Starobinsky inflation

Starobinsky inflation can be formulated as a metric theory of gravity with a squared Ricci scalar in the action. In a convention with positive de Sitter Ricci scalar, use natural units, reduced Planck mass $M$ and mass scale $m>0$:The higher-curvature term carries an extra scalar degree of freedom, the scalaron. Set $F=1+R/(3m^2)>0$ and perform a conformal transformation to $g_{\mu \nu }^E=F{g}_{\mu \nu }$, with canonical field $\varphi =\sqrt{3/2}M\log F$. Its Einstein frame scalar potential isThis scalar representation explains the flat large-field plateau without assuming an unrelated matter potential. The pure $R+R^2$ contribution has this scalar representation.^[ref].

Near $\varphi =0$, $V\simeq m^2{\varphi }^2/2$, allowing an oscillatory post-inflationary phase. On the plateau, the leading slow-roll approximation gives ${ϵ}_V\simeq 3/(4N^2)$ and ${\eta }_V\simeq -1/N$ for $N$ remaining number of e-folds. Hence $n_s\simeq 1-2/N$ and $r\simeq 12/N^2$. These are model predictions, not an observational measurement; the mapping from $N$ to a measured scale also depends on reheating.