A Cosmological Model Based on a Quadratic Equation of State Unifying Vacuum Energy, Radiation, and Dark Energy

Abstract

We consider a cosmological model based on a quadratic equation of state 𝑝/𝑐²=−4𝜌²/3𝜌_𝑃+𝜌/3−4𝜌_Λ/3 (where 𝜌_𝑃 is the Planck density and 𝜌_Λ is the cosmological density) “unifying” vacuum energy, radiation, and dark energy. For 𝜌→𝜌_𝑃, it reduces to 𝑝=−𝜌𝑐² leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density 𝜌_𝑃=5.16⋅10⁹⁹ g/m³ (vacuum energy). For 𝜌_Λ≪𝜌≪𝜌_𝑃, we recover the equation of state of radiation 𝑝=𝜌𝑐²/3. For 𝜌→𝜌_Λ, we get 𝑝=−𝜌𝑐² leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological density 𝜌_Λ=7.02⋅10⁻²⁴ g/m³ (dark energy). The temperature is determined by a generalized Stefan-Boltzmann law. We show a nice “symmetry” between the early universe (vacuum energy + radiation) and the late universe (radiation + 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 𝜌_𝑃 in the early universe plays a role similar to that of the cosmological density 𝜌_Λ in the late universe. They represent fundamental upper and lower density bounds differing by 122 orders of magnitude. We add the contribution of baryonic matter and dark matter considered as independent species and obtain a simple cosmological model describing the whole evolution of the universe. We study the evolution of the scale factor, density, and temperature. This model gives the same results as the standard ΛCDM model for 𝑡>23.3𝑡_𝑃, where 𝑡_𝑃=5.39⋅10⁻⁴⁴s is the Planck time and completes it by incorporating the phase of early inflation in a natural manner. Furthermore, this 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). Our study suggests that vacuum energy, radiation, and dark energy may be the manifestation of a unique form of “generalized radiation.” By contrast, the baryonic and dark matter components of the universe are treated as different species. This is at variance with usual models (quintessence, Chaplygin gas, ...) trying to unify dark matter and dark energy.

1. Introduction

The evolution of the universe may be divided into four main periods [1]. In the vacuum energy era (Planck era), the universe undergoes a phase of early inflation that brings it from the Planck size 𝑙_𝑃=1.62⋅10⁻³⁵ m to an almost “macroscopic” size 𝑎∼10⁻⁶ m in a tiniest fraction of a second [2–5]. The universe then enters in the radiation era and, when the temperature cools down below approximately 10³ K, in the matter era [6]. Finally, in the dark energy era (de Sitter era), the universe undergoes a phase of late inflation [7]. The early inflation is necessary to solve notorious difficulties such as the singularity problem, the flatness problem, and the horizon problem [2–5]. The late inflation is necessary to account for the observed accelerating expansion of the universe [8–11]. At present, the universe is composed of approximately 5% baryonic matter, 20% dark matter, and 75% dark energy [1]. Despite the success of the standard model, the nature of vacuum energy, dark matter, and dark energy remains very mysterious and leads to many speculations.

The phase of inflation in the early universe is usually described by some hypothetical scalar field 𝜙 with its origin in quantum fluctuations of vacuum [2–5]. This leads to an equation of state 𝑝=−𝜌𝑐², implying a constant energy density, called the vacuum energy. This energy density is usually identified with the Planck density 𝜌_𝑃=5.16⋅10⁹⁹g/m³. As a result of the vacuum energy, the universe expands exponentially rapidly on a timescale of the order of the Planck time  𝑡_𝑃=5.39⋅10⁻⁴⁴s. This phase of early inflation is followed by the radiation era described by an equation of state 𝑝=𝜌𝑐²/3. In the vacuum energy era, the density has a constant value 𝜌≃𝜌_𝑃, while it decreases as 𝜌∝𝑎⁻⁴ during the radiation era.

The phase of acceleration in the late universe is usually ascribed to the cosmological constant Λ which is equivalent to a constant energy density 𝜌_Λ=Λ/(8𝜋𝐺)=7.02⋅10⁻²⁴g/m³ called the dark energy. This acceleration can be modeled by an equation of state 𝑝=−𝜌𝑐², implying a constant energy density identified with the cosmological density 𝜌_Λ. As a result of the dark energy, the universe expands exponentially rapidly on a timescale of the order of the cosmological time 𝑡_Λ=1.46⋅10¹⁸s (de Sitter solution). This leads to a phase of late inflation. Inspired by the analogy with the early inflation, some authors have represented the dark energy by a scalar field called quintessence [12–24]. As an alternative to the quintessence, other authors have proposed to model the acceleration of the universe by an exotic fluid with an equation of state of the form 𝑝/𝑐²=𝐵𝜌−𝐴/𝜌^𝑎 with 𝐴≥0 and 𝑎≥−1 called the Chaplygin gas when (𝐵,𝑎)=(0,1) and the generalized Chaplygin gas otherwise [25–32]. This equation of state may be rewritten as 𝑝=𝛼𝜌+𝑘𝜌^1+1/𝑛𝑐², (1) with 𝑘≤0 in order to emphasize its analogy with the polytropic equation of state [33–36]. Indeed, it may be viewed as the sum of a linear equation of state 𝑝=𝛼𝜌𝑐² describing dust matter (𝛼=0) or radiation (𝛼=1/3) and a polytropic equation of state 𝑝=𝑘𝜌^1+1/𝑛𝑐². We shall call (1) a generalized polytropic equation of state. At late times, this equation of state with 𝑛<0 leads to a constant energy density implying an exponential growth of the scale factor that is similar to the effect of the cosmological constant. At earlier times, and for 𝛼=0, this equation of state returns the results of the dust matter model (𝑝≃0). Therefore, this hydrodynamic model provides a unification of matter (𝜌∝𝑎⁻³) and dark energy (𝜌=𝜌_Λ) in the late universe [25–36].

In [34–36], we remarked that a constant pressure 𝑝=−𝜌_Λ𝑐² has the same virtues as the Chaplygin gas model. (Actually, this is a particular case of the generalized Chaplygin gas model corresponding to 𝐵=0 and 𝑎=0.) This equation of state may be viewed as a generalized polytropic equation of state (1) with polytropic index 𝑛=−1, polytropic constant 𝑘=−𝜌_Λ, and linear coefficient 𝛼=0. At late times, this equation of state leads to a constant energy density equal to 𝜌_Λ implying an exponential growth of the scale factor (late inflation). At earlier times, this equation of state returns the results of the dust matter model (𝑝≃0). Therefore, as any Chaplygin gas model with 𝛼=0 and 𝑛<0, this equation of state provides a unification of matter (𝜌∝𝑎⁻³) and dark energy (𝜌=𝜌_Λ) in the late universe. Moreover, this equation of state gives the same results as the standard ΛCDM model where matter and dark energy are treated as independent species. This is interesting since the ΛCDM model is consistent with observations and provides a good description of the late universe. (By contrast, the Chaplygin gas model is not consistent with observations unless 𝑎 is close to zero [37]. Therefore, observations tend to select the constant pressure equation of state 𝑝=−𝜌_Λ𝑐² (𝑎=0, 𝑛=−1) among the whole family of Chaplygin gas models of the form 𝑝=−𝐴/𝜌^𝑎.) In [34–36], we also proposed to describe the transition between the vacuum energy era and the radiation era in the early universe by a unique equation of state of the form 𝑝/𝑐²=−4𝜌²/3𝜌_𝑃+𝜌/3. This equation of state may be viewed as a generalized polytropic equation of state (1) with polytropic index 𝑛=1, polytropic constant 𝑘=−4/(3𝜌_𝑃), and linear coefficient 𝛼=1/3. At early times, this equation of state leads to a constant energy density equal to 𝜌_𝑃 implying an exponential growth of the scale factor (early inflation). At later times, and for 𝛼=1/3, this equation of state returns the results of the radiation model (𝑝=𝜌𝑐²/3). Therefore, this hydrodynamic model provides a unification of vacuum energy (𝜌=𝜌_𝑃) and radiation (𝜌∝𝑎⁻⁴) in the early universe. This equation of state may be viewed as an extension of the Chaplygin (or polytropic) gas model to the early universe. In this approach, the late universe is described by negative polytropic indices 𝑛<0 (e.g., 𝑛=−1) and the early universe by positive polytropic indices 𝑛>0 (e.g., 𝑛=+1) [34–36]. Indeed, for 𝑛<0, the polytropic equation of state dominates in the late universe where the density is low, and for 𝑛>0, the polytropic equation of state dominates in the early universe where the density is high.

In the viewpoint just exposed, one tries to “unify” vacuum energy + radiation on the one hand and dust matter + dark energy on the other hand. (It is oftentimes argued that the dark energy (cosmological constant) corresponds to the vacuum energy. This leads to the so-called cosmological problem [38] since the cosmological density 𝜌_Λ=7.02⋅10⁻²⁴g/m³ and the Planck density 𝜌_𝑃=5.16⋅10⁹⁹g/m³ differ by about 122 orders of magnitude. We think that it is a mistake to identify the dark energy with the vacuum energy. In this paper, we shall regard the vacuum energy and the dark energy as two distinct entities. We shall call vacuum energy the energy associated with the Planck density and dark energy the energy associated with the cosmological density. The vacuum energy is responsible for the inflation in the early universe and the dark energy for the inflation in the late universe. In this viewpoint, the vacuum energy is due to quantum mechanics and the dark energy is an effect of general relativity. The cosmological constant Λ is interpreted as a fundamental constant of nature applying to the cosmophysics in the same way the Planck constant ℏ applies to the microphysics.) In this paper, we adopt a different point of view. We propose that vacuum energy, radiation, and dark energy may be the manifestation of a unique form of “generalized radiation”. We propose to describe it by a quadratic equation of state 𝑝=−4𝜌²3𝜌_𝑃𝑐²+13𝜌𝑐²−43𝜌_Λ𝑐², (2) involving the Planck density and the cosmological density. In the early universe, we recover the equation of state 𝑝/𝑐²=−4𝜌²/3𝜌_𝑃+𝜌/3 unifying vacuum energy and radiation. In the late universe, we obtain an equation of state 𝑝/𝑐²=𝜌/3−4𝜌_Λ/3 unifying radiation and dark energy. This equation of state may be viewed as a generalized polytropic equation of state (1) with polytropic index 𝑛=−1, polytropic constant 𝑘=−4𝜌_Λ/3, and linear coefficient 𝛼=1/3. The quadratic equation of state (2) “unifies” vacuum energy, radiation, and dark energy in a natural and simple manner. It also allows us to obtain a generalization of the Stefan-Boltzmann law for the evolution of the temperature [see (69)] that is valid during the whole evolution of the universe. In this approach, baryonic matter and dark matter are treated as different species. This is at variance with the usual scenario where vacuum energy + radiation on the one hand and dark matter + dark energy on the other hand are treated as different species.

The paper is organized as follows. In Section 2, we recall the basic equations of cosmology that will be needed in our study. In Section 3, we describe the transition between the vacuum energy era and the radiation era in the early universe. In Section 4, we describe the transition between the radiation era and the dark energy era in the late universe. In Section 5, we introduce the general model where vacuum energy + radiation + dark energy are described by a quadratic equation of state while baryonic matter and dark matter are considered as different species. We determine the general equations giving the evolution of the scale factor, density, and temperature (Section 5.1). We consider particular limits of these equations in the early universe (Section 5.2) and in the late universe (Sections 5.3 and 5.4). We connect these two limits and describe the whole evolution of the universe (Section 5.5). This model gives the same results as the standard ΛCDM model for 𝑡>23.3𝑡_𝑃 and completes it by incorporating the early inflation in a natural manner [see (100)]. It also reveals a nice “symmetry” between the early universe (vacuum energy + radiation) and the late universe (radiation + dark energy). These two phases are described by polytropic equations of state with index 𝑛=+1 and 𝑛=−1, respectively. Furthermore, the cosmological density 𝜌_Λ in the late universe plays a role similar to the Planck density 𝜌_𝑃 in the early universe. They represent fundamental upper and lower density bounds differing by 122 orders of magnitude.

2. Basic Equations of Cosmology

In a space with uniform curvature, the line element is given by the Friedmann-Lemaître-Roberston-Walker (FLRW) metric 𝑑𝑠²=𝑐²𝑑𝑡²−𝑎(𝑡)²𝑑𝑟²1−𝑘𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜙², (3) where 𝑎(𝑡) represents the radius of curvature of the 3-dimensional space, or the scale factor. (In our conventions, 𝑟 and 𝑘 are dimensionless, while the scale factor 𝑎(𝑡) has the dimension of a length. It is defined such that 𝑎₀=𝑐/𝐻₀ (where 𝑐 is the speed of light and 𝐻₀ is the Hubble constant) at the present epoch. This corresponds to the cosmological horizon. The usual (dimensionless) scale factor is 𝑎(𝑡)=𝑎(𝑡)/𝑎₀. It satisfies 𝑎₀=1 at the present epoch.) By an abuse of language, we shall sometimes call it the “radius of the universe.” On the other hand, 𝑘 determines the curvature of space. The universe may be closed (𝑘>0), flat (𝑘=0), or open (𝑘<0).

If the universe is isotropic and homogeneous at all points in conformity with the line element (3) and contains a uniform perfect fluid of energy density 𝜖(𝑡)=𝜌(𝑡)𝑐² and isotropic pressure 𝑝(𝑡), the energy-momentum tensor 𝑇^𝑖_𝑗 is 𝑇⁰₀=𝜌𝑐²,𝑇¹₁=𝑇²₂=𝑇³₃=−𝑝. (4) The Einstein equations 𝑅^𝑖_𝑗−12𝑔^𝑖_𝑗𝑅−Λ𝑔^𝑖_𝑗=−8𝜋𝐺𝑐²𝑇^𝑖_𝑗 (5) relate the geometrical structure of the spacetime (𝑔_𝑖𝑗) to the material content of the universe (𝑇_𝑖𝑗). For the sake of generality, we have accounted for a possibly nonzero cosmological constant Λ. Given Equations (3) and (4), these equations reduce to 𝑎̇8𝜋𝐺𝜌+Λ=3²+𝑘𝑐²𝑎²,8𝜋𝐺𝑐²̈𝑎̇𝑝−Λ=−2𝑎𝑎+²+𝑘𝑐²𝑎², (6) where dots denote differentiation with respect to time. These are the well-known cosmological equations describing a non-static universe first derived by Friedmann [6].

The Friedmann equations are usually written in the form 𝑑𝜌𝑎̇𝑑𝑡+3𝑎𝑝𝜌+𝑐²𝑎̈=0,𝑎=−4𝜋𝐺3𝜌+3𝑝𝑐²+Λ3,𝐻²=𝑎̇𝑎²=8𝜋𝐺3𝜌−𝑘𝑐²𝑎²+Λ3, (7)(8)(9) where we have introduced the Hubble parameter 𝐻=𝑎/̇𝑎. Among these three equations, only two are independent. The first equation, which can be viewed as an “equation of continuity”, can be directly derived from the conservation of the energy momentum tensor 𝜕_𝑖𝑇^𝑖𝑗=0, which results from the Bianchi identities. For a given barotropic equation of state 𝑝=𝑝(𝜌), it determines the relation between the density and the scale factor. Then, the temporal evolution of the scale factor is given by (9).

Equivalent expressions of the “equation of continuity” are 𝑎𝑑𝜌𝑝𝑑𝑎=−3𝜌+𝑐²,𝑑𝑑𝑎𝜌𝑎³𝑝=−3𝑐²𝑎²,𝑎³𝑑𝑝=𝑑𝑑𝑡𝑎𝑑𝑡³𝑝+𝜌𝑐². (10)(11)(12) Introducing the volume 𝑉=(4/3)𝜋𝑎³ and the energy 𝐸=𝜌𝑐²𝑉, (11) becomes 𝑑𝐸=−𝑝𝑑𝑉. This can be viewed as the first principle of thermodynamics for an adiabatic evolution of the universe 𝑑𝑆=0.

From the first principle of thermodynamics in the form 𝑑𝑆(𝑉,𝑇)=(1/𝑇)[𝑑(𝜌(𝑇)𝑐²𝑉)+𝑝(𝑇)𝑑𝑉], one can derive the thermodynamical equation [6] 𝑑𝑝=1𝑑𝑇𝑇𝜌𝑐²+𝑝. (13) For a given barotropic equation of state 𝑝=𝑝(𝜌), this equation can be integrated to obtain the relation 𝑇=𝑇(𝜌) between the temperature and the density. Combining (12) with (13), we get [6] that 𝑆̇=0,where𝑆=𝑎³𝑇𝑝+𝜌𝑐² (14) is the entropy of the universe in a volume 𝑎³. This confirms that the Friedmann equations (7)–(9) imply the conservation of the entropy.

In this paper, we consider a flat universe (𝑘=0) in agreement with the observations of the cosmic microwave background (CMB) [1]. On the other hand, we set Λ=0 because the effect of the cosmological constant will be taken into account in the equation of state. The Friedmann equations then reduce to 𝑑𝜌𝑎̇𝑑𝑡+3𝑎𝑝𝜌+𝑐²𝑎̈=0,𝑎=−4𝜋𝐺3𝜌+3𝑝𝑐²,𝐻²=𝑎̇𝑎²=8𝜋𝐺3𝜌. (15)(16)(17) The deceleration parameter is defined by 𝑞(𝑡)̈=−𝑎𝑎𝑎̇². (18) The universe is decelerating when 𝑞>0 and accelerating when 𝑞<0. Introducing the equation of state parameter 𝑤=𝑝/𝜌𝑐² and using the Friedmann equations (16) and (17), we obtain for a flat universe 𝑞(𝑡)=1+3𝑤(𝑡)2. (19) We see from (19) that the universe is decelerating if 𝑤>−1/3 (strong energy condition) and accelerating if 𝑤<−1/3. On the other hand, according to (15), the density decreases with the scale factor if 𝑤>−1 (null dominant energy condition) and increases with the scale factor if 𝑤<−1.

3. The Early Universe

In [34–36], we have proposed to describe the transition between the vacuum energy era and the radiation era in the early universe by a single equation of state of the form 1𝑝=3𝜌𝑐²−4𝜌²3𝜌_𝑃𝑐², (20) where 𝜌_𝑃=𝑐⁵/𝐺²ℏ=5.16⋅10⁹⁹g/m³ is the Planck density. This equation of state corresponds to a generalized polytropic equation of state (1) with 𝛼=1/3, 𝑛=1, and 𝑘=−4/(3𝜌_𝑃). For 𝜌≪𝜌_𝑃, we recover the equation of state of the pure radiation 𝑝∼𝜌𝑐²/3. For 𝜌→𝜌_𝑃, we get 𝑝→−𝜌_𝑃𝑐² corresponding to the vacuum energy.

The continuity equation (15) may be integrated into 𝜌𝜌=^𝑃𝑎/𝑎₁⁴,+1 (21) where 𝑎₁ is a constant of integration. This characteristic scale marks the transition between the vacuum energy era and the radiation era. The equation of state (20) interpolates smoothly between the vacuum energy era (𝑝=−𝜌𝑐² and 𝜌=𝜌_𝑃) and the radiation era (𝑝=𝜌𝑐²/3 and 𝜌_rad∼𝜌_𝑃𝑎⁴₁/𝑎⁴). It provides therefore a “unified” description of vacuum energy and radiation in the early universe. This amounts to summing the inverse of the densities of these two phases. Indeed, (21) may be rewritten as 1𝜌=1𝜌_rad+1𝜌_𝑃. (22) At 𝑎=𝑎₁ we have 𝜌_rad=𝜌_𝑃 so that 𝜌=𝜌_𝑃/2.

The equation of state parameter 𝑤=𝑝/𝜌𝑐² and the deceleration parameter 𝑞 are given by 1𝑤=3−4𝜌3𝜌_𝑃𝜌,𝑞=1−2𝜌_𝑃. (23) The velocity of sound 𝑐²_𝑠=𝑝^′(𝜌) is given by 𝑐²_𝑠𝑐²=13−83𝜌𝜌_𝑃. (24) As the universe expands from 𝑎=0 to 𝑎=+∞, the density decreases from 𝜌_𝑃 to 0, the equation of state parameter 𝑤 increases from −1 to 1/3, the deceleration parameter 𝑞 increases from −1 to 1, and the ratio (𝑐_𝑠/𝑐)² increases from −7/3 to 1/3.

3.1. The Vacuum Energy Era: Early Inflation

When 𝑎≪𝑎₁, the density tends to a maximum value 𝜌=𝜌_max=𝜌_𝑃, (25) and the pressure tends to 𝑝=−𝜌_𝑃𝑐². The Planck density 𝜌_𝑃=5.16⋅10⁹⁹g/m³ (vacuum energy) represents a fundamental upper bound for the density. A constant value of the density 𝜌≃𝜌_𝑃 gives rise to a phase of early inflation. From the Friedmann equation (17), we find that the Hubble parameter is constant 𝐻=(8𝜋/3)^1/2𝑡_𝑃⁻¹ where we have introduced the Planck time 𝑡_𝑃=1/(𝐺𝜌_𝑃)^1/2=(ℏ𝐺/𝑐⁵)^1/2. Numerically, 𝐻=5.37⋅10⁴³s⁻¹. Therefore, the scale factor increases exponentially rapidly with time as 𝑎(𝑡)∼𝑙_𝑃𝑒^{(8𝜋/3)1/2𝑡/𝑡𝑃}. (26) The timescale of the exponential growth is the Planck time 𝑡_𝑃=5.39⋅10⁻⁴⁴s. We have defined the “original” time 𝑡=0 such that 𝑎(0) is equal to the Planck length 𝑙_𝑃=𝑐𝑡_𝑃=(𝐺ℏ/𝑐³)^1/2=1.62⋅10⁻³⁵m. Mathematically speaking, the universe exists at any time in the past (𝑎→0 and 𝜌→𝜌_𝑃 for 𝑡→−∞), so there is no primordial singularity. However, when 𝑎→0, we cannot ignore quantum fluctuations associated with the spacetime. In that case, we cannot use the classical Einstein equations anymore and a theory of quantum gravity is required. It is not known whether quantum gravity will remove, or not, the primordial singularity. Therefore, we cannot extrapolate the solution (26) to the infinite past. However, this solution may provide a semiclassical description of the phase of early inflation when 𝑎>𝑙_𝑃.

3.2. The Radiation Era

When 𝑎≫𝑎₁, we recover the equation 𝜌_rad∼𝜌_𝑃𝑎⁴₁𝑎⁴, (27) corresponding to the pure radiation described by an equation of state 𝑝=𝜌𝑐²/3. The conservation of 𝜌_rad𝑎⁴ implies that 𝜌_𝑃𝑎⁴₁=𝜌_rad,0𝑎⁴₀ where 𝜌_rad,0 is the present density of radiation and 𝑎₀=𝑐/𝐻₀=1.32⋅10²⁶m the present distance of cosmological horizon determined by the Hubble constant 𝐻₀=2.27⋅10⁻¹⁸s⁻¹ (the Hubble time is 𝐻₀⁻¹=4.41⋅10¹⁷s). Writing  𝜌_rad,0=Ω_rad,0𝜌₀ where 𝜌₀=3𝐻²₀/8𝜋𝐺=9.20⋅10⁻²⁴g/m³ is the present density of the universe given by (17) and Ω_rad,0=8.48⋅10⁻⁵ is the present fraction of radiation in the universe [1], we obtain 𝑎₁=2.61⋅10⁻⁶m. (28) This scale is intermediate between the Planck length 𝑙_𝑃 and the present size of the universe 𝑎₀ (𝑎₁/𝑙_𝑃=1.61⋅10²⁹ and 𝑎₁/𝑎₀=1.97⋅10⁻³²). It gives the typical size of the universe at the end of the inflationary phase (or at the beginning of the radiation era). When 𝑎≫𝑎₁, the Friedmann equation (17) yields 𝑎𝑎₁∼32𝜋3^1/4𝑡𝑡_𝑃^1/2. (29) We then have 𝜌𝜌_𝑃∼3𝑡32𝜋𝑡^𝑃2. (30) During the radiation era, the density decreases algebraically as the universe expands.

3.3. The General Solution

The general solution of the Friedmann equation (17) with (21) is 𝑎/𝑎₁⁴+1−ln1+𝑎/𝑎₁⁴+1𝑎/𝑎₁²=28𝜋3^1/2𝑡𝑡_𝑃+𝐶, (31) where  𝐶  is a constant of integration. It is determined such that 𝑎=𝑙_𝑃 at 𝑡=0. Setting  𝜖=𝑙_𝑃/𝑎₁, we get 𝐶(𝜖)=√𝜖⁴√+1−ln1+𝜖⁴+1𝜖². (32) Numerically,  𝜖=6.20⋅10⁻³⁰ and 𝐶=−134. For 𝑡→−∞, we have the exact asymptotic result 𝑎(𝑡)∼𝑎₁𝑒^{(8𝜋/3)1/2𝑡/𝑡𝑃+𝐷}, (33) with 𝐷=(𝐶+ln2−1)/2. Due to the smallness of 𝜖, a very good approximation of 𝐶 and 𝐷 is given by 𝐶≃1−ln2+2ln𝜖 and 𝐷≃ln𝜖. With this approximation, (33) returns (26). Using (23), the universe is accelerating when 𝑎<𝑎_𝑐 (i.e., 𝜌>𝜌_𝑐) and decelerating when 𝑎>𝑎_𝑐 (i.e., 𝜌<𝜌_𝑐) where 𝑎_𝑐=𝑎₁ and 𝜌_𝑐=𝜌_𝑃/2. The time 𝑡_𝑐 at which the universe starts decelerating is given by 𝑡_𝑐=(3/32𝜋)^1/2[√2−ln(1+√2)−𝐶]𝑡_𝑃. This corresponds to the time at which the curve 𝑎(𝑡) presents an inflexion point. It turns out that this inflexion point 𝑎_𝑐 coincides with 𝑎₁ so it also marks the end of the inflation (𝑡_𝑐=𝑡₁). Numerically, 𝜌_𝑐=𝜌₁=0.5𝜌_𝑃=2.58⋅10⁹⁹g/m³, 𝑎_𝑐=𝑎₁=2.61⋅10⁻⁶m, and 𝑡_𝑐=𝑡₁=23.3𝑡_𝑃=1.25⋅10⁻⁴²s. At 𝑡=𝑡_𝑖=0, 𝑎_𝑖=𝑙_𝑃 and 𝜌_𝑖≃𝜌_𝑃. Numerically, 𝑎_𝑖=𝑙_𝑃=1.62⋅10⁻³⁵m and 𝜌_𝑖≃𝜌_𝑃=5.16⋅10⁹⁹g/m³.

3.4. The Temperature and the Pressure

The thermodynamical equation (13) with the equation of state (20) may be integrated into 𝑇=𝑇_𝑃15𝜋^21/4𝜌1−𝜌_𝑃^7/4𝜌𝜌_𝑃^1/4, (34) where 𝑇_𝑃 is a constant of integration. Using (21), we get 𝑇=𝑇_𝑃15𝜋^21/4𝑎/𝑎₁⁷𝑎/𝑎₁⁴+1². (35) Equation (34) may be seen as a generalized Stefan-Boltzmann law in the early universe. In the radiation era 𝜌≪𝜌_𝑃, it reduces to the ordinary Stefan-Boltzmann law 𝑇=𝑇_𝑃15𝜋^21/4𝜌𝜌_𝑃^1/4. (36) This allows us to identify the constant of integration 𝑇_𝑃 with the Planck temperature 𝑇_𝑃=(ℏ𝑐⁵/𝐺𝑘²_𝐵)^1/2=1.42⋅10³²K. On the other hand, in the inflationary phase 𝜌→𝜌_𝑃, the temperature is related to the density by 𝑇𝑇_𝑃≃15𝜋^21/4𝜌1−𝜌_𝑃^7/4. (37) According to (35), 𝑇∼𝑇_𝑃(15/𝜋²)^1/4(𝑎/𝑎₁)⁷ for 𝑎≪𝑎₁ and 𝑇∼𝑇_𝑃(15/𝜋²)^1/4(𝑎₁/𝑎) for 𝑎≫𝑎₁.

The temperature starts from 𝑇=0 at 𝑡=−∞, increases exponentially rapidly during the inflation as 𝑇𝑇_𝑃∼15𝜋^21/4𝑙_𝑃𝑎₁⁷𝑒^{7(8𝜋/3)1/2𝑡/𝑡𝑃} (38) reaches a maximum value 𝑇_𝑒 and decreases algebraically as 𝑇𝑇_𝑃∼4532𝜋^31/4𝑡_𝑃𝑡^1/2, (39) during the radiation era. Using (34) and (35), we find that the point corresponding to the maximum temperature is 𝜌_𝑒=𝜌_𝑃/8, 𝑎_𝑒=7^1/4𝑎₁, 𝑇_𝑒=(7/8)^7/4(15/8𝜋²)^1/4𝑇_𝑃. It is reached at a time 𝑡_𝑒=(3/32𝜋)^1/2[√8−ln(1+√8)+ln(7)/2−𝐶]𝑡_𝑃. Numerically, 𝜌_𝑒=0.125𝜌_𝑃=6.44⋅10⁹⁸g/m³, 𝑎_𝑒=1.63𝑎₁=4.24⋅10⁻⁶m, 𝑇_𝑒=0.523𝑇_𝑃=7.40⋅10³¹K, and 𝑡_𝑒=23.6𝑡_𝑃=1.27⋅10⁻⁴²s. At 𝑡=𝑡_𝑖=0, 𝑇_𝑖≃𝑇_𝑃(15/𝜋²)^1/4(𝑙_𝑃/𝑎₁)⁷. Numerically, 𝑇_𝑖=3.91⋅10⁻²⁰⁵𝑇_𝑃=5.54⋅10⁻¹⁷³K. At 𝑡=𝑡_𝑐=𝑡₁, 𝑇_𝑐=(1/4)(15/𝜋²)^1/4𝑇_𝑃. Numerically, 𝑇_𝑐=𝑇₁=0.278𝑇_𝑃=3.93⋅10³¹K.

The pressure is given by (20). Using (21), we get 1𝑝=3𝑎/𝑎₁⁴−3𝑎/𝑎₁⁴+1²𝜌_𝑃𝑐². (40) The pressure starts from 𝑝=−𝜌_𝑃𝑐² at 𝑡=−∞, remains approximately constant during the inflation, becomes positive, reaches a maximum value 𝑝_𝑒, and decreases algebraically during the radiation era. At 𝑡=𝑡_𝑖=0, 𝑝_𝑖≃−𝜌_𝑃𝑐². Numerically, 𝑝_𝑖=−4.64⋅10¹¹⁶g/ms². The point at which the pressure vanishes is 𝑎_𝑤=3^1/4𝑎₁, 𝜌_𝑤=𝜌_𝑃/4, 𝑇_𝑤=(9/16)(5/𝜋²)^1/4𝑇_𝑃. This corresponds to a time 𝑡_𝑤=(3/32𝜋)^1/2[2−(1/2)ln(3)−𝐶]𝑡_𝑃. Numerically, 𝑎_𝑤=1.32𝑎₁=3.43⋅10⁻⁶m, 𝜌_𝑤=0.25𝜌_𝑃=1.29⋅10⁹⁹g/m³, 𝑇_𝑤=0.475𝑇_𝑃=6.74⋅10³¹K, and 𝑡_𝑤=23.4𝑡_𝑃=1.26⋅10⁻⁴²s. On the other hand, the pressure reaches its maximum 𝑝_𝑒=𝜌_𝑃𝑐²/48 at the same point as the one at which the temperature reaches its maximum. Numerically, 𝑝_𝑒=9.66⋅10¹¹⁴g/ms². At 𝑡=𝑡_𝑐=𝑡₁, we have 𝑝_𝑐=𝑝₁=−𝜌_𝑃𝑐²/6. Numerically, 𝑝_𝑐=𝑝₁=−7.73⋅10¹¹⁵g/ms².

Using (20), (21), and (35), we find that the entropy (14) is given by 4𝑆=3𝜋²15^1/4𝜌_𝑃𝑐²𝑇_𝑃𝑎³₁, (41) and we check that it is a constant. Numerically, 𝑆/𝑘_𝐵=5.04⋅10⁸⁷.

3.5. The Evolution of the Early Universe

In our model, the universe starts at 𝑡=−∞ with a vanishing radius 𝑎=0, a finite density 𝜌=𝜌_𝑃=5.16⋅10⁹⁹g/m³, a finite pressure 𝑝=−𝜌_𝑃𝑐²=−4.64⋅10¹¹⁶g/ms², and a vanishing temperature 𝑇=0. The universe exists at any time in the past and does not present any singularity. For 𝑡<0, the radius of the universe is less than the Planck length 𝑙_𝑃=1.62⋅10⁻³⁵m. In the Planck era, quantum gravity should be taken into account and our semiclassical approach is probably not valid in the infinite past. At 𝑡=𝑡_𝑖=0, the radius of the universe is equal to the Planck length 𝑎_𝑖=𝑙_𝑃=1.62⋅10⁻³⁵m. The corresponding density, temperature, and pressure are 𝜌_𝑖≃𝜌_𝑃=5.16⋅10⁹⁹g/m³, 𝑇_𝑖=3.91⋅10⁻²⁰⁵𝑇_𝑃=5.54⋅10⁻¹⁷³K, and 𝑝_𝑖≃−𝜌_𝑃𝑐²=−4.64⋅10¹¹⁶g/ms². We note that quantum mechanics regularizes the finite time singularity present in the standard Big Bang theory. This is similar to finite size effects in second-order phase transitions (see Section 3.6). The Big Bang theory is recovered for ℏ=0. We also note that the universe is very cold at 𝑡=𝑡_𝑖=0, contrary to what is predicted by the Big Bang theory (a naive extrapolation of the law 𝑇∝𝑡^−1/2 leads to 𝑇(0)=+∞). The universe first undergoes a phase of inflation during which its radius and temperature increase exponentially rapidly, while its density and pressure remain approximately constant. The inflation “starts” at 𝑡_𝑖=0 and ends at 𝑡₁=23.3𝑡_𝑃=1.25⋅10⁻⁴²s. During this very short lapse of time, the radius of the universe grows from 𝑎_𝑖=𝑙_𝑃=1.62⋅10⁻³⁵m to 𝑎₁=2.61⋅10⁻⁶m, and the temperature grows from 𝑇_𝑖=3.91⋅10⁻²⁰⁵𝑇_𝑃=5.54⋅10⁻¹⁷³K to 𝑇₁=0.278𝑇_𝑃=3.93⋅10³¹K. By contrast, the density and the pressure do not change significatively: they go from 𝜌_𝑖≃𝜌_𝑃=5.16⋅10⁹⁹g/m³ and 𝑝_𝑖≃−𝜌_𝑃𝑐²=−4.64⋅10¹¹⁶g/ms² to 𝜌₁=0.5𝜌_𝑃=2.58⋅10⁹⁹g/m³ and 𝑝₁=−0.167𝜌_𝑃𝑐²=−7.73⋅10¹¹⁵g/ms². The pressure passes from negative values to positive values at 𝑡_𝑤=23.4𝑡_𝑃=1.26⋅10⁻⁴²s corresponding to 𝜌_𝑤=0.25𝜌_𝑃=1.29⋅10⁹⁹g/m³, 𝑎_𝑤=1.32𝑎₁=3.43⋅10⁻⁶m, and 𝑇_𝑤=0.475𝑇_𝑃=6.74⋅10³¹K. After the inflation, the universe enters in the radiation era and, from that point, we recover the standard model [6]. The radius increases algebraically as 𝑎∝𝑡^1/2, while the density and the temperature decrease algebraically as 𝜌∝𝑡⁻² and 𝑇∝𝑡^−1/2. The temperature and the pressure achieve their maximum values 𝑇_𝑒=0.523𝑇_𝑃=7.40⋅10³¹K and 𝑝_𝑒=2.08⋅10⁻²𝜌_𝑃𝑐²=9.66⋅10¹¹⁴g/ms² at 𝑡=𝑡_𝑒=23.6𝑡_𝑃=1.27⋅10⁻⁴²s. At that moment, the density is 𝜌_𝑒=0.125𝜌_𝑃=6.44⋅10⁹⁸g/m³ and the radius 𝑎_𝑒=1.63𝑎₁=4.24⋅10⁻⁶m. During the inflation, the universe is accelerating and during the radiation era it is decelerating. The transition (marked by an inflexion point) takes place at a time 𝑡_𝑐=𝑡₁=23.3𝑡_𝑃=1.25⋅10⁻⁴²s coinciding with the end of the inflation (𝑎_𝑐=𝑎₁). The evolution of the scale factor, density, and temperature as a function of time are represented in Figures 1, 2, 3, 4, 5, 6, and 7 in logarithmic and linear scales.

Figure 1 

Evolution of the scale factor 𝑎 with the time 𝑡 in logarithmic scales. This figure clearly shows the phase of inflation connecting the vacuum energy era to the radiation era. During the inflation, the scale factor increases by 29 orders of magnitude in less than 10−42s. In the radiation era, 𝑎 increases algebraically.

Figure 2 

Evolution of the scale factor 𝑎 with the time 𝑡 in linear scales. The dashed line corresponds to the standard radiation model of Section 3.2 exhibiting a finite time singularity at 𝑡=0 (Big Bang). When quantum mechanics is taken into account (as in our simple semiclassical model), the initial singularity is smoothed-out and the scale factor at 𝑡=0 is equal to the Planck length 𝑙𝑃=1.62⋅10−35m. This is similar to a second order phase transition where the Planck constant plays the role of finite size effects (see Section 3.6 for a development of this analogy).

Figure 3 

Evolution of the density 𝜌 with the time 𝑡 in logarithmic scales. During the inflation, the density remains approximately constant with the Planck value 𝜌max=𝜌𝑃 which represents an upper bound. In the radiation era, 𝜌 decreases algebraically.

Figure 4 

Evolution of the density 𝜌 with the time 𝑡 in linear scales. The dashed line corresponds to the standard radiation model of Section 3.2 exhibiting a finite time singularity at 𝑡=0. Quantum mechanics limits the rise of the density to the Planck value 𝜌𝑃=5.16⋅1099g/m3.

Figure 5 

Evolution of the temperature 𝑇 with the time 𝑡 in logarithmic scales. For 𝑡≤0, the universe is extremely cold (𝑇<10−173K). During the inflation, the temperature increases by 204 orders of magnitude in less than 10−42s. During the radiation era, the temperature decreases algebraically.

Figure 6 

Evolution of the temperature 𝑇 with the time 𝑡 in linear scales. The dashed line corresponds to the standard radiation model of Section 3.2 exhibiting a finite time singularity at 𝑡=0. Quantum mechanics limits the rise of the temperature to about half the Planck value 𝑇𝑃=1.42⋅1032K. In the vacuum energy era, the temperature drops to zero exponentially rapidly for 𝑡→−∞.

Figure 7 

Evolution of the pressure 𝑝 with the time 𝑡 in linear scales. The dashed line corresponds to the standard radiation model of Section 3.2 exhibiting a finite time singularity at 𝑡=0. Quantum mechanics limits the rise of the pressure to 𝜌𝑃𝑐2/48=9.66⋅10114g/ms2. In the vacuum energy era, the pressure decreases exponentially rapidly, becomes negative, and tends to −𝜌𝑃𝑐2=−4.64⋅10116g/ms2 for 𝑡→−∞.

We note that the inflationary process described previously is relatively different from the usual inflationary scenario [2–5]. In standard inflation, the universe is radiation dominated up to 𝑡_𝑖=10⁻³⁵s but expands exponentially by a factor 10³⁰ in the interval 𝑡_𝑖<𝑡<𝑡_𝑓 with 𝑡_𝑓=10⁻³³s. For 𝑡>𝑡_𝑓, the evolution is again radiation dominated. At 𝑡=𝑡_𝑖, the temperature is about 10²⁷K (this corresponds to the epoch at which most “grand unified theories” have a significant influence on the evolution of the universe). During the exponential inflation, the temperature drops drastically; however, the matter is expected to be reheated to the initial temperature of 10²⁷K by various high energy processes [39]. In the inflationary process described previously, the evolution of the temperature, given by a generalized Stefan-Boltzmann law, is different. It is initially very low and increases during the inflation.

3.6. Analogy with Phase Transitions

The standard Big Bang theory is a classical theory in which quantum effects are neglected. In that case, it exhibits a finite time singularity: the radius of the universe is equal to zero at 𝑡=0, while its density and its temperature are infinite. For 𝑡<0, the solution is not defined and we may take 𝑎=0. For 𝑡>0, the radius of the universe increases as 𝑎∝𝑡^1/2. This is similar to a second-order phase transition if we view the time 𝑡 as the control parameter (e.g., the temperature 𝑇) and the scale factor 𝑎 as the order parameter (e.g., the magnetization 𝑀). The exponent 1/2 is the same as in mean field theories of second-order phase transitions (i.e., 𝑀∼(𝑇_𝑐−𝑇)^1/2), but this is essentially a coincidence.

When quantum mechanics effects are taken into account, as in our simple semiclassical model, the singularity at 𝑡=0 disappears and the curves 𝑎(𝑡), 𝜌(𝑡) and 𝑇(𝑡) are regularized. In particular, we find that 𝑎=𝑙_𝑃>0 at 𝑡=0, instead of 𝑎=0, due to the finite value of ℏ. This is similar to the regularization due to finite size effects (e.g., the system size 𝐿 or the number of particles 𝑁) in ordinary phase transitions. In this sense, the classical limit ℏ→0 is similar to the thermodynamic limit (𝐿→+∞ or 𝑁→+∞) in ordinary phase transitions.

To study the convergence toward the classical Big Bang solution when ℏ→0, it is convenient to use a proper normalization of the parameters. We call ℏ₀=1.05⋅10⁻³⁴Js the true value of the Planck constant (in this section, the subscript 0 refers to true values of the parameters). Then, we express time in terms of 𝑡⁰_𝑃 and lengths in terms of 𝑎⁰₁. We introduce 𝑎=𝑎/𝑎⁰₁ and 𝑡̂=𝑡/𝑡⁰_𝑃. Using 𝑙_𝑃=𝑐𝑡_𝑃 and 𝜌_𝑃𝑎⁴₁=𝜌_rad,0𝑎⁴₀, we obtain 𝑙_𝑃𝑙⁰_𝑃=𝑡_𝑃𝑡⁰_𝑃=𝑎₁𝑎⁰₁²=ℏ^1/2, (42) where ℏ=ℏ/ℏ₀. Therefore 𝜖𝜖₀=𝑙_𝑃/𝑎₁𝑙⁰_𝑃/𝑎⁰₁=𝑎₁𝑎⁰₁=ℏ^1/4, (43) where 𝜖₀=𝑙⁰_𝑃/𝑎⁰₁=6.20⋅10⁻³⁰. Using these relations and (31), we get 𝑎⁴ℏ+1−ln1+𝑎⁴/ℏ+1𝑎²/ℏ^1/2=28𝜋3^1/2𝑡̂ℏ^1/2+𝐶𝜖₀ℏ^1/4, (44) where 𝐶(𝜖) is defined by (32). This equation describes a phase transition between the vacuum energy era and the radiation era. The normalized Planck constant ℏ plays the role of finite size effects. Finite size scalings are explicit in (44). For ℏ=1, we recover the nonsingular model of Section 3.3. For ℏ=0, we recover the singular radiation model of Section 3.2 (Big Bang). The convergence towards this singular solution as ℏ→0 is shown in Figure 8.

Figure 8 

Effect of quantum mechanics (finite value of the Planck constant) on the regularization of the singular Big Bang solution (ℏ=0, dashed line) in our simple semi-classical model. The singularity at 𝑡=0 is replaced by an inflationary expansion from the vacuum energy era to the radiation era. We can draw an analogy with second order phase transitions where the Planck constant plays the role of finite size effects (see the text for details).

3.7. Scalar Field Theory

The phase of inflation in the very early universe is usually described by a scalar field [5]. The ordinary scalar field minimally coupled to gravity evolves according to the equation 𝜙̈+3𝐻𝜙̇+𝑑𝑉𝑑𝜙=0, (45) where 𝑉(𝜙) is the potential of the scalar field. The scalar field tends to run down the potential towards lower energies. The density and the pressure of the universe are related to the scalar field by 𝜌𝑐²=12𝜙̇²+𝑉𝜙,𝑝=12𝜙̇²−𝑉𝜙. (46) Using standard technics [7], we find that the potential corresponding to the equation of state (20) is [34–36] 𝑉𝜓=13𝜌_𝑃𝑐²cosh²𝜓+2cosh⁴𝜓,𝑎𝜓≥0𝑎₁²=sinh𝜓,𝜓=8𝜋𝐺𝑐^21/2𝜙. (47) In the vacuum energy era (𝑡→−∞), using (26), we get 𝑙𝜓∼^𝑃𝑎₁²𝑒^{32𝜋/3𝑡/𝑡𝑃}⟶0. (48) In the radiation era (𝑡→+∞), using (29), we get 𝑡𝜓≃ln𝑡_𝑃+12ln8𝜋3+2ln(2)⟶+∞. (49) More generally, using (31), the evolution of the scalar field 𝜓(𝑡) in the early universe is given by cosh𝜓−ln1+cosh𝜓sinh𝜓=28𝜋3^1/2𝑡𝑡_𝑃+𝐶. (50) These results are illustrated in Figures 9 and 10.

Figure 9 

Potential of the scalar field in the early universe. The field tends to run down the potential.

Figure 10 

Evolution of the scalar field as a function of time in the early universe. It displays a phase of early inflation.

4. The Radiation and the Dark Energy in the Late Universe

We propose to describe the transition between the radiation era and the dark energy era in the late universe by a single equation of state of the form 1𝑝=3𝜌𝑐²−43𝜌_Λ𝑐², (51) where 𝜌_Λ=Λ/8𝜋𝐺=7.02⋅10⁻²⁴g/m³ is the cosmological density. This equation of state corresponds to a generalized polytropic equation of state (1) with 𝛼=1/3, 𝑛=−1, and 𝑘=−(4/3)𝜌_Λ. It may be viewed as the “symmetric” version of the equation of state (20) in the early universe. For 𝜌≫𝜌_Λ, we recover the equation of state of the pure radiation 𝑝∼𝜌𝑐²/3. For 𝜌→𝜌_Λ, we get 𝑝≃−𝜌_Λ𝑐² corresponding to the dark energy.

The continuity equation (15) may be integrated into 𝜌=𝜌_Λ𝑎₃𝑎⁴+1, (52) where 𝑎₃ is a constant of integration. This characteristic scale marks the transition between the radiation era and the dark energy era. The equation of state (51) interpolates smoothly between the radiation era (𝑝=𝜌𝑐²/3 and 𝜌_rad∼𝜌_Λ(𝑎₃/𝑎)⁴) and the dark energy era (𝑝=−𝜌𝑐² and 𝜌=𝜌_Λ). It provides therefore a “unified” description of radiation and dark energy in the late universe. This amounts to summing the density of these two phases. Indeed, (52) may be rewritten as 𝜌=𝜌_rad+𝜌_Λ. (53) At 𝑎=𝑎₃ we have 𝜌_rad=𝜌_Λ so that 𝜌=2𝜌_Λ.

4.1. The Dark Energy Era: Late Inflation

When 𝑎≫𝑎₃, the density tends to a minimum value 𝜌=𝜌_min=𝜌_Λ, (54) and the pressure tends to 𝑝=−𝜌_Λ𝑐². The cosmological density 𝜌_Λ=Λ/8𝜋𝐺=7.02⋅10⁻²⁴g/m³ (dark energy) represents a fundamental lower bound for the density. A constant value of the density 𝜌≃𝜌_Λ gives rise to a phase of late inflation. It is convenient to define a cosmological time 𝑡_Λ=1/(𝐺𝜌_Λ)^1/2=(8𝜋/Λ)^1/2=1.46⋅10¹⁸s and a cosmological length 𝑙_Λ=𝑐𝑡_Λ=(8𝜋𝑐²/Λ)^1/2=4.38⋅10²⁶m. These are the counterparts of the Planck scales for the late universe. From the Friedmann equation (17), we find that the Hubble parameter is constant 𝐻=(8𝜋/3)^1/2𝑡_Λ⁻¹. Numerically, 𝐻=1.98⋅10⁻¹⁸s⁻¹. Therefore, the scale factor increases exponentially rapidly with time as 𝑎(𝑡)∝𝑒^{(8𝜋/3)1/2𝑡/𝑡Λ}. (55) This exponential growth corresponds to the de Sitter solution [1]. The timescale of the exponential growth is the cosmological time 𝑡_Λ=1.46⋅10¹⁸s. This solution exists at any time in the future (𝑎→+∞ and 𝜌→𝜌_Λ for 𝑡→+∞), so there is no future singularity. This is not the case of all cosmological models. In a “phantom universe” [40–57], violating the null dominant energy condition (𝑤<−1), the density increases as the universe expands. This may lead to a future singularity called Big Rip (the density becomes infinite in a finite time). The possibility that we live in a phantom universe is not ruled out by observations.

4.2. The Radiation Era

When 𝑎≪𝑎₃, we recover the equation 𝜌_rad∼𝜌_Λ𝑎⁴₃𝑎⁴, (56) corresponding to the pure radiation described by an equation of state 𝑝=𝜌𝑐²/3. The conservation of 𝜌_rad𝑎⁴ implies that 𝜌_Λ𝑎⁴₃=𝜌_rad,0𝑎⁴₀. Using 𝜌_Λ=Ω_Λ,0𝜌₀ and 𝜌_rad,0=Ω_rad,0𝜌₀ with Ω_Λ,0=0.763 (dark energy) and Ω_rad,0=8.48⋅10⁻⁵ (radiation) [1], we obtain 𝑎₃/𝑎₀=(Ω_rad,0/Ω_Λ,0)^1/4 hence 𝑎₃=1.36⋅10²⁵m. (57) This can be rewritten as 𝑎₃=0.103𝑎₀=3.09⋅10⁻²𝑙_Λ where we have used 𝑙_Λ/𝑎₀=𝐻₀𝑡_Λ=3.31. This scale gives the typical size of the universe at the transition between the radiation era and the dark energy era. When 𝑎≪𝑎₃, the Friedmann equation (17) yields 𝑎𝑎₃∼32𝜋3^1/4𝑡𝑡_Λ^1/2. (58) We then have 𝜌𝜌_Λ∼3𝑡32𝜋𝑡^Λ2. (59)

4.3. The Temperature and the Pressure

The thermodynamical equation (13) with the equation of state (51) may be integrated into 𝑇=𝑇_𝑃15𝜋^21/4𝜌−𝜌_Λ𝜌_𝑃^1/4, (60) where 𝑇_𝑃 is a constant of integration. Using (52), we get 𝑇=𝑇_𝑃15𝜋^21/4𝜌_Λ𝜌_𝑃^1/4𝑎₃𝑎. (61) Equation (60) may be seen as a generalized Stefan-Boltzmann law in the late universe. In the radiation era 𝜌≫𝜌_Λ, it reduces to the ordinary Stefan-Boltzmann law (36). This allows us to identify the constant of integration 𝑇_𝑃 with the Planck temperature 𝑇_𝑃=(ℏ𝑐⁵/𝐺𝑘²_𝐵)^1/2=1.42⋅10³²K. We note that 𝑇∝𝑎⁻¹ both in the radiation and dark energy eras. This suggests that radiation and dark energy belong to the same “family”. In the radiation era the temperature decreases algebraically as 𝑇∼𝑇_𝑃(45/32𝜋³)^1/4(𝜌_Λ/𝜌_𝑃)^1/4(𝑡_Λ/𝑡)^1/2, and in the dark energy era, it decreases exponentially rapidly as 𝑇∝𝑒^{−(8𝜋/3)1/2𝑡/𝑡Λ}.

The pressure is given by (51). Using (52), we get 1𝑝=3𝑎₃𝑎⁴−1𝜌_Λ𝑐². (62) It decreases algebraically during the radiation era and tends to a constant negative value 𝑝=−𝜌_Λ𝑐² for 𝑡→+∞. Numerically, 𝑝=−6.31⋅10⁻⁷g/ms². The point at which the pressure vanishes is 𝑎^′_𝑤=𝑎₃/3^1/4, 𝜌^′_𝑤=4𝜌_Λ, 𝑇^′_𝑤=𝑇_𝑃(45/𝜋²)^1/4(𝜌_Λ/𝜌_𝑃)^1/4. Numerically, 𝑎^′_𝑤=0.760𝑎₃=1.03⋅10²⁵m, 𝜌^′_𝑤=4𝜌_Λ=2.81⋅10⁻²³g/m³, 𝑇^′_𝑤=2.81⋅10⁻³¹𝑇_𝑃=39.9K.

5. The General Model

5.1. The Quadratic Equation of State

We propose to describe the vacuum energy, radiation, and dark energy by a unique equation of state 𝑝_𝑔=−4𝜌²_𝑔3𝜌_𝑃𝑐²+13𝜌_𝑔𝑐²−43𝜌_Λ𝑐². (63) For 𝜌_𝑔→𝜌_𝑃, 𝑝_𝑔→−𝜌_𝑃𝑐² (vacuum energy); for 𝜌_Λ≪𝜌_𝑔≪𝜌_𝑃, 𝑝_𝑔∼𝜌_𝑔𝑐²/3 (radiation); for 𝜌_𝑔→𝜌_Λ, 𝑝_𝑔→−𝜌_Λ𝑐² (dark energy). This quadratic equation of state combines the properties of the equation of state (20) valid in the early universe and of the equation of state (51) valid in the late universe. A nice feature of this equation of state is that both the Planck density (vacuum energy) and the cosmological density (dark energy) explicitly appear. Therefore, this equation of state reproduces both the early inflation and the late inflation. On the other hand, this description suggests that vacuum energy, radiation, and dark energy may be the manifestation of a unique form of “generalized radiation”. This is, however, just a speculation before a justification of this equation of state has been given from first principles. In this paper, we limit ourselves to studying the properties of this equation of state.

Using the equation of continuity (15), we obtain 𝜌_𝑔=𝜌_𝑃𝑎/𝑎₁⁴+1+𝜌_Λ, (64) where 𝑎₁=2.61⋅10⁻⁶m (see Section 3). To obtain this expression, we have used the fact that 𝜌_𝑃≫𝜌_Λ so that 𝑝_𝑔/𝑐²+𝜌_𝑔≃(4/3𝜌_𝑃)(𝜌_𝑔−𝜌_Λ)(𝜌_𝑃−𝜌_𝑔). When 𝑎→0, 𝜌_𝑔→𝜌_𝑃 (vacuum energy); when 𝜌_Λ≪𝜌_𝑔≪𝜌_𝑃, 𝜌_𝑔∼𝜌_𝑃(𝑎/𝑎₁)⁻⁴ (radiation); when 𝑎→+∞, 𝜌_𝑔→𝜌_Λ (dark energy). In the early universe, the contribution of dark energy is negligible and we recover (21). In the late universe, the contribution of vacuum energy is negligible and we recover (52) with 𝜌_𝑃𝑎⁴₁=𝜌_Λ𝑎⁴₃.

We shall view the “generalized radiation”, the baryonic matter, and the dark matter as different species. In the general case, these species could interact. However, in this paper, we assume, for simplicity, that they do not. The generalized radiation (vacuum energy + radiation + dark energy) is described by the quadratic equation of state (63). Using 𝜌_𝑃𝑎⁴₁=𝜌_rad,0𝑎⁴₀ (see Section 3), we can rewrite (64) in the form 𝜌_𝑔=𝜌_rad,0𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴+𝜌_Λ, (65) where 𝑎₁/𝑎₀=1.97⋅10⁻³². The baryonic matter and the dark matter are described as pressureless fluids with 𝑝_𝐵=0 and 𝑝_DM=0. The equation of continuity (15) for each species leads to the relations𝜌_𝐵=𝜌_𝐵,0𝑎/𝑎₀³,𝜌_DM=𝜌_DM,0𝑎/𝑎₀³. (66a)(66b)

The total density 𝜌=𝜌_𝑔+𝜌_𝐵+𝜌_DM of generalized radiation, baryonic matter, and dark energy can be written as 𝜌𝜌=^rad,0𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴+𝜌_𝐵,0𝑎/𝑎₀³+𝜌_DM,0𝑎/𝑎₀³+𝜌_Λ. (67) Substituting this relation in the Friedmann equation (17) and writing 𝜌_rad,0=Ω_rad,0𝜌₀, 𝜌_𝐵,0=Ω_𝐵,0𝜌₀, 𝜌_DM,0=Ω_DM,0𝜌₀, and 𝜌_Λ=Ω_Λ,0𝜌₀, we obtain 𝐻𝐻₀=Ω_rad,0𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴+Ω_𝐵,0𝑎/𝑎₀³+Ω_DM,0𝑎/𝑎₀³+Ω_Λ,0. (68) From observations, Ω_rad,0=8.48⋅10⁻⁵, Ω_𝐵,0=0.0455, Ω_DM,0=0.1915, Ω_𝑚,0=0.237, and Ω_Λ,0=0.763 [1].

The thermodynamic equation (13) with the equation of state (63) may be integrated into 𝑇=𝑇_𝑃15𝜋^21/4𝜌1−^𝑔𝜌_𝑃^7/4𝜌_𝑔𝜌_𝑃−𝜌_Λ𝜌_𝑃^1/4. (69) Using (64), we get 𝑇=𝑇_𝑃15𝜋^21/4𝑎/𝑎₁⁷𝑎/𝑎₁⁴+1². (70) Equation (70) may also be written as 𝑇=𝑇₀𝑎/𝑎₀⁷𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴², (71) with 𝑇₀=𝑇_𝑃(15/𝜋²)^1/4(𝑎₁/𝑎₀). Numerically, 𝑇₀=3.11K. Since 𝑎₁≪𝑎₀, 𝑇₀ can be identified with the present temperature of radiation. Indeed, after the vacuum energy era, (71) reduces to 𝑇∼𝑇₀𝑎₀𝑎. (72)

Finally, using (63) and (64), the pressure of generalized radiation can be written in very good approximation as 𝑝_𝑔=13𝑎/𝑎₁⁴−3𝑎/𝑎₁⁴+1²𝜌_𝑃𝑐²−𝜌_Λ𝑐². (73)

5.2. The Early Universe

In the early universe, we can neglect the contribution of baryonic matter, dark matter, and dark energy in the density. We only consider the contribution of vacuum energy and radiation. Equation (67) then reduces to 𝜌𝜌=^rad,0𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴. (74) The Friedmann equation (68) becomes 𝐻𝐻₀=Ω_rad,0𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴. (75) It has the analytical solution given by (31). On the other hand, the temperature of the generalized radiation in the early universe is given by (34) and (35). The transition between the vacuum energy era and the radiation era corresponds to 𝜌_rad=𝜌_𝑃. This yields 𝑎₁/𝑎₀=(𝜌_rad,0/𝜌_𝑃)^1/4, 𝜌₁=𝜌_𝑃/2, 𝑇₁=(1/4)(15/𝜋²)^1/4𝑇_𝑃. Numerically, 𝑎₁=1.97⋅10⁻³²𝑎₀=2.61⋅10⁻⁶m, 𝜌₁=0.5𝜌_𝑃=2.58⋅10⁹⁹g/m³, 𝑇₁=0.278𝑇_𝑃=3.93⋅10³¹K. This takes place at a time 𝑡₁=23.3𝑡_𝑃=1.25⋅10⁻⁴²s. This time also corresponds to the inflexion point of the curve 𝑎(𝑡). The universe is accelerating for 𝑡<𝑡₁ and decelerating for 𝑡>𝑡₁.

In the vacuum energy era (𝑎≪𝑎₁), the density is constant: 𝜌≃𝜌_𝑃. (76) Therefore, the Hubble parameter is also a constant 𝐻=(8𝜋/3)^1/2𝑡_𝑃⁻¹. In that case, the scale factor and the temperature increase exponentially rapidly according to (26) and (38).

In the radiation era (𝑎≫𝑎₁), we have 𝜌𝜌∼^rad,0𝑎⁴₀𝑎⁴. (77)

The Friedmann equation (75) reduces to 𝐻𝐻₀=Ω_rad,0𝑎/𝑎₀⁴ (78) and yields 𝑎𝑎₀∼Ω^1/4_rad,02𝐻₀𝜌𝑡,𝜌₀∼12𝐻₀𝑡²,𝑇𝑇₀∼𝑎₀𝑎∼1Ω^1/4_rad,02𝐻₀𝑡. (79)

5.3. The Late Universe

In the late universe, we can neglect the contribution of vacuum energy and radiation in the density. We only consider the contribution of baryonic matter, dark matter, and dark energy. Equation (67) then reduces to 𝜌𝜌=^𝑚,0𝑎/𝑎₀³+𝜌_Λ, (80) where 𝜌_𝑚=𝜌_𝐵+𝜌_DM. This coincides with the ΛCDM model (see Section 5.5). The Friedmann equation (68) becomes 𝐻𝐻₀=Ω_𝑚,0𝑎/𝑎₀³+Ω_Λ,0, (81) with Ω_𝑚,0+Ω_Λ,0≃1. It has the analytical solution 𝑎𝑎₀=Ω_𝑚,0Ω_Λ,0^1/3sinh^2/332Ω_Λ,0𝐻₀𝑡. (82) The density evolves as 𝜌𝜌₀=Ω_Λ,0tanh²(3/2)Ω_Λ,0𝐻₀𝑡. (83) On the other hand, the temperature of the generalized radiation in the late universe is given by 𝑇=𝑇_𝑃15𝜋^21/4𝜌_𝑔𝜌_𝑃−𝜌_Λ𝜌_𝑃^1/4,𝑇=𝑇₀𝑎₀𝑎. (84) Using the expression (82) of the scale factor, we obtain 𝑇𝑇=⁰Ω_𝑚,0/Ω_Λ,0^1/3sinh^2/3(3/2)Ω_Λ,0𝐻₀𝑡. (85) Finally, the pressure of the generalized radiation in the late universe may be written as 𝑝_𝑔=13𝜌_rad,0𝑐²𝑎/𝑎₀⁴−𝜌_Λ𝑐². (86)

Setting 𝑎=𝑎₀ in (82), we find the age of the universe 𝑡₀=1𝐻₀231Ω_Λ,0sinh⁻¹Ω_Λ,0Ω_𝑚,0^1/2. (87) Numerically, 𝑡₀=1.03𝐻₀⁻¹=0.310𝑡_Λ=4.53⋅10¹⁷s∼14Gyrs. It is rather fortunate that the age of the universe almost coincides with the Hubble time 𝐻₀⁻¹.

The universe starts accelerating when 𝜌_𝑚=2𝜌_Λ [see (16) with 𝜌≃𝜌_𝑚+𝜌_Λ and 𝑝≃−𝜌_Λ𝑐²]. Using 𝜌_𝑚,0=Ω_𝑚,0𝜌₀ and 𝜌_Λ=Ω_Λ,0𝜌₀ with Ω_𝑚,0=0.237 (matter) and Ω_Λ,0=0.763 (dark energy), we obtain 𝑎=𝑎^′_𝑐 and 𝑡=𝑡^′_𝑐 with 𝑎^′_𝑐𝑎₀=Ω_𝑚,02Ω_Λ,0^1/3,𝑡^′_𝑐=1𝐻₀231Ω_Λ,0sinh⁻¹1√2. (88) At that point 𝜌=𝜌^′_𝑐=3𝜌_Λ and 𝑇=𝑇^′_𝑐=𝑇₀𝑎₀/𝑎^′_𝑐. Numerically, 𝑎^′_𝑐=0.538𝑎₀=0.162𝑙_Λ=7.11⋅10²⁵m, 𝑡^′_𝑐=0.504𝐻₀⁻¹=0.152𝑡_Λ=2.22⋅10¹⁷s, 𝜌^′_𝑐=3𝜌_Λ=2.11⋅10⁻²³g/m³, and 𝑇^′_𝑐=1.86𝑇₀=5.78K. The universe is decelerating for 𝑡<𝑡^′_𝑐 and accelerating for 𝑡>𝑡^′_𝑐.

The transition between the matter era and the dark energy era corresponds to 𝜌_𝑚=𝜌_Λ. This leads to 𝑎=𝑎₂ and 𝑡=𝑡₂ with 𝑎₂𝑎₀=Ω_𝑚,0Ω_Λ,0^1/3,𝑡₂=1𝐻₀231Ω_Λ,0sinh⁻¹(1). (89) At that point 𝜌=𝜌₂=2𝜌_Λ and 𝑇=𝑇₂=𝑇₀𝑎₀/𝑎₂. Numerically, 𝑎₂=0.677𝑎₀=0.204𝑙_Λ=8.95⋅10²⁵m, 𝑡₂=0.674𝐻₀⁻¹=0.203𝑡_Λ=2.97⋅10¹⁷s, 𝜌₂=2𝜌_Λ=1.40⋅10⁻²³g/m³, and 𝑇₂=1.48𝑇₀=4.59K. We also note that 𝑎^′_𝑐=(1/2)^1/3𝑎₂=0.794𝑎₂. Of course, the value 𝑎₂=8.95⋅10²⁵m (90) coincides with the one obtained in [34–36] using a different viewpoint.

In the dark energy era (𝑎≫𝑎₂), the density is constant: 𝜌≃𝜌_Λ. (91) Therefore, the Hubble parameter is also a constant 𝐻=𝐻₀Ω_Λ,0. Using (82), we find that the scale factor increases exponentially rapidly as 𝑎𝑎₀∼Ω_𝑚,04Ω_Λ,0^1/3𝑒^{ΩΛ,0𝐻0𝑡}. (92) This corresponds to the de Sitter solution. According to (85), the temperature decreases exponentially rapidly as 𝑇𝑇₀∼4Ω_Λ,0Ω_𝑚,0^1/3𝑒^{−ΩΛ,0𝐻0𝑡}. (93) In the matter era (𝑎≪𝑎₂), we have 𝜌𝜌∼^𝑚,0𝑎³₀𝑎³. (94) The Friedmann equation (81) reduces to 𝐻𝐻₀=Ω_𝑚,0𝑎/𝑎₀³, (95) and we have 𝑎𝑎₀=Ω^1/3_𝑚,032𝐻₀𝑡^2/3,𝜌𝜌₀=1(3/2)𝐻₀𝑡²,𝑇𝑇₀∼1Ω^1/3_𝑚,0(3/2)𝐻₀𝑡^2/3. (96)(97) This corresponds to the Einstein-de Sitter (EdS) universe.

5.4. The Evolution of the Late Universe

In the matter era (𝑎≪𝑎₂), the radius increases algebraically as 𝑎∝𝑡^2/3, while the density decreases algebraically as 𝜌∝𝑡⁻² (EdS). When 𝑎≫𝑎₂, the universe enters in the dark energy era. It undergoes a late inflation (de Sitter) during which its radius increases exponentially rapidly, while its density remains constant and equal to the cosmological density 𝜌_Λ. The transition takes place at 𝑎₂=0.677𝑎₀=0.204𝑙_Λ=8.95⋅10²⁵m, 𝜌₂=2𝜌_Λ=1.40⋅10⁻²³g/m³, 𝑇₂=1.48𝑇₀=4.59K, and 𝑡₂=0.674𝐻₀⁻¹=0.203𝑡_Λ=2.97⋅10¹⁷s. In the matter era, the universe is decelerating, and in the dark energy era, it is accelerating. The time at which the universe starts accelerating is 𝑡^′_𝑐=0.504𝐻₀⁻¹=0.152𝑡_Λ=2.22⋅10¹⁷s, corresponding to a radius 𝑎^′_𝑐=0.538𝑎₀=0.162𝑙_Λ=0.794𝑎₂=7.11⋅10²⁵m, a density 𝜌^′_𝑐=3𝜌_Λ=2.11⋅10⁻²³g/m³ and a temperature 𝑇^′_𝑐=1.86𝑇₀=5.78K. In the late universe, the temperature of generalized radiation always scales as 𝑇∝𝑎⁻¹. It decreases algebraically rapidly (𝑇∝𝑡^−2/3) in the matter era and exponentially rapidly in the dark energy era. The evolution of the scale factor, density, temperature, and pressure of radiation as a function of time is represented in Figures 11, 12, 13, 14, 15, and 16 in logarithmic and linear scales.

Figure 11 

Evolution of the scale factor 𝑎 with the time 𝑡 in logarithmic scales. This figure clearly shows the transition between the matter era (EdS) and the dark energy era (de Sitter). In the dark energy era, the radius increases exponentially rapidly on a timescale of the order of the cosmological time 𝑡Λ=1.46⋅1018s. This corresponds to a phase of late inflation. The universe is decelerating for 𝑎<𝑎′𝑐 and accelerating for 𝑎>𝑎′𝑐 with 𝑎′𝑐=7.11⋅1025m. The transition between the matter era and the dark energy era takes place at 𝑎2=8.95⋅1025m. Coincidentally, the present universe turns out to be close to the transition point (𝑎0∼𝑎2).

Figure 12 

Evolution of the scale factor 𝑎 with the time 𝑡 in linear scales.

Figure 13 

Evolution of the density 𝜌 with the time 𝑡 in logarithmic scales. During the late inflation, the density remains approximately constant with the cosmological value 𝜌min=𝜌Λ representing a lower bound.

Figure 14 

Evolution of the density 𝜌 with the time 𝑡 in linear scales. General relativity (cosmological constant) limits the decay of the density to the cosmological value 𝜌Λ=7.02⋅10−24g/m3.

Figure 15 

Evolution of the temperature of radiation 𝑇 with the time 𝑡 in linear scales.

Figure 16 

Evolution of the pressure of radiation 𝑝𝑔 with the time 𝑡 in linear scales.

The present size of the universe 𝑎₀=0.302𝑙_Λ=1.32⋅10²⁶m is precisely of the order of the scale 𝑎₂=0.204𝑙_Λ=8.95⋅10²⁵m (𝑎₀=1.48𝑎₂). We have 𝑎₀∼𝑎₂∼𝑙_Λ and 𝑡₀∼𝑡₂∼𝑡_Λ. Therefore, we live just at the transition between the matter era and the dark energy era (see bullets in Figures 11, 12, 13, 14, 15, and 16). (In the context of the standard ΛCDM model, the way to state this observation is to say that the present ratio Ω_Λ,0/Ω_𝑚,0=3.22 between dark energy and matter is of order unity. Since the matter density changes as 𝑎⁻³ the ratio between matter and dark energy is of order unity only during a brief period of time in the evolution of the universe. It turns out that we live precisely in this period. This coincidence is intriguing and often referred to as the “cosmic coincidence problem” [58]. Several theories have been proposed to explain why Ω_Λ,0/Ω_𝑚,0∼1 [59–61]. However, this may be just a pure coincidence without deeper reason. Life (and researchers inquiring about cosmology) may have emerged at this special moment of the history of the universe, ~14 Gyrs after the Big Bang, precisely at the epoch where Ω_Λ/Ω_𝑚∼1. Life cannot have emerged much earlier because galaxies, stars, and planets are not created, and it may not persist much later because planets, stars, and galaxies will die. Therefore, this coincidence is not really a “problem.”) The present density of the universe is 𝜌₀=1.31𝜌_Λ=9.20⋅10⁻²⁴ g/m³ and the age of the universe is 𝑡₀=1.03𝐻₀⁻¹=0.310𝑡_Λ=4.54⋅10¹⁷s∼14Gyrs. The present values of the equation of state parameter and deceleration parameter are 𝑤₀=−Ω_Λ,0 and 𝑞₀=(1−3Ω_Λ,0)/2 [since 𝑝≃−𝜌_Λ𝑐²]. Numerically, 𝑤₀=−0.763 and 𝑞₀=−0.645.

5.5. The Whole Evolution of the Universe

In the standard ΛCDM model, the radiation, baryonic matter, dark matter, and dark energy are treated as four independent species characterized by different equations of state. The equation of state is 𝑝=𝜌𝑐²/3 for the radiation, 𝑝=0 for the baryonic matter and dark matter, and 𝑝=−𝜌𝑐² for the dark energy. One then solves the equation of continuity (15) for these four species in order to obtain the relation between their density and the scale factor, that is, 𝜌_rad=𝜌_rad,0/(𝑎/𝑎₀)⁴, 𝜌_𝐵=𝜌_𝐵,0/(𝑎/𝑎₀)³, 𝜌_DM=𝜌_DM,0/(𝑎/𝑎₀)³, and 𝜌_Λ=𝜌_Λ,0. Finally, we add their individual densities 𝜌=𝜌_rad+𝜌_𝐵+𝜌_DM+𝜌_Λ, that is, 𝜌𝜌=^rad,0𝑎/𝑎₀⁴+𝜌_𝐵,0𝑎/𝑎₀³+𝜌_DM,0𝑎/𝑎₀³+𝜌_Λ,0 (98) and solve the Friedmann equation (17). This leads to the equation 𝐻𝐻₀=Ω_rad,0𝑎/𝑎₀⁴+Ω_𝐵,0𝑎/𝑎₀³+Ω_DM,0𝑎/𝑎₀³+Ω_Λ,0, (99) with Ω_rad+Ω_𝐵+Ω_DM+Ω_Λ=1. This standard model provides a good description of the universe after the Planck era. However, it exhibits a primordial singularity at 𝑡=0 (Big Bang). Furthermore, it does not describe the phase of early inflation. This phase is usually described by another theory, for example, a scalar field theory, and it is then connected to the standard model.

In the present model, only three species coexist: baryonic matter (see (66a)), dark matter (see (66b)), and a “generalized radiation” unifying vacuum energy, radiation, and dark energy (see (65)). As a result, the equation describing the whole evolution of the universe is (In the standard model, the radiation density diverges as 𝑎⁻⁴ when 𝑎→0 while the matter density diverges as 𝑎⁻³ which is subdominant. Therefore, we can safely extrapolate the matter density to 𝑎=0 (its contribution is negligible anyway). In the present model, the generalized radiation density (65) does not diverge anymore when 𝑎→0 since 𝑎₁≠0. Therefore, it becomes necessary to state explicitly that matter appears at sufficiently late time in order to avoid a spurious divergence of the matter density when 𝑎→0. To that purpose, it is sufficient to add a Heaviside function 𝐻(𝑎−𝑎₁) in the matter density and write Ω^∗_𝑚,0=Ω_𝑚,0𝐻(𝑎−𝑎₁).) 𝐻𝐻₀=Ω_rad,0𝑎/𝑎₀⁴+𝑎₁/𝑎₀⁴+Ω^∗_𝐵,0𝑎/𝑎₀³+Ω^∗_DM,0𝑎/𝑎₀³+Ω_Λ,0, (100) with Ω_𝐵+Ω_rad+Ω_DM+Ω_Λ=1. For 𝑎₁=0, we obtain the same equation as in the standard ΛCDM model where the contributions of radiation, baryonic matter, dark matter, and dark energy are added individually [1]. However, as we have previously mentioned, the ΛCDM model does not describe the phase of early inflation and it presents a singularity at 𝑡=0 (Big Bang). For 𝑎₁/𝑎₀=1.97⋅10⁻³², we obtain a generalized model which does not present a primordial singularity and which displays a phase of early inflation. In this model, the universe always exists in the past but, for 𝑡<0, it has a very small radius, smaller than the Planck length. (As already stated in Section 3, our model certainly breaks down in this period since it does not take quantum fluctuations into account. The Planck era may not be described in terms of an equation of state 𝑝(𝜌) or even in terms of the Einstein equations, as we have assumed. It probably requires the development of a theory of quantum gravity that does not exist for the moment. An interesting description of the early inflation has been given by Monerat et al. [62] in terms of a quantized model based on a simplified Wheeler-DeWitt equation. In that model, a quantum tunneling process explains the birth of the universe with a well-defined size after tunneling. Therefore, other inflationary scenarios are possible in addition to the one based on the generalized equation of state (63).) At 𝑡=0, it undergoes an inflationary expansion in a very short lapse of time of the order of 23.3 Planck times. For 𝑡>23.3𝑡_𝑃, this model gives the same results as the standard ΛCDM model: the universe first undergoes an algebraic expansion in the radiation and matter eras, then an exponential expansion (second inflation) in the dark energy era. A nice feature of this model is its simplicity since it incorporates a phase of early inflation in a very simple manner. We just have to add a term +(𝑎₁/𝑎₀)⁴ in the standard equation (99). Therefore, the modification implied by (65) to describe the early inflation is very natural. On the other hand, this model gives the same results in the late universe as the standard model, so this does not bring any modification to the usual equation (99). Therefore, our simple model completes the standard ΛCDM model by incorporating the phase of early inflation in a natural manner. This is an interest of this description since the standard model gives results that agree with observations at late times.

The evolution of the scale factor with time is obtained by solving the first-order differential equation (100). This yields 𝑑𝑥𝑥^𝑎/𝑎0_{𝑎𝑖/𝑎0}Ω_rad,0𝑥⁴+𝑎₁/𝑎₀⁴+Ω^∗_𝐵,0𝑥³+Ω^∗_DM,0𝑥³+Ω_Λ,0⁻¹=𝐻₀𝑡, (101) where 𝑎_𝑖=𝑎₁=0 in the standard model and 𝑎_𝑖=𝑙_𝑃 in the generalized model. The age of the universe is 𝑡₀=1𝐻₀×¹₀𝑑𝑥𝑥Ω_rad,0/𝑥⁴+Ω_𝐵,0/𝑥³+Ω_DM,0/𝑥³+Ω_Λ,0. (102) Of course, for the determination of the age of the universe, we can neglect the contribution of vacuum energy in the early universe and take 𝑎₁=0 (strictly speaking, the age of the universe is infinite since it has no origin; however, we define the age of the universe from the time 𝑡=0 at which 𝑎=𝑙_𝑃). We obtain the standard result 𝑡₀=1.03𝐻₀⁻¹=4.53⋅10¹⁷s=14.4Gyr. (The Hubble constant is usually written as 𝐻₀=2.268ℎ₇10⁻¹⁸s⁻¹ where the dimensionless parameter ℎ₇ is about 10% of unity [1]. For simplicity, we have taken ℎ₇=1 in the numerical applications. The current value is ℎ₇=1.05±0.05. If we take ℎ₇=1.05, the age of the universe is 𝑡₀=13.7Gyr.) Actually, we find the same result if we neglect radiation and use the analytical expression (87) instead of (102).

In Figures 17–19, we have represented the evolution of the radius, density, and temperature of the universe as a function of time. The universe exhibits two types of inflations: an early inflation due to the Planck density 𝜌_𝑃=5.16⋅10⁹⁹g/m³ (vacuum energy) and a late inflation due to the cosmological density 𝜌_Λ=7.02⋅10⁻²⁴g/m³ (dark energy). There exists a striking “symmetry” between the early and the late evolution of the universe, and the cosmological constant in the late universe plays the same role as the Planck constant in the early universe. In particular, Figure 18 shows that the density varies between two bounds 𝜌_max=𝜌_𝑃 and 𝜌_min=𝜌_Λ that are fixed by fundamental constants. These values differ by a factor of the order 10¹²². The early universe is governed by quantum mechanics (ℏ) and the late universe by general relativity (Λ).

Figure 17 

Evolution of the scale factor 𝑎 as a function of time in logarithmic scales. The early universe undergoes a phase of inflation due to the vacuum energy before entering into the radiation era. During the early inflation, the scale factor increases by 29 orders of magnitude in less than 10−42s. This is followed by the matter era and the dark energy era responsible for the accelerated expansion of the universe. The universe exhibits two types of inflation: an early inflation corresponding to the Planck density 𝜌𝑃 (vacuum energy) due to quantum mechanics (Planck constant) and a late inflation corresponding to the cosmological density 𝜌Λ (dark energy) due to general relativity (cosmological constant). The evolution of the early and late universe is remarkably symmetric. We have represented in dashed line the standard model leading to a primordial singularity (Big Bang). The dotted line corresponds to the model of Section 5.3 where the radiation is neglected. This analytical model provides a good description of the late universe. We have also represented the location of the present universe (bullet). It happens to be just at the transition between the matter era and the dark energy era.

Figure 18 

Evolution of the density 𝜌 as a function of time in logarithmic scales. The density goes from a maximum value 𝜌max=𝜌𝑃 determined by the Planck constant (quantum mechanics) to a minimum value 𝜌min=𝜌Λ determined by the cosmological constant (general relativity). These two bounds, which are fixed by fundamental constants of physics, are responsible for the early and late inflation of the universe. In between, the density decreases as 𝑡−2.

Figure 19 

Evolution of the temperature 𝑇 of the “generalized” radiation as a function of time in logarithmic scales. Before the early inflation, the universe is extremely cold (𝑇<10−173K). During the early inflation, the temperature increases by 204 orders of magnitude in less than 10−42s. During the radiation era, the temperature decreases algebraically. The present value of the temperature is 𝑇≃2.7K (bullet). During the late inflation, the temperature decreases exponentially rapidly.

Remark 1. We have described dark matter as a presureless fluid with 𝑝=0. There are some indications that dark matter may be described by an isothermal equation of state 𝑝=𝛼𝜌𝑐² [63]. In that case, the term Ω^∗_DM,0/(𝑎/𝑎₀)³ in (100) should be replaced by Ω^∗_DM,0/(𝑎/𝑎₀)^3(1+𝛼). Another generalization may be introduced if dark matter is made of Bose-Einstein condensates (BECs) instead of being pressureless. In that case, the term Ω^∗_DM,0/(𝑎/𝑎₀)³ in (100) should be replaced by Ω^∗_DM,0/[(𝑎/𝑎₀)³∓(𝑎_BEC/𝑎₀)³] as proposed in [64, 65]. It is shown in these papers that the presence of BECs can substantially accelerate the formation of the large-scale structures of the universe.

6. Conclusion

There are different manners to describe the evolution of the universe.

In the standard ΛCDM model [1], the radiation, baryonic matter, dark matter, and dark energy are treated as four independent species characterized by different equations of state. The equation of state is 𝑝=𝜌𝑐²/3 for the radiation, 𝑝=0 for the baryonic matter and dark matter, and 𝑝=−𝜌𝑐² for the dark energy. One then solves the equation of continuity (15) for each species in order to obtain the relation between their density and the scale factor. Finally, we add their densities 𝜌=𝜌_rad+𝜌_𝐵+𝜌_DM+𝜌_Λ and solve the Friedmann equation (17). This leads to (99). This standard model provides a good description of the universe after the Planck era. However, it exhibits a primordial singularity (Big Bang) and does not display a phase of early inflation.

In recent papers [34–36], we have proposed a different description. We have described the early universe (vacuum energy + radiation) and the late universe (matter + dark energy) by two symmetric equations of state. The transition between the vacuum energy era and the radiation era in the early universe has been described by a single equation of state of the form 𝑝=−(4/3)(𝜌²/𝜌_𝑃)𝑐²+𝜌𝑐²/3. Similarly, the transition between the matter era and the dark energy era in the late universe has been described by a single equation of state 𝑝=−𝜌_Λ𝑐². These equations of state may be viewed as two polytropic equations of state with index 𝑛=1 and 𝑛=−1, respectively. This description leads to very symmetric mathematical formulae in the early and late universe; the cosmological density in the late universe plays a role similar to that of the Planck density in the early universe. In order to obtain a single equation describing the whole evolution of the universe, we have just added the density of these two phases 𝜌=𝜌_early+𝜌_late in the Friedmann equation (17). This leads to (100) with 𝑎₁/𝑎₀=1.97⋅10⁻³². For 𝑡>23.3𝑡_𝑃, the evolution is the same as in the ΛCDM model. However, this model incorporates a phase of early inflation in a natural manner and yields a model without primordial singularity (although we should not extrapolate it in the infinite past due to our neglect of quantum fluctuations). This may be called “aioniotic universe” since it has no origin nor end. As mentioned in the Introduction, this model may be viewed as a generalization of the Chaplygin gas model.

Finally, in this paper, we have adopted a different point of view. We have described the vacuum energy + radiation + dark energy as a single species called “generalized radiation” characterized by a quadratic equation of state 𝑝_𝑔=−(4/3)(𝜌²_𝑔/𝜌_𝑃)𝑐²+𝜌_𝑔𝑐²/3−𝜌_Λ𝑐². From this equation of state, we have obtained a generalization of the Stefan-Boltzmann law (see (69)). The baryonic matter and dark matter are treated as different species with a pressureless equation of state 𝑝_𝐵=𝑝_DM=0. We have solved the equation of continuity (15) to obtain the relation between the density and the scale factor for these three species. In order to obtain a single equation describing the whole evolution of the universe, we have just added their densities 𝜌=𝜌_𝑔+𝜌_𝐵+𝜌_DM in the Friedmann equation (17). This again leads to (100), as in our previous approach. However, the point of view is different. The present approach suggests that vacuum energy and dark energy may correspond to a form of “generalized radiation.” This may open new perspectives in cosmology. Of course, more work is needed to justify this quadratic equation of state. However, the present paper shows that this equation of state is able to reproduce the main properties of our universe.

Appendices

A. The Intermediate Universe: Transition between the Radiation Era and the Matter Era

In this Appendix, we consider the epoch where the evolution of the universe is dominated by radiation and matter. The total density can be written as 𝜌𝜌=^rad,0𝑎/𝑎₀⁴+𝜌_𝑚,0𝑎/𝑎₀³. (A.1) The transition between the radiation era and the matter era takes place when 𝜌_rad=𝜌_𝑚 leading to 𝑎=𝑎₄=(Ω_rad,0/Ω_𝑚,0)𝑎₀. Numerically, 𝑎₄=3.58⋅10⁻⁴𝑎₀=1.08⋅10⁻⁴𝑙_Λ, hence 𝑎₄=4.73⋅10²²m. (A.2) This corresponds to a density 𝜌₄=2(Ω⁴_𝑚,0/Ω³_rad,0)𝜌₀ and a temperature 𝑇₄=𝑇₀𝑎₀/𝑎₄. Numerically, 𝜌₄=1.03⋅10¹⁰𝜌₀=1.36⋅10¹⁰𝜌_Λ=9.52⋅10⁻¹⁴g/m³ and 𝑇₄=2.79⋅10³𝑇₀=8.69⋅10³K.

The Friedmann equation (100) simplifies into 𝐻𝐻₀=Ω_rad,0𝑎/𝑎₀⁴+Ω_𝑚,0𝑎/𝑎₀³. (A.3)

The evolution of the scale factor with time is obtained by solving the first-order differential equation (A.3). This yields ^𝑎/𝑎0₀𝑑𝑥𝑥Ω_rad,0/𝑥⁴+Ω_𝑚,0/𝑥³=𝐻₀𝑡. (A.4) The integral can be performed analytically leading to 𝐻₀4𝑡=3Ω_rad,0^3/2Ω_𝑚,0²−231Ω_𝑚,0^1/22Ω_rad,0Ω_𝑚,0−𝑎𝑎₀Ω_rad,0Ω_𝑚,0+𝑎𝑎₀. (A.5) In the radiation era (𝑎≪𝑎₄), we recover (79), and in the matter era (𝑎≫𝑎₄), we recover (96). The transition between the radiation era and the matter era takes place at a time 𝐻₀𝑡₄=(1/3)(4−2√2)(Ω_rad,0)^3/2/(Ω_𝑚,0)². Numerically, 𝑡₄=5.43⋅10⁻⁶𝐻₀⁻¹=1.64⋅10⁻⁶𝑡_Λ = 2.39⋅10¹²s.

B. Analytical Solution of the Friedmann Equations for a Toy Universe without Matter

For the sake of completeness, we give the solution of the Friedmann equations for a universe described by the pure equation of state (63). This solution is interesting mathematically because it is analytical, and it exhibits two phases of inflation separated by a radiation era. However, it is unphysical in the intermediate phase because it ignores the contribution of baryonic and dark matter. The numerical solution of (100) studied in the core of the paper is more physical.

In the early universe, the equation of state (63) reduces to (20). The density is given as a function of the scale factor by (21). The evolution of the scale factor is given by (31). This solution is satisfactory physically since matter can be neglected in the early universe.

In the late universe, the equation of state (63) reduces to (51). The density is given as a function of the scale factor by (52). If we neglect the contribution of matter, the solution of the Friedmann equation (17) with the density (52) is 𝑎𝑎₃=sinh^1/232𝜋3^1/2𝑡𝑡_Λ. (B.1) For 𝑡→0, it reduces to (58), and for 𝑡→+∞, we obtain (55) with a prefactor 1/√2. Furthermore, we can associate to the equation of state (51) a scalar field as in Section 3.7. The potential of this scalar field is 𝑉𝜓=13𝜌_Λ𝑐²cosh²,𝑎𝜓+2𝜓≤0₃𝑎²=−sinh𝜓,𝜓=8𝜋𝐺𝑐^21/2𝜙. (B.2) For 𝑡→0, using (58), we get 𝑡𝜓≃ln𝑡_Λ+12ln3𝜋24+ln3. (B.3) For 𝑡→+∞, using (55) with a prefactor 1/√2, we get 𝜓∼−2𝑒^{−(32𝜋/3)1/2𝑡/𝑡Λ}. (B.4) More generally, using (B.1), the evolution of the scalar field is given by 𝜓=−sinh⁻¹1sinh(32𝜋/3)𝑡/𝑡_Λ. (B.5)

For the quadratic equation of state (63), the density is related to the scale factor by (64). If we neglect the contribution of matter, it is possible to solve the Friedmann equation (17) with the density-radius relation (64) analytically. Introducing 𝑅=𝑎/𝑎₁ and 𝜆=𝜌_Λ/𝜌_𝑃≪1, we obtain 1+𝑅⁴𝑅1+𝜆𝑅⁴𝑑𝑅=8𝜋3^1/2𝑡𝑡_𝑃+𝐶, (B.6) which can be integrated into 1√𝜆ln1+2𝜆𝑅⁴+2𝜆1+𝑅⁴+𝜆𝑅⁸−ln2+𝑅⁴+21+𝑅⁴+𝜆𝑅⁸𝑅⁴=48𝜋3^1/2𝑡𝑡_𝑃+𝐶, (B.7) where the constant 𝐶 is determined such that 𝑎=𝑙_𝑃 at 𝑡=0. This solution is interesting mathematically because it describes analytically a phase of early inflation and a phase of late inflation connected by a power-law evolution corresponding to the radiation. However, since (B.7) does not take into account the contribution of baryonic matter and dark matter, the intermediate phase is unphysical: the solution (B.7) describes a universe without matter! The correct solution is obtained by solving the complete Friedmann equation (100) as in Section 5.5.