Physics at Future Colliders: the Interplay Between Energy and Luminosity

In this note, as an input to the Snowmass studies, we provide a broad-brush picture of the physics output of future colliders as a function of their center of mass energies and luminosities. Instead of relying on precise projections of physics reaches, which are lacking in many cases, we mainly focused on simple benchmarks of physics yields, such as the number of Higgs boson produced. More detailed considerations for lepton colliders are given since there have been various recent proposals. A brief summary for hadron colliders based on a simple scaling estimate of the physics reaches is also included.