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The microlensing optical depth is the probability, when small, that a source lies within one Einstein radius of a foreground lens. For point lenses and compact-object mass density ${\rho }_l$ in nearby Euclidean geometry,The individual lens mass cancels because $n_l={\rho }_l/m$ and $R_E^2\propto m$. Detection efficiencies and event durations still depend on individual masses and survey cadence.

Objects near the derived cutoff are too faint for their absence in ordinary star counts alone to exclude a massive dark matter halo. The decisive test is gravitational microlensing: compact objects passing in front of background stars produce transient magnification even if they emit no detectable light. For a lens at distance $D_l$ and a source at $D_s$, the physical Einstein radius and event timescale areFor $D_l=10\mathrm{kpc}$, $D_s=50\mathrm{kpc}$ and $v_{\perp }=200\mathrm{km}{\mathrm{s}}^{-1}$, a $10^{-3}{M}_{\odot }$ lens gives $t_E\simeq 2.2$ days, within the regime tested by suitably sampled gravitational microlensing surveys. The microlensing optical depth isso at fixed compact-object mass density it is independent of the individual lens mass: reducing individual mass changes the event durations rather than hiding the total lensing probability.

EROS observations of the Magellanic Clouds found far fewer candidate events than a halo dominated by compact objects would predict and excluded such objects as the principal Galactic halo component over $6\times 10^{-8}<m/M_{\odot }<15$, under their tested halo models. Their often-quoted eight-percent bound is specifically near $0.4M_{\odot }$, not a uniform bound at every lens mass. A halo dominated by the approximately Jupiter-mass objects needed here is incompatible with those microlensing constraints. Deep optical and infrared counts also restrict a large population of luminous low-mass stars and brown dwarfs, while the cosmic baryon fraction provides a separate constraint on explaining all cosmological dark matter with baryonic objects.

The primordial-element abundances from Big Bang nucleosynthesis, together with the cosmic microwave background, constrain the total baryonic density. massive compact halo objects such as ordinary faint stars, brown dwarfs and stellar remnants are baryonic whether or not they shine. With the stipulated density fractions, the entire baryonic budget is only one fifth of the required dark-matter density, and much of it is already accounted for by ordinary gas and stars.

A second test does not require a compact object to emit light: gravitational microlensing temporarily magnifies a background star as the object crosses the line of sight. Monitoring many Magellanic-Cloud stars constrains the microlensing optical depth and event-duration distribution. Surveys such as EROS-2 found too few suitable events for a standard halo predominantly composed of compact objects over their tested mass range.^[ref] Artificial-event injection, cadence and blending corrections are needed because very short or long events are easily missed; lensing by stars within the Clouds must also be separated from halo lensing. These are mass-range and halo-model constraints, not a proof excluding every compact-object mass. Primordial black holes are not limited by the ordinary-baryon budget in the same way as stellar remnants; their viability requires their own observational bounds.