Minimal $k$-inflation in light of the conformal metric-affine geometry

We motivate a minimal realization of slow-roll $k$-inflation by incorporating the local conformal symmetry and the broken global $\mathrm{SO}(1,1)$ symmetry in the metric-affine geometry. With use of the metric-affine geometry where both the metric and the affine connection are treated as independent variables, the local conformal symmetry can be preserved in each term of the Lagrangian and thus higher derivatives of scalar fields can be easily added in a conformally invariant way. Predictions of this minimal slow-roll $k$-inflation, $n_\mathrm{s}\sim 0.96$, $r\sim 0.005$, and $c_\mathrm{s}\sim 0.03$, are not only consistent with current observational data but also have a prospect to be tested by forthcoming observations.