Power-law inflation with minimal and nonminimal coupling

Abstract

Power-law inflationary scenarios based on minimal and non-minimal coupling of a scalar field to gravity with exponential- and power-law potential, respectively, are studied using the symmetry-based approach. In particular, we obtained only one parameter Lie point symmetry for both the minimal and non-minimal coupling cases and it is interesting to note that the non-minimal coupled equation is invariant under a scale transformation. We find the exact analytical group invariant solutions from invariant curve condition for both the minimal and non-minimal cases of the power-law inflationary model. The solutions obtained are consistent with the Friedmann equations subject to constraints on the inflationary potential parameter $ \lambda$ for the minimal case and on the coupling parameter $ \zeta$ for the non-minimal case. In this scenario, we find transformation relations for various inflationary parameters e.g. amplitude of the scalar power spectrum, spectral index, slow-roll parameters (SRP), tensor-to-scalar perturbation ratio, equation of state parameters, non-Gaussianity parameter as well as the form of the potential in two different frames, namely the Jordan and Einstein frames by making use of the conformal transformation. The results for various inflationary parameters for the non-minimal case are presented in the background of Planck2015 and Planck2018 and are in good agreement with the cosmological observations if the non-minimal coupling parameter is chosen properly. We treat minimally and non-minimally coupled scalar field equations by the dynamical system theory and present critical point analysis. By checking the stability of the critical points in the phase space for both cases we have shown that the solutions obtained from the Lie symmetry approach are the stable attractor solutions.