Distinguishing between extra natural inflation and natural inflation after BICEP2

In this paper, we carefully calculated the tensor-to-scalar ratio, the running spectral index, and the running of running spectrum for (extra) natural inflation in order to compare with recent BICEP2 data, PLANCK satellite data and future 21 cm data. We discovered that the prediction for running spectral index and the running of running spectrum in natural inflation is different from that in the case of extra natural inflation. Near future observation for the running spectral index can only provide marginal accuracy which may not allow us distinguishing between extra natural inflation from natural inflation clearly unless the experimental accuracy can be further improved.