Longitudinal coupled-bunch instability evaluation for FCC-hh

High-order modes (HOM) of the accelerating rf structures and other machine elements, if not sufficiently damped, can drive longitudinal coupled-bunch instabilities (CBI). Their thresholds can be accurately obtained from macro-particle simulations using the detailed impedance model containing many different contributions. This method, however, is very difficult to apply for synchrotrons with a large number of bunches, as it is the case for the Future Circular hadron-hadron Collider (FCC-hh) with up to 10400 circulating bunches per beam. In this paper the semi-analytical approach is used for calculations of the instability thresholds during the acceleration cycle of the FCC-hh. As the result, we define requirements for the HOM damping that would be sufficient to prevent development of longitudinal CBI in the presence of weak synchrotron radiation damping.