Inflationary attractors in metric-affine gravity

Abstract

We study single-field slow-roll inflation in metric-affine gravity (MAG). We consider the most general action up to operators linear in curvature and quadratic in the distortion, without propagating torsion and non-metricity. We first impose projective invariance and promote every coefficient to a function of the scalar field. After integrating out the non-dynamical distortion, the net contribution is a correction $k(ϕ)$ to the scalar kinetic term. We then consider the class of $ξ$-attractors for which the non-minimal couplings and the scalar potential are fixed by a single function $f(ϕ)$. It is well-known that in the metric formulation the strong-coupling regime is universally the Starobinsky attractor. In the Palatini framework, instead, this attractor is lost and the predictions become model-dependent. Here, we show that the MAG framework restores universality: in the strong coupling limit with finite ratios of the distortion couplings to $ξ$ we have an $α$-attractor, $n_s=1-2/N_*$ and $r=12α/N_*^2$, with $α=1+ζ/(6ξ)$. The ratio $ζ/ξ$ encodes the distortion contribution. Starobinsky and Palatini inflation are recovered as specific limits. Through a numerical parameter scan we show that almost all the healthy points lie on one of the two attractors : either Nieh-Yan-like inflation (studied in a previous work and extended here) or the new MAG $α$-attractor found here. Finally, we explicitly drop projective invariance and analyse the general MAG theory. The same large-field asymptotic structure survives. We conclude that no extra inflationary phenomenology is reached by the non-projectively invariant operators: the $α$-attractor is a general prediction of MAG.