Weyl Scaling Invariant $R^2$ Gravity for Inflation and Dark Matter

Inflation in the early universe can generate the nearly conformal invariant fluctuation that leads to the structures we observe at the present. The simple viable Starobinsky $R^2$ inflation has an approximate global scale symmetry. We study the conformal symmetric Weyl $\hat{R}^2$ and general $F(\hat{R})$ theories and demonstrate their equivalence to Einstein gravity coupled with a scalar and a Weyl gauge field. The scalar field in Weyl $\hat{R}^2$ gravity can be responsible for inflation with Starobinsky model as the attractor, potentially distinguishable from the latter by future experiments. The intrinsic Weyl gauge boson becomes massive once the Einstein frame is fixed, and constitutes as a dark matter candidate with mass up to $\sim 5\times 10^{16}$GeV.