Detection and Characterization of Microlensing Events due to Isolated Black Holes towards the Magellanic Clouds in the Rubin Observations

Abstract: Gravitational microlensing surveys potentially discover isolated black holes (IBHs) at large distances through their gravitational effects on apparent brightness and motion of collinear stars. Here, we study detecting and characterizing IBHs within the mass range $[3,~5000]M_{\odot}$ with either stellar or dark matter origins in the upcoming observations by Vera~C.~Rubin Observatory towards the Large and Small Magellanic Clouds (LMC and SMC). We consider four lens mass functions (MFs:~$dN/dM\propto M<sup>{-β}$ for $β=0,~0.5,~1,~2$), and generate long-duration microlensing events due to IBHs detectable by Rubin. By assuming the fraction of IBHs's mass in total Galactic mass as $\mathcal{F}$, Rubin potentially detects $\sim0.3,~\rm{and}~2$ microlensing events towards SMC and LMC due to IBHs if $β=1,~\rm{and}~\mathcal{F}=5\times10<sup>{-3}$. These IBHs are inside our galaxy with the probability $\gtrsim70\%$. These events have on average $θ<em>{\rm{E}}\sim30-50~\rm{mas}$, so that the probabilities of resolving their lensing-induced images through the Rubin astrometric observations are $\sim1.4,~\rm{and}~16.3\%$ towards LMC and SMC. In $\sim1,~5.2\%$ of these events their astrometric deflections are realizable. The probabilities of discerning their parallax are $\sim40,~25\%$. We evaluate relative errors with simulating synthetic data points and by assuming the true models as the best-fitted ones. We conclude for a log-uniform MF for $\lesssim0.1,~\rm{and}~0.3\%$ of photometrically detectable microlensing events towards in LMC and SMC the relative errors in the lens mass, distance and velocity are $\lesssim3\%$. We calculate the number of detectable IBHs versus by assuming their complete contribution of compact objects in halos' darkmatter, and conclude Rubin specifies $95\%$~C.L. upper-limit on IBHs exclusion by masses $\lesssim208,~8M</em>{\odot}$ for $\mathcal{F}\simeq4\times10<sup>{-3}$.

• The paper forecasts Rubin microlensing detections using Monte Carlo simulations that model black-hole masses, Galactic and Magellanic Cloud structure, blending, and the baseline 10-year observing cadence.
• The study predicts only a few detectable events for plausible black-hole dark-matter fractions, with 67–93% of events arising from Milky Way halo lenses but precise characterization possible for fewer than 1%.
• The analysis shows that a null detection could exclude isolated black holes below about 208 solar masses as the entire dark-matter halo toward the LMC when the compact-object fraction is 0.004, subject to model assumptions.

Overview and motivation

Sajadian and Makler present Monte Carlo simulations of microlensing events caused by isolated black holes (IBHs) with masses in the range $[3, 5000]\,M_\odot$, observable by the Vera C. Rubin Observatory toward the Large and Small Magellanic Clouds (LMC, SMC) over its 10-year mission (2608.16448). The lens population includes both stellar-origin black holes and compact objects of dark-matter origin (MACHOs or primordial black holes). The study addresses four questions: the Rubin detection efficiency for such lenses, the probabilities of measuring parallax, astrometric deflection, and image resolution, the expected number of detectable events, and the resulting 95% C.L. exclusion limits on IBHs as halo dark matter.

The simulations adopt four lens mass functions $dN/dM \propto M^{-\beta}$ with $\beta = 0, 0.5, 1, 2$, spatial density models from the Besançon Galaxy model and published LMC/SMC structure models, velocity distributions from Gaia-informed kinematics, and synthetic observing cadences from the Rubin OpSim baseline_v5.1 strategy. Blending is treated explicitly via PSF-based star counts, and finite-source effects are neglected—justified by the large angular Einstein radii ($\theta_{\rm E} \sim 30$–$50$ mas on average) of these long-duration events.

Simulation methodology

Source stars are drawn from combined Galactic (thin/thick disk, bulge, stellar halo) and Magellanic Cloud (disk, bar/bulge, halo) density profiles, with photometric properties assigned from the Besançon model. Lens distances are sampled proportionally to the event rate $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$, with $f = 5\%$ of the Galactic and MC dark-matter halos assumed to be in compact form. Light curves include annual microlensing parallax computed numerically, and astrometric source trajectories include the lensing-induced centroid shift $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$.

Three source-detectability criteria are applied: detectability at peak in at least two Rubin filters against single-visit depths, weighting by the $r$-band blending parameter, and at least three data points per event. Photometric event detection requires $\Delta\chi^2 \geq 2N_{\rm data}$, at least three points above baseline by $dN/dM \propto M^{-\beta}$0, and FWHM $dN/dM \propto M^{-\beta}$1 (10 yr). Under baseline_v5.1, Rubin will collect roughly 170–450 visits toward the LMC and 70–390 toward the SMC.

Properties of detectable events

Steeper mass functions shift the detectable population toward lower masses: the average lens mass decreases from $dN/dM \propto M^{-\beta}$2 ($dN/dM \propto M^{-\beta}$3) to $dN/dM \propto M^{-\beta}$4 ($dN/dM \propto M^{-\beta}$5), with corresponding Einstein crossing times dropping from $dN/dM \propto M^{-\beta}$6 yr to $dN/dM \propto M^{-\beta}$7 yr. Very long events with $dN/dM \propto M^{-\beta}$8 comparable to the survey window risk confusion with long-period variables such as Miras—a classification ambiguity the paper notes but does not resolve.

A key structural result concerns the halo-lensing versus self-lensing dichotomy. Overall, $dN/dM \propto M^{-\beta}$9–$\beta = 0, 0.5, 1, 2$0 of detectable events have lenses inside the Milky Way, but this fraction falls to only 20–30% toward the dense central parts of the MCs, where self-lensing dominates. Self-lensing events have $\beta = 0, 0.5, 1, 2$1 and $\beta = 0, 0.5, 1, 2$2 smaller than halo events by one to two orders of magnitude, which propagates directly into poorer parameter recovery; consequently, physical characterization is concentrated in halo-lensing events away from the MC centers.

The Rubin star detection efficiency is $\beta = 0, 0.5, 1, 2$3 (LMC) and $\beta = 0, 0.5, 1, 2$4 (SMC), while the efficiency for recognizing lensing signals among detected stars is $\beta = 0, 0.5, 1, 2$5–$\beta = 0, 0.5, 1, 2$6. The lensing efficiency peaks for $\beta = 0, 0.5, 1, 2$7–4 yr, low-mass lenses ($\beta = 0, 0.5, 1, 2$8), small impact parameters ($\beta = 0, 0.5, 1, 2$9), and high blending parameters; blending alone changes the efficiency from zero to $\theta_{\rm E} \sim 30$0 across its full range.

Event rates and expected yields

Scaling out the uncertain IBH mass fraction $\theta_{\rm E} \sim 30$1, the predicted number of detectable events over 10 years from $\theta_{\rm E} \sim 30$2 fields scales linearly:

Mass function	LMC ($\theta_{\rm E} \sim 30$3)	SMC ($\theta_{\rm E} \sim 30$4)
$\theta_{\rm E} \sim 30$5	160	20
$\theta_{\rm E} \sim 30$6	318	44
$\theta_{\rm E} \sim 30$7	490	68
$\theta_{\rm E} \sim 30$8	792	116

For a fiducial $\theta_{\rm E} \sim 30$9, Rubin would detect only $50$0 and $50$1 IBH microlensing events toward the LMC and SMC respectively (for $50$2). This is a sobering yield: individual-event detections will be rare, and the primary scientific return may instead come from statistical non-detections. The authors estimate $50$3 from the initial stellar mass function plus a 5% MACHO halo fraction, an order of magnitude above their fiducial value, so the actual yield is highly sensitive to this assumption.

Parallax, astrometry, and image resolution

Among photometrically detected events, the probability of discerning parallax is $50$4–48% toward the LMC and $50$5–33% toward the SMC, increasing with steeper mass functions (lower masses give larger $50$6). Astrometric deflections are realizable in only $50$7 (LMC) and up to $50$8 (SMC) of events under the most favorable mass function, dropping essentially to zero for $50$9. Image resolution requires both sufficient separation relative to the two-star astrometric accuracy factor $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$0 and detectable fluxes; it succeeds in $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$1 (LMC) and up to $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$2 (SMC) for $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$3. Notably, toward the SMC image resolution outperforms astrometric-deflection detection because the sparser visit count (~200–300) makes the $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$4 criterion harder to satisfy than the three-point image-resolution requirement.

Since measuring any two of $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$5, $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$6, and $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$7 breaks the microlensing degeneracy, these results imply that unique mass and distance determinations will be possible for only a small minority of detected events—consistent with the historical pattern in which most BH microlensing candidates required Bayesian mass inference.

Parameter recovery accuracy

Relative errors are evaluated via Fisher/Covariance matrices assuming the best-fit model equals the true model. For halo-lensing events, errors decrease with lens mass, blending parameter, and source brightness; for self-lensing events they are higher by one to two orders of magnitude in $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$8 owing to the tiny $$\Gamma \propto R_{\rm E}\,\rho_{\rm tot}(D_{\rm l})$$9. An instructive exception is that errors in $f = 5\%$0 decrease with impact parameter, since the astrometric deflection transitions from a nearly straight line to a closed ellipse as $f = 5\%$1 grows.

The headline characterization numbers are modest: for a log-uniform mass function, all three lens parameters (mass, distance, proper motion) are recovered to better than 3% in only $f = 5\%$2 (LMC) and $f = 5\%$3 (SMC) of photometrically detectable events; the fraction rises to $f = 5\%$4 and $f = 5\%$5 at a 20% error threshold. The paper concedes that real astrometric recovery will likely be worse than simulated, because motions of blended stars—which were not included in the trajectory noise model—will contaminate the centroid shifts.

Dark-matter exclusion limits

Assuming IBHs constitute 100% of the halo dark matter and using discrete logarithmically spaced lens masses, the expected event count declines steeply with mass. If no events are detected, the Poisson 95% C.L. upper limit corresponds to three expected events, yielding exclusion of IBHs with masses below approximately:

$f = 5\%$6	LMC limit ($f = 5\%$7)	SMC limit ($f = 5\%$8)
$f = 5\%$9	3.7	—
$$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$0	9.1	—
$$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$1	208	8
$$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$2	1032	87

Thus, for $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$3, Rubin observations toward the LMC could exclude IBHs with $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$4 from fully comprising the halo dark matter at 95% C.L., while SMC observations constrain only $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$5. These limits depend entirely on the assumed $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$6 normalization and on the assumption that the IBH spatial distribution traces the total stellar-plus-dark-matter density scaled by $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$7—an assumption the authors acknowledge is not uniquely determined observationally.

Limitations and open questions

Several assumptions bound the applicability of these predictions. Finite-source effects are ignored throughout, appropriate for the large $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$8 values considered but untested for the lowest-mass, closest lenses. Blending-star proper motions are excluded from the astrometric error budget despite being acknowledged as a significant systematic. The IBH spatial distribution is assumed to follow the total mass density, and the mass-function family is restricted to power laws; no bimodal or physically motivated stellar-remnant mass function is explored. The degeneracy between very long-duration microlensing events and Mira-like variables is flagged but not quantified. Finally, whether Rubin's cadence toward the MCs in the final adopted survey strategy will match baseline_v5.1 remains an external dependency of all quantitative results.

Conclusion

This work provides a quantitative forecast for IBH microlensing toward the Magellanic Clouds in the Rubin survey, combining realistic Galaxy/MC structural models, OpSim cadences, blending, parallax, and astrometry. The central findings are that (i) detectable events will be rare—of order unity for plausible $$\boldsymbol{\delta\theta}_{\rm c} = \boldsymbol{u}\,\theta_{\rm E}/(u^2+2)$$9—but predominantly halo-lensing ($r$0); (ii) complete lens characterization to $r$1 precision will be achievable in well under 1% of events; and (iii) a null detection would set 95% C.L. exclusion limits on IBH dark matter down to $r$2 (LMC) for $r$3. The results position Rubin primarily as a statistical constraint instrument for massive compact halo objects rather than a prolific discoverer of individually characterized isolated black holes.