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 [Formula: see text] 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 [Formula: see text] parameter of the kinetic term, as well as to the first slow-roll parameter and the constant-roll parameter [Formula: see text]. Under the assumption of a fixed [Formula: see text] field, we demonstrate that the early universe has the ability to experience a two-stage inflation, as described by the EoS parameter.