Gravitational reheating formulas and bounds in oscillating backgrounds II: Constraints on the spectral index and gravitational dark matter production

The reheating temperature plays a crucial role in the early universe's evolution, marking the transition from inflation to the radiation-dominated era. It directly impacts the number of $e$-folds and, consequently, the observable parameters of inflation, such as the spectral index of scalar perturbations. By establishing a relationship between the gravitational reheating temperature and the spectral index, we can derive constraints on inflationary models. Specifically, the range of viable reheating temperatures imposes bounds on the spectral index, which can then be compared with observational data, such as those from the Planck satellite, to test the consistency of various models with cosmological observations. Additionally, in the context of dark matter production, we demonstrate that gravitational reheating provides a viable mechanism when there is a relationship between the mass of the dark matter particles and the mass of the particles responsible for reheating. This connection offers a pathway to link dark matter genesis with inflationary and reheating parameters, allowing for a unified perspective on early universe dynamics.