Abstract In this letter we shall demonstrate that the viable F ( R ) gravities can be classified mainly into two classes of inflationary attractors, either the R 2 attractors or the α-attractors. To show this, we shall derive the most general relation between the tensor-to-scalar ratio r and the spectral index of primordial curvature perturbations n s , namely the r − n s relation, by assuming that the slow-roll condition constrains the values of the slow-roll indices. As we show, the relation between the tensor-to-scalar ratio and the spectral index of the primordial curvature perturbations has the form r ≃ 48 ( 1 − n s ) 2 ( 4 − x ) 2 , where the dimensionless parameter x contains higher derivatives of the F ( R ) gravity function with respect to the Ricci scalar, and it is a function of the e-foldings number N and may also be a function of the free parameters of the various F ( R ) gravity models. For F ( R ) gravities which have a spectral index compatible with the observational data and also yield x ≪ 1 , these belong to the R 2 -type of attractors, with r ∼ 3 ( 1 − n s ) 2 , and these are viable theories. Moreover, in the case that x takes larger values in specific ranges and is constant for a given F ( R ) gravity, the resulting r − n s relation has the form r ∼ 3 α ( 1 − n s ) 2 , where α is a constant. Thus we conclude that the viable F ( R ) gravities may be classified into two limiting types of r − n s relations, one identical to the R 2 model at leading order in x, and one similar to the α-attractors r − n s relation, for the F ( R ) gravity models that yield x constant. Finally, we also discuss the case that x is not constant.

中文翻译：

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F(R) 引力中的通货膨胀吸引子

摘要 在这封信中，我们将证明可行的 F ( R ) 引力可以主要分为两类膨胀吸引子，即 R 2 吸引子或 α 吸引子。为了证明这一点，我们将通过假设慢滚条件约束 ns 的值，推导出张量与标量比 r 与原始曲率扰动的谱指数 ns 之间的最一般关系，即 r - ns 关系。慢速指标。正如我们所展示的，张量与标量之比与原始曲率扰动的谱指数之间的关系具有形式 r ≃ 48 ( 1 − ns ) 2 ( 4 − x ) 2 ，其中无量纲参数 x 包含更高的导数相对于 Ricci 标量的 F ( R ) 重力函数，它是 e 折叠数 N 的函数，也可能是各种 F ( R ) 重力模型的自由参数的函数。对于具有与观测数据兼容的光谱指数并且也产生 x ≪ 1 的 F ( R ) 重力，它们属于 R 2 型吸引子，具有 r ∼ 3 ( 1 − ns ) 2 ，这些是可行的理论. 此外，如果 x 在特定范围内取更大的值并且对于给定的 F ( R ) 重力是常数，则产生的 r − ns 关系具有形式 r ∼ 3 α ( 1 − ns ) 2 ，其中 α 是常数. 因此，我们得出结论，可行的 F ( R ) 引力可以分为两种限制类型的 r − ns 关系，一种与 x 中前导阶的 R 2 模型相同，另一种类似于 α-吸引子 r − ns 关系，对于产生 x 常数的 F ( R ) 重力模型。最后，