Strong Lensing Gravitational Waves

• Strong lensing gravitational waves is the splitting of a GW signal into multiple images with distinct arrival times and magnifications due to intervening massive galaxy clusters or halos.
• This phenomenon enhances detection by amplifying faint signals and enables precise measurements of time delays, which are critical for refining cosmological parameter estimates.
• Advanced detectors like LISA and third-generation observatories exploit these lensing effects to constrain lens populations, source statistics, and probe fundamental physics such as the speed of gravity.

Strong lensing of gravitational waves refers to the splitting of a gravitational wave (GW) signal into multiple distinct images, each arriving at the detector with a different amplitude and time delay due to intervening massive structures such as galaxies and clusters. In the context of space-based missions like LISA and third-generation ground-based detectors (Einstein Telescope, Cosmic Explorer), strong lensing is both a source of invaluable astrophysical and cosmological information and a unique probe of gravity in regimes inaccessible to electromagnetic (EM) observations. The phenomenon is particularly significant for GW sources at high redshift, where the lensing cross section is dominated by massive galactic halos and clusters. For essentially all astrophysical lensing configurations in GW astronomy, geometric optics applies, with negligible diffraction except very close to lens caustics.

1. Lensing Formalism: Geometric Optics, Lens Models, and Image Properties

In strong GW lensing, the lensing potential of an intervening galaxy or cluster bends the wavefront of an incoming GW, producing multiple images that arrive at different times and with different magnifications. The vast majority of optical depth for lensing in the LISA frequency band is contributed by massive galactic halos, primarily modeled as Singular Isothermal Spheres (SIS). The Einstein radius for an SIS lens is:

$$R_E = 4\pi \left( \frac{\sigma}{c} \right)^2 \frac{D_d D_{ds}}{D_s}$$

where $\sigma$ is the velocity dispersion, $c$ the speed of light, and $D_d$, $D_{ds}$, $D_s$ are the angular diameter distances. A source inside $R_E$ gives rise to two images at $x_\pm = y \pm 1$ (with $y$ the normalized source position). The magnifications of these images are:

$\mu_\pm = \frac{1}{y} \pm 1$

$A_\pm = \sqrt{ \mu_\pm }$

These amplify the GW signal amplitude, potentially raising otherwise sub-threshold sources above the detection limit. For strong lenses in LISA's regime, the time delay between images is:

$\Delta t = \Delta t_z \cdot y$

$$\Delta t_z = \frac{32 \pi^2}{c} \left( \frac{\sigma}{c} \right)^4 \frac{D_d D_{ds}}{D_s} (1+z_d)$$

with typical values on the order of $10^2$ days for $\sigma \sim 200$ km/s, $z_d \sim 5$, $z_s \sim 10$ (Sereno et al., 2010).

2. Lensing Statistics, Detection Probabilities, and Amplification

The lensing probability (‘optical depth’) is determined by integrating the differential lensing cross section over the distribution of lens properties, governed by a modified Schechter function for the halo population:

$$\frac{d^2\tau}{dz_d d\sigma} = \frac{dn}{d\sigma}(\sigma, z_d) \; s_\text{cr}(\sigma, z_d) \; \frac{cdt}{dz_d}$$

For future missions, the optical depth for high-redshift GW sources (e.g., MBHBs at $z \sim 10$) is significant; in optimistic astrophysical scenarios, up to four multiple lensed GW events (SNR $\geq 8$) are expected during a LISA five-year mission (Sereno et al., 2010). Importantly, some fainter signals are amplified above the detection threshold solely due to lensing—potentially up to two "extra" detectable events per year.

For ET-class ground-based detectors, SNR thresholding imposes a maximal impact parameter $y_\text{max}$ for detection of the fainter image; thus, only signals with

$$A_\pm > \frac{\text{SNR}_\text{threshold}}{\text{SNR}_\text{int}}$$

are brought above threshold. This lensing bias has implications for observed source populations and their inferred redshift/mass distributions (Piórkowska et al., 2013).

3. Time Delays and Scientific Applications

The time delay $\Delta t$ between lensed images is an observable of particular importance. It enables cosmological parameter estimation (e.g., measurement of $H_0$ and paper of dark energy) by relating $\Delta t$ to geometric distances via the lens equation. Since GWs offer sub-millisecond timing, and with EM assistance (e.g., host galaxy redshift), strong-lensing time delay measurements provide an independent and precise standard-siren cosmography—immune to distance ladder systematics (Sereno et al., 2010, Piórkowska et al., 2013).

Moreover, strong-lensed GW time delays can be exploited to test the speed of GWs relative to light with extremely high precision ($\sim10^{-7}$), independently of EM/modeling uncertainties. The test is based on the arrival times of multiple GW and EM images from the same lensed transient, requiring only the observed time separation and not the intrinsic lag—thus securely probing GR and alternatives (Collett et al., 2016).

4. Statistical Forecasts, Lensing Rates, and Survey Design

Detection rates are sensitive to detector sensitivity, survey duration, source rates, and lens population properties. For example, with typical assumptions:

Detector	Lensed Events/Yr	Most Probable Image Number	Quad Fractions
aLIGO	$1~\text{yr}^{-1}$	Double/Quad	$\sim30\%$ quads
Einstein Telescope(ET)	$40$-$80~\text{yr}^{-1}$	Double	$\sim6\%$ quads

Moreover, most lensed events originate from high-redshift sources ($z \sim 1$–$4$) and lensing time delays are generally $<1$ month, so finite-length surveys miss only a negligible portion of lensed pairs (Li et al., 2018, Wierda et al., 2021).

Magnification and magnification bias, when incorporated into lensing statistics, can modify both the source population and the fraction of quads/doubles in the observed sample (Li et al., 2018). A strong dependence on the galaxy velocity dispersion $\sigma_*$ implies that measured rates and time delay distributions can be used to measure the velocity dispersion function of lens galaxies to $\sim21\%$ precision after several years with 3G detectors (Xu et al., 2021).

5. Lensing-Induced Phase Effects and Template Construction

Each lensed GW image acquires a fixed Morse phase: type I (minimum) $n_j=0$, type II (saddle) $n_j=1$, type III (maximum) $n_j=2$. In Fourier space, the amplification factor for each image is:

$$F(\omega) \simeq \sum_j |\mu(\theta_j)|^{1/2} \exp\left[i \omega t_d(\theta_j) - i \operatorname{sign}(\omega) n_j \frac{\pi}{2}\right]$$

For circular, non-precessing binaries dominated by $m=2$, the addition of a $\pi/2$ phase shift is degenerate with a change in coalescence phase or (in some cases) the detector-dependent orientation angle. The degeneracy is broken by higher harmonics, orbital precession, or eccentricity. For present detectors, waveform mismatches remain below 1–5% for most astrophysical parameters, but with sufficiently loud sources, such phase effects might become observable. Importantly, an exact lensed template can always be built by applying a constant phase shift in Fourier space, facilitating the identification and joint inference of multiple images (Ezquiaga et al., 2020).

6. Bayesian Identification and Population Inference

A Bayesian statistical framework compares the probability that detected GW events are (a) independent sources or (b) lensed images of a single event. The joint likelihood incorporates common intrinsic parameters (masses, spins, sky location) and image-specific extrinsic parameters (time delay, magnification, Morse phase shift). Hierarchical methods marginalize over source redshift and selection biases. Inclusion of astrophysical priors (e.g., upper mass cutoff) can dramatically strengthen the evidence for lensing when pairs of observed events conflict with expected population properties under the unlensed hypothesis.

Lensing-induced phase shifts (Morse indices), if detectable (i.e., when higher multipoles or multiple images are observed), can flag image identities. Furthermore, simultaneous analysis of multiple images sharpens sky localization, improves source parameter estimation, and, for high-redshift lensed events, provides unique access to the high-z GW universe (Lo et al., 2021).

7. Implications for Cosmology and Fundamental Physics

Strong lensing of GWs establishes a new cosmological probe independent of the traditional distance ladder. Accurate time-delay measurements between images allow constraints on $H_0$ and the dark energy equation of state via standard-siren methodology, particularly when paired with EM redshift data of host galaxies or lenses. If multiple lensed GW–EM events are detected, the method also enables precise tests of the speed of gravity and the presence of additional GW polarizations—since relative amplification and polarization content can, in principle, be compared among images.

In a broader context, the statistics of strong GW lensing events constrain the mass distribution and evolution of lens galaxies, probe the abundance of high-redshift binaries, and provide complementary insights into galaxy/structure formation and the fundamental nature of gravity (Sereno et al., 2010, Piórkowska et al., 2013, Collett et al., 2016, Li et al., 2018, Xu et al., 2021).

Summary Table: Key Quantitative Relations

Quantity	Formula / Expression	Context
Einstein radius (SIS)	$R_E = 4\pi (\sigma/c)^2 (D_d D_{ds} / D_s)$	Lens geometry, image formation
Image positions	$x_\pm = y \pm 1$	SIS model, $y <$ Einstein radius
Magnification	$\mu_{\pm} = 1/y \pm 1$	SIS, yields amplification factor $A_\pm$
Time delay	$$\Delta t = (32\pi^2/c)(\sigma/c)^4 (D_d D_{ds}/D_s)(1+z_d) y$$	SIS, image separation in arrival time
Amplification factor	$A_\pm = \sqrt{\mu_\pm}$	SNR boost, detection threshold
Phase shift (Fourier)	$$F(\omega) \sim |\mu|^{1/2} e^{i \omega t_d - i n \pi/2}$$	Image Morse index ($n=0$, $1$, $2$)

Strong lensing of gravitational waves is therefore a powerful intersection of astrophysical, cosmological, and fundamental physics. It provides not only an enhancement in GW detection rates and access to distant sources, but also a new means of measuring cosmological parameters, testing general relativity, probing halo distributions, and characterizing the high‐redshift universe (Sereno et al., 2010, Piórkowska et al., 2013, Collett et al., 2016, Li et al., 2018, Lo et al., 2021, Xu et al., 2021).