Probing the spin of compact objects with gravitational microlensing of gravitational waves

TL;DR

The paper investigates wave-optics gravitational microlensing of gravitational waves by slowly rotating compact objects, deriving the spin-dependent lensing potential and the corresponding waveforms in the weak-field limit. It introduces a Fresnel-based magnification function $F(w)$ and a dimensionless frequency $w$, then computes lensed GW signals numerically, including spin-induced asymmetries and caustic structures. A Bayesian, high-SNR framework shows that lens spin parameters can be recovered within $90\%$ credibility for ET-like detectors at SNRs of 50–100, with stronger constraints at higher SNR and notable $M_\ell$–$y$ correlations. Because astrophysical BHs have tiny spin corrections in this regime, the method remains especially relevant for exotic or naked-singularity-like spin configurations, highlighting the potential of future GW lensing observations to probe fundamental physics and the spin content of compact objects.

Abstract

Propagating gravitational waves (GWs) can encounter a massive object (lens) whose gravitational radius is comparable to the wavelength of the GWs (wave-optics regime). The resulting `microlensed' signal contains imprints about the properties of the lens. In this work, we compute the GW waveforms microlensed by a rotating compact object in weak-field gravity. Using these waveforms, for the first time, we assess how well the parameters of the rotating lens can be inferred from GW lensing observations. We find that if we allow for naked singular solutions within general relativity or beyond, which in principle can have spins that are not bounded to be extremal, our method can be used to extract the rotating lens parameters using observations of microlensed GWs with future detectors. As a result, we find that the lens parameters for such lenses are well recovered within $90\%$ confidence for signal-to-noise ratio (SNR) 50 and especially well for SNR$=100$ with Einstein Telescope.

Figures (4)

• Figure 1: Schematic describing the coordinate system for lensing of a GW due to a compact object with spin $\bar{a}$ in the thin lens approximation. The GW propagating in the direction $\hat{n}$ strikes the lens plane ($\xi_1-\xi_2$) at $\bar{\xi}$, and travels to the detector after being lensed. The line of sight is shown as the dashed-dotted line perpendicular to the lens plane. The spin $\bar{a}$ is assumed to lie entirely in the lens plane, making an angle $\phi$ with the $\xi_1$ axis. $D_{\ell}$, $D_{s}$ and $D_{\ell s}$ are the angular diameter distances from the detector to the lens, from the detector to the source and from the lens to the source, respectively, along the line of sight.
• Figure 2: Caustic curves in the source plane ($y_1-y_2$) for a non-rotating (left) versus a rotating lens (middle $\&$ right) with dimensionless spin parameter $\hat{a}=0.3$ for $\phi$ fixed at $\phi=0^\circ$ and $\phi=75^\circ$, respectively. The caustic for a non-rotating point mass lens is a single point at the origin (black dot), and we always have two images (orange dots) for the source (orange cross). In the case of a rotating lens, if the source is inside the grey shaded region (blue cross), lensing produces five images (blue dots), whereas if the source lies anywhere outside the shaded region (orange cross), there are three images (orange dots). Also shown are the GO image magnifications for individual images (black numbers). As the angle $\phi$ between the $\mathrm{y}_1$ axis and $\hat{a}$ changes, the caustic for the rotating lens rotates, and the corresponding image locations shift.
• Figure 3: Top panel: Time-domain magnification function $\widetilde{\mathcal{F}}(t)$ as a function of $t$ (in units of $4M_\ell (1+z_\ell)$, with the global minima at $t=0$). The columns indicate various values of the spin parameter $\hat{a}=\{0, 0.3\}$ and the spin angle $\phi=\{0^\circ, 75^\circ \}$. Colours indicate the source position $\mathrm{y}=\{0.1, 0.5, 1.0\}$. Second panel: Amplitude of the magnification function, $|F(\mathit{w})|$ as a function of the dimensionless frequency $w$. For the cases with non-zero spin, $|F(\mathit{w})|$ is significantly modulated away from the non-rotating case. Third panel: Phase of the amplification function arg$F(\mathit{w})$, as a function of $w$. Bottom panel: Amplitude of the lensed GW strain as a function of frequency $f$ in Hz. In all cases, the lens mass is $M_\ell=300M_\odot$ located at redshift $z_\ell=0.5$. The unlensed waveform is also shown for reference, which is a quasi-circular waveform from a $20-20 M_\odot$ binary at redshift $z_s=2$.
• Figure 4: Posterior distributions for the case of rotating (top row) and non-rotating (bottom row) lensed waveforms, with $\text{SNR}=50$ (left column) and $\text{SNR}=100$ (right column), considering ET-D sensitivity. For the rotating case, the true lens parameters (red lines) are [$\hat{a}=0.22,\,M_\ell=300M_{\odot},\, \mathrm{y}=0.5$], with the same $M_\ell$, and $\,y$ values for the zero-spin case. In all cases, the projected spin-angle is fixed at $\phi=0$. The vertical black dashed lines show the 90% confidence interval, while the vertical black solid lines represent the median value. The unlensed waveform is a quasi-circular signal from a $20-20 M_\odot$ binary at redshift $z_s=2$.