# Frame Covariant Nonminimal Multifield Inflation

## TL;DR

This paper develops a frame-covariant geometric formalism for multifield inflation, enabling frame-invariant predictions of cosmological observables and analyzing the impact of field space curvature and boundary conditions.

## Contribution

It introduces a fully frame-covariant approach to multifield inflation, extending slow-roll parameters and analyzing isocurvature effects with a geometric perspective.

## Key findings

- Isocurvature effects are significant in nonminimal models.
- Frame invariance ensures consistent predictions across conformal frames.
- Boundary conditions influence inflationary observables.

## Abstract

We introduce a frame-covariant formalism for inflation of scalar-curvature theories by adopting a differential geometric approach which treats the scalar fields as coordinates living on a field-space manifold. This ensures that our description of inflation is both conformally and reparameterization covariant. Our formulation gives rise to extensions of the usual Hubble and potential slow-roll parameters to generalized fully frame-covariant forms, which allow us to provide manifestly frame-invariant predictions for cosmological observables, such as the tensor-to-scalar ratio $r$, the spectral indices $n_{\cal R}$ and $n_T$, their runnings $\alpha_{\cal R}$ and $\alpha_T$, the non-Gaussianity parameter $f_{NL}$, and the isocurvature fraction $\beta_{\rm iso}$. We examine the role of the field space curvature in the generation and transfer of isocurvature modes, and we investigate the effect of boundary conditions for the scalar fields at the end of inflation on the observable inflationary quantities. We explore the stability of the trajectories with respect to the boundary conditions by using a suitable sensitivity parameter. To illustrate our approach, we first analyze a simple minimal two-field scenario before studying a more realistic nonminimal model inspired by Higgs inflation. We find that isocurvature effects are greatly enhanced in the latter scenario and must be taken into account for certain values in the parameter space such that the model is properly normalized to the observed scalar power spectrum $P_{\cal R}$. Finally, we outline how our frame-covariant approach may be extended beyond the tree-level approximation through the Vilkovisky-De Witt formalism, which we generalize to take into account conformal transformations, thereby leading to a fully frame-invariant effective action at the one-loop level.