The Parameter Dependence of $\mathbf{n_{s}}$ and $\mathbf{r}$ of the Scalar Power Spectrum during Single-Field Slow-Roll Inflation: A Comparative Study of Inflationary Potentials

Abstract: Inflation in cosmology is a specific stage preceding the Big Bang, aimed at solving both old background problems and new perturbation issues. Single-field inflation is a candidate to illustrate the picture of the initial universe, and various potential functions lead to different scenarios during the inflationary stage. This paper introduces two essential parameters: the spectral index and the tensor-to-scalar ratio detected from the initial power spectrum, derived from the action of the scalar field and using approximation that the potential is flat. A brief overview of the origins of Starobinsky Inflation, Chaotic Inflation, Small Field Inflation, and Natural Inflation is also presented, along with their mathematical representations. Finally, the results derived from various inflation models regarding the index and ratio are tested using the Planck data, and the deviations in each model are analyzed.