  A high-energy electron-positron collider has been widely recognized by the
particle physics community to be the next crucial step for detailed studies of
the Higgs boson and other fundamental particles and processes. Several
proposals for such colliders, either linear or circular, are currently under
evaluation. Any such collider will be required to reach high lumimosities, in
order to collect enough data at a reasonable time scale, while at the same time
coping with high rates of background particles produced from beam-beam
interactions during the collisions. In this paper, we analyze the luminosity
and beam-beam interaction characteristics of the Cool Copper Collider (C$^3$)
and perform a comparison with other linear collider proposals. We conclude that
C$^3$ can reach the same or higher collision rates as the other proposals,
without having to cope with higher beam-induced background fluxes. Thus, C$^3$
emerges as an attractive option for a future electron-positron collider,
benefiting from the collective advancements in beam delivery and final focus
system technologies developed by other linear collider initiatives.
