The metric-affine ${\mathring{R}}^2$ scalaron: parameter-free inflation

We present a model of slow-roll inflation in metric-affine gravity, where the connection is independent of the metric. We require that the Einstein-Hilbert term is the only operator that breaks three out of four shift-symmetries of the connection, and impose them on the metric-affine Starobinsky term $\alpha R(\mathrm{\Gamma }{)}^2$. This replaces it by $\alpha {\mathring{R}}^2$, the square of the invariant part of the scalar curvature. The theory propagates only the graviton and one healthy scalar, the metric-affine ${\mathring{R}}^2$ scalaron. Its potential approaches a plateau polynomially, $U\simeq U_0(1-{\mu }^2\mathrm{/}{\chi }^2)$, with $\mu =\sqrt{6}{M}_P$ fixed by the curvature of the connection. Once $\alpha \simeq 2\times {10}^8$ is set by the scalar amplitude, the model has no free parameters. It predicts $n_s\simeq 1-3\mathrm{/}(2N_{\ast })$ and $r\simeq 2\sqrt{3}{N}_{\ast }^{-3\mathrm{/}2}$ at leading order in $N_{\ast }$, and $n_s=0.968$–$0.974$ and $r=0.011$-$0.008$ with exact slow-roll for $N_{\ast }=50$-$60$ $e$-folds. These values agree with current data, and $r$ is within reach of the next generation of $B$-mode measurements.