Running of the spectral index and inflationary dynamics of F(R) gravity

In this work we shall provide a model-independent general calculation of the running spectral index for vacuum $F(R)$ gravities. We exploit functional form and tensor-to-scalar ratio in order to present $n_s-r$ relation gravity theories. As show, viable theories can be classified two classes models regarding their prediction index. The $R^2$-attractor predict range $-10^{-3}<a_s<-10^{-4}$, which classifies them same universality class that most inflationary scalar field belong to. three sort, verify our claims detail. However there exist with outside $-10^{-3}<a_s<-10^{-4}$ some cases it positive. also an $R^2$-corrected model, predicts $-10^{-3}<a_s<-10^{-4}$. For all studied, found no evidence phenomenologically interesting scenario having $r<10^{-4}$ $a_s<-10^{-3}$, principle could realized.