Generality of Starobinsky and Higgs inflation in the Jordan frame

TL;DR

This paper compares the initial conditions for Starobinsky and Higgs inflation in both the Jordan and Einstein frames, revealing similarities and peculiarities in the inflationary zones and the effects of conformal transformations.

Contribution

It extends the analysis of initial conditions for inflation to arbitrary energies directly in the Jordan frame, highlighting differences and connections with the Einstein frame.

Findings

Similar initial condition zones for Higgs inflation in both frames

Irregularities in the transformation between frames for Starobinsky inflation

Conformal transformation affects the initial conditions in the $(H, R)$ plane

Abstract

We revisit the problem of generality of Starobinsky and Higgs inflation. The known results obtained in the Einstein frame are generalized for the case of an arbitrary initial energy of the scalar field. These results are compared with the results obtained directly in the Jordan frame, which, to our knowledge, has not been thoroughly explored in the literature previously. We demonstrate that the qualitative picture of initial conditions zone in the $(\phi, \dot \phi)$ plane, which leads to sufficient amount of inflation, is quite similar for both the frames in the case of Higgs inflation. For Starobinsky inflation, the conformal transformation between the frames relates the geometrical variables in the Jordan frame with the properties of an effective scalar field in the Einstein frame. We show that the transformation $(H, R) \to (\phi, \dot \phi)$ is not regular everywhere, leading to some peculiarities in the zone of good initial conditions in the $(H, R)$ plane.