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.
Highlights
The evolution of the universe may be divided into four main periods [1]
The usual equation of state p = −ρc2 used to model the primordial inflation or the late acceleration of the universe leads to a constant density but the value of this density cannot be read from the equation of state itself and it has to be adapted, rather arbitrarily, to the situation
We have proposed in this paper to unify three phases with the quadratic equation of state (2), or more generally with Equation (82), which can be viewed as a generalized Chaplygin gas model
Summary
In the vacuum energy era (Planck era), the universe undergoes a phase of early inflation that brings it from the Planck size lP = 1.62 × 10−35 m to an almost “macroscopic” size a ∼ 10−6 m in a tiniest fraction of a second [2,3,4,5]. The universe enters in the radiation era and, when the temperature cools down below approximately 103 K, in the matter era [6]. In the dark energy era (de Sitter era), the universe undergoes a phase of late inflation [7]. The universe is composed of approximately 5% baryonic matter, 20% dark matter, and 75% dark energy [1]. Despite the success of the standard ΛCDM model, the nature of vacuum energy, dark matter, and dark energy remains very mysterious and leads to many speculations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
