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

Authors: Pierre-Henri Chavanis
Journal: Journal of Gravity
 (2013)

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## Abstract

We consider a cosmological model based on a quadratic equation of statep/c2=-4ρ2/3ρP+ρ/3-4ρΛ/3 (where ρP is the Planck density and ρΛ is the cosmological density) “unifying” vacuum energy, radiation, and dark energy. For ρ→ρP, it reduces to p=-ρc2 leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density ρP=5.16·1099 g/m3 (vacuum energy). For ρΛ≪ρ≪ρP, we recover the equation of state of radiation p=ρc2/3. For ρ→ρΛ, we get p=-ρc2 leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological density ρΛ=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 ρP in the early universe plays a role similar to that of the cosmological density ρΛ 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>23.3tP, where tP=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.


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## Body

## 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 lP=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 fieldϕ 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 ρP=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  tP=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≃ρP, 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 constantΛ which is equivalent to a constant energy density ρΛ=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 ρΛ. As a result of the dark energy, the universe expands exponentially rapidly on a timescale of the order of the cosmological time tΛ=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/ρa 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
(1)p=(αρ+kρ1+1/n)c2,
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=kρ1+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=ρΛ) in the late universe [25–36].In [34–36], we remarked that a constant pressure p=-ρΛc2 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=-ρΛ, and linear coefficient α=0. At late times, this equation of state leads to a constant energy density equal to ρΛ 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=ρΛ) 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=-ρΛc2 (a=0, n=-1) among the whole family of Chaplygin gas models of the form p=-A/ρa.) 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=-4ρ2/3ρP+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/(3ρP), and linear coefficient α=1/3. At early times, this equation of state leads to a constant energy density equal to ρP 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=ρP) 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-calledcosmological problem [38] since the cosmological density ρΛ=7.02·10-24g/m3 and the Planck density ρP=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
(2)p=-4ρ23ρPc2+13ρc2-43ρΛc2,
involving the Planck density and the cosmological density. In the early universe, we recover the equation of state p/c2=-4ρ2/3ρP+ρ/3 unifying vacuum energy and radiation. In the late universe, we obtain an equation of state p/c2=ρ/3-4ρΛ/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=-4ρΛ/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 Section2, 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>23.3tP 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 ρΛ in the late universe plays a role similar to the Planck density ρP 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(3)ds2=c2dt2-a(t)2{dr21-kr2+r2(dθ2+sin2θdϕ2)},
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 a~0=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 Tji is
(4)T00=ρc2,T11=T22=T33=-p.
The Einstein equations
(5)Rji-12gjiR-Λgji=-8πGc2Tji
relate the geometrical structure of the spacetime (gij) to the material content of the universe (Tij). For the sake of generality, we have accounted for a possibly nonzero cosmological constant Λ. Given Equations (3) and (4), these equations reduce to
(6)8πGρ+Λ=3a˙2+kc2a2,8πGc2p-Λ=-2aa¨+a˙2+kc2a2,
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(7)dρdt+3a˙a(ρ+pc2)=0,(8)a¨a=-4πG3(ρ+3pc2)+Λ3,(9)H2=(a˙a)2=8πG3ρ-kc2a2+Λ3,
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 ∂iTij=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(10)adρda=-3(ρ+pc2),(11)dda(ρa3)=-3pc2a2,(12)a3dpdt=ddt[a3(p+ρc2)].
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 formdS(V,T)=(1/T)[d(ρ(T)c2V)+p(T)dV], one can derive the thermodynamical equation [6]
(13)dpdT=1T(ρc2+p).
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
(14)S˙=0,whereS=a3T(p+ρc2)
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
(15)dρdt+3a˙a(ρ+pc2)=0,(16)a¨a=-4πG3(ρ+3pc2),(17)H2=(a˙a)2=8πG3ρ.
The deceleration parameter is defined by
(18)q(t)=-a¨aa˙2.
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
(19)q(t)=1+3w(t)2.
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
(20)p=13ρc2-4ρ23ρPc2,
where ρP=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/(3ρP). For ρ≪ρP, we recover the equation of state of the pure radiation p~ρc2/3. For ρ→ρP, we get p→-ρPc2 corresponding to the vacuum energy.The continuity equation (15) may be integrated into
(21)ρ=ρP(a/a1)4+1,
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=ρP) and the radiation era (p=ρc2/3 and ρrad~ρPa14/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
(22)1ρ=1ρrad+1ρP.
At a=a1 we have ρrad=ρP so that ρ=ρP/2.The equation of state parameterw=p/ρc2 and the deceleration parameter q are given by
(23)w=13-4ρ3ρP,q=1-2ρρP.
The velocity of sound cs2=p′(ρ) is given by
(24)cs2c2=13-83ρρP.
As the universe expands from a=0 to a=+∞, the density decreases from ρP 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 (cs/c)2 increases from -7/3 to 1/3.


### 3.1. The Vacuum Energy Era: Early Inflation

Whena≪a1, the density tends to a maximum value
(25)ρ=ρmax=ρP,
and the pressure tends to p=-ρPc2. The Planck density ρP=5.16·1099g/m3 (vacuum energy) represents a fundamental upper bound for the density. A constant value of the density ρ≃ρP gives rise to a phase of early inflation. From the Friedmann equation (17), we find that the Hubble parameter is constant H=(8π/3)1/2tP-1 where we have introduced the Planck time tP=1/(GρP)1/2=(ℏG/c5)1/2. Numerically, H=5.37·1043s-1. Therefore, the scale factor increases exponentially rapidly with time as
(26)a(t)~lPe(8π/3)1/2t/tP.
The timescale of the exponential growth is the Planck time tP=5.39·10-44s. We have defined the “original” time t=0 such that a(0) is equal to the Planck length lP=ctP=(Gℏ/c3)1/2=1.62·10-35m. Mathematically speaking, the universe exists at any time in the past (a→0 and ρ→ρP 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>lP.


### 3.2. The Radiation Era

Whena≫a1, we recover the equation
(27)ρrad~ρPa14a4,
corresponding to the pure radiation described by an equation of state p=ρc2/3. The conservation of ρrada4 implies that ρPa14=ρrad,0a04 where ρrad,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  ρrad,0=Ωrad,0ρ0 where ρ0=3H02/8πG=9.20·10-24g/m3 is the present density of the universe given by (17) and Ωrad,0=8.48·10-5 is the present fraction of radiation in the universe [1], we obtain
(28)a1=2.61·10-6m.
This scale is intermediate between the Planck length lP and the present size of the universe a0 (a1/lP=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
(29)aa1~(32π3)1/4(ttP)1/2.
We then have
(30)ρρP~332π(tPt)2.
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
(31)(a/a1)4+1-ln(1+(a/a1)4+1(a/a1)2)=2(8π3)1/2ttP+C,
where  C  is a constant of integration. It is determined such that a=lP at t=0. Setting  ϵ=lP/a1, we get
(32)C(ϵ)=ϵ4+1-ln(1+ϵ4+1ϵ2).
Numerically,  ϵ=6.20·10-30 and C=-134. For t→-∞, we have the exact asymptotic result
(33)a(t)~a1e(8π/3)1/2t/tP+D,
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<ac (i.e., ρ>ρc) and decelerating when a>ac (i.e., ρ<ρc) where ac=a1 and ρc=ρP/2. The time tc at which the universe starts decelerating is given by tc=(3/32π)1/2[2-ln(1+2)-C]tP. This corresponds to the time at which the curve a(t) presents an inflexion point. It turns out that this inflexion point ac coincides with a1 so it also marks the end of the inflation (tc=t1). Numerically, ρc=ρ1=0.5ρP=2.58·1099g/m3, ac=a1=2.61·10-6m, and tc=t1=23.3tP=1.25·10-42s. At t=ti=0, ai=lP and ρi≃ρP. Numerically, ai=lP=1.62·10-35m and ρi≃ρP=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
(34)T=TP(15π2)1/4(1-ρρP)7/4(ρρP)1/4,
where TP is a constant of integration. Using (21), we get
(35)T=TP(15π2)1/4(a/a1)7[(a/a1)4+1]2.
Equation (34) may be seen as a generalized Stefan-Boltzmann law in the early universe. In the radiation era ρ≪ρP, it reduces to the ordinary Stefan-Boltzmann law
(36)T=TP(15π2)1/4(ρρP)1/4.
This allows us to identify the constant of integration TP with the Planck temperature TP=(ℏc5/GkB2)1/2=1.42·1032K. On the other hand, in the inflationary phase ρ→ρP, the temperature is related to the density by
(37)TTP≃(15π2)1/4(1-ρρP)7/4.
According to (35), T~TP(15/π2)1/4(a/a1)7 for a≪a1 and T~TP(15/π2)1/4(a1/a) for a≫a1.The temperature starts fromT=0 at t=-∞, increases exponentially rapidly during the inflation as
(38)TTP~(15π2)1/4(lPa1)7e7(8π/3)1/2t/tP
reaches a maximum value Te and decreases algebraically as
(39)TTP~(4532π3)1/4(tPt)1/2,
during the radiation era. Using (34) and (35), we find that the point corresponding to the maximum temperature is ρe=ρP/8, ae=71/4a1, Te=(7/8)7/4(15/8π2)1/4TP. It is reached at a time te=(3/32π)1/2[8-ln(1+8)+ln(7)/2-C]tP. Numerically, ρe=0.125ρP=6.44·1098g/m3, ae=1.63a1=4.24·10-6m, Te=0.523TP=7.40·1031K, and te=23.6tP=1.27·10-42s. At t=ti=0, Ti≃TP(15/π2)1/4(lP/a1)7. Numerically, Ti=3.91·10-205TP=5.54·10-173K. At t=tc=t1, Tc=(1/4)(15/π2)1/4TP. Numerically, Tc=T1=0.278TP=3.93·1031K.The pressure is given by (20). Using (21), we get
(40)p=13(a/a1)4-3[(a/a1)4+1]2ρPc2.
The pressure starts from p=-ρPc2 at t=-∞, remains approximately constant during the inflation, becomes positive, reaches a maximum value pe, and decreases algebraically during the radiation era. At t=ti=0, pi≃-ρPc2. Numerically, pi=-4.64·10116g/ms2. The point at which the pressure vanishes is aw=31/4a1, ρw=ρP/4, Tw=(9/16)(5/π2)1/4TP. This corresponds to a time tw=(3/32π)1/2[2-(1/2)ln(3)-C]tP. Numerically, aw=1.32a1=3.43·10-6m, ρw=0.25ρP=1.29·1099g/m3, Tw=0.475TP=6.74·1031K, and tw=23.4tP=1.26·10-42s. On the other hand, the pressure reaches its maximum pe=ρPc2/48 at the same point as the one at which the temperature reaches its maximum. Numerically, pe=9.66·10114g/ms2. At t=tc=t1, we have pc=p1=-ρPc2/6. Numerically, pc=p1=-7.73·10115g/ms2.Using (20), (21), and (35), we find that the entropy (14) is given by
(41)S=43(π215)1/4ρPc2TPa13,
and we check that it is a constant. Numerically, S/kB=5.04·1087.


### 3.5. The Evolution of the Early Universe

In our model, the universe starts att=-∞ with a vanishing radius a=0, a finite density ρ=ρP=5.16·1099g/m3, a finite pressure p=-ρPc2=-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 lP=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=ti=0, the radius of the universe is equal to the Planck length ai=lP=1.62·10-35m. The corresponding density, temperature, and pressure are ρi≃ρP=5.16·1099g/m3, Ti=3.91·10-205TP=5.54·10-173K, and pi≃-ρPc2=-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=ti=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 ti=0 and ends at t1=23.3tP=1.25·10-42s. During this very short lapse of time, the radius of the universe grows from ai=lP=1.62·10-35m to a1=2.61·10-6m, and the temperature grows from Ti=3.91·10-205TP=5.54·10-173K to T1=0.278TP=3.93·1031K. By contrast, the density and the pressure do not change significatively: they go from ρi≃ρP=5.16·1099g/m3 and pi≃-ρPc2=-4.64·10116g/ms2 to ρ1=0.5ρP=2.58·1099g/m3 and p1=-0.167ρPc2=-7.73·10115g/ms2. The pressure passes from negative values to positive values at tw=23.4tP=1.26·10-42s corresponding to ρw=0.25ρP=1.29·1099g/m3, aw=1.32a1=3.43·10-6m, and Tw=0.475TP=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∝t1/2, 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 Te=0.523TP=7.40·1031K and pe=2.08·10-2ρPc2=9.66·10114g/ms2 at t=te=23.6tP=1.27·10-42s. At that moment, the density is ρe=0.125ρP=6.44·1098g/m3 and the radius ae=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 tc=t1=23.3tP=1.25·10-42s coinciding with the end of the inflation (ac=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 factora 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 factora 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 densityρ 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 densityρ 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 temperatureT 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 temperatureT 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 pressurep 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 ti=10-35s but expands exponentially by a factor 1030 in the interval ti<t<tf with tf=10-33s. For t>tf, the evolution is again radiation dominated. At t=ti, 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 att=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∝t1/2. 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~(Tc-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 att=0 disappears and the curves a(t), ρ(t) and T(t) are regularized. In particular, we find that a=lP>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 whenℏ→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 tP0 and lengths in terms of a10. We introduce a^=a/a10 and t^=t/tP0. Using lP=ctP and ρPa14=ρrad,0a04, we obtain
(42)lPlP0=tPtP0=(a1a10)2=ℏ^1/2,
where ℏ^=ℏ/ℏ0. Therefore
(43)ϵϵ0=lP/a1lP0/a10=a1a10=ℏ^1/4,
where ϵ0=lP0/a10=6.20·10-30. Using these relations and (31), we get
(44)a^4ℏ^+1-ln(1+a^4/ℏ^+1a^2/ℏ^1/2)=2(8π3)1/2t^ℏ^1/2+C(ϵ0ℏ^1/4),
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
(45)ϕ¨+3Hϕ˙+dVdϕ=0,
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
(46)ρc2=12ϕ˙2+V(ϕ),p=12ϕ˙2-V(ϕ).
Using standard technics [7], we find that the potential corresponding to the equation of state (20) is [34–36]
(47)V(ψ)=13ρPc2cosh2ψ+2cosh4ψ(ψ≥0),(aa1)2=sinhψ,ψ=(8πGc2)1/2ϕ.
In the vacuum energy era (t→-∞), using (26), we get
(48)ψ~(lPa1)2e32π/3t/tP⟶0.
In the radiation era (t→+∞), using (29), we get
(49)ψ≃lnttP+12ln8π3+2ln(2)⟶+∞.
More generally, using (31), the evolution of the scalar field ψ(t) in the early universe is given by
(50)coshψ-ln(1+coshψsinhψ)=2(8π3)1/2ttP+C.
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.


## 3.1. The Vacuum Energy Era: Early Inflation

Whena≪a1, the density tends to a maximum value
(25)ρ=ρmax=ρP,
and the pressure tends to p=-ρPc2. The Planck density ρP=5.16·1099g/m3 (vacuum energy) represents a fundamental upper bound for the density. A constant value of the density ρ≃ρP gives rise to a phase of early inflation. From the Friedmann equation (17), we find that the Hubble parameter is constant H=(8π/3)1/2tP-1 where we have introduced the Planck time tP=1/(GρP)1/2=(ℏG/c5)1/2. Numerically, H=5.37·1043s-1. Therefore, the scale factor increases exponentially rapidly with time as
(26)a(t)~lPe(8π/3)1/2t/tP.
The timescale of the exponential growth is the Planck time tP=5.39·10-44s. We have defined the “original” time t=0 such that a(0) is equal to the Planck length lP=ctP=(Gℏ/c3)1/2=1.62·10-35m. Mathematically speaking, the universe exists at any time in the past (a→0 and ρ→ρP 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>lP.


## 3.2. The Radiation Era

Whena≫a1, we recover the equation
(27)ρrad~ρPa14a4,
corresponding to the pure radiation described by an equation of state p=ρc2/3. The conservation of ρrada4 implies that ρPa14=ρrad,0a04 where ρrad,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  ρrad,0=Ωrad,0ρ0 where ρ0=3H02/8πG=9.20·10-24g/m3 is the present density of the universe given by (17) and Ωrad,0=8.48·10-5 is the present fraction of radiation in the universe [1], we obtain
(28)a1=2.61·10-6m.
This scale is intermediate between the Planck length lP and the present size of the universe a0 (a1/lP=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
(29)aa1~(32π3)1/4(ttP)1/2.
We then have
(30)ρρP~332π(tPt)2.
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
(31)(a/a1)4+1-ln(1+(a/a1)4+1(a/a1)2)=2(8π3)1/2ttP+C,
where  C  is a constant of integration. It is determined such that a=lP at t=0. Setting  ϵ=lP/a1, we get
(32)C(ϵ)=ϵ4+1-ln(1+ϵ4+1ϵ2).
Numerically,  ϵ=6.20·10-30 and C=-134. For t→-∞, we have the exact asymptotic result
(33)a(t)~a1e(8π/3)1/2t/tP+D,
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<ac (i.e., ρ>ρc) and decelerating when a>ac (i.e., ρ<ρc) where ac=a1 and ρc=ρP/2. The time tc at which the universe starts decelerating is given by tc=(3/32π)1/2[2-ln(1+2)-C]tP. This corresponds to the time at which the curve a(t) presents an inflexion point. It turns out that this inflexion point ac coincides with a1 so it also marks the end of the inflation (tc=t1). Numerically, ρc=ρ1=0.5ρP=2.58·1099g/m3, ac=a1=2.61·10-6m, and tc=t1=23.3tP=1.25·10-42s. At t=ti=0, ai=lP and ρi≃ρP. Numerically, ai=lP=1.62·10-35m and ρi≃ρP=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
(34)T=TP(15π2)1/4(1-ρρP)7/4(ρρP)1/4,
where TP is a constant of integration. Using (21), we get
(35)T=TP(15π2)1/4(a/a1)7[(a/a1)4+1]2.
Equation (34) may be seen as a generalized Stefan-Boltzmann law in the early universe. In the radiation era ρ≪ρP, it reduces to the ordinary Stefan-Boltzmann law
(36)T=TP(15π2)1/4(ρρP)1/4.
This allows us to identify the constant of integration TP with the Planck temperature TP=(ℏc5/GkB2)1/2=1.42·1032K. On the other hand, in the inflationary phase ρ→ρP, the temperature is related to the density by
(37)TTP≃(15π2)1/4(1-ρρP)7/4.
According to (35), T~TP(15/π2)1/4(a/a1)7 for a≪a1 and T~TP(15/π2)1/4(a1/a) for a≫a1.The temperature starts fromT=0 at t=-∞, increases exponentially rapidly during the inflation as
(38)TTP~(15π2)1/4(lPa1)7e7(8π/3)1/2t/tP
reaches a maximum value Te and decreases algebraically as
(39)TTP~(4532π3)1/4(tPt)1/2,
during the radiation era. Using (34) and (35), we find that the point corresponding to the maximum temperature is ρe=ρP/8, ae=71/4a1, Te=(7/8)7/4(15/8π2)1/4TP. It is reached at a time te=(3/32π)1/2[8-ln(1+8)+ln(7)/2-C]tP. Numerically, ρe=0.125ρP=6.44·1098g/m3, ae=1.63a1=4.24·10-6m, Te=0.523TP=7.40·1031K, and te=23.6tP=1.27·10-42s. At t=ti=0, Ti≃TP(15/π2)1/4(lP/a1)7. Numerically, Ti=3.91·10-205TP=5.54·10-173K. At t=tc=t1, Tc=(1/4)(15/π2)1/4TP. Numerically, Tc=T1=0.278TP=3.93·1031K.The pressure is given by (20). Using (21), we get
(40)p=13(a/a1)4-3[(a/a1)4+1]2ρPc2.
The pressure starts from p=-ρPc2 at t=-∞, remains approximately constant during the inflation, becomes positive, reaches a maximum value pe, and decreases algebraically during the radiation era. At t=ti=0, pi≃-ρPc2. Numerically, pi=-4.64·10116g/ms2. The point at which the pressure vanishes is aw=31/4a1, ρw=ρP/4, Tw=(9/16)(5/π2)1/4TP. This corresponds to a time tw=(3/32π)1/2[2-(1/2)ln(3)-C]tP. Numerically, aw=1.32a1=3.43·10-6m, ρw=0.25ρP=1.29·1099g/m3, Tw=0.475TP=6.74·1031K, and tw=23.4tP=1.26·10-42s. On the other hand, the pressure reaches its maximum pe=ρPc2/48 at the same point as the one at which the temperature reaches its maximum. Numerically, pe=9.66·10114g/ms2. At t=tc=t1, we have pc=p1=-ρPc2/6. Numerically, pc=p1=-7.73·10115g/ms2.Using (20), (21), and (35), we find that the entropy (14) is given by
(41)S=43(π215)1/4ρPc2TPa13,
and we check that it is a constant. Numerically, S/kB=5.04·1087.


## 3.5. The Evolution of the Early Universe

In our model, the universe starts att=-∞ with a vanishing radius a=0, a finite density ρ=ρP=5.16·1099g/m3, a finite pressure p=-ρPc2=-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 lP=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=ti=0, the radius of the universe is equal to the Planck length ai=lP=1.62·10-35m. The corresponding density, temperature, and pressure are ρi≃ρP=5.16·1099g/m3, Ti=3.91·10-205TP=5.54·10-173K, and pi≃-ρPc2=-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=ti=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 ti=0 and ends at t1=23.3tP=1.25·10-42s. During this very short lapse of time, the radius of the universe grows from ai=lP=1.62·10-35m to a1=2.61·10-6m, and the temperature grows from Ti=3.91·10-205TP=5.54·10-173K to T1=0.278TP=3.93·1031K. By contrast, the density and the pressure do not change significatively: they go from ρi≃ρP=5.16·1099g/m3 and pi≃-ρPc2=-4.64·10116g/ms2 to ρ1=0.5ρP=2.58·1099g/m3 and p1=-0.167ρPc2=-7.73·10115g/ms2. The pressure passes from negative values to positive values at tw=23.4tP=1.26·10-42s corresponding to ρw=0.25ρP=1.29·1099g/m3, aw=1.32a1=3.43·10-6m, and Tw=0.475TP=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∝t1/2, 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 Te=0.523TP=7.40·1031K and pe=2.08·10-2ρPc2=9.66·10114g/ms2 at t=te=23.6tP=1.27·10-42s. At that moment, the density is ρe=0.125ρP=6.44·1098g/m3 and the radius ae=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 tc=t1=23.3tP=1.25·10-42s coinciding with the end of the inflation (ac=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 factora 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 factora 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 densityρ 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 densityρ 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 temperatureT 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 temperatureT 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 pressurep 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 ti=10-35s but expands exponentially by a factor 1030 in the interval ti<t<tf with tf=10-33s. For t>tf, the evolution is again radiation dominated. At t=ti, 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 att=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∝t1/2. 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~(Tc-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 att=0 disappears and the curves a(t), ρ(t) and T(t) are regularized. In particular, we find that a=lP>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 whenℏ→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 tP0 and lengths in terms of a10. We introduce a^=a/a10 and t^=t/tP0. Using lP=ctP and ρPa14=ρrad,0a04, we obtain
(42)lPlP0=tPtP0=(a1a10)2=ℏ^1/2,
where ℏ^=ℏ/ℏ0. Therefore
(43)ϵϵ0=lP/a1lP0/a10=a1a10=ℏ^1/4,
where ϵ0=lP0/a10=6.20·10-30. Using these relations and (31), we get
(44)a^4ℏ^+1-ln(1+a^4/ℏ^+1a^2/ℏ^1/2)=2(8π3)1/2t^ℏ^1/2+C(ϵ0ℏ^1/4),
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
(45)ϕ¨+3Hϕ˙+dVdϕ=0,
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
(46)ρc2=12ϕ˙2+V(ϕ),p=12ϕ˙2-V(ϕ).
Using standard technics [7], we find that the potential corresponding to the equation of state (20) is [34–36]
(47)V(ψ)=13ρPc2cosh2ψ+2cosh4ψ(ψ≥0),(aa1)2=sinhψ,ψ=(8πGc2)1/2ϕ.
In the vacuum energy era (t→-∞), using (26), we get
(48)ψ~(lPa1)2e32π/3t/tP⟶0.
In the radiation era (t→+∞), using (29), we get
(49)ψ≃lnttP+12ln8π3+2ln(2)⟶+∞.
More generally, using (31), the evolution of the scalar field ψ(t) in the early universe is given by
(50)coshψ-ln(1+coshψsinhψ)=2(8π3)1/2ttP+C.
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(51)p=13ρc2-43ρΛc2,
where ρΛ=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)ρΛ. It may be viewed as the “symmetric” version of the equation of state (20) in the early universe. For ρ≫ρΛ, we recover the equation of state of the pure radiation p~ρc2/3. For ρ→ρΛ, we get p≃-ρΛc2 corresponding to the dark energy.The continuity equation (15) may be integrated into
(52)ρ=ρΛ[(a3a)4+1],
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 ρrad~ρΛ(a3/a)4) and the dark energy era (p=-ρc2 and ρ=ρΛ). 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
(53)ρ=ρrad+ρΛ.
At a=a3 we have ρrad=ρΛ so that ρ=2ρΛ.


### 4.1. The Dark Energy Era: Late Inflation

Whena≫a3, the density tends to a minimum value
(54)ρ=ρmin=ρΛ,
and the pressure tends to p=-ρΛc2. The cosmological density ρΛ=Λ/8πG=7.02·10-24g/m3 (dark energy) represents a fundamental lower bound for the density. A constant value of the density ρ≃ρΛ gives rise to a phase of late inflation. It is convenient to define a cosmological time tΛ=1/(GρΛ)1/2=(8π/Λ)1/2=1.46·1018s and a cosmological length lΛ=ctΛ=(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)1/2tΛ-1. Numerically, H=1.98·10-18s-1. Therefore, the scale factor increases exponentially rapidly with time as
(55)a(t)∝e(8π/3)1/2t/tΛ.
This exponential growth corresponds to the de Sitter solution [1]. The timescale of the exponential growth is the cosmological time tΛ=1.46·1018s. This solution exists at any time in the future (a→+∞ and ρ→ρΛ 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

Whena≪a3, we recover the equation
(56)ρrad~ρΛa34a4,
corresponding to the pure radiation described by an equation of state p=ρc2/3. The conservation of ρrada4 implies that ρΛa34=ρrad,0a04. Using ρΛ=ΩΛ,0ρ0 and ρrad,0=Ωrad,0ρ0 with ΩΛ,0=0.763 (dark energy) and Ωrad,0=8.48·10-5 (radiation) [1], we obtain a3/a0=(Ωrad,0/ΩΛ,0)1/4 hence
(57)a3=1.36·1025m.
This can be rewritten as a3=0.103a0=3.09·10-2lΛ where we have used lΛ/a0=H0tΛ=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
(58)aa3~(32π3)1/4(ttΛ)1/2.
We then have
(59)ρρΛ~332π(tΛt)2.


### 4.3. The Temperature and the Pressure

The thermodynamical equation (13) with the equation of state (51) may be integrated into
(60)T=TP(15π2)1/4(ρ-ρΛρP)1/4,
where TP is a constant of integration. Using (52), we get
(61)T=TP(15π2)1/4(ρΛρP)1/4a3a.
Equation (60) may be seen as a generalized Stefan-Boltzmann law in the late universe. In the radiation era ρ≫ρΛ, it reduces to the ordinary Stefan-Boltzmann law (36). This allows us to identify the constant of integration TP with the Planck temperature TP=(ℏc5/GkB2)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~TP(45/32π3)1/4(ρΛ/ρP)1/4(tΛ/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
(62)p=[13(a3a)4-1]ρΛc2.
It decreases algebraically during the radiation era and tends to a constant negative value p=-ρΛc2 for t→+∞. Numerically, p=-6.31·10-7g/ms2. The point at which the pressure vanishes is aw′=a3/31/4, ρw′=4ρΛ, Tw′=TP(45/π2)1/4(ρΛ/ρP)1/4. Numerically, aw′=0.760a3=1.03·1025m, ρw′=4ρΛ=2.81·10-23g/m3, Tw′=2.81·10-31TP=39.9K.


## 4.1. The Dark Energy Era: Late Inflation

Whena≫a3, the density tends to a minimum value
(54)ρ=ρmin=ρΛ,
and the pressure tends to p=-ρΛc2. The cosmological density ρΛ=Λ/8πG=7.02·10-24g/m3 (dark energy) represents a fundamental lower bound for the density. A constant value of the density ρ≃ρΛ gives rise to a phase of late inflation. It is convenient to define a cosmological time tΛ=1/(GρΛ)1/2=(8π/Λ)1/2=1.46·1018s and a cosmological length lΛ=ctΛ=(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)1/2tΛ-1. Numerically, H=1.98·10-18s-1. Therefore, the scale factor increases exponentially rapidly with time as
(55)a(t)∝e(8π/3)1/2t/tΛ.
This exponential growth corresponds to the de Sitter solution [1]. The timescale of the exponential growth is the cosmological time tΛ=1.46·1018s. This solution exists at any time in the future (a→+∞ and ρ→ρΛ 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

Whena≪a3, we recover the equation
(56)ρrad~ρΛa34a4,
corresponding to the pure radiation described by an equation of state p=ρc2/3. The conservation of ρrada4 implies that ρΛa34=ρrad,0a04. Using ρΛ=ΩΛ,0ρ0 and ρrad,0=Ωrad,0ρ0 with ΩΛ,0=0.763 (dark energy) and Ωrad,0=8.48·10-5 (radiation) [1], we obtain a3/a0=(Ωrad,0/ΩΛ,0)1/4 hence
(57)a3=1.36·1025m.
This can be rewritten as a3=0.103a0=3.09·10-2lΛ where we have used lΛ/a0=H0tΛ=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
(58)aa3~(32π3)1/4(ttΛ)1/2.
We then have
(59)ρρΛ~332π(tΛt)2.


## 4.3. The Temperature and the Pressure

The thermodynamical equation (13) with the equation of state (51) may be integrated into
(60)T=TP(15π2)1/4(ρ-ρΛρP)1/4,
where TP is a constant of integration. Using (52), we get
(61)T=TP(15π2)1/4(ρΛρP)1/4a3a.
Equation (60) may be seen as a generalized Stefan-Boltzmann law in the late universe. In the radiation era ρ≫ρΛ, it reduces to the ordinary Stefan-Boltzmann law (36). This allows us to identify the constant of integration TP with the Planck temperature TP=(ℏc5/GkB2)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~TP(45/32π3)1/4(ρΛ/ρP)1/4(tΛ/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
(62)p=[13(a3a)4-1]ρΛc2.
It decreases algebraically during the radiation era and tends to a constant negative value p=-ρΛc2 for t→+∞. Numerically, p=-6.31·10-7g/ms2. The point at which the pressure vanishes is aw′=a3/31/4, ρw′=4ρΛ, Tw′=TP(45/π2)1/4(ρΛ/ρP)1/4. Numerically, aw′=0.760a3=1.03·1025m, ρw′=4ρΛ=2.81·10-23g/m3, Tw′=2.81·10-31TP=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(63)pg=-4ρg23ρPc2+13ρgc2-43ρΛc2.
For ρg→ρP, pg→-ρPc2 (vacuum energy); for ρΛ≪ρg≪ρP, pg~ρgc2/3 (radiation); for ρg→ρΛ, pg→-ρΛc2 (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
(64)ρg=ρP(a/a1)4+1+ρΛ,
where a1=2.61·10-6m (see Section 3). To obtain this expression, we have used the fact that ρP≫ρΛ so that pg/c2+ρg≃(4/3ρP)(ρg-ρΛ)(ρP-ρg). When a→0, ρg→ρP (vacuum energy); when ρΛ≪ρg≪ρP, ρg~ρP(a/a1)-4 (radiation); when a→+∞, ρg→ρΛ (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 ρPa14=ρΛa34.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 ρPa14=ρrad,0a04 (see Section 3), we can rewrite (64) in the form
(65)ρg=ρrad,0(a/a0)4+(a1/a0)4+ρΛ,
where a1/a0=1.97·10-32. The baryonic matter and the dark matter are described as pressureless fluids with pB=0 and pDM=0. The equation of continuity (15) for each species leads to the relations(66a)ρB=ρB,0(a/a0)3,(66b)ρDM=ρDM,0(a/a0)3.The total densityρ=ρg+ρB+ρDM of generalized radiation, baryonic matter, and dark energy can be written as
(67)ρ=ρrad,0(a/a0)4+(a1/a0)4+ρB,0(a/a0)3+ρDM,0(a/a0)3+ρΛ.
Substituting this relation in the Friedmann equation (17) and writing ρrad,0=Ωrad,0ρ0, ρB,0=ΩB,0ρ0, ρDM,0=ΩDM,0ρ0, and ρΛ=ΩΛ,0ρ0, we obtain
(68)HH0=Ωrad,0(a/a0)4+(a1/a0)4+ΩB,0(a/a0)3+ΩDM,0(a/a0)3+ΩΛ,0.
From observations, Ωrad,0=8.48·10-5, ΩB,0=0.0455, ΩDM,0=0.1915, Ωm,0=0.237, and ΩΛ,0=0.763 [1].The thermodynamic equation (13) with the equation of state (63) may be integrated into
(69)T=TP(15π2)1/4(1-ρgρP)7/4(ρgρP-ρΛρP)1/4.
Using (64), we get
(70)T=TP(15π2)1/4(a/a1)7[(a/a1)4+1]2.
Equation (70) may also be written as
(71)T=T0(a/a0)7[(a/a0)4+(a1/a0)4]2,
with T0=TP(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
(72)T~T0a0a.Finally, using (63) and (64), the pressure of generalized radiation can be written in very good approximation as
(73)pg=13(a/a1)4-3[(a/a1)4+1]2ρPc2-ρΛc2.


### 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
(74)ρ=ρrad,0(a/a0)4+(a1/a0)4.
The Friedmann equation (68) becomes
(75)HH0=Ωrad,0(a/a0)4+(a1/a0)4.
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 ρrad=ρP. This yields a1/a0=(ρrad,0/ρP)1/4, ρ1=ρP/2, T1=(1/4)(15/π2)1/4TP. Numerically, a1=1.97·10-32a0=2.61·10-6m, ρ1=0.5ρP=2.58·1099g/m3, T1=0.278TP=3.93·1031K. This takes place at a time t1=23.3tP=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:
(76)ρ≃ρP.
Therefore, the Hubble parameter is also a constant H=(8π/3)1/2tP-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
(77)ρ~ρrad,0a04a4.The Friedmann equation (75) reduces to
(78)HH0=Ωrad,0(a/a0)4
and yields
(79)aa0~Ωrad,01/42H0t,ρρ0~1(2H0t)2,TT0~a0a~1Ωrad,01/42H0t.


### 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
(80)ρ=ρm,0(a/a0)3+ρΛ,
where ρm=ρB+ρDM. This coincides with the ΛCDM model (see Section 5.5). The Friedmann equation (68) becomes
(81)HH0=Ωm,0(a/a0)3+ΩΛ,0,
with Ωm,0+ΩΛ,0≃1. It has the analytical solution
(82)aa0=(Ωm,0ΩΛ,0)1/3sinh2/3(32ΩΛ,0H0t).
The density evolves as
(83)ρρ0=ΩΛ,0tanh2((3/2)ΩΛ,0H0t).
On the other hand, the temperature of the generalized radiation in the late universe is given by
(84)T=TP(15π2)1/4(ρgρP-ρΛρP)1/4,T=T0a0a.
Using the expression (82) of the scale factor, we obtain
(85)T=T0(Ωm,0/ΩΛ,0)1/3sinh2/3((3/2)ΩΛ,0H0t).
Finally, the pressure of the generalized radiation in the late universe may be written as
(86)pg=13ρrad,0c2(a/a0)4-ρΛc2.Settinga=a0 in (82), we find the age of the universe
(87)t0=1H0231ΩΛ,0sinh-1[(ΩΛ,0Ωm,0)1/2].
Numerically, t0=1.03H0-1=0.310tΛ=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 whenρm=2ρΛ [see (16) with ρ≃ρm+ρΛ and p≃-ρΛc2]. Using ρm,0=Ωm,0ρ0 and ρΛ=ΩΛ,0ρ0 with Ωm,0=0.237 (matter) and ΩΛ,0=0.763 (dark energy), we obtain a=ac′ and t=tc′ with
(88)ac′a0=(Ωm,02ΩΛ,0)1/3,tc′=1H0231ΩΛ,0sinh-1(12).
At that point ρ=ρc′=3ρΛ and T=Tc′=T0a0/ac′. Numerically, ac′=0.538a0=0.162lΛ=7.11·1025m, tc′=0.504H0-1=0.152tΛ=2.22·1017s, ρc′=3ρΛ=2.11·10-23g/m3, and Tc′=1.86T0=5.78K. The universe is decelerating for t<tc′ and accelerating for t>tc′.The transition between the matter era and the dark energy era corresponds toρm=ρΛ. This leads to a=a2 and t=t2 with
(89)a2a0=(Ωm,0ΩΛ,0)1/3,t2=1H0231ΩΛ,0sinh-1(1).
At that point ρ=ρ2=2ρΛ and T=T2=T0a0/a2. Numerically, a2=0.677a0=0.204lΛ=8.95·1025m, t2=0.674H0-1=0.203tΛ=2.97·1017s, ρ2=2ρΛ=1.40·10-23g/m3, and T2=1.48T0=4.59K. We also note that ac′=(1/2)1/3a2=0.794a2. Of course, the value
(90)a2=8.95·1025m
coincides with the one obtained in [34–36] using a different viewpoint.In the dark energy era (a≫a2), the density is constant:
(91)ρ≃ρΛ.
Therefore, the Hubble parameter is also a constant H=H0ΩΛ,0. Using (82), we find that the scale factor increases exponentially rapidly as
(92)aa0~(Ωm,04ΩΛ,0)1/3eΩΛ,0H0t.
This corresponds to the de Sitter solution. According to (85), the temperature decreases exponentially rapidly as
(93)TT0~(4ΩΛ,0Ωm,0)1/3e-ΩΛ,0H0t.
In the matter era (a≪a2), we have
(94)ρ~ρm,0a03a3.
The Friedmann equation (81) reduces to
(95)HH0=Ωm,0(a/a0)3,
and we have
(96)aa0=Ωm,01/3(32H0t)2/3,ρρ0=1((3/2)H0t)2,(97)TT0~1Ωm,01/3((3/2)H0t)2/3.
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∝t2/3, 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 ρΛ. The transition takes place at a2=0.677a0=0.204lΛ=8.95·1025m, ρ2=2ρΛ=1.40·10-23g/m3, T2=1.48T0=4.59K, and t2=0.674H0-1=0.203tΛ=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 tc′=0.504H0-1=0.152tΛ=2.22·1017s, corresponding to a radius ac′=0.538a0=0.162lΛ=0.794a2=7.11·1025m, a density ρc′=3ρΛ=2.11·10-23g/m3 and a temperature Tc′=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 factora 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<ac′ and accelerating for a>ac′ with ac′=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 factora with the time t in linear scales.Figure 13
Evolution of the densityρ 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 densityρ 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 radiationT with the time t in linear scales.Figure 16
Evolution of the pressure of radiationpg with the time t in linear scales.The present size of the universea0=0.302lΛ=1.32·1026m is precisely of the order of the scale a2=0.204lΛ=8.95·1025m (a0=1.48a2). We have a0~a2~lΛ and t0~t2~tΛ. 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 ΩΛ,0/Ωm,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 ΩΛ,0/Ωm,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 ΩΛ/Ωm~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=1.31ρΛ=9.20·10-24 g/m3 and the age of the universe is t0=1.03H0-1=0.310tΛ=4.54·1017s~14Gyrs. The present values of the equation of state parameter and deceleration parameter are w0=-ΩΛ,0 and q0=(1-3ΩΛ,0)/2 [since p≃-ρΛc2]. Numerically, w0=-0.763 and q0=-0.645.


### 5.5. The Whole Evolution of the Universe

In the standardΛCDM 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, ρrad=ρrad,0/(a/a0)4, ρB=ρB,0/(a/a0)3, ρDM=ρDM,0/(a/a0)3, and ρΛ=ρΛ,0. Finally, we add their individual densities ρ=ρrad+ρB+ρDM+ρΛ, that is,
(98)ρ=ρrad,0(a/a0)4+ρB,0(a/a0)3+ρDM,0(a/a0)3+ρΛ,0
and solve the Friedmann equation (17). This leads to the equation
(99)HH0=Ωrad,0(a/a0)4+ΩB,0(a/a0)3+ΩDM,0(a/a0)3+ΩΛ,0,
with Ωrad+ΩB+ΩDM+ΩΛ=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 Ωm,0*=Ωm,0H(a-a1).)
(100)HH0=Ωrad,0(a/a0)4+(a1/a0)4+ΩB,0*(a/a0)3+ΩDM,0*(a/a0)3+ΩΛ,0,
with ΩB+Ωrad+ΩDM+ΩΛ=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>23.3tP, 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
(101)∫ai/a0a/a0dx(xΩrad,0x4+(a1/a0)4+ΩB,0*x3+ΩDM,0*x3+ΩΛ,0)-1‍=H0t,
where ai=a1=0 in the standard model and ai=lP in the generalized model. The age of the universe is
(102)t0=1H0×∫01‍dxx(Ωrad,0/x4)+(ΩB,0/x3)+(ΩDM,0/x3)+ΩΛ,0.
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=lP). 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 Figures17–19, 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 ρP=5.16·1099g/m3 (vacuum energy) and a late inflation due to the cosmological density ρΛ=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 ρmax=ρP and ρmin=ρΛ 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 factora 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 densityρ 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 temperatureT 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 withp=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 ΩDM,0*/(a/a0)3 in (100) should be replaced by ΩDM,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 ΩDM,0*/(a/a0)3 in (100) should be replaced by ΩDM,0*/[(a/a0)3∓(aBEC/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.


## 5.1. The Quadratic Equation of State

We propose to describe the vacuum energy, radiation, and dark energy by a unique equation of state(63)pg=-4ρg23ρPc2+13ρgc2-43ρΛc2.
For ρg→ρP, pg→-ρPc2 (vacuum energy); for ρΛ≪ρg≪ρP, pg~ρgc2/3 (radiation); for ρg→ρΛ, pg→-ρΛc2 (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
(64)ρg=ρP(a/a1)4+1+ρΛ,
where a1=2.61·10-6m (see Section 3). To obtain this expression, we have used the fact that ρP≫ρΛ so that pg/c2+ρg≃(4/3ρP)(ρg-ρΛ)(ρP-ρg). When a→0, ρg→ρP (vacuum energy); when ρΛ≪ρg≪ρP, ρg~ρP(a/a1)-4 (radiation); when a→+∞, ρg→ρΛ (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 ρPa14=ρΛa34.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 ρPa14=ρrad,0a04 (see Section 3), we can rewrite (64) in the form
(65)ρg=ρrad,0(a/a0)4+(a1/a0)4+ρΛ,
where a1/a0=1.97·10-32. The baryonic matter and the dark matter are described as pressureless fluids with pB=0 and pDM=0. The equation of continuity (15) for each species leads to the relations(66a)ρB=ρB,0(a/a0)3,(66b)ρDM=ρDM,0(a/a0)3.The total densityρ=ρg+ρB+ρDM of generalized radiation, baryonic matter, and dark energy can be written as
(67)ρ=ρrad,0(a/a0)4+(a1/a0)4+ρB,0(a/a0)3+ρDM,0(a/a0)3+ρΛ.
Substituting this relation in the Friedmann equation (17) and writing ρrad,0=Ωrad,0ρ0, ρB,0=ΩB,0ρ0, ρDM,0=ΩDM,0ρ0, and ρΛ=ΩΛ,0ρ0, we obtain
(68)HH0=Ωrad,0(a/a0)4+(a1/a0)4+ΩB,0(a/a0)3+ΩDM,0(a/a0)3+ΩΛ,0.
From observations, Ωrad,0=8.48·10-5, ΩB,0=0.0455, ΩDM,0=0.1915, Ωm,0=0.237, and ΩΛ,0=0.763 [1].The thermodynamic equation (13) with the equation of state (63) may be integrated into
(69)T=TP(15π2)1/4(1-ρgρP)7/4(ρgρP-ρΛρP)1/4.
Using (64), we get
(70)T=TP(15π2)1/4(a/a1)7[(a/a1)4+1]2.
Equation (70) may also be written as
(71)T=T0(a/a0)7[(a/a0)4+(a1/a0)4]2,
with T0=TP(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
(72)T~T0a0a.Finally, using (63) and (64), the pressure of generalized radiation can be written in very good approximation as
(73)pg=13(a/a1)4-3[(a/a1)4+1]2ρPc2-ρΛc2.


## 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
(74)ρ=ρrad,0(a/a0)4+(a1/a0)4.
The Friedmann equation (68) becomes
(75)HH0=Ωrad,0(a/a0)4+(a1/a0)4.
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 ρrad=ρP. This yields a1/a0=(ρrad,0/ρP)1/4, ρ1=ρP/2, T1=(1/4)(15/π2)1/4TP. Numerically, a1=1.97·10-32a0=2.61·10-6m, ρ1=0.5ρP=2.58·1099g/m3, T1=0.278TP=3.93·1031K. This takes place at a time t1=23.3tP=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:
(76)ρ≃ρP.
Therefore, the Hubble parameter is also a constant H=(8π/3)1/2tP-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
(77)ρ~ρrad,0a04a4.The Friedmann equation (75) reduces to
(78)HH0=Ωrad,0(a/a0)4
and yields
(79)aa0~Ωrad,01/42H0t,ρρ0~1(2H0t)2,TT0~a0a~1Ωrad,01/42H0t.


## 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
(80)ρ=ρm,0(a/a0)3+ρΛ,
where ρm=ρB+ρDM. This coincides with the ΛCDM model (see Section 5.5). The Friedmann equation (68) becomes
(81)HH0=Ωm,0(a/a0)3+ΩΛ,0,
with Ωm,0+ΩΛ,0≃1. It has the analytical solution
(82)aa0=(Ωm,0ΩΛ,0)1/3sinh2/3(32ΩΛ,0H0t).
The density evolves as
(83)ρρ0=ΩΛ,0tanh2((3/2)ΩΛ,0H0t).
On the other hand, the temperature of the generalized radiation in the late universe is given by
(84)T=TP(15π2)1/4(ρgρP-ρΛρP)1/4,T=T0a0a.
Using the expression (82) of the scale factor, we obtain
(85)T=T0(Ωm,0/ΩΛ,0)1/3sinh2/3((3/2)ΩΛ,0H0t).
Finally, the pressure of the generalized radiation in the late universe may be written as
(86)pg=13ρrad,0c2(a/a0)4-ρΛc2.Settinga=a0 in (82), we find the age of the universe
(87)t0=1H0231ΩΛ,0sinh-1[(ΩΛ,0Ωm,0)1/2].
Numerically, t0=1.03H0-1=0.310tΛ=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 whenρm=2ρΛ [see (16) with ρ≃ρm+ρΛ and p≃-ρΛc2]. Using ρm,0=Ωm,0ρ0 and ρΛ=ΩΛ,0ρ0 with Ωm,0=0.237 (matter) and ΩΛ,0=0.763 (dark energy), we obtain a=ac′ and t=tc′ with
(88)ac′a0=(Ωm,02ΩΛ,0)1/3,tc′=1H0231ΩΛ,0sinh-1(12).
At that point ρ=ρc′=3ρΛ and T=Tc′=T0a0/ac′. Numerically, ac′=0.538a0=0.162lΛ=7.11·1025m, tc′=0.504H0-1=0.152tΛ=2.22·1017s, ρc′=3ρΛ=2.11·10-23g/m3, and Tc′=1.86T0=5.78K. The universe is decelerating for t<tc′ and accelerating for t>tc′.The transition between the matter era and the dark energy era corresponds toρm=ρΛ. This leads to a=a2 and t=t2 with
(89)a2a0=(Ωm,0ΩΛ,0)1/3,t2=1H0231ΩΛ,0sinh-1(1).
At that point ρ=ρ2=2ρΛ and T=T2=T0a0/a2. Numerically, a2=0.677a0=0.204lΛ=8.95·1025m, t2=0.674H0-1=0.203tΛ=2.97·1017s, ρ2=2ρΛ=1.40·10-23g/m3, and T2=1.48T0=4.59K. We also note that ac′=(1/2)1/3a2=0.794a2. Of course, the value
(90)a2=8.95·1025m
coincides with the one obtained in [34–36] using a different viewpoint.In the dark energy era (a≫a2), the density is constant:
(91)ρ≃ρΛ.
Therefore, the Hubble parameter is also a constant H=H0ΩΛ,0. Using (82), we find that the scale factor increases exponentially rapidly as
(92)aa0~(Ωm,04ΩΛ,0)1/3eΩΛ,0H0t.
This corresponds to the de Sitter solution. According to (85), the temperature decreases exponentially rapidly as
(93)TT0~(4ΩΛ,0Ωm,0)1/3e-ΩΛ,0H0t.
In the matter era (a≪a2), we have
(94)ρ~ρm,0a03a3.
The Friedmann equation (81) reduces to
(95)HH0=Ωm,0(a/a0)3,
and we have
(96)aa0=Ωm,01/3(32H0t)2/3,ρρ0=1((3/2)H0t)2,(97)TT0~1Ωm,01/3((3/2)H0t)2/3.
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∝t2/3, 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 ρΛ. The transition takes place at a2=0.677a0=0.204lΛ=8.95·1025m, ρ2=2ρΛ=1.40·10-23g/m3, T2=1.48T0=4.59K, and t2=0.674H0-1=0.203tΛ=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 tc′=0.504H0-1=0.152tΛ=2.22·1017s, corresponding to a radius ac′=0.538a0=0.162lΛ=0.794a2=7.11·1025m, a density ρc′=3ρΛ=2.11·10-23g/m3 and a temperature Tc′=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 factora 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<ac′ and accelerating for a>ac′ with ac′=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 factora with the time t in linear scales.Figure 13
Evolution of the densityρ 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 densityρ 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 radiationT with the time t in linear scales.Figure 16
Evolution of the pressure of radiationpg with the time t in linear scales.The present size of the universea0=0.302lΛ=1.32·1026m is precisely of the order of the scale a2=0.204lΛ=8.95·1025m (a0=1.48a2). We have a0~a2~lΛ and t0~t2~tΛ. 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 ΩΛ,0/Ωm,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 ΩΛ,0/Ωm,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 ΩΛ/Ωm~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=1.31ρΛ=9.20·10-24 g/m3 and the age of the universe is t0=1.03H0-1=0.310tΛ=4.54·1017s~14Gyrs. The present values of the equation of state parameter and deceleration parameter are w0=-ΩΛ,0 and q0=(1-3ΩΛ,0)/2 [since p≃-ρΛc2]. Numerically, w0=-0.763 and q0=-0.645.


## 5.5. The Whole Evolution of the Universe

In the standardΛCDM 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, ρrad=ρrad,0/(a/a0)4, ρB=ρB,0/(a/a0)3, ρDM=ρDM,0/(a/a0)3, and ρΛ=ρΛ,0. Finally, we add their individual densities ρ=ρrad+ρB+ρDM+ρΛ, that is,
(98)ρ=ρrad,0(a/a0)4+ρB,0(a/a0)3+ρDM,0(a/a0)3+ρΛ,0
and solve the Friedmann equation (17). This leads to the equation
(99)HH0=Ωrad,0(a/a0)4+ΩB,0(a/a0)3+ΩDM,0(a/a0)3+ΩΛ,0,
with Ωrad+ΩB+ΩDM+ΩΛ=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 Ωm,0*=Ωm,0H(a-a1).)
(100)HH0=Ωrad,0(a/a0)4+(a1/a0)4+ΩB,0*(a/a0)3+ΩDM,0*(a/a0)3+ΩΛ,0,
with ΩB+Ωrad+ΩDM+ΩΛ=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>23.3tP, 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
(101)∫ai/a0a/a0dx(xΩrad,0x4+(a1/a0)4+ΩB,0*x3+ΩDM,0*x3+ΩΛ,0)-1‍=H0t,
where ai=a1=0 in the standard model and ai=lP in the generalized model. The age of the universe is
(102)t0=1H0×∫01‍dxx(Ωrad,0/x4)+(ΩB,0/x3)+(ΩDM,0/x3)+ΩΛ,0.
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=lP). 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 Figures17–19, 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 ρP=5.16·1099g/m3 (vacuum energy) and a late inflation due to the cosmological density ρΛ=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 ρmax=ρP and ρmin=ρΛ 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 factora 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 densityρ 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 temperatureT 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 withp=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 ΩDM,0*/(a/a0)3 in (100) should be replaced by ΩDM,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 ΩDM,0*/(a/a0)3 in (100) should be replaced by ΩDM,0*/[(a/a0)3∓(aBEC/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 standardΛCDM 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=ρrad+ρB+ρDM+ρΛ 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/ρP)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=-ρΛc2. 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=ρearly+ρlate in the Friedmann equation (17). This leads to (100) with a1/a0=1.97·10-32. For t>23.3tP, 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 aquadratic equation of state pg=-(4/3)(ρg2/ρP)c2+ρgc2/3-ρΛc2. 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 pB=pDM=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=ρg+ρB+ρDM 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.



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