a cosmological paradigm of the early universe, inflation has achieved many reputations thanks to its capability in addressing several problems of Big Bang cosmology [@Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi] such as horizon, flatness, unwanted relics and so on (see [@Starobinsky:1980te; @Fang:1980wi; @Sato:1980yn] for early works). Furthermore, inflation predicts a nearly scale-invariant primordial power spectrum, which was verified to high precision by Cosmic Microwave Background (CMB) observations [@Komatsu:2010fb; @Ade:2013lta]. Due to these salient features, more and more people has been paying attention to the study of inflation and more and more models has been explored. Observational data has also been developed to better and better precision in order to verify/falsify these models.

Among a variety of theoretical models of inflation, one interesting kind of models is that based on modified gravity. In these models, Einstein’s Gravity is modified in high energy level, which can lead to a period of inflationary expansion of the universe. One of the example is the well-known Starobinsky inflation model [@Starobinsky:1980te], where a squared term of Ricci scalar has been added to the normal Lagrangian of Gravity. The newest PLANCK constraints on inflation has shown us that the Starobinsky model is among the best fit models to the observations 