Can $n_s$ hint at a parity-odd sector of gravity?

Abstract

We study slow-roll inflation in the context of metric-affine gravity. We consider the most general projectively invariant metric-affine theory at most quadratic in the distortion and linear in curvature with no propagating torsion and non-metricity and couple it non-minimally to a scalar field. We focus on the class of $ξ$-attractors, in which a single function $f(ϕ)$ fixes the whole scalar sector. Integrating out the non-dynamical distortion generates a correction to the kinetic term $k(ϕ)$ which is $f'(ϕ)^2$ times a ratio of polynomials in $f(ϕ)$. The new kinetic term can develop poles at finite field values, where the distortion field equations degenerate. We classify the poles and derive the conditions on the effective couplings for poles of order $p=2$ and $p=3$, the maximal order allowed in this setting. We then proceed to study inflation. Second-order poles realize $α$-attractors with $α$ fixed by the residue and are already available in the parity-even sector of the theory. Third-order poles instead require the parity-odd sector: whenever the parity-odd quadratic couplings vanish, the kinetic function splits into two independent terms with poles of order at most $p=2$. Third-order pole-inflation, as we confirm numerically for $f(ϕ)\proptoϕ^2$, yields $r\propto N_*^{-3/2}$ with a tunable amplitude and $n_s\simeq 1-\tfrac{3}{2N_*}$, a spectral index mildly larger than the $α$-attractor prediction and well centered in the region preferred by the current combination of ACT, SPT and DESI data. Moreover, when inflation happens far away from the pole we also find a further universal attractor, an inverse-linear plateau, with $n_s \simeq 1-\frac{4}{3N_*}$ and $r\propto N_*^{-4/3}$, testable with the forthcoming BICEP/Keck observations.