Wave-optical effects in the microlensing of continuous gravitational waves by star clusters

The characteristic strains (1) are orders of magnitude lower than those due to the violent merger events that have thus far been detected. Persistent emissions, associated with magnetically (or otherwise) deformed neutron stars, have the advantage however that signal can be accrued over many cycles. In particular, if the star houses a mass or current quadrupole moment with a lifetime that exceeds the observational window $T_{\text{obs}}$, continual monitoring leads to an increase in detector sensitivity. In a fully-coherent search, a ground-based interferometer can detect a signal of amplitude

with $90\%$ confidence (Watts et al., 2008), where $S_{n}$ is the noise power spectral density of the detector. Although not shown here (see, for example, Figure 5 in Soldateschi & Bucciantini, 2021), it is likely that observations spanning at least a year would be necessary to detect continuous GWs from many of the known sources with existing instruments (see also Lasky, 2015; Suvorov, 2021).

Suppose however that the GWs from the system were lensed en route to the detector. In the case of burst-like signals associated with mergers, for example, where the bulk of the measurable GW luminosity is emitted within a time window spanning a few seconds, relative motion between the lens and source is negligible. Still, wave-like effects are likely to be important here because the source frequency sweeps (‘chirps’) through a wide band, and the lensing-induced amplification is an oscillatory function of $f_{\text{GW}}$ (Nakamura & Deguchi, 1999; Takahashi & Nakamura, 2003; Christian et al., 2018).

By contrast, while continuous GWs are expected to be roughly monochromatic (though see below), the relative motion between the lens and source cannot be ignored when considering observational windows spanning some months. For a phase-coherent search lasting $\gtrsim$ one year, the neutron star would have travelled a (relative) distance of $\sim 10^{15}\times(v/300\text{ km s}^{-1})(T_{\text{obs}}/\text{ yr})$ cm, where $v=300\text{ km s}^{-1}$ is a typical transverse velocity for a millisecond pulsar (Hobbs et al., 2005). This distance, while negligible compared to $D_{\text{OS}}\sim 10$ kpc, comfortably exceeds the Einstein radius associated with a solar-mass microlens, viz.

for a lens located $D_{\text{OL}}=5$ kpc from the observer and a further¹¹1We ignore cosmological corrections to all distance quantities since $z\ll 1$ for the sources considered here. $D_{\text{LS}}=5$ kpc from the source. In rare cases, the emitter may thus cross multiple Einstein rings over a long $T_{\text{obs}}$ when located behind particularly dense regions of the Galaxy (see de Paolis et al., 2001; Jow et al., 2020, for some rate estimates). Crossing interference fringes will lead to modulations of the GW signal, and could noticeably affect parameter estimation in the event of a detection, depending on the location of the neutron star and its environs.

Moreover, wave-optical lensing will generally cause the phase of the signal to drift over time. This may be problematic for phase-coherent GW searches, as matched filtering generally requires the (noisy) detector output, multiplied by a template waveform, to remain in phase with the signal to within $\lesssim$ 1 rad (e.g., Jones, 2004). This problem is well known in the case of searches directed at sources within active binaries, where the GW frequency can drift due to accretion-induced spin evolution and it is necessary to analyse the signal semi-coherently over segments shorter than the full $T_{\text{obs}}$ (Dreissigacker et al., 2018). The spins of isolated neutron stars may also wander over $\sim$ year-long timescales for a variety of reasons (e.g., glitches); see Suvorova et al. (2016) for a discussion. One aspect we can explore with wave-optical calculations is, for a given macrolens, the maximum interval over which matched filtering can be reliably applied (see Sec. 5).

As it stands, the Fresnel-Kirchhoff integral (8) contains a non-local Green’s function. We introduce the thin-lens approximation to reduce the dimensionality of the problem and to eliminate the unwieldy denominator term; this amounts to projecting each microlens onto a single 2-dimensional screen, the mathematical details of which can be found, for example, in Takahashi et al. (2005). The end result is that the amplification factor, $F=\phi/\phi^{(0)}$, can be written as [see equation (2.11) in Nakamura & Deguchi (1999)]

where $\boldsymbol{x}$ (lens-plane-projected coordinates) and $\boldsymbol{x}_{s}$ (source-plane-projected coordinates) are expressed in units of

respectively, and the Einstein radius $R_{E}$ is defined in expression (4). For a lens that is equidistant between the source and the observer, we have $\eta_{0}=2\xi_{0}$. The (normalised) time delay $t_{d}$, up to a constant factor, is a sum of the geometric and Shapiro delays,

where $\psi(\boldsymbol{x})$ is the dimensionless deflection potential (i.e., the projection of $U$ onto the 2D lens screen). For a collection of $n$ point lenses, we have

where the bodies of mass $M_{k}$ are located at $(x_{k},y_{k})$ on the lens plane.

We close this section by noting that in microlensing calculations, one generally also includes convergence ($\kappa$) and shear $(\gamma)$ components related to ‘off-screen’ elements within the time-delay function (11) (see, e.g., Paczyński, 1986a; Meena & Bagla, 2020; Lewis, 2020). The former optical scalar accounts for magnifications due to a smooth mass component (e.g., a background Galactic contribution) while the latter is induced by large-scale anisotropies (e.g., tidal distortions). While the formalism presented in the next section can also be used to study cases with non-zero convergence or shear, we defer such a calculation to a future work.

For sources out to $\gtrsim 10$ kpc, the likelihood that any emitted GWs non-negligibly interact with the gravitational field of a perturber en route to Earth is relatively low: the lensing probability by stars for a source within the bulge may reach a few times $10^{-6}$ (Paczyński, 1986b; de Paolis et al., 2001). If, however, a non-negligible fraction²²2Woan et al. (2018) have provided population-based evidence that all millisecond pulsars house a minimum ellipticity of $\varepsilon\sim 10^{-9}$, comparable to expression (2), a fraction $f_{l}$ of which may be observable in GWs. Although dependent on recycling and star formation assumptions, Zhu et al. (2015) estimate that the millisecond neutron star birth rate is $\sim 10^{-4}\text{ yr}^{-1}$ from the combined core collapse and accretion-induced collapse channels. A rough estimate for the number of observable sources is then $\sim 10^{6}\times f_{l}$, further multiplied by the fraction of the source’s lifetime ($\lesssim$Gyr, Zhu et al., 2015) relative to the age of the Milky Way. ($\approx 10^{-5}$, Liao, Biesiada, & Fan, 2019) of the $\sim 10^{9}$ neutron stars within the bulge emit appreciable GWs, microlensing events may conceivably be observed by the next generation of interferometers over long $T_{\text{obs}}$. Furthermore, Kıroğlu et al. (2021) have recently suggested that for neutron stars in the globular clusters 47 Tuc and M22, ‘self-lensing’ (i.e., lensing by a fellow member of the cluster) rates may be as high as $2\times 10^{-3}\text{ yr}^{-1}$ and $4\times 10^{-5}\text{ yr}^{-1}$, respectively. The single-lens scenario has been considered in detail by Liao, Biesiada, & Fan (2019), who made use of the fact that the Fresnel-Kirchhoff integral can be evaluated analytically in the case of an individual point-mass lens (see Appendix B). Here, we generalise their scenario by considering instead macrolenses consisting of $n\gtrsim 10^{2}$ stars at various distances from the line of sight.

For concreteness, we consider two distinct cluster distributions in this work. Defining an ‘optical depth’, $\tau$, as the ratio of the area covered by the individual Einstein rings to a fiducial screen area $A$, expressed in units of $R_{E}$, we have $\tau\approx n\pi/A$. If we consider a $50\times 50$ screen (say), we then have that $\tau\approx 10^{-3}n$. We consider two cases, one with $n=25$ ($\tau\approx 0.03$), henceforth the low optical depth case, and $n=250$ ($\tau\approx 0.3$), which we call the high optical depth case. On a practical level, these two individual distributions are constructed by plucking points from bivariate Gaussians with relatively large variances over a disc of diameter $50R_{E}$. For each of these two cases, we used the available sampling functions in MATHEMATICA®  to define the macrolens. The resulting initialisations are shown in Figures 1 and 2, respectively. In these Figures, the overlaid colour scale shows the projected potential $\psi(\boldsymbol{x})$ from expression (12), which is naturally stronger by a factor $\sim 10$ in the higher $\tau$ case.

It should be stressed that the distributions considered here are not meant to represent any particular astrophysical system. They are constructed primarily to illustrate the mathematical machinery and to qualitatively explore what the wave-optical impact of lensing by $n\gg 1$ stars may be for continuous GWs in the $\sim$kHz band.

In practice, there are several numerical obstacles encountered when applying the above ideas. The first major difficulty concerns the fact that the Morse flow terminates at points of stationary phase. After performing the complex decomposition of the integrand, we locate the roots, parameterised by the angle $\theta$ and the screen parameters $\boldsymbol{x}_{s}$, of the first derivative of $t_{d}$ plus the logarithm of the Jacobian⁴⁴4Such a step is not strictly necessary, as one could instead monitor the flow to check if an image has been met (i.e., if the velocity falls below a threshold), and automatically issue a perturbation in the manner described to continue the thimble. For the $n\lesssim 250$ cases considered here, this root solving stage is relatively inexpensive, though for $n\gg 10^{2}$ a subroutine approach would be faster.. This entails solving high-order polynomial equations, which we achieve through an exhaustive search using different initial guesses and the Newton-Raphson method within MATHEMATICA®. Once an image is encountered, the flow velocity is then artificially perturbed, $\boldsymbol{\gamma}^{\prime}(\lambda)\rightarrow\boldsymbol{\gamma}^{\prime}(\lambda)+\epsilon$, to ‘kick’ the flow onto the next thimble. In practice, we set $\epsilon=10^{-4}$.

Not all images are of relevance, however, since some may be topologically disconnected from the overall flow (see Appendix A). For example, if a saddle occurs at a place of negative real component $[\text{Re}(\boldsymbol{p})<0]$, the flow cannot encounter it and the associated thimble is irrelevant. Classifying such irrelevant images and related Stokes transitions is in general a difficult problem; see Feldbrugge, Pen, & Turok (2019) for a thorough discussion. We employ something of a trial-and-error approach in this paper, where images are flowed from, but only connected (relative to the origin) components are kept to ensure topological continuity.

The root solver is initialised over a grid of $\theta$ and $\boldsymbol{x}_{s}$ values, and a set of thimbles (in $\boldsymbol{r}$) is then constructed at each grid point. For each figure produced here, $512$ angular steps are used, i.e., a spacing of $2\pi/\Delta\theta=2\pi/512$ is used. The results obtained with this resolution differ by at most $\sim 1\%$ from those obtained with $256$ steps. Some additional tests on convergence with respect to resolution are given in Appendix B. Generally, higher radiation frequencies require larger $\Delta\theta$ (by a factor $\sim f/f_{\text{GW}}$) because the integrand varies more rapidly, and thus $\sim 128$ steps is already adequate for frequencies $\lesssim 1$kHz. Simpson’s method is then used to sum the $\theta$-integrals together to build the full, 2D diffraction integral (9).

The Morse flow equations (14) are sequentially solved using a Runge-Kutta method up to some maximum radius $\text{Re}(\boldsymbol{r})$. Formally speaking, this radius must extend to infinity, else Cauchy’s theorem (13) cannot be applied. However, because the Morse flow is also a contour of steepest descent, high accuracy can be achieved even with relatively early truncations that depend on $\boldsymbol{x}_{s}$; see Appendix A.

Finally, it is important to note that the function $t_{d}$ is not analytic over the complex plane, as the $\psi$ piece introduces logarithmic singularities at each microlens position $(r_{k}\cos\theta_{k},r_{k}\sin\theta_{k})$. This is a problem firstly because Cauchy’s theorem cannot be strictly applied for $\theta=\theta_{k}$, though small arcs around the branch points $r_{k}$ can be constructed to restore analyticity if such an angle is within the numerical grid. Additionally, the flow tends to infinitely wind around singular points (i.e., the velocity blows up), causing the integral to diverge. We have explored several possibilities for tempering the singularities:

• •

  Introducing an additional $n-1$ lens planes can circumvent the appearance of more than one singularity on any given plane, as discussed by Feldbrugge (2020) (see also Ramesh et al., 2021). In particular, if each microlens resides on a different plane, screen-adapted coordinates can be used to center each individual singularity away from the relevant regions, so that it may be ignored. This approach introduces considerable computational demand however, as the Fresnel-Kirchhoff integral effectively becomes $2n$ dimensional.

• •

  Various regularisation techniques are possible, including (i) approximating $\psi$ near the singularities by a different function that is regular there (cf. Christian et al., 2018; Guo & Lu, 2020), and (ii) cutting out regions of some size surrounding the singularities. These approaches can be difficult to tune however because if too much of the contour is cut, or if $\psi$ is poorly approximated, significant errors can be introduced.

• •

  In the building of the total contour $\gamma_{3}$ and applying (13), it is not necessary that the entire portion (or even any portion) be a Morse flow. Therefore, if singularities lie within the Morse-constructed contour, one can deform the path to restore analyticity, as is typically done when computing inverse Laplace transforms, for example. These deformed segments could be fittingly called ‘suboptimal’ thimbles.

A thorough exploration of the above possibilities lies beyond the scope of this paper. Nevertheless, the third option is employed here for concreteness, as some testing with low $n$ cases suggests the results agree with the multi-plane approach to within a few percent. Since it is only ever necessary to suboptimally flow over short segments, the oscillations introduced are not fatal for the numerical method.

Figure 4 shows the intensity pattern, as a function of the normalised screen parameters $x_{s}$ and $y_{s}$, associated with the low-optical depth microlens distribution shown in Fig. 1, where we have fixed $f_{\text{GW}}=1$ kHz. Figure 5 instead shows the same intensity profile but for $f_{\text{GW}}=2$ kHz, i.e., for a star spinning twice as quickly. Although no neutron stars with such a high rotational velocity have been directly observed, this latter case provides a useful comparison to illustrate how the radiation frequency impacts on the overall intensity map. Furthermore, theoretical models of accretion-induced spin-up allow, in principle, for the rotation rate to reach this level (e.g., Glampedakis & Suvorov, 2021), and GW searches at this frequency range have been conducted (Dergachev & Papa, 2021).

For $f_{\text{GW}}=1$ kHz, the maximum intensity over the screen is relatively low, $|F|_{\text{max}}^{2}=1.8$, owing primarily to the sparse nature of the microlens distribution and, hence, the weakness of the bulk gravitational potential $U$. For the faster star with $f_{\text{GW}}=2$ kHz, we instead have $|F|_{\text{max}}^{2}=2.3$. Monotonicity in $|F|_{\text{max}}$ as a function of frequency is generally expected, since as $f_{\text{GW}}\rightarrow\infty$ we approach the geometric optics limit, where the amplification becomes formally infinite along the caustic surface(s) where $\boldsymbol{x}+\tfrac{1}{2}\nabla_{\boldsymbol{x}}\psi=0$ (Jow et al., 2021). To put these maxima in perspective, consider the case of a single solar-mass lens, where one finds maximum values of $|F_{n=1}|^{2}=1.21$ and $|F_{n=1}|^{2}=1.44$ for $f_{\text{GW}}=1$ kHz and $f_{\text{GW}}=2$ kHz, respectively [see equation (B1)]. These maxima occur when the source is oriented directly behind the perturbing body, i.e., at $(x_{s},y_{s})=(x_{1},y_{1})$. It is unsurprising therefore that the maxima in our simulations occur when the source aligns itself behind the centre of mass of the densest mini-cluster located at $(x_{s},y_{s})\approx(0,15)$; see Fig. 1. In particular, a collection of point masses in close proximity to one another generally behave as one larger lens around the centre of mass, and since $|F|$ scales with $M_{L}$, as can be seen from (9), the amplification is larger there. In the local vicinity (i.e., within a few Einstein radii) of isolated stars in our distribution, such as at $(x_{s},y_{s})\approx(-20,0)$, the intensity strongly resembles that of the single point-lens case (Liao, Biesiada, & Fan, 2019; Meena & Bagla, 2020).

The oscillatory nature of the intensity along any given line is also the hallmark of interference, as expected in a wave-optics calculation. The variability is more extreme in the higher frequency case in Fig. 5, as the dimensionless exponent $f_{\text{GW}}t_{d}$ varies over length-scales exactly half as long. The emergence of interference fringes is also more obvious in this case, especially surrounding isolated members of the cluster, e.g., near $(x_{s},y_{s})\approx(-20,0)$. The bulk influence is minimal in the vicinity of these regions, and the amplification roughly matches the analytic profile for the single lens case described above, as does the spacing between interference fringes computable from the Fourier spectrum (Nakamura & Deguchi, 1999). For lower GW frequencies, the amplitudes of the oscillations become virtually invisible since the wavelength is so long that the GWs hardly experience the lens, implying that lensing by stellar-mass bodies is unimportant for ‘ordinary’ neutron stars with $\nu_{\star}\lesssim 10^{2}$ Hz.

To better illustrate the wave-like nature of the Fresnel-Kirchhoff integral, we consider also flux, $|F(t)|^{2}$, and phase, $\theta_{F}(t)=-i\ln[F(t)/|F(t)|]$, variations along a hypothetical trajectory of the source. To this end, suppose that the neutron star begins at the origin $(x_{s},y_{s})=(0,0)$ at $t=0$, and then moves, relative to the macrolens, in the $+y_{s}$ direction with velocity $v=600\text{ km s}^{-1}$; a value not unreasonable for millisecond pulsars (Hobbs et al., 2005). (Note, however, that a smaller $v$ but longer $T_{\text{obs}}$ or $D_{\text{OL}}$ yields the same qualitative picture). Within a year, the star therefore travels $\sim 14\eta_{0}$ [see equation (10)], crossing a large number of interference fringes. Figures 6 and 7 show the 1D variations in flux and phase, respectively, experienced along this path as a function of time, where we implicitly ignore variations in $D_{\text{OS}}$. The red (blue) data points correspond to $f_{\text{GW}}=1(2)\text{ kHz}$, with the size of the symbols roughly characterising the error bars present in the calculation ($\lesssim$ percent level).

The oscillatory nature of the modulation is evident in the flux, especially near $t=0$ where the source is caught between two neighbouring elements of the cluster (the origin in Fig. 1). In regions of low density, i.e., at early or late times, the $f_{\text{GW}}=2$ kHz case oscillates roughly twice as often, as expected from the relative decrease in spacing between interference fringes. The maximum flux achieved by the higher-frequency case is greater than in the lower frequency case by $\sim 25\%$, in accord with Figs. 4 and 5. These maxima are achieved after approximately one year of travel time in this setup (or two years with $v\sim 300\text{ km s}^{-1}$), whereupon the neutron star enters into the densest region shown in the top half of Fig. 1.

Similar oscillatory patterns are observed in the phase, which covers a wide range of values ($-2\lesssim\theta_{F}\lesssim 1.5$) over $T_{\text{obs}}$. Because phase drifts exceeding a sizeable fraction of unity are likely to inhibit a fully coherent search (Jones, 2004; Dreissigacker et al., 2018), variations on the order seen in Fig. 7 are potentially detrimental for detection prospects, despite the flux enhancement (Fig. 6). Within $\sim$ 6 (3) months, the phase wanders by more than $0.3$ radian for the $f_{\text{GW}}=1(2)$ kHz source in this scenario, though the gradient $d\theta_{F}/dt$ increases more rapidly in the dense parts of the cluster and coherence is lost at a faster rate. As such, it is likely that the data would need to be analysed semi-coherently in this scenario over (at most) $\sim$ month-long segments, especially in the higher frequency case, even in the absence of (accretion-induced or otherwise) spin wandering. A thorough examination of the trade-off between lensing-induced amplification and decoherence lies beyond the scope of this work; we refer the reader to Suvorova et al. (2016); Dreissigacker et al. (2018) for a comparison between fully- and semi-coherent sensitivities. Given a lens model however, such phase errors may be corrected for in the template waveform using the PL methodology described here. Either way, issues related to phase modulation may be further compounded by the Earth’s diurnal and orbital motions if the sky location or the relative velocity of the source is poorly constrained.

Similar to the previous section, Figure 8 shows the intensity pattern with $f_{\text{GW}}=1$ kHz, as a function of the screen parameters, for the $n=250$ case shown in Fig. 2. Because $\psi$ is everywhere larger by a factor $\sim 10$ than in the previous case, and more importantly because the distribution is highly concentrated around the origin, the bulk magnifications in the heart of the cluster are considerably larger ($|F|_{\text{max}}^{2}=73$). By contrast, the analytic formula (see Appendix B) for a point-mass lens with $M_{L}=250M_{\odot}$ gives $|F_{n=1}|^{2}=97$ at the origin for the same radiation frequency. For $n\gtrsim 10^{2}$ and the Gaussian distributions we use, the overall mass density on the screen appears similar to that of a grainy sphere with $\psi$ decaying with radius as a power-law. Over- and under-dense regions in the magnification profile are clearly visible however even for $n=250$, especially at $(x_{s},y_{s})\approx(2,-2)$ where there is angle-dependent structure within the second peak ‘ring’. Interference fringes are also abundant, as can be seen from the network of graininess near the origin that extends out to $|\boldsymbol{x}_{s}|\lesssim 20$. Overall, the intensity profile is compactified over the screen relative to the $n=25$ case; in the limit $n\rightarrow\infty$, $\psi$ tends to that of a point-lens with ever-growing $M_{L}$ (Nakamura & Deguchi, 1999).

To better visualise the oscillations above the ambient contribution, suppose that the source moves in the $+y_{s}$ direction with velocity $v=300\text{ km s}^{-1}$ (or greater $v$ and proportionately contracted time axis). Similar to Fig. 6, the variation in flux for the $n=250$ cluster is shown in Figure 9, where the red (blue) symbols correspond to $f_{\text{GW}}=1(2)$ kHz. The emergence of interference fringes is readily apparent as the flux begins at a maximum as the line of sight intersects with the densest region at the origin, descends to an order-unity trough after $\lesssim 2(1)$ months in the $f_{\text{GW}}=1(2)$ kHz case, and then rises on a similar timescale to reach a second peak. Each subsequent peak after the first, the total number of which is doubled in the higher frequency case as expected, is generally of lower amplitude because the stellar density decreases as a function of radius (see Fig. 2). This pattern is not monotonic however, especially around $t\sim 4$ months for the $2$ kHz case where only a mild peak is reached ($|F|^{2}=4$), because the lens distribution is grainy. In this case, the phase drifts are so extreme that coherence is likely to be lost even over timescales of $\sim$ weeks. Once the line of sight encounters only the outskirts of the cluster after $\sim 1.5$ yr, the flux closely resembles that seen in Fig. 6, though oscillates more frequently as there are a greater number of interference fringes produced by the microlenses.

Unlike in the case of a merging binary, where the bulk of the GW luminosity is produced within a few seconds, continuous GW emissions from a deformed neutron star are longer lived – potentially persisting for its entire lifetime – though much fainter; see equation (1). With present-day (future) instruments, it is likely therefore that a detection of these sources requires monitoring for a year (month) or more (Suvorov, 2021; Soldateschi & Bucciantini, 2021). If the line of sight linking the neutron star to the detector intersects with a dense cluster during the observation, lensing effects may modulate the strain, as explored in Figs. 6 and 9. What impact could this be expected to have on parameter estimation?

For magnetic deformations, the gravitational waveform is expected to have a sinusoidal profile of the form $h(t)\sim h_{0}\sin(2\nu_{\star}t)$ plus some subdominant harmonics. The lensed waveform will be similar but with a time-dependent and complex prefactor, so that the signal is amplified by a real factor while its phase is shifted. Because the variations in $h(t)$ are much more rapid than the interference-induced variations in $F(t)$, it may be possible to disentangle the effects of lensing via ‘beat’ patterns (Jung & Shin, 2019; Meena & Bagla, 2020; Cheung et al., 2021). At the simplest level however, we note that only mean values for the (upper-limit) quadrupole moments of neutron stars are typically reported from search efforts (Dergachev & Papa, 2021; The LIGO Scientific Collaboration et al., 2022). This is because the coherent nature of continuous GW-searches necessitates that averaging processes between individual interferometers be carried out to properly subtract noise (Jaranowski et al., 1998; Owen, 2010). As such, a detection of the mean strain $\langle h_{0}\rangle$ implies that the true strain would be lower by a factor $1/\langle|F|\rangle$, where the angled brackets denote a statistical average over $T_{\text{obs}}$. For small clusters, the amplification will not exceed $\sim 20\%$ or so (see Fig. 6), implying that, since $h_{0}\propto B_{\star}^{2}$, the difference in the $B$-field estimate would likely be no more than $\sim 10\%$. For $n\gg 25$, the amplifications could potentially be much larger (see Fig. 9 and Mishra et al., 2021).

The signal-to-noise ratio (SNR) of the system, given by (Jaranowski et al., 1998)

may non-negligibly increase for large amplifications. Considering the trajectory defined within Fig. 6, for example, we estimate that the relative increase in the SNR due to lensing for $f_{\text{GW}}=1$ kHz and $T_{\text{obs}}=2$ yr reads

For the first $T_{\text{obs}}=1$ yr, we find instead $\text{SNR}(F\neq 1)/\text{SNR}(F=1)=1.07$. While these SNR increases are marginal, they could be sufficient to propel an otherwise borderline source into the detectable threshold (Lasky, 2015). As seen in Fig. 7 however, these increases may be offset by the fact that coherence is lost due to lensing-induced phase wandering. Depending on the gradient $d\theta_{F}/dt$, it may be necessary to break the data into many shorter segments, thereby actually reducing the overall sensitivity (Suvorova et al., 2016; Dreissigacker et al., 2018). For greater values of the total lensing mass over a fixed area, we generally find that the SNR increase is higher.

For large SNR ($\gtrsim 10$), parameter-estimation errors that may result if lensing were not taken into account can be calculated through a Fisher analysis (Takahashi & Nakamura, 2003). Given the inner product $(\cdot|\cdot)$, well approximated by the integral in expression (16) for a monochromatic source (Jaranowski et al., 1998), one can define the Fisher matrix, $\Gamma_{ij}=\left(\frac{\partial h}{\partial q^{i}}|\frac{\partial h}{\partial q^{j}}\right)$, where $\boldsymbol{q}$ is a vector of parameters, which includes the source parameters (e.g., $\varepsilon$), location parameters (e.g., position relative to solar system barycenter), and lensing parameters (e.g., $M_{k}$). In general, these are not all independent. For example, the intensity pattern depends intimately on both $f_{\text{GW}}$ and the $M_{k}$. Regardless, the relative error on any given $q^{k}$ is estimable through $(\Delta q^{k})^{2}=(\Gamma^{-1})_{kk}$. Unfortunately, a reliable calculation of $\Gamma_{ij}$ requires one to build many amplification profiles so that derivatives with respect to the lens parameters can be taken, which is beyond our current computational capacity. Nevertheless, using the methodology described herein, a thorough exploration of the parameter space and relative errors could be performed; see Appendix A in Takahashi & Nakamura (2003) for a Fisher analysis in the case of a single point-mass lens.

Finally, we note that a misinterpretation of a time-varying $h_{0}$ as being intrinsic to the source, as opposed to being a result of lensing, may have important consequences for the evolutionary modelling of the system. For example, in actively accreting systems that nevertheless show little variation in the spin frequency, it has been argued that GW backreaction may be a key factor in maintaining spin equilibrium (Bildsten, 1998; Gittins & Andersson, 2019). The validity of this scenario can be tested, for any given system, if the braking index, $n_{b}=\nu_{\star}\ddot{\nu}_{\star}/\dot{\nu}_{\star}^{2}$, can be independently measured: GW-dominated spindown implies $n_{b}\approx 5$. However, if $\varepsilon$ were to intrinsically vary over sufficiently short timescales, $n_{b}$ could differ significantly from 5, even for GW-dominated emission (de Araujo et al., 2016). Simultaneous measurements of $n_{b}$ and $\varepsilon$ could then be used as an independent means to quantify the wave-like effects of lensing.

Although we have considered $n\leq 250$ here, the PL approach can, in principle, be used to study wave-optical microlensing for arbitrary $n$. While lensing by a large number of stars is unlikely for a Galactic source, the amplification can be substantially increased if the perturbers are embedded into larger mass distributions (Mishra et al., 2021; Cheung et al., 2021), the extent to which is quantified by the optical scalars $\kappa$ and $\gamma$. For extragalactic sources, these quantities may be sizeable fractions of unity (Garsden & Lewis, 2010; Lewis, 2020). As noted in Sec. 5.2, and although dependent on the source kinematics, phase drifts exceeding $\sim 1$ rad can theoretically occur within the space of a few weeks for $n=250$. For even larger $n$, the loss of coherence will be quicker, which may present a significant impediment to narrow-band searches as the signal is scrambled.