$R^2$ corrections to the cosmological dynamics of inflation in the Palatini formulation

TL;DR

This paper studies how adding an $R^2$ term in the Palatini formulation affects inflationary cosmology, showing that it leaves the Friedmann equation unchanged during potential energy domination but modifies it during kinetic energy domination, with implications for initial conditions.

Contribution

It provides a detailed analysis of $R^2$ corrections in the Palatini formulation, revealing conditions on initial kinetic energy and modifications to cosmological dynamics during inflation.

Findings

Friedmann equation remains unchanged during potential energy domination.

Modified Friedmann equation arises during kinetic energy domination.

Initial kinetic energy density must be sub-Planckian in closed and flat universes.

Abstract

We investigate the corrections to the inflationary cosmological dynamics due to a $R^2$ term in the Palatini formulation which may arise as quantum corrections to the effective Lagrangian in early universe. We found that the standard Friedmann equation will not be changed when the scalar field is in the potential energy dominated era. However, in the kinetic energy dominated era, the standard Friedmann equation will be modified and in the case of closed and flat universe, the Modified Friedmann equation will automatically require that the initial kinetic energy density of the scalar field must be in sub-Planckian scale.