  The hunt for compact objects is on. Rarely seen massive binaries with a
compact object are a crucial phase in the evolution towards compact object
mergers. In Gaia data release 3 (DR3), the first Gaia astrometric orbital
solutions for binary sources will become available. We investigate how many
black holes (BH) with massive main-sequence dwarf companions (OB+BH binaries)
are expected to be detected as binaries in DR3 and at the end of the nominal
5-yr mission (DR4). We estimate the fraction of identifiable OB+BH binaries and
discuss the distributions of the masses of both components and the orbital
periods. We study the impact of different BH-formation scenarios. Using
tailored models for the massive star population, which assume a direct collapse
and no kick upon BH formation (the fiducial case), we estimate the fraction of
OB+BH systems that Gaia will detect as binaries. A distance distribution
according to that of the second Alma Luminous Star catalogue (ALSII) is
assumed. We investigate how many of the systems detected as binaries are
identifiable as OB+BH binaries, using a method based on astrometric data. In
the fiducial case we conservatively estimate that 77% of the OB+BH binaries in
ALSII will be detected as binaries in DR3, of which 89% are identifiable as
OB+BH binaries. This leads to a total of around 190 OB+BH binaries, a 20-fold
increase in the known sample of OB+BH binaries, covering an uncharted parameter
space of long-period binaries. The size and properties of the identifiable
OB+BH population will contain crucial observational constraints to improve our
understanding of BH formation. In DR4, the detected fraction will increase to
85%, of which 82% will be identifiable. Hence, an additional ~5 systems could
be identified, which are expected to have either very short or long periods.
The fractions become smaller for different BH-formation scenarios. (truncated)
