Abstract
We use a mathematical approach based on the constraints systems in order to reconstruct the equation of state and the inflationary potential for the inflaton field from the observed spectral indices for the density perturbations n_{s} and the tensor to scalar ratio r. From the astronomical data, we can observe that the measured values of these two indices lie on a two-dimensional surface. We express these indices in terms of the Hubble slow-roll parameters and we assume that n_{s}-1=hleft( rright) . For the function hleft( rright) , we consider three cases, where hleft( rright) is constant, linear and quadratic, respectively. From this, we derive second-order equations whose solutions provide us with the explicit forms for the expansion scale-factor, the scalar-field potential, and the effective equation of state for the scalar field. Finally, we show that for there exist mappings which transform one cosmological solution to another and allow new solutions to be generated from existing ones.
Highlights
An ‘inflaton’ is a scalar field that can drive a period of acceleration in the early universe
Using the slow-roll expressions for these indices, we find ordinary differential equations whose solutions provide us with the inflationary scalar field potentials and the equation of state for the energy density and the pressure of the scalar field while the density perturbations to tensor-to-scalar ration diagrams are presented for the analytical solutions that we derive
In scalar-field cosmology, the dark-energy equation of state parameter (EoS) and the inflationary scalar-field potential have been reconstructed from the spectral index, ns
Summary
An ‘inflaton’ is a scalar field that can drive a period of acceleration in the early universe. The slow-roll parameters and their relations to the spectral indices have been reconstructed in closed-form [43,44,45] This is the approach that we will follow here to find the equation of state for the effective perfect fluid which corresponds to the scalar field with a self-interaction potential. While this approach is not so accurate as the previous approaches (because it depends on approximate relations between the spectral indices and the slow-roll parameters [38]) it can more reconstruct closed-form solutions for the inflationary potential and the expansion scale factor expansion. For a spatially-flat FLRW universe, with scale factor, a (t), the field equations (1–3) are
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