Gravitational Microlensing: Theory and Applications

• Gravitational microlensing is a transient amplification phenomenon where compact masses deflect and magnify light from distant sources, producing characteristic achromatic light curves.
• Advanced high-cadence surveys and space-based parallax observations enable precise measurement of key parameters such as the Einstein radius, event timescale, and lens mass to break inherent degeneracies.
• This technique plays a critical role in probing dark matter, mapping exoplanet demographics, studying stellar atmospheres, and identifying compact remnants across the Galaxy.

Gravitational microlensing is the transient amplification or modulation of radiation from a background source caused by the gravitational deflection of light (or other signals) by one or more compact masses as they pass near-alignment with a distant source and observer. While general gravitational lensing can produce multiple, resolvable images or continuous distortions, microlensing arises in regimes where the image separation is much smaller than the instrumental PSF, resulting in unresolved but temporally variable magnification. The phenomenon was originally proposed as a means to detect dark compact objects in the Galactic halo, but has evolved into a multi-faceted probe of exoplanets, Galactic structure, black holes, and new physics. The canonical microlensing event is characterized by the formation of multiple unresolved images, a distinctive achromatic light-curve, and, in more complex lens configurations, additional perturbation signatures due to planetary or binary companions. Microlensing is observationally challenging due to its rarity, short timescales, and intricate degeneracies in parameter inference, but advances in high-cadence monitoring and space-based follow-up have led to transformative applications across astrophysics.

1. Theoretical Framework and Core Equations

The mathematical basis of gravitational microlensing is the lens equation derived in the thin-lens approximation, relating the source angular position $\boldsymbol{\beta}$, the image angular position(s) $\boldsymbol{\theta}$, and the deflection angle: $$\boldsymbol{\beta} = \boldsymbol{\theta} - \frac{D_{LS}}{D_S}\hat{\boldsymbol{\alpha}}(\boldsymbol{\theta}),$$ where $\hat{\boldsymbol{\alpha}}$ is the physical lens deflection angle, $D_{LS}$, $D_S$ are the angular-diameter distances from lens to source and observer to source, respectively. For a single point mass $M$,

$$\hat{\boldsymbol{\alpha}}(\boldsymbol{\theta}) = \frac{4GM}{c^2 |\boldsymbol{\theta}|} \hat{\boldsymbol{\theta}}$$

and the scalar lens equation becomes: $$\beta = \theta - \frac{\theta_E^2}{\theta}, \quad \theta_E = \left[\frac{4GM}{c^2}\frac{D_{LS}}{D_L D_S}\right]^{1/2}$$ which defines the angular Einstein radius, the characteristic scale at which lensing is significant (Rahvar, 2015, Gaudi, 2010, Mao, 2012).

The magnification $A$ as a function of (normalized) impact parameter $u = \beta/\theta_E$ is

$A(u) = \frac{u^2 + 2}{u \sqrt{u^2 + 4}}$

producing the familiar symmetric "Paczynski curve" for the light curve of a point-source point-lens (PSPL) event (Gaudi, 2010, Rahvar, 2015, Rektsini et al., 9 Jul 2024). For $u \ll 1$, $A \sim 1/u$; as $u\to\infty$, $A \to 1 + 2/u^4$.

The Einstein radius in the lens plane is $R_E = D_L \theta_E$, typically $\sim 1.5$ AU for solar or sub-solar-mass lenses in the Galactic bulge (Tsapras, 2018). The event timescale is $t_E = R_E / v_\perp$ where $v_\perp$ is the lens–source relative transverse velocity, commonly of order weeks to months for stellar lenses (Mao, 2012).

For binary or higher-order lenses, the lens equation generalizes to

$$w = z - \sum_{i=1}^N \frac{\epsilon_i}{\bar z - \bar z_i}$$

in complex notation. The magnification for each image $z_j$ is $A_j = 1/|\det J(z_j)|$ with Jacobian $\det J = 1 - |\partial w / \partial\bar z|^2$ (Rektsini et al., 9 Jul 2024, George et al., 2021).

2. Observational Strategies and Degeneracy Breaking

Microlensing events are inherently rare, with optical depths toward the Galactic bulge of $\tau \sim 10^{-6}$–$10^{-5}$ and event rates per monitored star $\Gamma \sim 2\times 10^{-6}~\text{yr}^{-1}$ (Mao, 2012, Rahvar, 2015). Large-area, high-cadence photometric surveys (OGLE, MOA, EROS, KMTNet) monitor millions to billions of stars to capture these events (Tsapras, 2018).

Parameter degeneracies are a central challenge: the observed Einstein crossing time $t_E$ is a degenerate function of lens mass $M$, lens geometry (distance ratio), and $v_\perp$. To resolve these, secondary observables are exploited:

• Annual Parallax: Earth's orbital motion produces characteristic light-curve distortions for long-duration events, yielding the microlens parallax parameter $\pi_E \equiv \mathrm{AU}/\tilde r_E$, and, with $\theta_E$, enables unique mass and distance measurements (Rahvar, 2015, Han et al., 2017).
• Space-based Parallax: Simultaneous ground and satellite photometry (e.g., Spitzer, K2, Roman) yields strong constraints on $\pi_E$ (Rektsini et al., 9 Jul 2024).
• Finite-source Effects: Non-negligible source angular diameters round off the light-curve peak or caustic crossing, enabling measurement of $\rho_* = \theta_*/\theta_E$ and thus a determination of $\theta_E$ and absolute scaling (Rahvar, 2015).
• Astrometric Microlensing: The light centroid traces an astrometric ellipse during the event, with displacement $\delta\theta = \theta_E u/(u^2+2)$. Space astrometry (Gaia, WFIRST) enables direct measurement of $\theta_E$ and the direction of motion (Sajadian et al., 2014).
• Polarimetry: Microlensing-induced symmetry breaking in the source yields transient linear polarization signatures, sensitive to the source brightness profile and trajectory (Sajadian et al., 2014).

Three-point information—$t_E$, $\pi_E$, and $\theta_E$—suffices for full recovery of the lens mass, distance, and transverse kinematics (Han et al., 2017, Rahvar, 2019).

3. Microlensing for Astrophysical and Cosmological Applications

Dark Matter and Compact Object Demographics

Microlensing surveys toward the Magellanic Clouds placed powerful constraints on the population of Massive Compact Halo Objects (MACHOs). Aggregate optical depths measured were insufficient to support MACHOs as the dominant component of the Galactic dark matter halo. Derived upper limits are $\lesssim 20\%$ of the halo mass in MACHOs of $10^{-6}$–$10\,M_\odot$ (Mao, 2012, Rahvar, 2015).

Within the inner Galaxy ($R<R_0$), microlensing-inferred stellar mass, when combined with gas-disk estimates, is consistent with the dynamical mass inferred from rotation curves, implying little room for additional non-baryonic matter in this regime (Sikora et al., 2011).

Microlensing yields constraints on new physics, including limits on dark compact populations (primordial black holes with/without dark matter halos, Ellis wormholes, global-monopole–type modifications), variants of gravity (e.g., emergent gravity; note the $\sim \mu$as-scale astrometric signatures predicted in such scenarios), and the existence of exotic objects (Abe, 2010, Liu et al., 2016, Cai et al., 2022).

Exoplanet Demographics and Characterization

Microlensing is uniquely sensitive to low-mass exoplanets at separations near the snow line (1–10 AU), including those beyond the detection range of radial velocity and transit methods (Tsapras, 2018, Rektsini et al., 9 Jul 2024, Gaudi, 2010). Planet discovery is based on detection of short-duration anomalies (caustic crossings, cusp approaches) in otherwise symmetric light curves. The planet detection efficiency peaks at projected separations $s \sim 1$ (Einstein radii) and scales approximately as $\sqrt{q}$ with mass ratio $q$. Recent statistical analyses yield a planet mass-ratio distribution with a break at $q_{br}\sim1.7\times 10^{-4}$; cold Neptunes are $\sim 10\times$ more common than cold Jupiters (Rektsini et al., 9 Jul 2024).

Constant-monitoring ground-based surveys (MOA-II, OGLE-IV, KMTNet) coupled with space-based follow-up have led to detection of over 200 planets—spanning cold super-Earths, Neptunes, Jupiter/Saturn analogs, circumbinaries, free-floating candidates, and stellar remnants (Rektsini et al., 9 Jul 2024, Gaudi, 2010).

Stellar Astrophysics

High-magnification microlensing events enable resolution of stellar atmospheres, allowing measurement of limb-darkening (by modeling finite-source effects) and detection of stellar spots (using polarimetric and astrometric microlensing) across kiloparsec distances (Sajadian et al., 2014, Rahvar, 2015). This provides empirical tests of atmospheric models otherwise unattainable for remote stars.

Black Hole and Remnant Census

Microlensing events with $t_E \gtrsim 100$ days are highly sensitive to stellar-mass black holes due to the $M^2$ scaling of event rates at long timescales (Abrams et al., 2020). The LSST, with monitoring of $\sim2\times 10^{10}$ bulge sources over 10 years, is forecast to discover $\sim 6\times 10^5$ black hole events, providing a critical complement to gravitational-wave searches.

4. Extensions: Multi-lens Systems and Gravitational Waves

Multi-lens microlensing (binary, triple, circumbinary, planetary) is modeled via complex polynomial lens equations. The magnification topology exhibits critical-curve caustics—regions of formally infinite magnification—which can have intricate morphologies (resonant caustics, fold/cusp structures) and yield abrupt or subtle anomalies in observed light curves. Numerical techniques (inverse ray-shooting, Green's theorem/ray-bundle integration, resultant polynomials for triple lenses) are deployed to generate magnification maps and fit observed data (George et al., 2021, Rektsini et al., 9 Jul 2024).

Microlensing applies also to non-EM signals. For gravitational waves (GWs), wave-optics effects (diffraction, interference) dominate when the GW wavelength approaches the Einstein radius of stellar-mass lenses (Yeung et al., 2021). In strongly-lensed systems, microlensing can imprint frequency-dependent modulation on type-II (saddle-point) GW macroimages, producing mismatches $\gtrsim3\%$ for $M_\text{lens} \gtrsim 20\,M_\odot$—detectable in current and next-generation GW detectors.

5. Novel Observational Modalities and Technological Challenges

Spectroscopy: The relative transverse motion of lens and source produces a general-relativistic, Doppler-like frequency shift between the two unresolved micro-images. This frequency shift is minute—of order $\Delta\nu/\nu \sim 10^{-11}$ for a $0.5 M_\odot$ lens at typical Galactic velocities—but, in combination with microlens parallax (from space-based photometry) and astrometric proper-motion (GAIA), enables direct, degeneracy-free determination of all lens parameters (mass, distance, 2D velocity). Instruments like ESPRESSO (VLT) approach $\sim0.1$ m/s radial velocity sensitivity, making this observable marginally accessible for stellar-mass black holes and a future target in the era of 30-m-class telescopes (Rahvar, 2019).

Astrometry: Sub-milliarcsecond astrometry (Gaia, VLTI/GRAVITY, WFIRST) is now capable of tracing the centroid motion induced by microlensing and provides a mass scale independent of photometric parameters (Sajadian et al., 2014).

Polarimetry: Breaks in the circular symmetry of stellar sources—caused by asymmetric magnification during high-magnification and caustic-crossing events—generate transient polarization signals, which, in combination with simultaneous astrometric data, uniquely reveal the lens–source geometry and can disentangle model degeneracies (Sajadian et al., 2014).

Radio Amplification and SETI: The transient magnification during a microlensing event can act as a "natural telescope," potentially amplifying leakage of radio emission from Earth-analog extraterrestrial technologies to detectable levels for instruments like SKA, especially during caustic crossings in binary systems (Rahvar, 2015).

AGN Torus Microlensing: Extended sources such as AGN dusty tori are typically resistant to microlensing, but near-IR flux ratios can be moderately amplified ($\mu\sim1.2$–$1.4$) on multi-decade timescales, an effect relevant for interpreting lensed quasar flux ratios (Stalevski et al., 2012).

6. Statistical Outcomes and Prospects

Current microlensing surveys have matured into robust statistical tools for mapping the frequency and properties of exoplanets throughout the Milky Way, probing the compact object population (white dwarfs, neutron stars, black holes) over $>10$ order-of-magnitude mass ranges, and constraining (or excluding) non-luminous baryonic and non-baryonic candidates for dark matter (Mao, 2012, Rahvar, 2015, Rektsini et al., 9 Jul 2024). Free-floating planetary-mass objects, brown dwarf distributions, and localized stellar mass functions have all been surveyed or constrained.

Next-generation facilities—including Roman Space Telescope (WFIRST), Euclid, KMTNet, LSST, and high-cadence, synoptic ground-space campaigns—are forecast to yield microlensing samples of $>10^4$ exoplanet discoveries (down to Mars mass), direct parallax and astrometric mass measurements for thousands of lenses, and the first statistical census of free-floating and wide-orbit planet populations (Rektsini et al., 9 Jul 2024).

Instrumental and observational frontiers include improved crowding/blending correction, ultra-stable spectrographs for frequency-shift detection, rapid-response high-resolution imaging for follow-up, and multi-messenger microlensing of gravitational waves. Advanced modeling must address complex lens geometries, time-variable systematics, degeneracies near planetary caustics, limb-darkened extended sources, and population-inference under nontrivial survey selection.

Microlensing thus provides a critical, model-independent window on the dark and cold universe, constraining both baryonic and non-baryonic compact object populations, exoplanet demographics, and fundamental tests of gravity at intermediate scales.