  Testing the true nature of black holes - the no-hair hypothesis - will become
increasingly more precise in the next few years as new observational data is
collected in both the gravitational wave channel and the electromagnetic
channel. In this paper we consider numerically generated spacetimes of Kerr
black holes with synchronised scalar hair and build stationary models of
magnetized thick disks (or tori) around them. Our approach assumes that the
disks are not self-gravitating, they obey a polytropic equation of state, the
distribution of their specific angular momentum is constant, and they are
marginally stable, i.e. the disks completely fill their Roche lobe. Moreover,
contrary to existing approaches in the literature, our models are
thermodinamically relativist, as the specific enthalpy of the fluid can adopt
values significantly larger than unity. We study the dependence of the
morphology and properties of the accretion tori on the type of black hole
considered, from purely Kerr black holes with varying degrees of spin
parameter, namely from a Schwarzschild black hole to a nearly extremal Kerr
case, to Kerr black holes with scalar hair with different ADM mass and horizon
angular velocity. Comparisons between the disk properties for both types of
black holes are presented. The sequences of magnetized, equilibrium disks
models discussed in this study can be used as initial data for numerical
relativity codes to investigate their dynamical (non-linear) stability and used
in tandem with ray-tracing codes to obtain synthetic images of black holes
(i.e. shadows) in astrophysically relevant situations where the light source is
provided by an emitting accretion disk.
