Motivated by the ultraviolet complete theory of quantum gravity, for example the string theory, we investigate a polynomial $f(R )$ inflation model in detail. We calculate the spectral index and tensor-to-scalar ratio in the $f(R )$ inflation model with the form of $f(R )=R+{ R^2\over 6M^2}+{ \lambda_n\over 2n}{ R^n\over (3M^2)^{ n-1}}$. Compared to Planck 2013, we find that $R^n$ term should be exponentially suppressed, i.e. $|\lambda_n|\lesssim 10^{ -2n+2.6}$.
Categories: hep-th astro-ph.CO gr-qc hep-ph