Parametric resonance in the Einstein frame: The Jordan-frame Doppelgänger

Abstract

Modified $f\left(R\right)$ theories of gravity have been investigated for quite a long time in the literature as a possible explanation for the inflationary period of the universe. The correspondence to General Relativity with an extra scalar field $\tilde{\varphi }$ in the so-called Einstein Frame via a conformal transformation is a major tool in this class of theories. Since the trivial forms of $f\left(R\right)$ theory of gravity have already been ruled out from observations, the quest for the “correct” $f\left(R\right)$ remains unfinished. We argue that the function $f\left(R\right)$ can be more soundly justified if one starts from its corresponding potential $V\left(\tilde{\varphi }\right)$ in the Einstein Frame. To shed some light on the subject, we focus on the phase of preheating in this work. We assume three suitably chosen potentials in the Einstein Frame and a conformal-like coupling $\sim R{\psi }^2$ in the Jordan Frame, from which we then obtain both a non-trivial $f\left(R\right)$ and a non-trivial coupling between the fields $\tilde{\varphi }$ and $\tilde{\psi }$ that yield Parametric Resonance in the Einstein Frame. We propose that a corresponding so-called vacuum awakening mechanism in the Jordan frame explains the exponential amplification of the $\psi$ field. This study is a first step towards building an important relationship between the two frames that will allow further investigations on the production of primordial gravitational waves and black holes during preheating.