Gravitational Microlensing: Theory & Applications

• Gravitational microlensing is the transient, achromatic amplification of background light by a foreground compact object, producing symmetric light curves that encode lens mass and distance.
• This effect enables precise detection of non-luminous objects such as exoplanets, stellar remnants, and dark matter candidates through careful analysis of light curve morphology and anomalies.
• Advanced modeling incorporating binary lenses, finite source size, and environmental effects improves mass measurement and expands applications in probing planetary systems and gravitational wave sources.

Gravitational microlensing is the transient, achromatic amplification (or, in special environments, modulation) of a background source’s flux as an intervening compact object (the microlens) passes near the observer–source line-of-sight. In the simplest scenario—stellar lenses and sources in the Galactic bulge or nearby galaxies—microlensing produces a symmetric, time-variable light curve whose morphology and timescale encode the mass and distance of the lens. Microlensing has been foundational for the detection and characterization of dark, dim, or otherwise non-luminous astrophysical objects, and serves as a powerful astrophysical probe for planets, stellar remnants, compact dark matter, and, more recently, gravitational wave signals.

1. Single-Point Mass Microlensing: Formalism and Observable Effects

The classic microlensing geometry involves a point mass $M_L$ at distance $D_L$, with a source at $D_S$ ($D_L < D_S$), and observer at $D_O$. The crucial angular scale is the Einstein radius,

$$\theta_E = \sqrt{\frac{4GM_L}{c^2}\frac{D_{LS}}{D_L D_S}},$$

with $D_{LS}=D_S-D_L$. In angular units normalized by $\theta_E$, the single-lens equation is quadratic: $$\theta^2 = \beta\theta + 1, \qquad \textrm{or equivalently}\qquad \theta - \beta = \frac{1}{\theta}.$$ Here, $\beta$ is the angular separation between source and lens on the sky. The two image solutions,

$$\theta_\pm = \frac12\left[\beta \pm \sqrt{\beta^2 + 4}\right],$$

are “positive-parity” (major) and “negative-parity” (minor) images’s positions, with corresponding magnifications: $$A_\pm = \frac{1}{2}\left[ \frac{\beta^2 + 2}{\beta\sqrt{\beta^2 + 4}} \pm 1 \right].$$ The total magnification for unresolved images simplifies to

$$A(\beta) = A_+ + A_- = \frac{\beta^2 + 2}{\beta\sqrt{\beta^2 + 4}}.$$

This framework yields a characteristic symmetric light curve (“Paczynski curve” (Gaudi, 2010)) as the lens and source undergo relative transverse motion.

At large impact parameter ($\beta \gg 1$), the negative-parity image ($\theta_-$) approaches the lens, and its magnification $A_- \sim 1/\beta^4$ becomes negligible. This proximity underpins rare eclipsing events in certain microlensing regimes (Rahvar, 2016).

2. Image Parity, Negative-Parity Eclipse, and Mass Determination

In the general case, the positive-parity image forms outside the Einstein ring and dominates the observable magnification. The negative-parity image, always fainter and located inside the Einstein ring, may lie so close to the lens star that it can be obscured by an extended lens (e.g., a red giant).

The eclipse of the negative-parity image by a red-giant lens leads to a characteristic flux decrement: $\Delta F/F \simeq - \frac{1}{u^4} b$ where $u$ is the (dimensionless) impact parameter, and $b=F_S/(F_S+F_L)$ is the blending parameter. The predicted amplitude is $\Delta F/F \sim 10^{-3}$ for optimal lens–source combinations (red giant lens and source, $b \sim 0.5$, $u\sim5$), making this a feasible observational target for high-precision photometry. The eclipse duration directly yields the normalized source size $\rho_*$, while a contemporaneous analogy with annual or satellite parallax measurements provides the microlens parallax $\pi_E$. The combination enables direct measurement of the lens mass via

$$M_L = \frac{\theta_E}{\kappa\pi_E}, \quad \kappa = 4G/(c^2\,\mathrm{au}) \approx 8.144\,\mathrm{mas}\, M_\odot^{-1}.$$

A feasibility study of seven actual events demonstrates that such eclipses can realistically be targeted by dedicated campaigns using high-spatial-resolution imaging and precision photometry (Rahvar, 2016).

3. Light Curve Modifications: Planetary Perturbations and Second-order Effects

Deviations from the symmetric single-lens light curve are strong indicators of non-trivial lens structure or environmental effects. For binary or planetary lenses, the lens equation in complex notation generalizes to

$$\zeta = z - \frac{1}{1+q}\frac{1}{\bar z} - \frac{q}{1+q}\frac{1}{\bar z - \bar z_p}$$

where $q = m_\mathrm{planet} / M_\mathrm{host}$ and $z_p$ represents the suitably scaled planetary position (Gaudi, 2010). These setups generate critical curves and caustics with topologically distinct regimes (“close,” “resonant,” “wide” (Rektsini et al., 2024)), and the resulting light curve may exhibit sharp spikes, “U-shapes,” or other anomalous features if the source crosses a caustic.

The detection and modeling of such anomalies enables measurement of planetary mass ratios down to $q\sim 10^{-5}$ (particularly for cold planets near or beyond the snow line) and supports the inference of multi-planet systems and free-floating planets, with event durations $t_p \sim \sqrt{q}\, t_E$ (Tsapras, 2018, Rektsini et al., 2024).

Second-order effects, such as finite source size, limb darkening, microlens parallax, orbital motion, xallarap, and blending, further modulate the light curve and are now systematically incorporated into light curve analysis to break degeneracies and improve mass and distance measurements (Rektsini et al., 2024).

4. Environmental and Exotic Microlensing Regimes

The standard geometric-optics microlensing theory assumes a vacuum and a simple mass profile for the lens. However, environmental factors drastically modulate the lensing signature. For instance, a power-law electron plasma environment induces chromatic, frequency-dependent refraction which can nearly cancel the gravitational deflection, yielding “hill–hole” transitions—absent images at perfect alignment leading to dips rather than peaks in the light curve (Tsupko et al., 2019). These effects are most pronounced in low-frequency (radio) bands and for high plasma densities.

For dark compact objects embedded in extended halos (e.g., “dressed” primordial black holes with minihalos), the microlensing observables are governed by the relative scale of halo size and Einstein radius. In the intermediate regime ($r_h \sim R_E$), the light curve departs sharply from the point-mass Paczynski form due to the halo’s density gradient and may exhibit caustics or extended high-magnification features (Cai et al., 2022, Croon et al., 2020).

Microlensing by structureless dark matter (NFW halos, fuzzy dark matter cores, boson stars) or compact exotic objects is described in full wave optics in the frequency domain. For gravitational wave sources, the Kirchhoff diffraction integral captures frequency-dependent amplification and phase shifts, enabling direct inference of lens mass and distinguishing between geometric and wave-optics regimes (Fairbairn et al., 2022).

5. Gravitational Microlensing of Gravitational Waves

Strong lensing of gravitational wave (GW) transients may be subject to microlensing by a foreground galaxy’s stellar content or substructures. The formalism uses the complex amplification factor $F(f)$, encoding both amplitude and phase modifications, and is computed via the wave-optics diffraction integral: $$F(w, y) = \frac{w}{2i\pi} \int d^2x\, e^{iwT(x, y)},$$ with $w$ a dimensionless frequency parameter and $T(x, y)$ the normalized time-delay surface (Fairbairn et al., 2022).

In the LIGO/Virgo frequency bands, for lens masses $m\lesssim100 M_\odot$, diffractive effects generally suppress the observable microlensing signature, rendering most stellar microlenses undetectable in GW data. However, high macro-magnification, special source–lens alignments, or more massive compact objects ($\gtrsim 300 M_\odot$) can produce frequency-dependent fringes and significant mismatches in waveform templates, particularly for third-generation GW observatories (Cheung et al., 2020, Mishra et al., 2021, Meena et al., 2022).

6. Applications, Observational Strategies, and Astrophysical Insights

Microlensing has enabled the detection of over 200 exoplanets (notably cold Neptunes), free-floating planets, and several isolated stellar-mass black hole candidates (Tsapras, 2018, Rektsini et al., 2024). Its utility extends to probing Galactic structure, stellar remnant populations, and compact dark matter candidates. High-cadence, high-precision photometry (OGLE, MOA, KMTNet, Gaia, WFIRST/Roman) and wide-field monitoring underpin decade-long surveys for microlensing events (Rektsini et al., 2024).

Critical detection strategies include:

• Difference-imaging photometry in crowded fields (Tsapras, 2018).
• Multi-telescope follow-up for anomaly confirmation.
• High spatial resolution to quantify blending.
• AO/HST imaging years after events for direct host–lens separation (Rektsini et al., 2024).
• Time-series spectroscopy (with precision down to $\sim0.01$ m/s) and astrometry for unique parameter extraction and degeneracy-breaking (Rahvar, 2019).

Microlensing is unique in its reach: sensitivity to dim objects, non-luminous planets at wide separations, and the potential to probe compact objects in the local and extragalactic Universe through both electromagnetic and GW observations.

7. Future Directions and Challenging Regimes

The next decade will see a dramatic expansion in microlensing science, enabled by next-generation facilities (Nancy Grace Roman Space Telescope, LSST, ELT HIRES, Euclid). Roman will routinely deliver mass measurements for planets and their hosts, and catalog free-floating low-mass planets.

Increasing photometric and spectroscopic precision will allow measurement of previously inaccessible effects: chromatic signatures of plasma lensing (Tsupko et al., 2019), relativistic frequency shifts in high-magnification microlensing events (Rahvar, 2019), and rare higher-order image eclipses (Rahvar, 2016). In the GW regime, systematic implementation of wave-optics templates will advance lens population inference and sensitivity constraints on small-scale dark matter (Fairbairn et al., 2022).

Microlensing signatures in high-resolution VLBI images of black hole shadows—such as shift, asymmetry, and size changes induced by lenses in the line of sight—may eventually be detectable as baselines and imaging precision extend below the $1\mu$as scale (Verma et al., 2023).

Precise modeling and analysis of extended and structured lenses, inclusion of environmental effects (blending, plasma, halos), and joint photometric–spectroscopic–astrometric fitting will further improve the robustness of microlensing parameter determination, distinguish among models for planetary, stellar, and dark matter populations, and drive advances in time-domain astrophysics at all scales.