2.1 Galactic Centre $\gamma$ -ray excess

The Fermi Gamma-Ray Space Telescope has performed an all-sky survey since 2009, measuring $\gamma$ -rays with energies between $20\,\mathrm{MeV}-300\,{\mathrm{GeV}}$ (Atwood et al. Reference Atwood2009). Its primary instrument, the Large Area Telescope (LAT), has revealed an excess of $\gamma$ -rays within $10^\circ$ of the Galactic Centre, which departs from the diffuse $\gamma$ -ray background and peaks at $\sim1-3\,\mathrm{GeV}$ (Ajello et al. Reference Ajello2016).

The origin of the GCE has long been a subject of debate. Hooper & Goodenough (Reference Hooper and Goodenough2011) suggested that the excess is consistent with annihilating dark matter with cross section $\langle\sigma v\rangle\sim 10^{-26}\,\mathrm{cm}^3/\mathrm{s}$ and mass $m_\chi\sim7-10\,\mathrm{GeV}$ . Since then, many other works have suggested a dark matter origin (Hooper & Linden Reference Hooper and Linden2011; Gordon & Macas Reference Gordon and Macas2013; Abazajian et al. Reference Abazajian, Canac, Horiuchi and Kaplinghat2014; Daylan et al. Reference Daylan2016; Linden et al. Reference Linden, Rodd, Safdi and Slatyer2016). This is supported by the fact that the GCE is spectrally uniform across the bulge with a spectrum compatible with predicted self-annihilation spectra (Calore, Cholis, & Weniger Reference Calore, Cholis and Weniger2015). Spatially, the morphology of the excess can be fit to a modified Navarro–Frenk–White profile (see, Navarro et al. Reference Navarro, Frenk and White1997), given by,

(1) \begin{equation} \rho(r) = \frac{\rho_0}{\left(\frac{r}{r_s}\right)^\gamma\left[1+\frac{r}{r_s}\right]^{3-\gamma}}.\end{equation}

Here, r is the radius from the Galactic Centre, $r_s$ is the scale radius, and $\rho_0$ is a characteristic density. More specifically, for dark matter pair annihilation, the resulting emission would be described by the line-of-sight integral of the squared profile from the sources to the Fermi satellite. The $\gamma$ parameter describes the slope of the inner profile. Past studies have shown that $\gamma$ ranges from $1.1$ to $1.4$ (Leane & Slatyer Reference Leane and Slatyer2020a).

Shortly after the GCE’s discovery, a faint population of astrophysical sources was proposed as an alternative explanation. For instance, Abazajian (Reference Abazajian2011) found that the GCE is spatially and spectrally consistent with known globular clusters in the Galactic Centre. These clusters are thought to host large populations of unresolved MSPs. Although earlier papers found a discrepancy with MSP models in the spectrum at sub-GeV energies, more recent work suggests that a ‘typical’ population of (potentially as many as $10^4$ − $10^5$ ) MSPs can well explain the GCE (Ploeg et al. Reference Ploeg, Gordon, Crocker and Macias2020). Regarding the spatial morphology of the GCE, it remains debated whether a (stellar) bulge may provide a better fit than an NFW profile (Bartels et al. Reference Bartels, Storm, Weniger and Calore2018; Clark et al. Reference Clark, Scott, Trotta and Lewis2018; McDermott et al. Reference McDermott, Zhong and Cholis2023; Song et al. Reference Song2024), and the preference has been shown to sensitively depend on the choice of employed templates.

In 2015, two independent papers suggested that the distribution is spatially clustered, indicating the presence of astrophysical sources, using Non-Poissonian template fitting (NPTF; Lee et al. Reference Lee, Lisanti, Safdi, Slatyer and Xue2016; Mishra-Sharma, Rodd, & Safdi Reference Mishra-Sharma, Rodd and Safdi2017) and a wavelet-based analysis (Bartels et al. Reference Bartels, Krishnamurthy and Weniger2016). In later works using the NPTF, however, these results were called into question. Leane & Slatyer (Reference Leane and Slatyer2019) demonstrated that a smooth GCE component artificially injected into the Fermi data was mistakenly ascribed to the point-like GCE component, indicating mismodelling (Buschmann et al. Reference Buschmann2020; Chang et al. Reference Chang2020). In Leane & Slatyer (Reference Leane and Slatyer2020b); Leane & Slatyer (Reference Leane and Slatyer2020a), it was further shown that giving the GCE templates the freedom to float independently in the northern and southern hemispheres entirely removes the preference for point-like emission. As for the wavelet technique, Zhong et al. (Reference Zhong, McDermott, Cholis and Fox2020) found that when using an updated mask of resolved point sources from the 4FGL catalogue, the preference for bright GCE sources disappears (see also Zhong & Cholis Reference Zhong and Cholis2024 for a recent assessment of the robustness against masking).

More recently, convolutional neural networks have been used to disentangle the smooth and point-like components of the GCE (List et al. Reference List, Rodd, Lewis and Bhat2020; List et al. Reference List, Rodd and Lewis2021; Mishra-Sharma & Cranmer Reference Mishra-Sharma and Cranmer2022). Although mismodelling remains a concern with these methods, the ‘patch-based’ way in which they analyse photon-count maps (contrasting with the product-over-pixels likelihood employed by traditional statistical methods) results in a different response to mismodelling, offering an independent cross-check on previous analyses. The GCE source population preferred by these analyses would need to be significantly dimmer and consist of many more sources than the few hundred estimated by earlier works such as Lee et al. (Reference Lee, Lisanti, Safdi, Slatyer and Xue2016). When spectral information is additionally exploited alongside the spatial photon-count map, the GCE appears to be essentially consistent with Poisson emission (List et al. Reference List, Park, Rodd, Schoen and Wolf2025).

In summary, 15 yr after its detection, the question as to whether the GCE originates from an extended source like dark matter, or point sources such as MSPs, is still unresolved. Most analyses of the GCE to date have involved spatial template fitting, which is particularly sensitive to the background distribution template used (Calore et al. Reference Calore, Cholis and Weniger2015). Investigating the excess temporally may offer an unbiased way to determine the nature of the source.

2.2 Gravitational microlensing

Paczynski (Reference Paczynski1986) suggested that gravitational microlensing could be used to probe the nature of dark matter with the Galactic Halo. If dark matter is in the form of stellar mass compact objects, such as black holes, their gravitational influence will magnify more distant sources, resulting in a brightening and fading as the lens passes near the line of sight. However, even if all dark matter is in the form of stellar mass compact objects, the optical depth for significant magnification events is small ( $\tau \lesssim 10^{-4}$ ), and it was proposed that a significant number of stellar sources, namely the Galactic Bulge and Magellanic Clouds, be targeted. Unfortunately, after substantial effort, microlensing surveys failed to identify the expected signature of compact dark matter comprising the majority of dark matter, leading to the current focus on a particle nature. However, the optical depth through the Galactic Halo proved to be non-zero due to the presence of other astrophysical masses, comprising mainly of main-sequence stars, with an optical depth towards the Galactic Centre of order $\tau \sim 10^{-6}$ .

To understand the impact of gravitational microlensing, it is important to consider the relevant length scale, namely the Einstein radius in the source plane. This is given by

(2) \begin{equation}R_E = \sqrt{ \frac{4 G M}{ c^2} \frac{D_{os} D_{ls} }{D_{ol}} },\end{equation}

where M is the mass of the lensing object, and $D_{ij}$ are the relative distances between the observer (o), lens (l), and source (s). Given the typical lensing configuration about the Galactic Bulge, $D_{os} = 8\,\mathrm{kpc}$ and $D_{ol} = 4\,\mathrm{kpc}$ , this scale is

(3) \begin{equation}R_E \approx 8.1 \sqrt{\frac{M}{M_\odot}} \mathrm{au}.\end{equation}

For typical halo velocities of $\sim 200 \, \mathrm{km/s}$ (projected into the plane of the source), the crossing time of the Einstein radius is of order $\sim 35\,\mathrm{days}$ , and hence this is the average time scale of significant magnification events.

For a typical lens of stellar mass along the line of sight to the Galactic Bulge of $(0.1-1)\,{M}_{\odot}$ , this scale is eight orders of magnitude larger than the $\sim 10\,\mathrm{km}$ radius of our sources – the MSPs. Hence, in terms of microlensing, these MSPs are effectively point-like and are sensitive to extreme gravitational lensing magnification. Emission from dark matter self-annihilation, which is expected to be smooth on au scales, will be immune to the influence of gravitational microlensing.

Figure 1. First and third columns (Left) – binary lens configurations. The lenses are shown as black dots, with their relative sizes representing their mass. The source trajectories are the coloured straight lines. The critical curve is in red, and the caustic is the spiky blue curve. Second and fourth columns (Right) – lightcurve for the respective lens configuration. The coloured lines represent the corresponding source trajectory’s magnitude over time. The peaks when the source crosses a caustic are limited by the time resolution of our simulation; however, these are formally infinite.

As a lens passes along the line of sight to a source, two micro-images of the source are observed. A third image is also formed, but in such a scenario, this image is often within the lens’ radius and strongly demagnified, implying that its contribution to the amplification is negligible. As is the nature of microlensing, these micro-images are too spatially clustered to be resolved. Instead, we observe a periodic brightening of the source, which is the contribution of the source and each micro-image. For an isolated, point-like microlensing mass, the magnification $\mu$ is given by

(4) \begin{equation} \mu=\frac{y^2+2}{y\sqrt{y^2+4}},\end{equation}

where y is the impact parameter (see. Schneider et al. Reference Schneider, Ehlers and Falco1992). That is the distance between the source and the lens, projected on the source plane and scaled to the Einstein radius in Equation (2).

The maximum magnification occurs with perfect alignment $(y=0)$ , which results in the formation of an Einstein ring. For a true point source, this magnification is formally infinite; however, our magnification is limited by the non-zero radius of the source. Taking into account the conservation of surface brightness, the magnification can be approximated as the ratio between the area of the lensed image and the area of the source. For a point source with radius $r_s \ll R_E$ , the Einstein ring of radius $R_E$ has thickness $r_s$ , and the ratio of areas can be approximated as

(5) \begin{equation}\mu_{\max}\sim \frac{2 \pi R_E r_s}{\pi r_s^2} = 2 \frac{R_E}{r_s}.\end{equation}

For the scenario considered in the previous paragraph, the magnification will be of order

(6) \begin{equation}\mu_{\max} \sim 2.4 \times 10^8 \sqrt{\frac{M}{M_\odot}}.\end{equation}

Of course, this is the maximum magnification, and typical magnifications will be significantly lower than this upper limit. However, it is clear from stellar microlensing surveys that a proportion of microlenses within the Galactic Halo exist in binary systems, and there has even been a recent claim of a triple microlensing event towards the Galactic Bulge (Han et al. Reference Han2023). In a binary system, the region for magnification unfolds and becomes spatially extensive. Furthermore, these regions are bounded by caustics in the source plane, which are locations for formally infinite magnification for a truly point-like source, but will be limited to a very high value due to the finite nature of the source.

2.2.1 Binary lenses

Microlensing by a binary system will have a much stronger amplification than a single lens (SL). We consider the situation in which the stellar lens is not isolated, but in a system with another star, or perhaps a planet. To describe this system, we need to understand the mapping from the lens plane to the source plane, that is, the binary lens (BL) equation. In complex notation taken from Witt & Mao (Reference Witt and Mao1995),

(7) \begin{equation} \zeta=z +\frac{m_1}{\bar{w}_1-\bar{z}} +\frac{m_2}{\bar{w}_2-\bar{z}}. \end{equation}

Here, the known lens positions are $w_1$ and $w_2$ , and their masses $m_1$ and $m_2$ . The source position is $\zeta$ and its image is at $z$ . All length scales are normalised to the Einstein radius corresponding to an arbitrary mass. We will assume later that the source positions are known, and we wish to determine the total magnification of the image.

Inverting (7) to determine the image positions and the magnification is non-trivial, returning a complex 5th order polynomial (Witt Reference Witt1990). This polynomial will return five solutions, of which two may be imaginary. Each of the real roots corresponds to one of the five, or three, observable micro-images. As with the SL case, these images are too spatially clustered to be resolved; however, the contribution from each image is observed as the total amplification. This is the sum of the absolute amplification of each of the real solutions,

(8) \begin{equation} \mu_{\mathrm{tot}}=\sum_{i=1}^n\mu_i=\sum_{i=1}^n|\det{J}|^{-1}|_{z=z_i}\quad\mu_i\gt0,\end{equation}

and

(9) \begin{equation} \det{J}=1-\frac{d\zeta}{d\bar{z}}\frac{\overline{d\zeta}}{d\bar{z}}.\end{equation}

Here, $z_i$ for $i = 1, \ldots, n$ denotes the position of the $i_{\text{th}}$ micro-image (i.e. the $i_{\text{th}}$ real root), where n is 3 or 5, depending on the number of real solutions. Clearly, in the case that $\det{J}=0$ , we obtain formally infinite amplification. This solution forms closed curves called critical curves. We can then use (7) to trace the image of this curve back to the source plane, obtaining the caustics.

Outside of the region bounded by the caustic, the number of real solutions to our lens equation is three. As the source and lenses move in their respective planes, the source may cross the caustic curve. When the source enters the caustic region, two things happen. Firstly, all solutions to our polynomial are real, and we observe five micro-images instead of three. Depending on the source/lens configuration, the magnification within the caustic region may be significantly larger than outside of this region. Secondly, we observe a spike in the amplification as the source crosses the caustic fold and $\det{J}=0$ . This formally infinite peak is limited by the non-zero radius of the source, but given the small radius of MSPs, we expect to see a very strong peak at this point.

To describe these configurations, the lenses are parametrised by two quantities, the mass ratio between the lenses (q), and the separation of the lenses (s) in units of Einstein radii. Typical examples of BL configurations are given in Figure 1 where the lenses are the black dots, the critical curves are displayed in red, and the caustics blue. The source trajectories in the source plane are shown by the coloured lines, and the respective lightcurves are shown on their right. The three types of topologies are evident: the close BL – which returns a central caustic and two off-axis caustics (top two panels in the left column); the intermediate BL – with a single central caustic (bottom panel in the left column, and top two in the right column), and the wide BL – with a central caustic and a secondary caustic (final panel). We will only consider a caustic crossing (CC) event in which a source enters and exits the caustic region once.

As in (5), the maximum magnification at the CCs is determined by the radius of the source. Given the limiting amplitude of catalogued microlensing events, we can estimate the limiting magnification of our MSPs. Taking the maximum magnification of observed events as $\mu_\mathrm{star}$ , from Gaudi & Petters (Reference Gaudi and Petters2002) the limiting magnification of our pulsars $\mu_\mathrm{MSP}$ is,

(10) \begin{equation}\mu_\mathrm{MSP}\sim\sqrt{\frac{r_\mathrm{star}}{r_\mathrm{MSP}}}\mu_\mathrm{star}.\end{equation}

Considering the microlensing events from the 2006 $-$ 2008 seasons in the OGLE catalogue from Jaroszyński et al. (Reference Jaroszyński2010), we can measure the change in amplitude for the CC events. This magnitude variation is converted to a luminosity ratio, and we see an average factor of 21.6 increase in luminosity during these CC events. Of these events, some have a measured source radius (in Einstein radii). Assuming these have typical halo velocities of $200\,\mathrm{km/s}$ , we can approximate that the average diameter of these sources is $\sim2\times10^9\,\mathrm{m}$ . Using (10) we obtain a limiting magnification of $\sim8\,700$ . These events have an average impact parameter of $0.53\,R_E$ ; however, the impact parameter can be as large as $2\,R_E$ .

Earlier microlensing surveys identified BL and CC events by visual inspection. Alcock et al. (Reference Alcock2000) found $6.0\%$ of microlensing events from the MACHO survey to be caused by BLs, $66\%$ of these being CC. In the same year, findings from the OGLE catalogue found $9.3\%$ of events were attributed to BLs, $70\%$ being CC events. In subsequent years, the binary fraction has been reported between $1.5\%$ to $5.1\%$ (Jaroszynski Reference Jaroszynski2002; Jaroszynski et al. Reference Jaroszynski2004; Jaroszynski et al. Reference Jaroszynski2006; Skowron et al. Reference Skowron2007; Jaroszyński et al. Reference Jaroszyński2010). It is typical to assume $6\%$ of microlensing events are attributed to BLs (Sumi et al. Reference Sumi2006; Sumi et al. Reference Sumi2013). More recently, Abrams et al. (Reference Abrams2025) used population fit models to identify BL events in lightcurves that were previously well fit by SL models, finding instead that $23.3\%$ of events are attributed to BLs. However, we will assume that $6\%$ of our lenses are BLs, and that $\sim70\%$ of these events are our CC events (so $3.5\%$ of all events are CC).

While detailed simulations are required to quantify their impact fully, we can predict that the inclusion of a third lens, due to the more intricate caustic structures, may increase the number of CCs in one microlensing event. This effect is especially pronounced in non-hierarchical configurations, where all three lenses have comparable separations, leading to more complex and potentially overlapping caustics that could enhance detectability in rare cases (Kuang et al. Reference Kuang, Mao, Wang, Zang and Long2021); these events are rare and we will not consider them further in this work.

If MSPs are the sources of the $\gamma$ -ray excess at the Galactic Centre, their small size means that they would be susceptible to high magnification due to the presence of galactic halo microlensing masses. However, given the low optical depth, the frequency of any microlensing signal will depend critically on the surface density of MSPs. We turn to this in the next section.

4.1 Generating a signal

Our method of generating a lightcurve is summarised in Figure 2 and proceeds in the following stages.

1. 1. Source distribution. We simulate $N_s$ sources and randomise their distribution based on the 2D projection of the spherically symmetric NFW profile described in (1). A realistic MSP population will have a slightly different spatial distribution from the NFW profile (e.g. Coleman et al. Reference Coleman, Paterson, Gordon, Macias and Ploeg2020), but for this proof of concept, this profile is sufficient. The three cases we used for $N_s$ are described in Section 3.

2. 2. Lens population. We simulate a uniform distribution of $6.3\times10^4$ stellar mass lenses, each assigned a 2D velocity vector with random direction and magnitude $200 \, \mathrm{km/s}$ . This corresponds to $0.28\,R_E\mathrm{/day}$ . The number of lenses is set by the optical depth $\tau=3\times10^{-6}$ within a lensing region with radius $10^\circ$ , corresponding to $4\times10^7\,R_E$ .

3. 3. Microlensing detection. For each source, we compute the distance to all neighbouring lenses. Based on the known magnification profile, we set a selection criterion: any lens passing within an impact parameter $y \leq 4R_E$ is considered to generate a detectable microlensing event. This threshold corresponds to a maximum peak magnification of approximately $6\%$ , which we take as a conservative detection limit.

4. 4. Lightcurve generation. Each qualifying event produces a magnification lightcurve that modulates the flux of the corresponding source. Since we assume an identical brightness for all our sources, the absolute luminosity scale is arbitrary. We sum the contribution from all $N_s$ sources to produce the total observed signal over the sky.

5. 5. Event classification. In the low optical depth regime, we assume each source is lensed at most once. Of all microlensing events, we randomly designate $6\%$ as originating from BLs. Following the breakdown in Section 2.2.1, $70\%$ as classified as CC events. For simplicity, all CC events have identical lens configurations with mass ratio $q=1$ and separation $s=1$ . As a result, all CC share the same lightcurve shape shown in Figure 3. This assumption also implies uniformity in MSP radius and corresponding peak magnification. The remaining $94\%$ are classified as SL events.

Figure 3. Examples of time-scaled lightcurves for SL (top) and BL (bottom) events. Depending on the impact parameter of the lensing event, we should see different maximum magnifications for the SL case.

A summary of simulation parameters can be seen in Table 1. An example output for $N_s=10^5$ is shown in Figure 4. In the next section, we examine how well such a signal can be resolved by LAT, and what improvements in detector sensitivity would be needed.

4.2 Detecting the signatures

To define our region of interest (ROI), we consider the distribution of sources in the Galactic Centre, based on the three cases introduced in Section 3, and the angular resolution of our detector. The LAT has an angular resolution $\sim0.15^{\circ}$ at $1$ − $3\,\mathrm{GeV}$ , assuming only the highest-graded count quartile in terms of the reconstructed direction is considered.Footnote ^a Hence, the smallest region in which we can search for a microlensing event is given by a circular region with diameter $0.15^\circ$ . Depending on our selected value for $N_s$ , this resolution element may contain more than 1 source. We note that other event selections are also possible; for instance, lowering the requirements on the reconstructed direction would result in a larger photon number, at the expense of a worse angular resolution.

Table 1. Key parameters used in the microlensing simulations.

Figure 4. Simulated lightcurve of the entire Galactic Centre containing $10^5$ sources. The lightcurve is normalised to the total luminosity of $10^5$ identical sources. The inset shows the two consecutive CC events at $\sim9\,500$ days.

For $N_s=10^5$ , we randomly sample 15 populations of $N_s$ MSPs. We then count the number of sources contained within one resolution element at different distances away from the Galactic Centre, at a random angle. The average number of sources across all 15 populations within a resolution element, as a function of distance from the galactic centre, is shown in Figure 5, with the radii containing 25% and 50% of the sources indicated. For the $N_s=500$ and $N_s=5\,000$ cases, we find that a single resolution element is extremely likely to contain only one source, so we do not apply the same treatment.

Anticipating that the LAT does not provide the sensitivity to observe these time variable signatures, we define a parameter a which scales the number of photons detected by the LAT to characterise some hypothetical detector for which,

(18) \begin{equation}\lambda_s= a\lambda_{\mathrm{LAT}}.\end{equation}

In our analysis, we consider only improving detector sensitivity, as next-generation $\gamma$ -ray instruments are expected to improve more in this respect than in angular resolution. Improved angular resolution would reduce the number of sources within each resolution element. For a large, faint MSP population, this would lead to more isolated microlensing events per element, resembling what we will show for the $N_s=500$ and $N_s=5\,000$ cases. However, the individual events would still be too faint to be detected without substantially higher sensitivity. For a brighter, sparser MSP population, higher resolution would instead facilitate direct source detection.

Figure 5. Average number of sources per resolution element as a function of distance from the Galactic Centre for a population of $10^5$ MSPs. The shaded region is the standard error. The dashed (dotted) line indicates the radius beyond which 75% (50%) of MSPs are located.

We generate test lightcurves to determine the minimum value of a that produces an observable signal. Our ROI consists of a single resolution element, populated with a realistic number of randomly distributed sources. The precise spatial distribution of sources within the ROI does not impact the statistical nature of the problem, and we assume that on these small scales, the distribution is approximately uniform anyway. We artificially modify the lightcurve of one source within the ROI to follow a microlensing profile, based on those shown in Figure 3. remaining sources remain unlensed, with constant magnification $\mu=1$ .

For each source, we scale the expected Poisson rate per day, $\lambda_s$ , by its respective magnification curve, $\mu(t)$ . We then generate the Poisson-realised lightcurve by drawing Poisson random variates for each source over the time window of interest, and combining the contributions from all sources to form the final lightcurve. The combined lightcurve is binned into 1-day-wide intervals.

To test whether an event is detectable, we compute the Poisson significance of the observed counts using the cumulative distribution function (CDF). For each 3-day sliding window, we calculate the total observed counts k and the expected background $\lambda_s$ and evaluate the one-sided significance $p=1-\mathrm{CDF}(k-1;\lambda)$ . This returns the probability of obtaining k or more counts given the expected background. We classify an event as a significant detection if the one-sided p-value satisfies $p\leq0.01$ .

In practice, detecting some peaks in real data would require computing the Poisson rate of each resolution element in the GC, and searching for a significant p-value over the entire observation period of the LAT. Of course, searching such a large parameter space significantly increases the chance of detecting high Poisson fluctuations not attributed to any microlensing events. To mitigate this so-called ‘look-elsewhere effect’, one could apply a Bonferroni correction by adjusting the significance threshold based on the number of independent trials, reducing the likelihood of false positives due to multiple comparisons (see Bayer & Seljak Reference Bayer and Seljak2020).

While our detection method uses the Poisson CDF to evaluate the significance of observed count excesses, this implicitly assumes a fixed background rate and a fixed number of sources in each resolution element. The former assumption holds because we treat the number of contributing sources and their Poisson rates as known, except for one potential anomaly. Regarding the latter assumption, a more careful treatment would adopt a compound Poisson likelihood, which accommodates for the scatter of sources located in each resolution element; however, the effect of this correction is expected to be modest. In practical applications, where the number of individual sources and their individual rates are not known, marginalising over the source-count distribution might be necessary.

4.2.1 Caustic crossing event detection

We start by considering the case in which one of our sources is lensed by a binary, and the source crosses the caustic curve. Its lightcurve is the bottom curve on Figure 3. We should observe an amplification of $\sim8\,700$ at the peaks when the source crosses the caustic. These peaks only last on the order of a few minutes and will not actually return a statistically significant increase in the number of photons arriving for small $\lambda_s$ . So, we require a higher photon rate at these peaks to observe a significant signal.

We use the method described above to generate the Poisson-realised lightcurve, where one source undergoes a CC event. This is done for a range of values for a. For each realisation, we compute the Poisson significance in each time bin and search for two peaks separated by approximately 70 days. In practice, most lens systems along the line of sight can be assumed to have similar velocities; however, the size of the caustics depends on the specific parameters of each binary system, and thus the resulting crossing times can range from tens to hundreds of days. A correct detection is defined as both peaks falling within 5 days of the true peaks in the microlensing lightcurve, with $p\leq0.01$ . For a given a, we classify this correct detection as 1; otherwise, the detection value is 0. We fit a logistic curve to this data, taking the inflection point of the curve as the minimum value for a, which is required to detect a CC event.

Of course, not all microlensing events will resemble the specific light curve we have simulated. Detecting such signals in real Fermi-LAT data would require placing additional constraints on the lensing configuration, including the expected peak separation and amplitude. However, such detections are not feasible with current instruments. Moreover, we do not consider the specific shape of the peaks in this analysis, as their short timescale requires a high photon count to resolve their structure. For this proof of concept, we therefore adopt an optimistic scenario to demonstrate the potential observability of the effect, and leave a more detailed treatment of the relevant parameters to future work.

Case 1: $N_s = 10^5$ , $\lambda_\mathrm{LAT}=0.01\,\mathrm{photons}/\mathrm{year}/\mathrm{source}$

Considering first the case from Gautam et al. (Reference Gautam2022) where $N_s$ is maximised, we show the logistic curves for detecting a CC event in our ROI containing 50 sources (Figure 6). The required increase in detector sensitivity is a factor of $\log_{10}(a)=4.26$ to resolve the peaks on the CCs. A planned future detector, such as the Advanced Particle-physics Telescope (Buckley et al. Reference Buckley2022), is expected to provide a magnitude increase in the effective area, and hence the detection sensitivity, with an angular resolution comparable to that of the LAT. But even then, the detection is still more than 3 orders of magnitude out of reach for the APT.

Figure 6. Correct peak detections for a CC event as a function of $\log_{10}(a)$ – the log increase in efficiency from the LAT, with the number of sources contributing to the GCE indicated on each plot. A logistic curve is fit to each of these, with the inflection defining the detector threshold; the marked inflection points from top to bottom have values $4.26$ , $1.41$ , $2.31$ .

Case 2: $N_s = 500$ , $\lambda_\mathrm{LAT}=2\,\mathrm{photons}/\mathrm{year}/\mathrm{source}$

We now consider the opposite extreme, minimising the number of sources as in Lee et al. (Reference Lee, Lisanti, Safdi, Slatyer and Xue2016). Similar to the $N_s=10^5$ case, we produce artificial lightcurves and generate photon arrival times within a certain ROI. Due to the low density of sources, we can assume a ROI to contain a single source and hence take this as the only source being microlensed. The logistic curve for detecting this event is shown in the middle panel of Figure 6. This gives a much more optimistic value of $\log_{10}(a)=1.41$ at the inflection point, which is less than an order of magnitude larger than the sensitivity of the APT.

This method, however, does not account for the increased magnification when the source is within the caustic region. In this case, we observe an order-of-magnitude increase in the number of photons, lasting for $\sim 70$ days (see Figure 3). While this small change would not be significant for the $N_s=10^5$ case, in this case, we observe 4 photons within this $\sim2$ month period, in addition to the background of $0.33$ . Taking now $\log_{10}(a)=1$ , we would actually see 40 photons during this period, compared to the background rate of 4 photons.

For a population of 500 sources, a future detector such as the APT might be able to detect the change in photon counts if a source were within the closed caustic region of a BL, although disentangling the microlensed GCE signal from the diffuse background might still be challenging in practice. However, the probability of observing this with the given source population is incredibly rare, the number of events being proportional to the number of sources. We predict one CC event will take place every $\sim1\,500$ yr.

Case 3: $N_s = 5\,000$ , $\lambda_\mathrm{LAT}=0.2\,\mathrm{photons}/\mathrm{year}/\mathrm{source}$

Figure 7. $\log_{10}(a)$ at the inflection point for the event identification curve of the SL lightcurve, as a function of maximum amplification of the SL peak. We also show a fit to these points, where the events with $\log_{10}(a)=0$ in the central panel are omitted from the fits. The colour bar indicates the predicted number of events at a given magnification over a survey of 100 years. For the left figure, the number of events is constrained to the outer 75% of sources, where a single resolution element contains no more than 50 sources.

With the statistics of the previous case being so low, let us consider an intermediate case. The counts per source are scaled down by 10 from the previous case, and we see a minimum $\log_{10}(a)=2.31$ required for the peak detection. To observe an increase from 4 to 40 photons when the source is within the caustic region, we need a $\log_{10}(a)=2$ increase in LAT’s sensitivity. With this source population, one event will be expected every $\sim150$ yr.