Reconstructions of the dark-energy equation of state and the inflationary potential

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 = h (r ). For the function h (r ), we consider three cases, where h (r ) 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.

Dynamically Generated Inflationary ΛCDM

“…Pl is way larger than the current value (∼ 10 −122 M 4 Pl ) of the cosmological constant Λ DE (42). So, as in the numerical example above for the numerical solution of the system for u(t), H(t) (61)-(62), we will take again the values for the parameters according to (63) meaning that we set the initial condition for the Hubble parameter to be according to (58)…”

Noether symmetries as a geometric criterion to select theories of gravity

“…It has been shown [24,25], that the Cartan symmetries for holonomic dynamical systems are equivalent to the generalized Noether symmetries. In cosmology there are many studies in this direction (see, for example [26,27,28] and references therein).…”

Dynamically Generated Inflationary Lambda-CDM

“…Pl is way larger than the current value (∼ 10 −122 M 4 Pl ) of the cosmological constant Λ DE (42). So, as in the numerical example above for the numerical solution of the system for u(t), H(t) (61)-(62), we will take again the values for the parameters according to (63) meaning that we set the initial condition for the Hubble parameter to be according to (58) H initial = Λ 0 3 = 50 3 .…”

“…according to (42) (in reality Λ DE is much smaller than 1/500 part of Λ 0 : Λ 0 ∼ To obtain plausible values for the observables -the scalar power spectral index n s and the tensor to scalar ratio r [53,105,171]. we need the functional dependence of the slow-roll parameters and η w.r.t.…”