The Minimally Coupled and Canonical Scalar Field Inflationary Cosmology with Negative Quadratic and Modified Higgs-like Potentials: A Symmetry Based Approach

The inflationary model based on a single scalar field with a quadratic potential V ( ϕ ) ∼ ϕ 2 is disfavored by the recent Planck constraints on the values of scalar spectral index, and the tensor-to-scalar ratio for cosmological density perturbations. In order to overcome the discrepancies we study the scalar field equations based on the minimally coupled and canonical Lagrangian densities with negative quadratic potential and modified Higgs-like potential. We use a Lie symmetry-based approach to study the homogeneous scalar field equations for both cases. In particular, we have investigated the Lie symmetries of the scalar field equations and use them to find the exact analytical solutions. New exact analytical solution is obtained for an inflationary model with the modified Higgs-like potential. We have calculated the values of the inflationary parameters, namely the amplitude of scalar power-spectrum ( P S ) , scalar spectral index ( n S ), its running ( n S r u n ), tensor-to-scalar ratio ( r ) and non-Gaussinity parameters. We make useful checks whether the obtained parameters are supported by the observational constraints set by the Planck2018 data. We find that the model equation with negative quadratic potential is disfavored by the recent Planck2018 constraints on the tensor-to-scalar ratio for cosmological density perturbations but the scalar field model equation with the modified Higgs-like potential resolves the problem. We have predicted the values of Higgs self-coupling constant ( λ ) that lies within the 3.924 × 10 - 14 ≲ λ ≲ 6.56 × 10 - 14 and also the values of vacuum expectation value of the scalar inflaton field lies in the range 17.17 ≲ ϕ 0 ≲ 23.54 which give best fit the Planck2018 measured range of the scalar spectral index n S and upper bound of tensor-to-scalar ratio r . The small value of λ indicates that the inflaton field is extremely weakly coupled and gives the necessary condition for successful inflation. The values of tensor-to-scalar ratio and all the other important observable indices obtained using the model with the modified Higgs-like potential lie well inside the limits set in by the Planck2018 data. But we have checked that both the models generate very small values of non-Gaussianity parameters. We also explain the exit from the inflationary phase in both the cases successfully.