Gaia BH3 is the first observed dormant black hole (BH) with a mass of ≈30 M⊙, and it represents the first confirmation that such massive BHs are associated with metal-poor stars. Here, we explore the isolated binary formation channel for Gaia BH3, focusing on the old and metal-poor stellar population of the Milky Way halo. We used the MIST stellar models and our open-source population synthesis code SEVN to evolve 5.6 × 108 binaries, exploring 20 sets of parameters that encompass different natal kicks, metallicities, common envelope efficiencies and binding energies, and models for the Roche-lobe overflow. We find that systems such as Gaia BH3 form preferentially from binaries initially composed of a massive star (40- 60 M⊙) and a low-mass companion (<1 M⊙) in a wide (P > 103 days) and eccentric orbit (e > 0.6). Such progenitor binary stars do not undergo any Roche-lobe overflow episode during their entire evolution, so the final orbital properties of the BH-star system are essentially determined at the core collapse of the primary star. Low natal kicks (≲10 km/s) significantly favour the formation of Gaia BH3-like systems, but high velocity kicks up to ≈220 km/s are also allowed. We estimated the formation efficiency for Gaia BH3-like systems in old (t >10 Gyr) and metal-poor (Z < 0.01) populations to be ~4 × 10-8 M⊙-1 (for our fiducial model), representing ~3% of the whole simulated BH-star population. We expect up to ≈4000 BH-star systems in the Galactic halo formed through isolated evolution, of which ≈100 are compatible with Gaia BH3. Gaia BH3-like systems represent a common product of isolated binary evolution at low metallicity (Z < 0.01), but given the steep density profile of the Galactic halo, we do not expect more than one at the observed distance of Gaia BH3. Our models show that even if it was born inside a stellar cluster, Gaia BH3 is compatible with a primordial binary star that escaped from its parent cluster without experiencing significant dynamical interactions.
