Scalar-tensor gravity in the Palatini approach

TL;DR

This thesis explores scalar-tensor theories within the Palatini formalism, introducing a novel action functional, analyzing invariants, and examining equivalences with $f(R)$ theories, with applications to cosmology and inflation.

Contribution

It develops a new formulation of scalar-tensor theories in the Palatini approach, including frame-independent equations and analysis of invariants, filling a gap in existing literature.

Findings

Identified differences in invariants between metric and Palatini formalisms.

Established equivalence between $f(R)$ and scalar-tensor theories.

Derived Friedmann equations for a flat, empty universe.

Abstract

The main objective of this thesis is to discuss scalar-tensor theories in the Palatini approach. Both scalar-tensor theories and Palatini formalism are means of alternating classical theory of gravity, general relativity, in order to account for phenomena being seemingly unexplainable on the ground of the Einstein theory or to serve as toy models used to test limitations of the theory in question. In the literature, both Palatini approach and scalar-tensor theories have been widely discussed, but there are very few - if none - authors writing about a merge of these two ideas. The present paper is a result of an insufficient attention given to the topic of scalar-tensor theories in Palatini formalism. In the course of the thesis action functional for scalar-tensor theories of gravity will be introduced. This action functional differs significantly from the action defined in case of scalar-tensor theories in metric approach. We aim at analysing the theory using the language of invariants, allowing us to write down all equations in a frame-independent manner. We discover that invariants defined for the metric case do not always have their counterparts in Palatini formalism. Also, two frames most frequently used in the literature are discussed: Einstein and Jordan frame. Possible applications of the theory developed in the first part of the thesis are presented. We show the equivalence between $f(R)$ and scalar-tensor theories of gravity and exploit this fact by analysing the former using methods developed for scalar-tensor theories. We conclude the thesis with calculating the Friedmann equations for an empty universe of vanishing spatial curvature and preparing set-up for analysing inflationary behaviour.