Inflation and quintessence: theoretical approach of cosmological reconstruction

In the first part of this paper, we outline the construction of an inflationary cosmology in the framework where inflation is described by a universally evolving scalar field phi with potential V ( f). By considering a generic situation that inflaton attains a nearly constant velocity, during inflation, m(P)(-1) vertical bar d phi/dN vertical bar alpha + beta exp (beta N) (where N In a is the e-folding time), we reconstruct a scalar potential and find the conditions that have to be satisfied by the (reconstructed) potential to be consistent with the WMAP inflationary data. The consistency of our model with the WMAP result ( such as n(s) = 0.951(-0.019)(+0.015) and r < 0.3) would require 0.16 < alpha < 0.26 and beta < 0. The running of the spectral index, (alpha) over tilde = dn(s)/d ln k, is found to be small for a wide range of a. In the second part of this paper, we introduce a novel approach of constructing dark energy within the context of the standard scalar - tensor theory. The assumption that a scalar field might roll with a nearly constant velocity, during inflation, can also be applied to quintessence or dark energy models. For the minimally coupled quintessence, alpha(Q) = dA(Q)/d(kappa Q)= 0 ( where A( Q) is the standard matter - quintessence coupling), the dark energy equation of state in the range -1 <= w(DE) < -0.82 can be obtained for 0 <= alpha < 0.63. For alpha < 0.1, the model allows for only modest evolution of dark energy density with redshift. We also show, under certain conditions, that the alpha(Q) > 0 solution decreases the dark energy equation of state w(Q) with decreasing redshift as compared to the aQ = 0 solution. This effect can be opposite in the alpha(Q) < 0 case. The effect of the matter quintessence coupling can be significant only if vertical bar aQ vertical bar greater than or similar to 0.1, while a small coupling vertical bar aQ vertical bar < 0.1 will have almost no effect on cosmological parameters, including Omega(Q), wQ and H(z). The best fit value of aQ in our model is found to be alpha(Q) similar or equal to 0.06, but it may contain significant numerical errors, namely alpha(Q) = 0.06 +/- 0.35, which clearly implies the consistency of our model with general relativity ( for which alpha(Q) = 0) at 1 sigma level.