Reconstructing $f(R)$ gravity from the spectral index

Recent cosmological observations are in good agreement with the scalar spectral index $n_s$ with $n_s-1\simeq -2\mathrm{/}N$, where $N$ is the number of e-foldings. In the previous work, the reconstruction of the inflaton potential for a given $n_s$ was studied, and it was found that for $n_s-1=-2\mathrm{/}N$, the potential takes the form of either $\alpha$-attractor model or chaotic inflation model with ${\varphi }^2$ to the leading order in the slow-roll approximation. Here we consider the reconstruction of $f(R)$ gravity model for a given $n_s$ both in the Einstein frame and in the Jordan frame. We find that for $n_s-1=-2\mathrm{/}N$ (or more general $n_s-1=-p\mathrm{/}N$), $f(R)$ is given parametrically and is found to asymptote to $R^2$ for large $R$. This behavior is generic as long as the scalar potential is of slow-roll type.