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 nR and nT, their runnings αR and αT, the non-Gaussianity parameter fNL, and the isocurvature fraction βiso. 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 PR. 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.
Highlights
The framework of inflation, originally conceived as a way to resolve the flatness and horizon problems, has been extremely successful in explaining the origin of cosmological perturbations [1,2,3]
When dealing with nonminimal models, one has to necessarily contend with the socalled frame problem. This problem pertains to the question of whether inflationary models related by a frame transformation, namely a local rescaling of the metric followed by a field reparameterization, are physically equivalent [13, 14]
In analogy to our conformally covariant extension of inflation at the tree level, we extend the Vilkovisky–De Witt formalism to take into consideration conformal transformations, which we expect to be essential in future computations for fully frame-covariant radiative corrections to inflationary quantities
Summary
The framework of inflation, originally conceived as a way to resolve the flatness and horizon problems, has been extremely successful in explaining the origin of cosmological perturbations [1,2,3]. The outline of this paper is as follows: in Section 2, we present the classical action for the class of theories that we will be studying, specified by three model functions: (i) the nonminimal coupling f (φ), (ii) the multifield wavefunction kAB(φ), and (iii) the scalar potential V (φ), where φ collectively stands for all the scalar fields. By considering their properties under conformal transformations and field reparametrizations, we show that the functional form of the classical action remains invariant under frame transformations.
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