Inflation with non-minimal coupling: Metric vs. Palatini formulations

Abstract

We analyze non-minimally coupled scalar field theories in metric (second-order) and Palatini (first-order) formalisms in a comparative fashion. After contrasting them in a general setup, we specialize to inflation and find that the two formalisms differ in their predictions for various cosmological parameters. The main reason is that dependencies on the non-minimal coupling parameter are different in the two formalisms. For successful inflation, the Palatini approach prefers a much larger value for the non-minimal coupling parameter than the Metric approach. Unlike the Metric formalism, in Palatini, the inflaton stays well below the Planck scale whereby providing a natural inflationary epoch.

Higgs–Palatini inflation and unitarity

“…In the body of the work, we shall show that, Higgs inflation in the Palatini formulation is not only natural as was explicitly shown in [16] difference between the metric and Palatini formulations in regard to the ones given in [16].…”

“…This very setup, the so-called Palatini formulation or first-order formalism, does not necessarily admit the Levi-Civita connection if the matter sector depends explicitly on the affine connection [14,15]. In this formalism, connection and metric are independent geometrodynamical variables [16], and this very fact has physically interesting consequences for inflation [16][17][18][19][20][21] as well as other phenomena like the cosmological constant problem [22,23].…”

Comparing metric and Palatini approaches to vector Horndeski theory

“…Although some of the models may be plagued by ghosts and instabilities [18], their elegant mathematical structure [19,20] motivates the detailed study. The particular case of Horndeski prescription in Palatini approach remains barely covered by the investigations, though the theories with non-minimally coupled scalar field were extensively explored recently in both metric and Palatini formulations [21,22,23,24]. However, in Horndeski case the model with scalar field is quite complicated, because it contains several distinct types of coupling between matter and geometry.…”

Braneworld nonminimal inflation with induced gravity

“…Even if for the classical, unperturbed theory this non-minimal coupling vanishes, it is necessary for the renormalizability of the scalar field theory in curved space. In most theories used to describe inflationary scenarios, it turns out that a non-vanishing value of the coupling constant cannot be avoided [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55]. Nevertheless, incorporation of an explicit non-minimal coupling has disadvantage that it is harder to realize inflation even with potentials that are known to be inflationary in the minimal theory [29,30,31].…”

“…Nevertheless, incorporation of an explicit non-minimal coupling has disadvantage that it is harder to realize inflation even with potentials that are known to be inflationary in the minimal theory [29,30,31]. Using the conformal equivalence between gravity theories with minimally and non-minimally coupled scalar fields, for any inflationary model based on a minimally-coupled scalar field, it is possible to construct infinitely many conformally related models with a non-minimal coupling [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57]. However, an important question then arises: are these conformally related frames really equivalent from physics viewpoint?…”