Inflation with the Chern-Simons term in the Palatini formulation

TL;DR

This work analyzes a Chern–Simons (Pontryagin) term coupled to the inflaton in Palatini gravity, where an independent connection allows backreaction on the background and scalar perturbations. By solving for the connection at leading order and performing order reduction in a gradient expansion, the authors derive an effective theory in which the inflaton kinetic term is modified by a piece proportional to $P'^2 V^2$ for unconstrained and zero non-metricity cases, while the zero-torsion case leaves the kinetic term unchanged. Applying this framework to polynomial and Higgs-like inflation, they show the CS coupling can restore flatness for higher-power potentials and, in the Higgs scenario, allow a tensor-to-scalar ratio $r$ compatible with current bounds with a smaller non-minimal coupling to the Ricci scalar. The Palatini formulation also cures the tensor-mode instability found in the metric CS case, though sub-Hubble propagation can still be affected and stability issues remain an open question for further study. The results point to distinctive phenomenology in tensor modes and potential non-Gaussian signatures that could test the Palatini CS inflationary scenario.

Abstract

We consider the Chern--Simons term coupled to the inflaton in the Palatini formulation of general relativity. In contrast to the metric formulation, here the Chern--Simons term affects also the background evolution. We approximately solve for the connection, insert it back into the action, and reduce the order of the equations to obtain an effective theory in the gradient approximation. We consider three cases: when the connection is unconstrained, and when non-metricity or torsion is put to zero. In the first two cases, the inflaton kinetic term is modified with a term proportional to the square of the potential. For polynomial potentials dominated by the highest power of the field, the Chern--Simons term solves the problem that higher order corrections spoil the flatness of the potential. For Higgs inflation, the tensor-to-scalar ratio can be as large as the current observational bound, and the non-minimal coupling to the Ricci scalar can be as small as in the metric case. The Palatini contribution cures the known instability of the tensor modes due to the Chern--Simons term in the metric formulation.