Abstract
In a previous paper we introduced a cosmological model describing the early inflation, the intermediate decelerated expansion, and the late accelerating expansion of the universe in terms of a single barotropic fluid characterized by a quadratic equation of state. We obtained a scalar field representation of this fluid and determined the potential V(ϕ) connecting the inflaton potential in the early universe to the quintessence potential in the late universe. This scalar field has later been called the ‘vacuumon’ by other authors, in the context of the Running Vacuum model. In this paper, we study how the scalar field potential is modified by the presence of other cosmic components such as stiff matter, black-body radiation, baryonic matter, and dark matter. We also determine the mass m and the self-interaction constant λ of the scalar field given by the second and fourth derivatives of the potential at its extrema. We find that its mass is imaginary in the early universe with a modulus of the order of the Planck mass MP=(ℏc/G)1/2=1.22×1019GeV/c2 and real in the late universe with a value of the order of the cosmon mass mΛ=(Λℏ2/c4)1/2=2.08×10−33eV/c2 predicted by string theory. Although our model is able to describe the evolution of the homogeneous background for all times, it cannot account for the spectrum of fluctuations in the early universe. Indeed, by applying the Hamilton–Jacobi formalism to our model of early inflation, we find that the Hubble hierarchy parameters and the spectral indices lead to severe discrepancies with the observations. This suggests that the vacuumon potential is just an effective classical potential that cannot be directly used to compute the fluctuations in the early universe. A fully quantum field theory may be required to achieve that goal. Finally, we discuss the connection between our model based on a quadratic equation of state and the Running Vacuum model which assumes a variation of the cosmological constant with the Hubble parameter.
Highlights
The universe displays three main periods of evolution: an early phase of inflation during which the scale factor increases exponentially rapidly with time, an intermediate phase of decelerated expansion during which the scale factor increases algebraically, and a late phase of accelerating expansion during which the scale factor increases exponentially rapidly again.1 The idea that a period of accelerated expansion may have occured in the early universe was introduced by Guth [1] in 1981 to explain the observed isotropy, homogeneity, and flatness of the universe in a natural way
We proposed to describe the transition between the inflation and the stiff matter era in the primordial universe by an equation of state of the form of Equation (6) with α = 1, and we derived the corresponding scalar field potential (see Equation (140) in [45] and Equation (F.42) in [46]).4 (vii) In Refs. [39,40,42,44], we solved the general model described by the quadratic equation of state (5)
Using φ =Ha and the Friedmann Equation (19), we find that the relation between the scalar field and the scale factor is given by12 dφ = 3c2 1/2 1 + wφ da 8πG
Summary
The universe displays three main periods of evolution: an early phase of inflation during which the scale factor increases exponentially rapidly with time, an intermediate phase of decelerated expansion during which the scale factor increases algebraically, and a late phase of accelerating expansion during which the scale factor increases exponentially rapidly again. The idea that a period of accelerated expansion (early inflation) may have occured in the early universe was introduced by Guth [1] in 1981 to explain the observed isotropy, homogeneity, and flatness of the universe in a natural way. We obtained an exact analytical solution of the Friedmann equations giving the complete temporal evolution of the scale factor a(t) and energy density ρ(t) from t = −∞ to t = +∞ (see Equation (106) of [44]) This solution describes the early inflation, the intermediate decelerated expansion, and the late accelerating expansion of the universe. If we take α = 0, the first term in Equation (5) describes the early inflation and the last two terms describe the transition between the matter era and the late accelerating expansion of the universe This equation of state does not account for the radiation era.
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