Gravitational Dark Matter Production in Supergravity $\alpha$-Attractor Inflation

We consider gravitational particle production (GPP) of dark matter (DM) under a supergravity framework, where the $\alpha$-attractor inflation model is used. The particle spectrum is computed numerically and the DM number density is obtained. We show how the DM mass, gravitino mass and inflation model parameters modify the results, and find the reheating temperature which leads to sufficient DM production. In our setup, supergravity corrections suppress the efficiency of GPP, and make the isocurvature constraint much weaker compared with the normal case. With tensor-to-scalar ratio ranging from $10^{-3}-10^{-4}$ and DM mass from $10^{-2} m_\phi - m_\phi$, the required reheating temperature should be around $10^3 \textrm{GeV} - 10^7 \textrm{GeV}$.