The two stages of inflation with non-canonical scalar fields

This study aims to examine the phenomenological consequences of the constant-roll condition on the f(R,phi,X) gravity in the context of a non-canonical scalar field. In this regard, two periods of inflation are investigated, specifically the slow-roll and quintessence phases, within the context of the non-canonical scalar field. We will make two assumptions: first, the slow-roll condition is imposed on the first slow-roll parameters, and second, the scalar field evolves according to the constant-roll condition. We provide a comprehensive explanation of the gravitational equations of motion that govern the behavior of the cosmological system. We consider the constant-roll condition and use a specifically chosen potential to simplify analytical calculations. By doing so, we express the slow-roll indices and the resulting observational indices of the theory as functions of the e-foldings number. Our analysis demonstrates that the existence of constant-roll inflation with non-canonical scalar fields aligns with observed results across a wide range of model parameters. Furthermore, we establish an equation that combines the fundamental equations of the theory. By using a differential operator, we convert this equation into a linear form, yielding the effective equation of state (EoS) parameter. The value of this parameter is dependent upon the exponential alpha parameter of the kinetic term, as well as to the first slow-roll parameter and the constant-roll parameter sigma. Under the assumption of a fixed sigma field, we demonstrate that the early universe has the ability to experience a two-stage inflation, as described by the EoS parameter.