Realize cosmological inflation in supersymmetric Grand Unified models with $R$-symmetry breaking

In this paper, we present a discussion for cosmological inflation based on a general renormalizable supersymmetric model which can be naturally embedded into grand unified models. Successful hybrid inflation has been realized with an effective Mexican-hat potential, while avoiding the generation of extra massless multiplets so that gauge coupling unification is maintained in the supersymmetric grand unified models. This is achieved by including $R$-symmetry violating couplings. A relatively large tensor-to-scalar ratio $r\approx 0.03$ and a spectral index $n_s\approx 0.965$ are obtained, which are expected to be further tested by future experiments. Futhermore, The hierarchy of parameters in the model can be solved at the same time, and all dimensionless parameters are in the range from 0.3 to 3 by adjustment of $R$-charges.