Lense-Thirring Precession Modulates Repeated Lensing of Continues Gravitational Wave Source from AGN Disks

TL;DR

This work investigates how spin-induced Lense–Thirring precession (LT) of a supermassive black hole modulates the repeated gravitational lensing of continuous GW sources in AGN disks. Using a wave-optics treatment of lensing with a point-mass model and a CW source embedded in an AGN disk, the authors quantify how LT precession alters the lensing geometry over time, affecting crossing durations $T_{\rm Ein}$ and maximum magnifications $|F|$. They employ matched-filter analyses to assess detectability of LT-induced waveform modulations and show that higher SMBH spins, smaller orbital radii, higher GW frequencies, and larger misalignment angles increase the observable imprint and the probability of lensing within finite observing times, with potential to constrain SMBH spin. The results suggest that lensed CWs could serve as indirect probes of SMBH spin and AGN environments, extending GW lensing studies beyond transient events to continuous signals.

Abstract

Gravitational lensing of gravitational waves (GWs) offers a novel observational channel that complements traditional electromagnetic approaches and provides unique insights into the astrophysical environments of GW sources. In this work, we investigate the repeated lensing of continuous gravitational wave (CW) sources in active galactic nucleus (AGN) disks by central supermassive black holes (SMBHs), focusing on the imprint of SMBH spin via the Lense-Thirring (LT) effect. Although typically weak and challenging to observe, the spin-induced precession of source orbits can accumulate over time, thereby modulating the lensing geometry. Such modulations influence the magnification, duration, and waveform structure of each repeated lensing event, and enhance the overall probability of lensing occurrences. Using matched filtering, we demonstrate that spin-dependent signatures may be detectable, suggesting that lensed CW signals could serve as an indirect probe of SMBH spin in AGNs.

Figures (10)

• Figure 1: Geometry of the SMBH–CW source system. We define the $z$-axis to be aligned with the spin vector $\boldsymbol{\chi}$ of the SMBH. The reference plane is then the plane perpendicular to this spin direction. The CW source resides in the orbital plane with an orbital radius $a$. The orbital angular momentum $\boldsymbol{L}_{\mathrm{orb}}$ forms an angle $\alpha$ with $\boldsymbol{\chi}$. The observer’s line of sight is denoted by $\boldsymbol{N}$, and its projection onto the reference plane is $\boldsymbol{N}'$. The source direction is $\boldsymbol{S}$, located on the AGN disk. The angle between $\boldsymbol{N}$ and $\boldsymbol{L}_{\mathrm{orb}}$ is defined as $\iota$; the angle between $\boldsymbol{S}$ and $\boldsymbol{N}'$ is $\phi$; and the source position is $\beta$. A Cartesian coordinate system is defined in the reference plane with $z$-axis along $\boldsymbol{\chi}$; note that this differs from the orbital-plane coordinate system where $z$ is along $\boldsymbol{L}_{\mathrm{orb}}$. The projection of $\boldsymbol{L}_{\mathrm{orb}}$ onto the reference plane defines an angle $\Omega$ with respect to the $x$-axis. $D_{\rm L}$ and $D_{\rm LS}$ denote the distances from the lens to the observer and the projected distance from the lens to the source along $\boldsymbol{N}$, respectively.
• Figure 2: Variation of the transmission factor as a function of time for different inclination angles $\iota = 80^\circ$ to $89.9^\circ$ (six values), with frequency $f = 10^{-2}$ Hz ($\omega \sim 12$), where the transmission factor is computed using the full wave optics formalism Eq.\ref{['eq:transmission_factor']}. The source orbits a lens of mass $M_{\rm L} = 10^7\,M_\odot$ at an orbital radius $a = 100R_{\rm s}$. The red dashed line marks the time of $\phi = 0$. The lensing effect weakens as $\iota$ deviates from $90^\circ$ due to reduced alignment.
• Figure 3: Same setup as Fig. \ref{['fig:TF001']} but with a higher frequency of $f = 800$ Hz ($\omega \sim 10^6$), approaching the geometric optics limit. The transmission factor becomes significantly sharper and higher, exhibiting rapid oscillations, as shown in the magnified view in the upper left corner. We zoom into a narrow window of $1 \times 10^{-4}$ days around $-0.1$ days to clearly illustrate the fine-scale oscillatory behavior.
• Figure 4: Dynamical timescales as functions of the central SMBH mass $M_{\rm L}$, with the outer orbital separation fixed at $a = 100 R_{\rm S}$. The plotted periods include the Lense-Thirring precession ($P_{\rm LT}$), Lidov-Kozai cycles ($P_{\rm LK}$) assuming inner binaries emitting gravitational waves at 1 mHz and 10 mHz, de Sitter precession ($P_{\rm dS}$), the outer orbital period ($P_{\rm o}$), and Time through $R_{\rm Ein}$. The LT precession timescale is generally much longer than observation timescale. To amplify the subtle LT effect, we rely on the high positional sensitivity provided by lensing observations. Since the LT period increases with the lens mass $M_{\rm L}$, it is necessary to avoid excessively massive lenses to ensure that the LT modulation remains within observationally accessible timescales. Given the short orbital period, the LT-induced precession accumulates over many orbits. Meanwhile, lensing evolves on much shorter timescales, which justifies the neglect of other secular effects when modeling the lensing configuration.
• Figure 5: Characteristic timescales as functions of a in units of $R_{\rm S}$, with the SMBH mass fixed at $M_{\rm L} = 10^7 \,M_\odot$. The plotted dynamical timescales are the same as in Fig. \ref{['fig:period1']}. The LT precession period decreases rapidly with decreasing orbital radius. This suggests that smaller orbits enhance the effect and improve its detectability.