= Starobinsky inflation
{c}
{title2=$R+R^2/(6m^2)$}
{wiki}
<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 $m>0$:
$$
S=\frac{M^2}{2}\int d^4x\sqrt{-g}\left(R+\frac{R^2}{6m^2}\right).
$$
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 $g^E_{\mu\nu}=Fg_{\mu\nu}$, with canonical field $\phi=\sqrt{3/2}\,M\log F$. Its <Einstein frame> <scalar potential> is
$$
V(\phi)=\frac34M^2m^2\left(1-e^{-\sqrt{2/3}\,\phi/M}\right)^2.
$$
This scalar representation explains the flat large-field plateau without assuming an unrelated matter potential. The pure $R+R^2$ contribution has this scalar representation.<Embedding (R+R^2)-Inflation into Supergravity by Sergei V. Ketov and Alexei A. Starobinsky>[Section I and the first term of equation (26)]{ref}.
Near $\phi=0$, $V\simeq m^2\phi^2/2$, allowing an oscillatory post-inflationary phase. On the plateau, the leading <slow-roll approximation> gives $\epsilon_V\simeq3/(4N^2)$ and $\eta_V\simeq-1/N$ for $N$ remaining <number of e-folds>. Hence $n_s\simeq1-2/N$ and $r\simeq12/N^2$. These are model predictions, not an observational measurement; the mapping from $N$ to a measured scale also depends on <reheating>.
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