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

在这项工作中，我们将提供一个与模型无关的真空光谱指数运行的一般计算$F(R)$重力。我们将利用谱指数和张量-标量比的函数形式来呈现一个一般的$n_s-r$真空关系$F(R)$引力理论。正如我们所展示的，可行的$F(R)$关于它们对运行光谱指数的预测，引力理论可以分为两类模型。这$R^2$-吸引子模型预测光谱指数在该范围内的运行$-{10}^{-3}<{\mathrm{一个}}_s<-{10}^{-4}$，这将它们归为大多数暴胀标量场模型所属的同一普遍性类别。我们提供了三种此类模型，我们对此进行了详细验证。然而存在可行的$F(R)$光谱指数超出范围的重力模型$-{10}^{-3}<{\mathrm{一个}}_s<-{10}^{-4}$在某些情况下，它可能是积极的。我们还提出了一个$R^2$-校正的标量场模型，它还预测光谱指数在该范围内的运行$-{10}^{-3}<{\mathrm{一个}}_s<-{10}^{-4}$. 对于我们研究的所有案例，我们没有发现任何证据表明最有趣的现象是$r<{10}^{-4}$和跑步${\mathrm{一个}}_s<-{10}^{-3}$，原则上是可以实现的。