Supersymmetric Dark Matter and the Energy of a Linear Electron-Positron Collider

We suggest that supersymmetric dark matter be used to set the energy scale of a linear $e^+ e^-$ collider. Assuming that the lightest supersymmetric particle (LSP) is a stable neutralino $\chi$, as in many incarnations of the MSSM with conserved R parity, previous calculations that include coannihilation effects have delineated the region of the $(m_{1/2}, m_0)$ plane where the LSP cosmological relic density lies in the preferred range $0.1 \la \Omega_{\chi} h^2 \la 0.3$. We evaluate here the total cross section for $e^+ e^- \to$ visible pairs of supersymmetric particles, for different values of $m_{1/2}$ and $m_0$, and investigate how much of the dark matter region can be explored by $e^+ e^-$ colliders with different centre-of-mass energies $E_{CM}$. We find that a collider with $E_{CM} = 500$ GeV or 1 TeV can only explore part of the cosmological region, and that a collider with $E_{CM} = 1.5$ TeV with sufficient luminosity can explore all of the supersymmetric dark matter region.