How do OGLE-III and GAIA measure the mass of free microlensing black holes? What is the &quot;hypothesized lower mass gap&quot; between 2.5 and 5 solar masses? eventually links to Constraining the masses of microlensing black holes and the mass gap with Gaia DR2.
The angle of deflection of light passing a massive object is given by:
$$\theta = \frac{4GM}{r c^2}$$
where $r$ is the minimum distance from the mass that the light passes.
If two black holes pass by a line-of-sight to a distant object and their speeds and distances of closest approach $r$ both scale linearly with their mass, they produce the identical deflection magnitude and time dependence.
Question: How then can such an observation of a free-floating black hole be used to determine its mass? What additional information is necessary? I see from the article that parallax is involved, but without knowing the distance to the black hole I don't see how this is enough to work out a mass.
The introduction of the Wyrzykowski &amp; Mandel paper gives the following information about estimating the lens mass.

In order to obtain the mass of the lens (Gould 2000a), it is
necessary to measure both the angular Einstein radius of the lens
($\theta_\mathrm{E}$) and the microlensing parallax ($\pi_\mathrm{E}$)
$$M = \frac{\theta_\mathrm{E}}{\kappa \pi_\mathrm{E}}$$
where $\kappa = 4G / (c^2\ \mathrm{AU}) = 8.144\ \mathrm{mas/M_\odot}$; and $\pi_\mathrm{E}$ is the length
of the parallax vector $\mathbf{\pi_\mathrm{E}}$, defined as $\pi_\mathrm{rel}/\theta_\mathrm{E}$, where $\pi_\mathrm{rel}$
is relative parallax of the lens and the source. The microlensing parallax vector $\mathbf{\pi_\mathrm{E}}$ is measurable from the non-linear motion
of the observer along the Earth’s orbital plane around the Sun.
The effect of microlensing parallax often causes subtle deviations and asymmetries relative to the standard Paczynski light
curve in microlensing events lasting a few months or more, so
that the Earth’s orbital motion cannot be neglected. The parameter $\mathbf{\pi_\mathrm{E}}$ can also be obtained from simultaneous observations of
the event from the ground and from a space observatory located
∼1 AU away (e.g., Spitzer or Kepler, e.g., Udalski et al. 2015b,
Calchi Novati et al. 2015, Zhu et al. 2017).

In particular, the Gould 2000a paper gives a good summary of the various relationships between the quantities. The Udalski et al. 2015b notes that the distance between the Earth and Spitzer (which would also apply to Gaia) means that Spitzer would see differences in the light curve, allowing the parallax to be determined.
Note that things get more complicated if the source is a binary, in which case a &quot;reverse parallax&quot; effect from the source's orbital motion, usually called &quot;xallarap&quot; needs to be taken into account — but that's a matter for another question...
The other relevant quantity is the angular Einstein radius of the lens. In their discussion of measuring $\theta_\mathrm{E}$, Wyrzykowski &amp; Mandel reference Rybicki et al. 2018. That paper notes that precision astrometry can help measure $\theta_\mathrm{E}$ because microlensing also changes the apparent position of the source:

The positional change of the centroid depends on the
$\theta_\mathrm{E}$ and separation $u$. Contrary to the photometric case, the
maximum shift occurs at $u_0 = \sqrt{2}$ and reads (Dominik &amp; Sahu 2000)
$$\delta_\mathrm{max} = \frac{\sqrt{2}}{4} \theta_\mathrm{E} \approx 0.354 \theta_\mathrm{E}$$
Thus, for the relatively nearby lens at $D_l = 4\ \mathrm{kpc}$, source
in the bulge $D_s = 8\ \mathrm{kpc}$ and lensing by a stellar BH with
the mass $M = 4M_\odot$, the astrometric shift due to microlensing will be about 0.7 milliarcsecond.

The bulk of the paper goes on to determine that these shifts should be observable by Gaia.
Another way to measure the size of the lens is to measure the lens-source proper motion by searching for the lens several years after the event, this has been done for a couple of exoplanet-hosting lenses but would not be possible for a dark lens like a black hole.