Modelling optical emission of ultraluminous X-ray sources accreting above the Eddington limit

We study the evolution of binary systems of ultraluminous X-ray sources and compute their optical emission assuming accretion onto a black hole via a non-standard, advection-dominated slim disc with an outflow. We consider systems with black holes of 20-100 M_{⊙}, and donor masses between 8 and 25 M_{⊙}. Supercritical accretion has considerable effects on the optical emission. The irradiating flux in presence of an outflow remains considerably stronger than that produced by a standard disc. However, at very high accretion rates the contribution of X-ray irradiation becomes progressively less important in comparison with the intrinsic flux emitted from the disc. After main sequence the evolutionary tracks of the optical counterpart on the colour-magnitude diagram are markely different from those computed for Eddington-limited accretion. Systems with stellar-mass black holes and 12-20 M_{⊙} donors accreting supercritically are characterized by blue colours (F450W - F555W ≃ - 0.2: +0.1) and high luminosity (M_V ≃ -4: -6.5). Systems with more massive black holes accreting supercritically from evolved donors of similar mass have comparable colours but can reach M_V ≃ -8. We apply our model to NGC 1313 X-2 and NGC 4559 X-7. Both sources are well represented by a system accreting above Eddington from a massive evolved donor. For NGC 1313 X-2 the agreement is for an {∼ } 20 M_{⊙} black hole, while NGC 4559 X-7 requires a significantly more massive black hole.