Study of General Relativity and $f(R)$ Modified Gravity with Cosmological Constant for Einstein and Jordan Frames

TL;DR

This thesis explores a novel approach to inflation in modified gravity theories by starting from a simple scalar potential in the Einstein frame and deriving the corresponding $f(R)$ models, revealing new inflationary solutions and stability criteria.

Contribution

It introduces a method of deriving $f(R)$ gravity models from scalar potentials in the Einstein frame, providing new insights into inflationary solutions and stability in modified gravity.

Findings

Discovered a self-terminating inflationary solution with linear Lagrangian.

Established a stability criterion for $f(R)$ theories.

Provided a thermodynamical interpretation of the system with a cosmological constant.

Abstract

This thesis investigates a toy model for inflation in a class of modified theories of gravity in the metric formalism. Instead of the standard procedure -- assuming a non-linear Lagrangian $f(R)$ in the Jordan frame -- we start from a simple {\phi}^2 potential in the Einstein frame and investigate the corresponding $f(R)$ in the former picture. Such approach yields plenty of new pieces of information, namely a self-terminating inflationary solution with a linear Lagrangian, a robust criterion for stability of such theories, a dynamical effective potential for the Ricci scalar $R$, the addition of an ad-hoc Cosmological Constant in the Einstein frame leads to a Thermodynamical interpretation of this physical system, which allows further insight on its (meta)stability and evolution.