Inflationary Study of the Universe in $f\left(R\right)$ Gravity Theory of Brans-Dicke Assisted by a Scalar Field

In this paper, we have studied cosmic inflation in the context of modified $f\left(R\right)$ gravity, by integrating a scalar field. The general objective is to understand the primordial phases of the universe and to test whether alternative models to general relativity can better correspond to recent observations (especially those of the Planck satellite). In this work, we used an approach based on the Brans-Dicke hypothesis, which introduces a scalar field that is minimally coupled to gravity. Two types of inflation are explored: The first approach is inflation related to the power law. Here, we have chosen a scale factor of the form $a\left(t\right)=A{t}^b$ , we then derived the evolution equations for the scalar field and the inflation potential. We analytically expressed observational parameters such as the spectral index ${\eta }_s$ and the tensor-to-scalar ratio $r$ in terms of the parameter $b$ and the folds $N$ . Specific values of $b$ (between 60 and 70) correspond to observational data. Chaotic inflation for the second approach, we chose from the outset a specific form of potential defined as $V\left(？\right)=\frac{1}{2}{m}^2{？}^2$ . The objective set is to study the cosmological implications of this form and verify the consistency of the predictions with the current constraints (for example, Planck 2018). The results obtened prove that models with high values of the parameter $r$ lead to predictions consistent with observations (spectral index close to 0.965 and low ratio $$