Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation

Abstract

We study models where a scalar field has derivative and non-derivative couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation. We consider both the metric formulation and the Palatini formulation. In the Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the same result regardless of whether the connection is unconstrained or the non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is included, the unconstrained case and the zero torsion case are physically different. We reduce all the actions to the Einstein frame with minimally coupled matter, and find the leading order differences between the metric case and the Palatini cases.

Inflation and reheating in quadratic metric-affine gravity with derivative couplings

“…where, for brevity, we have omitted the tildes from the rescaled quantities and the arguments from the functions γ and γ 1 . Our action (2.12) aligns with the one presented in [122], with the sole distinction being the replacement given by equation (2.8). Note however that the presence of the auxiliary scalar χ will turn out to have important effects on the inflationary behaviour.…”

“…While actions constructed solely from the Ricci scalar can be transformed via a Weyl rescaling (or conformal transformation), since our action (2.5) involves derivative couplings of the scalar field to the Ricci tensors, we need to employ a broader set of transformations, namely the disformal transformations. These transformations are defined as (we have closely followed the notation of [122])…”

“…Following [122] we may replace the co-Ricci tensor with the average Ricci tensor, defined as R µν = (R µν + R µν )/2, which vanishes if the connection is metric compatible. In general metric-affine theories have non-zero torsion T µλν = 2C [µ|λ|ν] (see [131] for a recent review on gravitational theories with torsion) and non-metricity ∇ ρ g µν = −2C (µν)ρ .…”

“…This general feature of metric-affine R 2 models is maintained in the presence of non-minimal f (ϕ)R couplings . Nevertheless, the effect of derivative couplings ∼ (∂ϕ) 2 R or ∼ (∂ µ ϕ∂ ν ϕ)R µν on this behaviour [121,122], being an open issue, is JCAP06(2024)005 the main focus of the present paper. As working examples we consider the case of a scalar field with a quadratic potential and the case of a scalar field with a quartic self interaction.…”