Estimates of Collective Effects for the FCC-$e^+e^-$ Pre-Booster Ring

The FCC-e⁺e⁻ injector complex needs to produce and to transport high-intensity e⁺ and e⁻ beams at a fast repetition rate for topping up the collider at its collision energy. Two different options are under consideration as pre-accelerator before the bunches are transferred to the high-energy booster: either using the existing SPS machine or designing a completely new ring. The purpose of this paper is to present the studies of collective effects with analytical estimates for both the pre-booster ring design options including space charge (SC), longitudinal micro-wave instability (LMI), transverse mode coupling instability (TMCI), ion effects, electron cloud (e-cloud), coherent synchrotron radiation (CSR), and intra-beam scattering (IBS).