Scalar field evolution at background and perturbation levels for a broad class of potentials

In this paper, a non-interacting scalar field cosmology with an arbitrary potential using the f-deviser method that relies on the differentiability properties of the potential is investigated. Using this alternative mathematical approach, a unified dynamical system analysis at a scalar field's background and perturbation levels with arbitrary potentials is presented. For illustration, a monomial and double exponential potential is considered. These two classes of potentials comprise the asymptotic behavior of several classes of scalar field potentials, and, therefore, they provide the skeleton for the typical behavior of arbitrary potentials. Moreover, the linear cosmological perturbations in the matterless case by considering three scalar perturbations: the evolution of the Bardeen potentials, the comoving curvature perturbation, the so-called Sasaki-Mukhanov variable, or the scalar field perturbation in uniform curvature gauge is analyzed. Finally, an exhaustive dynamical system analysis for each scalar perturbation is presented, including the evolution of Bardeen potentials in the presence of matter. (© 2023 The Authors. Fortschritte der Physik published by Wiley‐VCH GmbH)