  Soon after the Gaia data release (DR) 3 in June 2022, some candidates (and
one confirmed) of detached black hole (BH) - luminous companion (LC) binaries
have been reported. Existing and future detections of astrometric BH-LC
binaries will shed light on the spatial distribution of these systems, which
can deepen our understanding of the natal kicks and the underlying formation
mechanism of BHs. By tracking Galactic orbits of BH-LC binaries obtained from
BSE, we find that distributions of BH mass and the height from the Galactic
plane |z| would help us give a constraint on supernova model. We also indicate
that the correlations of (i) orbital periods and eccentricities, and (ii) BH
mass and $|z|$ could be clues for the strength of natal kick, and that the
correlations of ($P$, $Z/Z_\odot$) may tell us a clue for common envelope (CE)
efficiency. We also discuss the possibility of forming BH-LC binaries like the
BH binary candidates reported in Gaia DR3 and Gaia BH 1, finding that if the
candidates as well as the confirmed binary originate from isolated binaries,
they favor models which produce low-mass BHs and have high CE efficiencies
exceeding unity.
