# A Cosmological Model Describing the Early Inflation, the Intermediate Decelerating Expansion, and the Late Accelerating Expansion of the Universe by a Quadratic Equation of State

Academic Editors: Kazuharu Bamba and Sergei D. Odintsov

Received: 28 June 2015 / Revised: 1 October 2015 / Accepted: 10 October 2015 / Published: 6 November 2015

(This article belongs to the Special Issue Modified Gravity Cosmology: From Inflation to Dark Energy)

# Abstract

We develop a cosmological model based on a quadratic equation of state \(p/c^2=-(\alpha+1){\rho^2}/{\rho_P}+\alpha\rho-(\alpha+1)\rho_ {\Lambda}\), where \(\rho_P\) is the Planck density and \(\rho_{\Lambda}\) the cosmological density, ``unifying'' vacuum energy and dark energy in the spirit of a generalized Chaplygin gas model. For \(\rho\rightarrow \rho_P\), it reduces to \(p=-\rho_P c^2\) leading to a phase of early accelerating expansion (early inflation) with a constant density equal to the Planck density \(\rho_P=5.16 \times 10^{99}\, {\rm g}/{\rm m}^3\) (vacuum energy). For \(\rho_{\Lambda}\ll\rho\ll \rho_P\), we recover the standard linear equation of state \(p=\alpha \rho c^2\) describing radiation (\(\alpha=1/3\)) or pressureless matter (\(\alpha=0\)) and leading to an intermediate phase of decelerating expansion. For \(\rho\rightarrow \rho_{\Lambda}\), we get \(p=-\rho_{\Lambda} c^2\) leading to a phase of late accelerating expansion (late inflation) with a constant density equal to the cosmological density \(\rho_{\Lambda}=7.02\times 10^{-24}\, {\rm g}/{\rm m}^3\) (dark energy). The pressure is successively negative (vacuum energy), positive (radiation and matter), and negative again (dark energy). We show a nice ``symmetry'' between the early universe (vacuum energy \(+\) \(\alpha\)-fluid) and the late universe (\(\alpha\)-fluid \(+\) dark energy). In our model, they are described by two polytropic equations of state with index \(n=+1\) and \(n=-1\) respectively. Furthermore, the Planck density \(\rho_P\) in the early universe plays a role similar to the cosmological density \(\rho_{\Lambda}\) in the late universe. They represent fundamental upper and lower density bounds differing by \(122\) orders of magnitude. The cosmological constant ``problem'' may be a false problem. We study the evolution of the scale factor, density, and pressure. Interestingly, our quadratic equation of state leads to a fully analytical model describing the evolution of the universe from the early inflation (Planck era) to the late accelerating expansion (de Sitter era). These two phases are bridged by a decelerating algebraic expansion (\(\alpha\)-era). Our model does not present any singularity at \(t=0\) and exists eternally in the past (although it may be incorrect to extrapolate the solution to the infinite past). On the other hand, it admits a scalar field interpretation based on an inflaton, quintessence, or tachyonic field. Our model generalizes the standard \(\Lambda\)CDM model by incorporating naturally a phase of early inflation that avoids the primordial singularity. Furthermore, it describes the early inflation, the intermediate decelerating expansion, and the late accelerating expansion of the universe simultaneously in terms of a single equation of state. We determine the corresponding scalar field potential that unifies the inflaton and quintessence potentials. View Full-Text*Keywords:*inflation; dark matter; dark energy; polytropic equation of state

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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