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

Abstract

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 $αR(Γ)^2$. This replaces it by $α\mathring{R}^2$, 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, $U\simeq U_0(1-μ^2/χ^2)$, with $μ=\sqrt{6}\,M_P$ fixed by the curvature of the connection. Once $α\simeq2\times10^{8}$ is set by the scalar amplitude, the model has no free parameters. It predicts $n_s\simeq1-3/(2N_*)$ and $r\simeq2\sqrt3\,N_*^{-3/2}$ at leading order in $N_*$, and $n_s=0.968$--$0.974$ and $r=0.011$-$0.008$ with exact slow-roll for $N_*=50$-$60$ $e$-folds. These values agree with current data, and $r$ is within reach of the next generation of $B$-mode measurements.