Consistency relation for $R^p$ inflation

We consider $R^p$ inflation with $p\approx 2$, allowing small deviation from $R^2$ inflation. Using the inflaton potential in the Einstein frame, we construct a consistency relation between the scalar spectral index, the tensor-to-scalar ratio, as well as the running of the scalar spectral index, which will be useful to constrain a deviation from $R^2$ inflation in future observations.

Figure 1

Potential for $R^p$ inflation with $p=2$ (blue solid), and 1.85, 1.90, 1.95, 2.05 (magenta dashed).

Figure 2

Time evolution of the scalaron $\varphi$ and the Hubble parameter $H$ for $p=2$ (blue solid), and 1.85, 1.90, 1.95, 2.05 (magenta dashed).

Figure 3

Left: Slow roll parameters, $\epsilon$ (blue solid), $|\eta |$ (magenta dashed), and $\xi$ (green dot-dashed). Right: Inflationary observables, $1-n_s$ (blue solid), $r$ (magenta dashed), and $-\alpha$ (green dot-dashed). The thick lines and thin lines are for $N_k=60$ and 50, respectively.

Figure 4

Scalar spectral index $n_s$, its running $\alpha$, and tensor-to-scalar ratio $r$ for $p=2$ (solid blue), and 1.95, 1.90, 1.85 (magenta dashed), where e-folds between $N_k=50$ and 60 are highlighted (red solid). Lines for fixed e-folds $N_k=40$, 50, 60, 70 (green dot-dashed) are also shown.