm LS}$. The MBH acts as a gravitational lens (L), affecting the GW signal, with arrives distorted at the observer (O) when S, L and O are close to alignment. Bottom: View from the observer, where the source travels behind the lens. The lensing effect is dominant behind the cross-section of radius $R_{\rm E}$ and, as we will see in Sec. \ref{['sec:mass']}, the amplitude of the signal is modulated by the transmission factor ($F$ in Eq. \ref{['eq:imprint-freq']}) shown by the oscillating pattern here. The source can have an offset $y_0 R_{\rm E}$ with respect to the perfect alignment (dashed line going through the center of the lens).
• Figure 2: Representation of a GW signal CBC "chirp", through the strain $h$ (relative deformation of space-time) as a function of time $t$. The amplitude of the Paczyński-like curve on the inspiral phase of the signal can be comparable to (or even larger than) the final merger. The parameters used here are a source "chirp mass" $M_{\rm chirp}=1 \,{\rm M}_\odot$, a distance from the observer $d_{\rm L}=1\,{\rm Gpc}$ (just affecting the overall amplitude of the strain), $y_0=0.1$ and $d_{\rm LS}\simeq3\times10^{10}\,{\rm m}$. In self-lensing, the width of the lensing curve ($2t_{\rm E}=4d_{\rm LS}/c$) is independent on $M_{\rm MBH}$ [Eq. \ref{['eq:tcross']}]. Instead, as we will see in Sec. \ref{['sec:mass']}, $M_{\rm MBH}$ can be obtained from the interference pattern modulations on the signal (Fig. \ref{['fig:zoom-waveform']}). For comparison purposes, the lensed signal is represented in the time reference of the arrival time of the first image, $t_1$. For simplicity, we represent the inspiral part of the signal from $f\gtrsim 12\,{\rm Hz}$, truncated at merger.
• Figure 3: Zoom-in view of Fig. \ref{['fig:lensed_inspiral_waveform']}, at the peak of the lensing curve at GW frequencies $f\simeq 15\,{\rm Hz}$. The interference pattern modulating the GW signal depends on the mass, (a) $M_{{\rm MBH},z}=10^5\,M_\odot$, (b) $M_{{\rm MBH},z}=10^6\,M_\odot$, and the distance $d_{\rm LS}\simeq 3\times 10^{10}\,{\rm m}$ (or equivalently $t_{\rm E}\simeq 200\,{\rm s}$). The modulation period far from the peak can be estimated with Eq. \ref{['eq:modulation-period']}, for example here in the segment $t\in(400-420)\,{\rm s}$.
• Figure 4: Amplitude of the signal (envelope curve) as a function of the closest source position $y_0$. The curve follows a Paczyński-like curve. The width of the curve is $2t_{\rm E}$: it is practically independent on $y_0$, and for self-lensing it only depends on $d_{\rm LS}$ [Eq. \ref{['eq:tcross']}].
• Figure 5: Parameter thresholds to detect $d_{\rm LS}$ and $M_{{\rm MBH},z}$ jointly. If $M_{{\rm MBH},z} \gtrsim 10^4\,M_\odot\,({\rm Hz}/f)$, the maximum distance $d_{\rm LS}$ given a value of $M_{\rm chirp}$ is limited by observing $t_{\rm E}$, shown by the red dashed line (Eq. \ref{['eq:dls-condition']}). If $M_{{\rm MBH},z} \lesssim 10^4\,M_\odot\,({\rm Hz}/f)$, the threshold lies in the blue region, where the exact curve depends on the different parameters in Eq. \ref{['eq:m-condition']}.