Microlensing and Compact Lenses

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Microlensing detects compact deflectors through a transient magnification, often without seeing the lens. Event duration alone mixes lens mass, distance, and transverse speed. Parallax, finite-source structure, astrometric motion, and lens flux can break parts of that degeneracy, but population membership remains conditional on Galactic priors and the survey detection efficiency. The evidence boundary here is 2026-08-19.

Required background. Geodesics and Jacobi Fields supplies point-lens deflection and image magnification; Probability and Statistical Inference supplies selection-normalized event and population likelihoods.

Helpful background. Weak-Field Gravity supplies the compact-lens bending law; Relativistic Cosmology supplies distance conventions and compact-object population context.

Mass information in a light curve

For an isolated point-mass lens in GR, let $D_l,D_s,D_{ls}$ be the observer–lens, observer–source, and lens–source angular-diameter distances. The Einstein angular radius and crossing time are

\theta_E= \left(\frac{4GM D_{ls}}{D_lD_s}\right)^{1/2}, \qquad t_E=\frac{\theta_E}{\mu_{\rm rel}},

with relative proper motion $\mu_{\rm rel}$ . All angles use the same unit; restoring SI units puts $c^2$ in the denominator of $\theta_E^2$ . For rectilinear relative motion, a point source has dimensionless angular separation $u=\beta/\theta_E$ and magnification

u^2(t)=u_0^2+\left(\frac{t-t_0}{t_E}\right)^2, \qquad A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}.

Here $t_0$ is closest approach and $u_0$ its separation in Einstein-radius units. Observed flux is $F(t)=F_sA[u(t)]+F_b$ , with unlensed source flux $F_s$ and constant blended flux $F_b$ . At $u=1$ , for example, $A=3/\sqrt5\simeq1.34$ ; at large $u$ it tends to unity. A well-measured $t_E$ still does not give $M$ because $D_l$ and $\mu_{\rm rel}$ are unknown. Finite-source structure measures $\rho=\theta_*/\theta_E$ when the lens crosses or approaches the source disk. A color–surface-brightness estimate of the angular stellar radius $\theta_*$ then gives $\theta_E$ .

For Galactic microlensing, where Euclidean distance relations apply, $\pi_{\rm rel}=\mathrm{AU}(D_l^{-1}-D_s^{-1})$ is the lens–source relative parallax in radians and AU is the astronomical unit. Annual or satellite parallax measures the vector $\boldsymbol{\pi}_E$ with magnitude $\pi_E=\pi_{\rm rel}/\theta_E$ . Combining $\theta_E^2=(4G/\mathrm{AU})M\pi_{\rm rel}$ with this definition gives

M=\frac{\theta_E}{\kappa\pi_E}, \qquad \pi_{\rm rel}=\pi_E\theta_E, \qquad \kappa=\frac{4G}{\mathrm{AU}},

in the present units. Directional parallax degeneracies, accelerated source motion, binary lenses, and orbital motion can yield multiple solutions. Resolved lens flux or an astrometric centroid shift supplies another constraint but requires a mass–luminosity relation, extinction, and a calibrated reference frame. The point-lens and parallax formalism is reviewed in Mao 2012, §§2–4.

For unresolved images of a point source and a dark point lens, with no additional blended light, the light-centroid displacement relative to the unlensed source is

\delta\boldsymbol{\theta}_c= \frac{\boldsymbol u}{u^2+2}\theta_E.

Astrometric sampling can therefore measure an angular scale even when finite-source structure is absent. Its long, shallow wings also make reference-frame drift, source proper motion, unresolved companions, and seasonal parallax consequential. Combining photometric and astrometric likelihoods requires their shared epochs and calibration covariance; treating a catalog proper motion as an independent prior can reuse the same images.

These relations use the GR bending law. For a weak-field phenomenological lens with unchanged $\Phi=-GM/r$ and constant $\Psi=\gamma\Phi$ , the Einstein-radius square is multiplied by $(1+\gamma)/2$ . Using the GR mass formula would then return the effective lensing mass $(1+\gamma)M/2$ . Thus a mass or abundance estimate conditional on GR is not yet a gravity test; an independent dynamical mass, or a complete alternative bending law fitted to the same data, is needed. The lensing–dynamics comparison derives that factor.

Catalog, calibration, and population likelihood

A reproducible event analysis identifies the versioned survey light curves, observatory and filter, cadence, photometric reduction, uncertainty rescaling, difference-imaging response, quality flags, and any follow-up data. The OGLE Early Warning System provides public event pages and survey-defined alerts; an event list is not a complete statistical sample until the detection pipeline, cadence gaps, masks, and injection-derived efficiency are supplied OGLE Collaboration, official Early Warning System archive, accessed 2026-08-19.

Selections based on magnification, duration, baseline stability, crowding, coverage, or model fit depend on the very parameters being inferred. For population hyperparameters $\Lambda$ , a rate-aware schematic is

p(\{d_i\}\mid\Lambda) \propto e^{-N_{\rm det}(\Lambda)} \prod_i\int d\lambda_i\, p(d_i\mid\lambda_i)\, \frac{dN}{d\lambda_i}(\lambda_i\mid\Lambda), \qquad N_{\rm det}(\Lambda)=\int d\lambda\, \epsilon(\lambda)\frac{dN}{d\lambda}(\lambda\mid\Lambda),

where $\lambda_i$ contains mass, distance, velocity, binarity, source properties, and blending, and $p(d_i\mid\lambda_i)$ is the unconditional measurement density evaluated for a detected datum. Detection efficiency enters the expected count $N_{\rm det}$ ; inserting it again in the event integral would double count selection. An equivalent likelihood may condition on the observed count, but its selection normalization must then be explicit. Priors for disk, bulge, halo, stellar-remnant, free-floating-planet, or primordial-compact-object populations can dominate membership odds. Posterior predictive duration, parallax, sky-position, and proper-motion distributions are more informative than labeling an isolated long event a black hole.

Systematics include photometric calibration, variable-source contamination, unresolved neighbors, finite-source limb darkening, parallax degeneracy, binary topology, lens luminosity, extinction, cadence, seasonal gaps, and efficiency simulation. Machine-readable light curves, injection code, selection rules, Galactic model, random seeds, and posterior samples define the reproduction boundary.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
Duration and symmetric magnification	Establish a lensing-like transient and constrain $t_E$	Mass, distance, speed, blending, and binary alternatives remain degenerate
Finite-source plus parallax signal	Can determine $\theta_E$ , $\pi_E$ , and hence lens mass	Multiple parallax solutions and source-size calibration can remain
Lens flux or astrometric deflection with survey efficiency	Sharpens individual mass and population membership	Mass–luminosity, reference-frame, Galactic-prior, and selection dependence persists

The conservative position treats compact-object membership probabilistically. A competing interpretation may identify a dark remnant from a long duration and absent flux, but ordinary low-speed or distant lenses can produce similar data. A null search constrains only the selected mass, distance, velocity, sky, and binary domain under the adopted population.

The adversarial check injects events spanning finite-source, parallax, binary, blending, and cadence regimes through the full detection pipeline. It must recover individual masses when enough observables exist and population coverage under alternative Galactic priors. The unresolved boundary is clear: without two independent angular or parallax scales, a light curve remains model-dependent evidence for mass.

What could change the conclusion. The interpretation could change after a superseding survey catalog, photometric or astrometric calibration, injection-efficiency release, or population analysis changes mass recovery or membership coverage across the declared parallax and finite-source regimes.

Exercise

An event has $\theta_E=1.0\,\mathrm{mas}$ and $\pi_E=0.10$ . Find the lens mass using

\kappa=8.14\,\mathrm{mas}\,M_\odot^{-1},

and explain what remains uncertain. Here mas denotes milliarcseconds.

Solution

Substitution gives

M=\frac{\theta_E}{\kappa\pi_E} =\frac{1.0}{8.14\times0.10}M_\odot \simeq1.23M_\odot.

This algebraic mass is meaningful only for the chosen parallax branch and calibrated source angular size. Alternative parallax solutions, unmodeled binary motion, or biased finite-source modeling would change it. Even if the mass is secure, population membership still follows from distance, velocity, luminosity limits, and selection-normalized Galactic priors rather than mass alone.

References

Mao, S. (2012), “Astrophysical Applications of Gravitational Microlensing,” Research in Astronomy and Astrophysics 12, 947–972. DOI.
OGLE Collaboration (n.d.), “OGLE-IV Early Warning System,” official survey event and alert archive, accessed 2026-08-19. Official site.