We use a mathematical approach based on the constraints systems in order to reconstruct the equation of state and the inflationary potential for the inflaton field from the observed spectral indices for the density perturbations $$n_{s}$$ns and the tensor to scalar ratio r. From the astronomical data, we can observe that the measured values of these two indices lie on a two-dimensional surface. We express these indices in terms of the Hubble slow-roll parameters and we assume that $$n_{s}-1=h\left( r\right) $$ns-1=hr. For the function $$h\left( r\right) $$hr, we consider three cases, where $$h\left( r\right) $$hr is constant, linear and quadratic, respectively. From this, we derive second-order equations whose solutions provide us with the explicit forms for the expansion scale-factor, the scalar-field potential, and the effective equation of state for the scalar field. Finally, we show that for there exist mappings which transform one cosmological solution to another and allow new solutions to be generated from existing ones.

中文翻译：

---

暗能量状态方程的重建和暴胀潜力

我们使用基于约束系统的数学方法，以便根据密度扰动 $$n_{s}$$ns 和张量与标量比的观测光谱指数重建膨胀场的状态方程和膨胀潜力河 从天文数据中，我们可以观察到这两个指数的测量值位于二维表面上。我们用哈勃慢滚参数来表达这些指数，我们假设 $$n_{s}-1=h\left(r\right) $$ns-1=hr。对于函数$$h\left(r\right)$$hr，我们考虑三种情况，其中$$h\left(r\right)$$hr 分别是常数、线性和二次。由此，我们推导出二阶方程，其解为我们提供了扩展比例因子、标量场势的显式形式，以及标量场的有效状态方程。最后，我们表明存在将一种宇宙学解转换为另一种并允许从现有解生成新解的映射。