research article a cosmological model based on a quadratic equation of state unifying vacuum

Hindawi Publishing CorporationJournal of GravityVolume 2013 Article ID 682451 20 pageshttpdxdoiorg1011552013682451

Research ArticleA Cosmological Model Based on a Quadratic Equation ofState Unifying Vacuum Energy Radiation and Dark Energy

Pierre-Henri Chavanis

Laboratoire de Physique Theorique IRSAMC CNRS UPS Universite de Toulouse 31062 Toulouse France

Correspondence should be addressed to Pierre-Henri Chavanis chavanisirsamcups-tlsefr

Received 26 March 2013 Accepted 17 May 2013

Academic Editor Kazuharu Bamba

Copyright copy 2013 Pierre-Henri Chavanis This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider a cosmological model based on a quadratic equation of state 1199011198882 = minus412058823120588119875+ 1205883 minus 4120588

Λ3 (where 120588

119875is the Planck

density and 120588Λis the cosmological density) ldquounifyingrdquo vacuum energy radiation and dark energy For 120588 rarr 120588

119875 it reduces to

119901 = minus1205881198882 leading to a phase of early accelerated expansion (early inflation) with a constant density equal to the Planck density

120588119875= 516 sdot 10

99 gm3 (vacuum energy) For 120588Λ≪ 120588 ≪ 120588

119875 we recover the equation of state of radiation 119901 = 120588119888

23 For 120588 rarr 120588

Λ

we get 119901 = minus1205881198882 leading to a phase of late accelerated expansion (late inflation) with a constant density equal to the cosmological

density 120588Λ= 702sdot10

minus24 gm3 (dark energy)The temperature is determined by a generalized Stefan-Boltzmann lawWe show a niceldquosymmetryrdquo between the early universe (vacuum energy + radiation) and the late universe (radiation + dark energy) In our modelthey are described by two polytropic equations of state with index 119899 = +1 and 119899 = minus1 respectively Furthermore the Planck density120588119875in the early universe plays a role similar to that of the cosmological density 120588

Λin 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 matterconsidered as independent species and obtain a simple cosmological model describing the whole evolution of the universe Westudy the evolution of the scale factor density and temperature This model gives the same results as the standard ΛCDM modelfor 119905 gt 233119905

119875 where 119905

119875= 539 sdot 10

minus44 s is the Planck time and completes it by incorporating the phase of early inflation in a naturalmanner Furthermore this model does not present any singularity at 119905 = 0 and exists eternally in the past (although it may beincorrect to extrapolate the solution to the infinite past) Our study suggests that vacuum energy radiation and dark energy maybe the manifestation of a unique form of ldquogeneralized radiationrdquo By contrast the baryonic and dark matter components of theuniverse are treated as different species This is at variance with usual models (quintessence Chaplygin gas ) trying to unify darkmatter and dark energy

1 Introduction

The evolution of the universe may be divided into four mainperiods [1] In the vacuum energy era (Planck era) the uni-verse undergoes a phase of early inflation that brings it fromthe Planck size 119897

119875= 162 sdot10

minus35m to an almost ldquomacroscopicrdquosize 119886 sim 10

minus6m in a tiniest fraction of a second [2ndash5] Theuniverse then enters in the radiation era and when thetemperature cools down below approximately 103 K in thematter era [6] Finally in the dark energy era (de Sitter era)the universe undergoes a phase of late inflation [7] The earlyinflation is necessary to solve notorious difficulties such asthe singularity problem the flatness problem and the horizon

problem [2ndash5] The late inflation is necessary to account forthe observed accelerating expansion of the universe [8ndash11]At present the universe is composed of approximately 5baryonic matter 20 dark matter and 75 dark energy [1]Despite the success of the standard model the nature ofvacuum energy dark matter and dark energy remains verymysterious and leads to many speculations

The phase of inflation in the early universe is usuallydescribed by some hypothetical scalar field 120601 with its originin quantum fluctuations of vacuum [2ndash5] This leads to anequation of state 119901 = minus120588119888

2 implying a constant energy den-sity called the vacuum energy This energy density is usuallyidentified with the Planck density 120588

119875= 516 sdot 10

99 gm3

2 Journal of Gravity

As a result of the vacuum energy the universe expandsexponentially rapidly on a timescale of the order of the Plancktime 119905

119875= 539 sdot 10

minus44 s This phase of early inflation isfollowed by the radiation era described by an equation ofstate 119901 = 120588119888

23 In the vacuum energy era the density has

a constant value 120588 ≃ 120588119875 while it decreases as 120588 prop 119886

minus4 duringthe radiation era

The phase of acceleration in the late universe is usuallyascribed to the cosmological constant Λ which is equivalentto a constant energy density120588

Λ= Λ(8120587119866) = 702sdot10

minus24 gm3called the dark energy This acceleration can be modeled byan equation of state 119901 = minus120588119888

2 implying a constant energydensity identified with the cosmological density 120588

Λ As a

result of the dark energy the universe expands exponentiallyrapidly on a timescale of the order of the cosmological time119905Λ

= 146 sdot 1018 s (de Sitter solution) This leads to a

phase of late inflation Inspired by the analogy with the earlyinflation some authors have represented the dark energy bya scalar field called quintessence [12ndash24] As an alternativeto the quintessence other authors have proposed to modelthe acceleration of the universe by an exotic fluid with anequation of state of the form119901119888

2= 119861120588minus119860120588

119886 with119860 ge 0 and119886 ge minus1 called the Chaplygin gas when (119861 119886) = (0 1) and thegeneralized Chaplygin gas otherwise [25ndash32] This equationof state may be rewritten as

119901 = (120572120588 + 1198961205881+1119899

) 1198882 (1)

with 119896 le 0 in order to emphasize its analogy with thepolytropic equation of state [33ndash36] Indeed it may be viewedas the sum of a linear equation of state 119901 = 120572120588119888

2 describingdust matter (120572 = 0) or radiation (120572 = 13) and a polytropicequation of state 119901 = 119896120588

1+11198991198882 We shall call (1) a generalized

polytropic equation of state At late times this equation ofstate with 119899 lt 0 leads to a constant energy density implyingan exponential growth of the scale factor that is similar to theeffect of the cosmological constant At earlier times and for120572 = 0 this equation of state returns the results of the dustmatter model (119901 ≃ 0) Therefore this hydrodynamic modelprovides a unification of matter (120588 prop 119886

minus3) and dark energy(120588 = 120588

Λ) in the late universe [25ndash36]

In [34ndash36] we remarked that a constant pressure 119901 =

minus120588Λ1198882 has the same virtues as the Chaplygin gas model

(Actually this is a particular case of the generalized Chap-lygin gas model corresponding to 119861 = 0 and 119886 = 0) Thisequation of state may be viewed as a generalized polytropicequation of state (1) with polytropic index 119899 = minus1 polytropicconstant 119896 = minus120588

Λ and linear coefficient 120572 = 0 At late

times this equation of state leads to a constant energy densityequal to 120588

Λimplying an exponential growth of the scale factor

(late inflation) At earlier times this equation of state returnsthe results of the dust matter model (119901 ≃ 0) Thereforeas any Chaplygin gas model with 120572 = 0 and 119899 lt 0 thisequation of state provides a unification of matter (120588 prop 119886

minus3)and dark energy (120588 = 120588

Λ) in the late universe Moreover

this equation of state gives the same results as the standardΛCDM model where matter and dark energy are treatedas independent species This is interesting since the ΛCDM

model is consistent with observations and provides a gooddescription of the late universe (By contrast the Chaplygingas model is not consistent with observations unless 119886 isclose to zero [37] Therefore observations tend to select theconstant pressure equation of state 119901 = minus120588

Λ1198882 (119886 = 0 119899 = minus1)

among the whole family of Chaplygin gas models of the form119901 = minus119860120588

119886) In [34ndash36] we also proposed to describe thetransition between the vacuum energy era and the radiationera in the early universe by a unique equation of state of theform 119901119888

2= minus4120588

23120588119875+ 1205883 This equation of state may be

viewed as a generalized polytropic equation of state (1) withpolytropic index 119899 = 1 polytropic constant 119896 = minus4(3120588

119875)

and linear coefficient 120572 = 13 At early times this equation ofstate leads to a constant energy density equal to 120588

119875implying

an exponential growth of the scale factor (early inflation) Atlater times and for 120572 = 13 this equation of state returnsthe results of the radiation model (119901 = 120588119888

23) Therefore

this hydrodynamic model provides a unification of vacuumenergy (120588 = 120588

119875) and radiation (120588 prop 119886

minus4) in the early universeThis equation of state may be viewed as an extension of theChaplygin (or polytropic) gas model to the early universeIn this approach the late universe is described by negativepolytropic indices 119899 lt 0 (eg 119899 = minus1) and the early universeby positive polytropic indices 119899 gt 0 (eg 119899 = +1) [34ndash36]Indeed for 119899 lt 0 the polytropic equation of state dominatesin the late universe where the density is low and for 119899 gt 0 thepolytropic equation of state dominates in the early universewhere the density is high

In the viewpoint just exposed one tries to ldquounifyrdquo vacuumenergy + radiation on the one hand and dust matter +dark energy on the other hand (It is oftentimes arguedthat the dark energy (cosmological constant) corresponds tothe vacuum energy This leads to the so-called cosmologicalproblem [38] since the cosmological density 120588

Λ= 702 sdot

10minus24 gm3 and the Planck density 120588

119875= 516sdot10

99 gm3 differby about 122 orders ofmagnitudeWe think that it is amistaketo identify the dark energy with the vacuum energy In thispaper we shall regard the vacuum energy and the dark energyas two distinct entities We shall call vacuum energy theenergy associatedwith the Planck density and dark energy theenergy associated with the cosmological densityThe vacuumenergy is responsible for the inflation in the early universe andthe dark energy for the inflation in the late universe In thisviewpoint the vacuum energy is due to quantum mechanicsand the dark energy is an effect of general relativity Thecosmological constant Λ is interpreted as a fundamentalconstant of nature applying to the cosmophysics in the sameway the Planck constant ℎ applies to the microphysics) Inthis paper we adopt a different point of view We proposethat vacuum energy radiation and dark energy may be themanifestation of a unique form of ldquogeneralized radiationrdquoWepropose to describe it by a quadratic equation of state

119901 = minus41205882

3120588119875

1198882+1

31205881198882minus4

3120588Λ1198882 (2)

involving the Planck density and the cosmological density Inthe early universe we recover the equation of state 119901119888

2=

minus412058823120588119875+ 1205883 unifying vacuum energy and radiation In

Journal of Gravity 3

the late universe we obtain an equation of state 1199011198882 = 1205883 minus

4120588Λ3 unifying radiation and dark energy This equation of

state may be viewed as a generalized polytropic equation ofstate (1) with polytropic index 119899 = minus1 polytropic constant119896 = minus4120588

Λ3 and linear coefficient 120572 = 13 The quadratic

equation of state (2) ldquounifiesrdquo vacuum energy radiation anddark energy in a natural and simple manner It also allows usto obtain a generalization of the Stefan-Boltzmann law for theevolution of the temperature [see (69)] that is valid during thewhole evolution of the universe In this approach baryonicmatter and dark matter are treated as different speciesThis isat variance with the usual scenario where vacuum energy +radiation on the one hand and dark matter + dark energy onthe other hand are treated as different species

The paper is organized as follows In Section 2 we recallthe basic equations of cosmology that will be needed inour study In Section 3 we describe the transition betweenthe vacuum energy era and the radiation era in the earlyuniverse In Section 4 we describe the transition between theradiation era and the dark energy era in the late universeIn Section 5 we introduce the general model where vacuumenergy + radiation + dark energy are described by a quadraticequation of state while baryonic matter and dark matter areconsidered as different species We determine the generalequations giving the evolution of the scale factor densityand temperature (Section 51) We consider particular limitsof these equations in the early universe (Section 52) andin the late universe (Sections 53 and 54) We connectthese two limits and describe the whole evolution of theuniverse (Section 55) This model gives the same results asthe standard ΛCDM model for 119905 gt 233119905

119875and completes it

by incorporating the early inflation in a natural manner [see(100)] It also reveals a nice ldquosymmetryrdquo between the earlyuniverse (vacuum energy + radiation) and the late universe(radiation + dark energy) These two phases are described bypolytropic equations of state with index 119899 = +1 and 119899 = minus1respectively Furthermore the cosmological density 120588

Λin the

late universe plays a role similar to the Planck density120588119875in the

early universe They represent fundamental upper and lowerdensity bounds differing by 122 orders of magnitude

2 Basic Equations of Cosmology

In a spacewith uniform curvature the line element is given bythe Friedmann-Lemaıtre-Roberston-Walker (FLRW) metric

1198891199042= 11988821198891199052minus 119886(119905)

2

1198891199032

1 minus 1198961199032+ 1199032(1198891205792+ sin21205791198891206012) (3)

where 119886(119905) represents the radius of curvature of the 3-dimensional space or the scale factor (In our conventions119903 and 119896 are dimensionless while the scale factor 119886(119905) has thedimension of a length It is defined such that 119886

0= 119888119867

0

(where 119888 is the speed of light and1198670is the Hubble constant)

at the present epoch This corresponds to the cosmologicalhorizon The usual (dimensionless) scale factor is 119886(119905) =

119886(119905)1198860 It satisfies 119886

0= 1 at the present epoch) By an abuse

of language we shall sometimes call it the ldquoradius of theuniverserdquo On the other hand 119896 determines the curvature of

space The universe may be closed (119896 gt 0) flat (119896 = 0) oropen (119896 lt 0)

If the universe is isotropic and homogeneous at all pointsin conformity with the line element (3) and contains a uni-form perfect fluid of energy density 120598(119905) = 120588(119905)119888

2 and isotro-pic pressure 119901(119905) the energy-momentum tensor 119879119894

119895is

1198790

0= 1205881198882 119879

1

1= 1198792

2= 1198793

3= minus119901 (4)

The Einstein equations

119877119894

119895minus1

2119892119894

119895119877 minus Λ119892

119894

119895= minus

8120587119866

1198882119879119894

119895(5)

relate the geometrical structure of the spacetime (119892119894119895) to the

material content of the universe (119879119894119895) For the sake of general-

ity we have accounted for a possibly nonzero cosmologicalconstant Λ Given Equations (3) and (4) these equationsreduce to

8120587119866120588 + Λ = 31198862+ 1198961198882

1198862

8120587119866

1198882119901 minus Λ = minus

2119886 119886 + 1198862+ 1198961198882

1198862

(6)

where dots denote differentiation with respect to time Theseare the well-known cosmological equations describing a non-static universe first derived by Friedmann [6]

The Friedmann equations are usually written in the form

119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (7)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) +

Λ

3 (8)

1198672= (

119886

119886)

2

=8120587119866

3120588 minus

1198961198882

1198862+Λ

3 (9)

where we have introduced the Hubble parameter 119867 = 119886119886Among these three equations only two are independentThe first equation which can be viewed as an ldquoequation ofcontinuityrdquo can be directly derived from the conservation ofthe energy momentum tensor 120597

119894119879119894119895= 0 which results from

the Bianchi identities For a given barotropic equation of state119901 = 119901(120588) it determines the relation between the densityand the scale factorThen the temporal evolution of the scalefactor is given by (9)

Equivalent expressions of the ldquoequation of continuityrdquo are

119886119889120588

119889119886= minus3 (120588 +

119901

1198882) (10)

119889

119889119886(1205881198863) = minus3

119901

11988821198862 (11)

1198863 119889119901

119889119905=

119889

119889119905[1198863(119901 + 120588119888

2)] (12)

Introducing the volume 119881 = (43)1205871198863 and the energy 119864 =

1205881198882119881 (11) becomes 119889119864 = minus119901119889119881 This can be viewed as the


first principle of thermodynamics for an adiabatic evolutionof the universe 119889119878 = 0

From the first principle of thermodynamics in the form119889119878(119881 119879) = (1119879)[119889(120588(119879)119888

2119881) + 119901(119879)119889119881] one can derive the

thermodynamical equation [6]

119889119901

119889119879=

1

119879(1205881198882+ 119901) (13)

For a given barotropic equation of state 119901 = 119901(120588) thisequation can be integrated to obtain the relation 119879 = 119879(120588)

between the temperature and the density Combining (12)with (13) we get [6] that

119878 = 0 where 119878 =1198863

119879(119901 + 120588119888

2) (14)

is the entropy of the universe in a volume 1198863 This confirmsthat the Friedmann equations (7)ndash(9) imply the conservationof the entropy

In this paper we consider a flat universe (119896 = 0) inagreement with the observations of the cosmic microwavebackground (CMB) [1] On the other hand we set Λ = 0

because the effect of the cosmological constant will be takeninto account in the equation of state The Friedmann equa-tions then reduce to

119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (15)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) (16)

1198672= (

119886

119886)

2

=8120587119866

3120588 (17)

The deceleration parameter is defined by

119902 (119905) = minus119886119886

1198862 (18)

The universe is decelerating when 119902 gt 0 and acceleratingwhen 119902 lt 0 Introducing the equation of state parameter119908 = 119901120588119888

2 and using the Friedmann equations (16) and (17)we obtain for a flat universe

119902 (119905) =1 + 3119908 (119905)

2 (19)

We see from (19) that the universe is decelerating if119908 gt minus13

(strong energy condition) and accelerating if 119908 lt minus13 Onthe other hand according to (15) the density decreases withthe scale factor if 119908 gt minus1 (null dominant energy condition)and increases with the scale factor if 119908 lt minus1

3 The Early Universe

In [34ndash36] we have proposed to describe the transitionbetween the vacuum energy era and the radiation era in theearly universe by a single equation of state of the form

119901 =1

31205881198882minus41205882

3120588119875

1198882 (20)

where 120588119875= 11988851198662ℎ = 516 sdot 10

99 gm3 is the Planck densityThis equation of state corresponds to a generalized polytropicequation of state (1) with 120572 = 13 119899 = 1 and 119896 = minus4(3120588

119875)

For 120588 ≪ 120588119875 we recover the equation of state of the pure

radiation 119901 sim 12058811988823 For 120588 rarr 120588

119875 we get 119901 rarr minus120588

1198751198882

corresponding to the vacuum energyThe continuity equation (15) may be integrated into

120588 =120588119875

(1198861198861)4

+ 1 (21)

where 1198861is a constant of integration This characteristic scale

marks the transition between the vacuum energy era and theradiation eraThe equation of state (20) interpolates smoothlybetween the vacuum energy era (119901 = minus120588119888

2 and 120588 = 120588119875) and

the radiation era (119901 = 12058811988823 and 120588rad sim 120588

1198751198864

11198864) It pro-

vides therefore a ldquounifiedrdquo description of vacuum energy andradiation in the early universeThis amounts to summing theinverse of the densities of these two phases Indeed (21) maybe rewritten as

1

120588=

1

120588rad+

1

120588119875

(22)

At 119886 = 1198861we have 120588rad = 120588

119875so that 120588 = 120588

1198752

The equation of state parameter 119908 = 1199011205881198882 and the dece-

leration parameter 119902 are given by

119908 =1

3minus

4120588

3120588119875

119902 = 1 minus 2120588

120588119875

(23)

The velocity of sound 1198882

119904= 1199011015840(120588) is given by

1198882

119904

1198882=1

3minus8

3

120588

120588119875

(24)

As the universe expands from 119886 = 0 to 119886 = +infin the densitydecreases from 120588

119875to 0 the equation of state parameter

119908 increases from minus1 to 13 the deceleration parameter 119902

increases from minus1 to 1 and the ratio (119888119904119888)2 increases from

minus73 to 13

31 The Vacuum Energy Era Early Inflation When 119886 ≪ 1198861

the density tends to a maximum value

120588 = 120588max = 120588119875 (25)

and the pressure tends to 119901 = minus1205881198751198882 The Planck density 120588

119875=

516 sdot 1099 gm3 (vacuum energy) represents a fundamental

upper bound for the density A constant value of the density120588 ≃ 120588

119875gives rise to a phase of early inflation From the

Friedmann equation (17) we find that the Hubble parameteris constant 119867 = (81205873)

12119905minus1

119875where we have introduced the

Planck time 119905119875= 1(119866120588

119875)12

= (ℎ1198661198885)12 Numerically119867 =

537 sdot 1043 sminus1 Therefore the scale factor increases expone-

ntially rapidly with time as

119886 (119905) sim 119897119875119890(81205873)

12119905119905119875 (26)


The timescale of the exponential growth is the Planck time119905119875= 539 sdot 10

minus44 s We have defined the ldquooriginalrdquo time 119905 = 0

such that 119886(0) is equal to the Planck length 119897119875

= 119888119905119875

=

(119866ℎ1198883)12

= 162 sdot 10minus35m Mathematically speaking the

universe exists at any time in the past (119886 rarr 0 and 120588 rarr 120588119875

for 119905 rarr minusinfin) so there is no primordial singularity Howeverwhen 119886 rarr 0 we cannot ignore quantum fluctuationsassociated with the spacetime In that case we cannot use theclassical Einstein equations anymore and a theory of quantumgravity is required It is not known whether quantum gravitywill remove or not the primordial singularity Therefore wecannot extrapolate the solution (26) to the infinite past How-ever this solution may provide a semiclassical description ofthe phase of early inflation when 119886 gt 119897

119875

32 The Radiation Era When 119886 ≫ 1198861 we recover the equa-

tion

120588rad sim1205881198751198864

1

1198864 (27)

corresponding to the pure radiation described by an equationof state 119901 = 120588119888

23 The conservation of 120588rad119886

4 implies that1205881198751198864

1= 120588rad0119886

4

0where 120588rad0 is the present density of radiation

and 1198860

= 1198881198670

= 132 sdot 1026m the present distance of

cosmological horizon determined by the Hubble constant1198670= 227 sdot 10

minus18 sminus1 (the Hubble time is119867minus10

= 441 sdot 1017 s)

Writing 120588rad0 = Ωrad01205880 where 1205880= 3119867

2

08120587119866 = 920 sdot

10minus24 gm3 is the present density of the universe given by (17)

andΩrad0 = 848 sdot 10minus5 is the present fraction of radiation in

the universe [1] we obtain

1198861= 261 sdot 10

minus6m (28)

This scale is intermediate between the Planck length 119897119875and

the present size of the universe 1198860(1198861119897119875= 161 sdot 10

29 and11988611198860= 197 sdot 10

minus32) It gives the typical size of the universe atthe end of the inflationary phase (or at the beginning of theradiation era) When 119886 ≫ 119886

1 the Friedmann equation (17)

yields

119886

1198861

sim (32120587

3)

14

(119905

119905119875

)

12

(29)

We then have120588

120588119875

sim3

32120587(119905119875

119905)

2

(30)

During the radiation era the density decreases algebraicallyas the universe expands

33 The General Solution The general solution of the Fried-mann equation (17) with (21) is

radic(1198861198861)4

+ 1 minus ln(1 + radic(119886119886

1)4

+ 1

(1198861198861)2

)

= 2(8120587

3)

12119905

119905119875

+ 119862

(31)

where 119862 is a constant of integration It is determined suchthat 119886 = 119897

119875at 119905 = 0 Setting 120598 = 119897

1198751198861 we get

119862 (120598) = radic1205984 + 1 minus ln(1 + radic1205984 + 1

1205982) (32)

Numerically 120598 = 620 sdot 10minus30 and 119862 = minus134 For 119905 rarr minusinfin

we have the exact asymptotic result

119886 (119905) sim 1198861119890(81205873)

12119905119905119875+119863

(33)

with 119863 = (119862 + ln 2 minus 1)2 Due to the smallness of 120598 a verygood approximation of119862 and119863 is given by119862 ≃ 1minusln 2+2 ln 120598and 119863 ≃ ln 120598 With this approximation (33) returns (26)Using (23) the universe is accelerating when 119886 lt 119886

119888(ie

120588 gt 120588119888) and decelerating when 119886 gt 119886

119888(ie 120588 lt 120588

119888) where

119886119888= 1198861and 120588119888= 1205881198752The time 119905

119888at which the universe starts

decelerating is given by 119905119888= (332120587)

12[radic2minusln(1+radic2)minus119862]119905

119875

This corresponds to the time at which the curve 119886(119905) presentsan inflexion point It turns out that this inflexion point 119886

119888

coincides with 1198861so it also marks the end of the inflation

(119905119888= 1199051) Numerically 120588

119888= 1205881= 05 120588

119875= 258 sdot 10

99 gm3119886119888= 1198861= 261 sdot 10

minus6m and 119905119888= 1199051= 233 119905

119875= 125 sdot 10

minus42 sAt 119905 = 119905

119894= 0 119886

119894= 119897119875and 120588

119894≃ 120588119875 Numerically 119886

119894= 119897119875=

162 sdot 10minus35m and 120588

119894≃ 120588119875= 516 sdot 10

99 gm3

34The Temperature and the Pressure The thermodynamicalequation (13) with the equation of state (20)may be integratedinto

119879 = 119879119875(15

1205872)

14

(1 minus120588

120588119875

)

74

(120588

120588119875

)

14

(34)

where 119879119875is a constant of integration Using (21) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (35)

Equation (34)may be seen as a generalized Stefan-Boltzmannlaw in the early universe In the radiation era 120588 ≪ 120588

119875 it

reduces to the ordinary Stefan-Boltzmann law

119879 = 119879119875(15

1205872)

14

(120588

120588119875

)

14

(36)

This allows us to identify the constant of integration 119879119875with

the Planck temperature 119879119875= (ℎ119888

51198661198962

119861)12

= 142 sdot 1032 K

On the other hand in the inflationary phase 120588 rarr 120588119875 the

temperature is related to the density by

119879

119879119875

≃ (15

1205872)

14

(1 minus120588

120588119875

)

74

(37)

According to (35) 119879 sim 119879119875(15120587

2)14

(1198861198861)7 for 119886 ≪ 119886

1and

119879 sim 119879119875(15120587

2)14

(1198861119886) for 119886 ≫ 119886

1

The temperature starts from 119879 = 0 at 119905 = minusinfin increasesexponentially rapidly during the inflation as

119879

119879119875

sim (15

1205872)

14

(119897119875

1198861

)

7

1198907(81205873)

12119905119905119875 (38)


reaches a maximum value 119879119890and decreases algebraically as

119879

119879119875

sim (45

321205873)

14

(119905119875

119905)

12

(39)

during the radiation era Using (34) and (35) we find that thepoint corresponding to the maximum temperature is 120588

119890=

1205881198758 119886119890= 714

1198861 119879119890= (78)

74(158120587

2)14

119879119875 It is reached

at a time 119905119890= (332120587)

12[radic8 minus ln(1 + radic8) + ln(7)2 minus 119862]119905

119875

Numerically 120588119890= 0125 120588

119875= 644 sdot 10

98 gm3 119886119890= 163119886

1=

424sdot10minus6m119879

119890= 0523 119879

119875= 740sdot10

31 K and 119905119890= 236 119905

119875=

127 sdot 10minus42 s At 119905 = 119905

119894= 0 119879

119894≃ 119879119875(15120587

2)14

(1198971198751198861)7

Numerically 119879119894= 391 sdot 10

minus205119879119875

= 554 sdot 10minus173 K At

119905 = 119905119888= 1199051 119879119888= (14)(15120587

2)14

119879119875 Numerically 119879

119888= 1198791=

0278 119879119875= 393 sdot 10

31 KThe pressure is given by (20) Using (21) we get

119901 =1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882 (40)

The pressure starts from 119901 = minus1205881198751198882 at 119905 = minusinfin remains

approximately constant during the inflation becomes posi-tive reaches amaximumvalue119901

119890 and decreases algebraically

during the radiation era At 119905 = 119905119894= 0 119901

119894≃ minus1205881198751198882 Numeri-

cally119901119894= minus464sdot10

116 gms2The point at which the pressurevanishes is 119886

119908= 314

1198861 120588119908= 1205881198754 119879119908= (916)(5120587

2)14

119879119875

This corresponds to a time 119905119908= (332120587)

12[2 minus (12) ln(3) minus

119862]119905119875 Numerically 119886

119908= 132119886

1= 343sdot10

minus6m120588119908= 025120588

119875=

129 sdot 1099 gm3 119879

119908= 0475 119879

119875= 674 sdot 10

31 K and 119905119908

=

234119905119875= 126sdot10

minus42 s On the other hand the pressure reachesits maximum 119901

119890= 120588119875119888248 at the same point as the one at

which the temperature reaches its maximum Numerically119901119890= 966 sdot 10

114 gms2 At 119905 = 119905119888= 1199051 we have 119901

119888= 1199011=

minus12058811987511988826 Numerically 119901

119888= 1199011= minus773 sdot 10

115 gms2Using (20) (21) and (35) we find that the entropy (14) is

given by

119878 =4

3(1205872

15)

14

1205881198751198882

119879119875

1198863

1 (41)

and we check that it is a constant Numerically 119878119896119861= 504 sdot

1087

35 The Evolution of the Early Universe In our model theuniverse starts at 119905 = minusinfin with a vanishing radius 119886 = 0a finite density 120588 = 120588

119875= 516 sdot 10

99 gm3 a finite pressure119901 = minus120588

1198751198882= minus464sdot10

116 gms2 and a vanishing temperature119879 = 0The universe exists at any time in the past and does notpresent any singularity For 119905 lt 0 the radius of the universeis less than the Planck length 119897

119875= 162 sdot 10

minus35m In thePlanck era quantum gravity should be taken into accountand our semiclassical approach is probably not valid in theinfinite past At 119905 = 119905

119894= 0 the radius of the universe is

equal to the Planck length 119886119894= 119897119875

= 162 sdot 10minus35m The

corresponding density temperature and pressure are 120588119894≃

120588119875= 516 sdot 10

99 gm3 119879119894= 391 sdot 10

minus205119879119875= 554 sdot 10

minus173 K

and 119901119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 We note that quantummechanics regularizes the finite time singularity present inthe standard Big Bang theory This is similar to finite sizeeffects in second-order phase transitions (see Section 36)The Big Bang theory is recovered for ℎ = 0 We also note thatthe universe is very cold at 119905 = 119905

119894= 0 contrary to what is pre-

dicted by the Big Bang theory (a naive extrapolation of the law119879 prop 119905

minus12 leads to 119879(0) = +infin) The universe first undergoesa phase of inflation during which its radius and temperatureincrease exponentially rapidly while its density and pressureremain approximately constantThe inflation ldquostartsrdquo at 119905

119894= 0

and ends at 1199051= 233 119905

119875= 125sdot10

minus42 s During this very shortlapse of time the radius of the universe grows from 119886

119894= 119897119875=

162sdot10minus35m to 119886

1= 261sdot10

minus6m and the temperature growsfrom119879

119894= 391sdot10

minus205119879119875= 554sdot10

minus173 K to1198791= 0278 119879

119875=

393 sdot 1031 K By contrast the density and the pressure do not

change significatively they go from 120588119894≃ 120588119875= 516sdot10

99 gm3and 119901

119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 to 1205881= 05 120588

119875=

258 sdot 1099 gm3 and 119901

1= minus0167120588

1198751198882= minus773 sdot 10

115 gms2The pressure passes from negative values to positive valuesat 119905119908

= 234119905119875

= 126 sdot 10minus42 s corresponding to 120588

119908=

025120588119875= 129 sdot 10

99 gm3 119886119908= 132 119886

1= 343 sdot 10

minus6m and119879119908= 0475 119879

119875= 674sdot10

31 KAfter the inflation the universeenters in the radiation era and from that point we recoverthe standard model [6] The radius increases algebraically as119886 prop 119905

12 while the density and the temperature decreasealgebraically as 120588 prop 119905

minus2 and 119879 prop 119905minus12 The temperature and

the pressure achieve their maximum values 119879119890= 0523 119879

119875=

740 sdot 1031 K and 119901

119890= 208 sdot 10

minus21205881198751198882= 966 sdot 10

114 gms2at 119905 = 119905

119890= 236 119905

119875= 127 sdot 10

minus42 s At that moment thedensity is 120588

119890= 0125 120588

119875= 644 sdot 10

98 gm3 and the radius119886119890= 163 119886

1= 424sdot10

minus6mDuring the inflation the universeis accelerating and during the radiation era it is deceleratingThe transition (marked by an inflexion point) takes place ata time 119905

119888= 1199051= 233 119905

119875= 125 sdot 10

minus42 s coinciding with theend of the inflation (119886

119888= 1198861)The evolution of the scale factor

density and temperature as a function of time are representedin Figures 1 2 3 4 5 6 and 7 in logarithmic and linear scales

We note that the inflationary process described pre-viously is relatively different from the usual inflationaryscenario [2ndash5] In standard inflation the universe is radiationdominated up to 119905

119894= 10minus35 s but expands exponentially by a

factor 1030 in the interval 119905119894lt 119905 lt 119905

119891with 119905

119891= 10minus33 s For 119905 gt

119905119891 the evolution is again radiation dominated At 119905 = 119905

119894 the

temperature is about 1027 K (this corresponds to the epoch atwhich most ldquogrand unified theoriesrdquo have a significant influ-ence on the evolution of the universe) During the exponen-tial inflation the temperature drops drastically however thematter is expected to be reheated to the initial temperature of1027 K by various high energy processes [39] In the inflation-

ary process described previously the evolution of the temper-ature given by a generalized Stefan-Boltzmann law is differ-ent It is initially very low and increases during the inflation

36 Analogy with Phase Transitions The standard Big Bangtheory is a classical theory in which quantum effects areneglected In that case it exhibits a finite time singularity


0

20

Inflation

Radiation era

minus20

minus40minus60 minus50 minus40 minus30 minus20 minus10 0 10

simlPsimtP

sima1

log [t (s)]

log [a

(m)]

Figure 1 Evolution of the scale factor 119886 with the time 119905 in logarith-mic scales This figure clearly shows the phase of inflation connect-ing the vacuum energy era to the radiation era During the inflationthe scale factor increases by 29 orders of magnitude in less than10minus42 s In the radiation era 119886 increases algebraically

0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1

Figure 2 Evolution of the scale factor 119886 with the time 119905 in linearscalesThe dashed line corresponds to the standard radiation modelof Section 32 exhibiting a finite time singularity at 119905 = 0 (Big Bang)When quantum mechanics is taken into account (as in our simplesemiclassical model) the initial singularity is smoothed-out and thescale factor at 119905 = 0 is equal to the Planck length 119897

119875= 162 sdot 10

minus35mThis is similar to a second order phase transition where the Planckconstant plays the role of finite size effects (see Section 36 for adevelopment of this analogy)

the radius of the universe is equal to zero at 119905 = 0 whileits density and its temperature are infinite For 119905 lt 0 thesolution is not defined and we may take 119886 = 0 For 119905 gt 0 theradius of the universe increases as 119886 prop 119905

12 This is similarto a second-order phase transition if we view the time 119905 asthe control parameter (eg the temperature 119879) and the scalefactor 119886 as the order parameter (eg the magnetization 119872)The exponent 12 is the same as in mean field theories ofsecond-order phase transitions (ie 119872 sim (119879

119888minus 119879)12) but

this is essentially a coincidenceWhen quantummechanics effects are taken into account

as in our simple semiclassical model the singularity at 119905 = 0

0

50

100

minus60 minus50 minus40 minus30 minus20 minus10 0 10

simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]

Figure 3 Evolution of the density 120588 with the time 119905 in logarithmicscales During the inflation the density remains approximatelyconstant with the Planck value 120588max = 120588

119875which represents an upper

bound In the radiation era 120588 decreases algebraically

0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P

Figure 4 Evolution of the density 120588 with the time 119905 in linear scalesThe dashed line corresponds to the standard radiation model ofSection 32 exhibiting a finite time singularity at 119905 = 0 Quantummechanics limits the rise of the density to the Planck value 120588

119875=

516 sdot 1099 gm3

disappears and the curves 119886(119905) 120588(119905) and 119879(119905) are regularizedIn particular we find that 119886 = 119897

119875gt 0 at 119905 = 0 instead of 119886 = 0

due to the finite value of ℎThis is similar to the regularizationdue to finite size effects (eg the system size 119871 or the numberof particles119873) in ordinary phase transitions In this sense theclassical limit ℎ rarr 0 is similar to the thermodynamic limit(119871 rarr +infin or119873 rarr +infin) in ordinary phase transitions

To study the convergence toward the classical Big Bangsolution when ℎ rarr 0 it is convenient to use a propernormalization of the parameters We call ℎ

0= 105 sdot 10

minus34 J sthe true value of the Planck constant (in this section thesubscript 0 refers to true values of the parameters) Thenwe express time in terms of 1199050

119875and lengths in terms of 1198860

1


0

minus60minus200

minus150

minus100

minus50 minus40 minus30 minus20 minus10 0 10log [t (s)]

log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP

Figure 5 Evolution of the temperature 119879 with the time 119905 inlogarithmic scales For 119905 le 0 the universe is extremely cold (119879 lt

10minus173 K) During the inflation the temperature increases by 204

orders of magnitude in less than 10minus42 s During the radiation era

the temperature decreases algebraically

0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P

Figure 6 Evolution of the temperature 119879 with the time 119905 in linearscalesThe dashed line corresponds to the standard radiation modelof Section 32 exhibiting a finite time singularity at 119905 = 0 Quantummechanics limits the rise of the temperature to about half the Planckvalue 119879

119875= 142 sdot 10

32 K In the vacuum energy era the temperaturedrops to zero exponentially rapidly for 119905 rarr minusinfin

We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)

where ℎ = ℎℎ0 Therefore

120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2

Figure 7 Evolution of the pressure 119901with the time 119905 in linear scalesThe dashed line corresponds to the standard radiation model ofSection 32 exhibiting a finite time singularity at 119905 = 0 Quantummechanics limits the rise of the pressure to 120588

119875119888248 = 966 sdot

10114 gms2 In the vacuum energy era the pressure decreases

exponentially rapidly becomes negative and tends to minus1205881198751198882

=

minus464 sdot 10116 gms2 for 119905 rarr minusinfin

where 1205980= 1198970

1198751198860

1= 620sdot10

minus30 Using these relations and (31)we get

radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)

where119862(120598) is defined by (32)This equation describes a phasetransition between the vacuum energy era and the radiationera The normalized Planck constant ℎ plays the role of finitesize effects Finite size scalings are explicit in (44) For ℎ = 1we recover the nonsingular model of Section 33 For ℎ = 0we recover the singular radiation model of Section 32 (BigBang) The convergence towards this singular solution asℎ rarr 0 is shown in Figure 8

37 Scalar Field Theory The phase of inflation in the veryearly universe is usually described by a scalar field [5] Theordinary scalar field minimally coupled to gravity evolvesaccording to the equation

120601 + 3119867 120601 +119889119881

119889120601= 0 (45)

where 119881(120601) is the potential of the scalar field The scalar fieldtends to run down the potential towards lower energies Thedensity and the pressure of the universe are related to thescalar field by

1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0

Figure 8 Effect of quantum mechanics (finite value of the Planckconstant) on the regularization of the singular Big Bang solution(ℎ = 0 dashed line) in our simple semi-classical model Thesingularity at 119905 = 0 is replaced by an inflationary expansion fromthe vacuum energy era to the radiation era We can draw an analogywith second order phase transitions where the Planck constant playsthe role of finite size effects (see the text for details)

Using standard technics [7] we find that the potentialcorresponding to the equation of state (20) is [34ndash36]

119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)

In the vacuum energy era (119905 rarr minusinfin) using (26) we get

120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)

In the radiation era (119905 rarr +infin) using (29) we get

120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)

More generally using (31) the evolution of the scalar field120595(119905) in the early universe is given by

cosh120595 minus ln(1 + cosh120595sinh120595

) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)

These results are illustrated in Figures 9 and 10

4 The Radiation and the Dark Energy inthe Late Universe

We propose to describe the transition between the radiationera and the dark energy era in the late universe by a singleequation of state of the form

119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2

Figure 9 Potential of the scalar field in the early universe The fieldtends to run down the potential

0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP

Figure 10 Evolution of the scalar field as a function of time in theearly universe It displays a phase of early inflation

where 120588Λ= Λ8120587119866 = 702 sdot 10

minus24 gm3 is the cosmologicaldensity This equation of state corresponds to a generalizedpolytropic equation of state (1) with 120572 = 13 119899 = minus1 and119896 = minus(43)120588

Λ It may be viewed as the ldquosymmetricrdquo version of

the equation of state (20) in the early universe For120588 ≫ 120588Λ we

recover the equation of state of the pure radiation 119901 sim 12058811988823

For 120588 rarr 120588Λ we get 119901 ≃ minus120588

Λ1198882 corresponding to the dark

energyThe continuity equation (15) may be integrated into

120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)


marks the transition between the radiation era and the darkenergy era The equation of state (51) interpolates smoothlybetween the radiation era (119901 = 120588119888

23 and 120588rad sim 120588

Λ(1198863119886)4)

and the dark energy era (119901 = minus1205881198882 and 120588 = 120588

Λ) It provides

therefore a ldquounifiedrdquo description of radiation and dark energy


in the late universe This amounts to summing the density ofthese two phases Indeed (52) may be rewritten as

120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ

41 The Dark Energy Era Late Inflation When 119886 ≫ 1198863 the

density tends to a minimum value

120588 = 120588min = 120588Λ (54)

and the pressure tends to 119901 = minus120588Λ1198882 The cosmological den-

sity 120588Λ= Λ8120587119866 = 702 sdot10

minus24 gm3 (dark energy) representsa fundamental lower bound for the density A constant valueof the density 120588 ≃ 120588

Λgives rise to a phase of late inflation It is

convenient to define a cosmological time 119905Λ= 1(119866120588

Λ)12

=

(8120587Λ)12

= 146 sdot1018 s and a cosmological length 119897

Λ= 119888119905Λ=

(81205871198882Λ)12

= 438 sdot 1026m These are the counterparts of

the Planck scales for the late universe From the Friedmannequation (17) we find that the Hubble parameter is constant119867 = (81205873)

12119905minus1

Λ Numerically 119867 = 198 sdot 10

minus18 sminus1Therefore the scale factor increases exponentially rapidlywith time as

119886 (119905) prop 119890(81205873)

12119905119905Λ (55)

This exponential growth corresponds to the de Sitter solution[1] The timescale of the exponential growth is the cosmolog-ical time 119905

Λ= 146 sdot 10

18s This solution exists at any time inthe future (119886 rarr +infin and 120588 rarr 120588

Λfor 119905 rarr +infin) so there is

no future singularity This is not the case of all cosmologicalmodels In a ldquophantom universerdquo [40ndash57] violating the nulldominant energy condition (119908 lt minus1) the density increasesas the universe expands This may lead to a future singularitycalled Big Rip (the density becomes infinite in a finite time)The possibility that we live in a phantom universe is not ruledout by observations

42 The Radiation Era When 119886 ≪ 1198863 we recover the equa-

tion

120588rad sim120588Λ1198864

3

1198864 (56)



4 implies that120588Λ1198864

3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0

= 0763 (dark energy) andΩrad0 = 848sdot10minus5 (radiation)

[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)

This can be rewritten as 1198863= 0103119886

0= 309 sdot 10

minus2119897Λwhere

we have used 119897Λ1198860= 1198670119905Λ= 331This scale gives the typical

size of the universe at the transition between the radiationera and the dark energy era When 119886 ≪ 119886

3 the Friedmann

equation (17) yields

119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)

43The Temperature and the Pressure The thermodynamicalequation (13) with the equation of state (51) may be integratedinto

119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)

Equation (60)may be seen as a generalized Stefan-Boltzmannlaw in the late universe In the radiation era 120588 ≫ 120588

Λ it reduces

to the ordinary Stefan-Boltzmann law (36) This allows usto identify the constant of integration 119879

119875with the Planck

temperature 119879119875

= (ℎ11988851198661198962

119861)12

= 142 sdot 1032 K We note

that 119879 prop 119886minus1 both in the radiation and dark energy eras

This suggests that radiation and dark energy belong to thesame ldquofamilyrdquo In the radiation era the temperature decreasesalgebraically as 119879 sim 119879

119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and

in the dark energy era it decreases exponentially rapidly as119879 prop 119890

minus(81205873)12119905119905Λ

The pressure is given by (51) Using (52) we get

119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)

It decreases algebraically during the radiation era and tendsto a constant negative value 119901 = minus120588

Λ1198882 for 119905 rarr +infin Numer-

ically 119901 = minus631 sdot 10minus7 gms2 The point at which the

pressure vanishes is 1198861015840119908

= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model

51 The Quadratic Equation of State We propose to describethe vacuum energy radiation and dark energy by a uniqueequation of state

119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)

For 120588119892rarr 120588119875 119901119892rarr minus120588

1198751198882 (vacuum energy) for 120588

Λ≪ 120588119892≪

120588119875119901119892sim 12058811989211988823 (radiation) for120588

119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark

energy) This quadratic equation of state combines the prop-erties of the equation of state (20) valid in the early universeand of the equation of state (51) valid in the late universe Anice feature of this equation of state is that both the Planckdensity (vacuum energy) and the cosmological density (darkenergy) explicitly appear Therefore this equation of state


reproduces both the early inflation and the late inflation Onthe other hand this description suggests that vacuum energyradiation and dark energy may be the manifestation of aunique form of ldquogeneralized radiationrdquo This is however justa speculation before a justification of this equation of statehas been given from first principles In this paper we limitourselves to studying the properties of this equation of state

Using the equation of continuity (15) we obtain

120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10

minus6m (see Section 3) To obtain this exp-ression we have used the fact that 120588

119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588

119875)(120588119892minus 120588Λ)(120588119875minus 120588119892) When 119886 rarr 0 120588

119892rarr 120588119875

(vacuum energy) when 120588Λ

≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4

(radiation) when 119886 rarr +infin 120588119892rarr 120588Λ(dark energy) In the

early universe the contribution of dark energy is negligibleand we recover (21) In the late universe the contribution ofvacuum energy is negligible and we recover (52) with 120588

1198751198864

1=

120588Λ1198864

3We shall view the ldquogeneralized radiationrdquo the baryonic

matter and the darkmatter as different species In the generalcase these species could interact However in this paperwe assume for simplicity that they do not The generalizedradiation (vacuum energy + radiation + dark energy) isdescribed by the quadratic equation of state (63) Using1205881198751198864

1= 120588rad0119886

4

0(see Section 3) we can rewrite (64) in the

form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10

minus32 The baryonic matter and the darkmatter are described as pressureless fluids with 119901

119861= 0 and

119901DM = 0 The equation of continuity (15) for each speciesleads to the relations

120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)

The total density 120588 = 120588119892+ 120588119861+ 120588DM of generalized radiation

baryonic matter and dark energy can be written as

120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)

Substituting this relation in the Friedmann equation (17) andwriting 120588rad0 = Ωrad01205880 1205881198610 = Ω

11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)

From observations Ωrad0 = 848 sdot 10minus5 Ω1198610

= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]

The thermodynamic equation (13) with the equation ofstate (63) may be integrated into

119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)

Equation (70) may also be written as

119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861

≪ 1198860 1198790can be identified with the present

temperature of radiation Indeed after the vacuum energyera (71) reduces to

119879 sim 1198790

1198860

119886 (72)

Finally using (63) and (64) the pressure of generalizedradiation can be written in very good approximation as

119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)

52 The Early Universe In the early universe we can neglectthe contribution of baryonic matter dark matter and darkenergy in the density We only consider the contribution ofvacuum energy and radiation Equation (67) then reduces to

120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)

The Friedmann equation (68) becomes

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)

It has the analytical solution given by (31) On the otherhand the temperature of the generalized radiation in the earlyuniverse is given by (34) and (35) The transition betweenthe vacuum energy era and the radiation era corresponds to120588rad = 120588

119875 This yields 119886

11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=

261 sdot 10minus6m 120588

1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=

393 sdot 1031 K This takes place at a time 119905

1= 233119905

119875= 125 sdot

10minus42 s This time also corresponds to the inflexion point of

the curve 119886(119905) The universe is accelerating for 119905 lt 1199051and

decelerating for 119905 gt 1199051


In the vacuum energy era (119886 ≪ 1198861) the density is

constant

120588 ≃ 120588119875 (76)

Therefore the Hubble parameter is also a constant 119867 =

(81205873)12

119905minus1

119875 In that case the scale factor and the temperature

increase exponentially rapidly according to (26) and (38)In the radiation era (119886 ≫ 119886

1) we have

120588 sim120588rad0119886

4

0

1198864 (77)

The Friedmann equation (75) reduces to

119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)

53The Late Universe In the late universe we can neglect thecontribution of vacuum energy and radiation in the densityWe only consider the contribution of baryonic matter darkmatter and dark energy Equation (67) then reduces to

120588 =1205881198980

(1198861198860)3+ 120588Λ (80)

where 120588119898= 120588119861+ 120588DM This coincides with the ΛCDMmodel

(see Section 55) The Friedmann equation (68) becomes

119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0

≃ 1 It has the analytical solution

119886

1198860

= (Ω1198980

ΩΛ0

)

13

sinh23 (32radicΩΛ0

1198670119905) (82)

The density evolves as

120588

1205880

=ΩΛ0

tanh2 ((32)radicΩΛ0

1198670119905) (83)

On the other hand the temperature of the generalized radia-tion in the late universe is given by

119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)

Using the expression (82) of the scale factor we obtain

119879 =1198790

(Ω1198980

ΩΛ0

)13sinh23 ((32)radicΩ

Λ01198670119905)

(85)

Finally the pressure of the generalized radiation in the lateuniverse may be written as

119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)

Setting 119886 = 1198860in (82) we find the age of the universe

1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim

14Gyrs It is rather fortunate that the age of the universealmost coincides with the Hubble time119867minus1

0

The universe starts accelerating when 120588119898= 2120588Λ[see (16)

with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763

(dark energy) we obtain 119886 = 1198861015840

119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ

= 211 sdot 10minus23 gm3 and

1198791015840

119888= 186119879

0= 578K The universe is decelerating for 119905 lt 119905

1015840

119888

and accelerating for 119905 gt 1199051015840

119888

The transition between thematter era and the dark energyera corresponds to 120588

119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)

At that point 120588 = 1205882= 2120588Λand 119879 = 119879

2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10

minus23 gm3 and 1198792=

1481198790= 459K We also note that 1198861015840

119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)

coincides with the one obtained in [34ndash36] using a differentviewpoint

In the dark energy era (119886 ≫ 1198862) the density is constant

120588 ≃ 120588Λ (91)


1198670radicΩΛ0

Using (82) we find that the scale factor increasesexponentially rapidly as

119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)


This corresponds to the de Sitter solution According to (85)the temperature decreases exponentially rapidly as

119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)

In the matter era (119886 ≪ 1198862) we have

120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)

This corresponds to the Einstein-de Sitter (EdS) universe

54The Evolution of the Late Universe In thematter era (119886 ≪

1198862) the radius increases algebraically as 119886 prop 119905

23 while thedensity decreases algebraically as 120588 prop 119905

minus2 (EdS) When 119886 ≫

1198862 the universe enters in the dark energy era It undergoes

a late inflation (de Sitter) during which its radius increasesexponentially rapidly while its density remains constant andequal to the cosmological density 120588

Λ The transition takes

place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ

= 297 sdot 1017 s In the matter era the universe is

decelerating and in the dark energy era it is acceleratingThe time at which the universe starts accelerating is 119905

1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10

17 s corresponding to a radius1198861015840

119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10

minus23 gm3 and a temperature1198791015840119888= 186119879

0=

578K In the late universe the temperature of generalizedradiation always scales as 119879 prop 119886

minus1 It decreases algebraicallyrapidly (119879 prop 119905

minus23) in the matter era and exponentiallyrapidly in the dark energy era The evolution of the scalefactor density temperature and pressure of radiation as afunction of time is represented in Figures 11 12 13 14 15 and16 in logarithmic and linear scales

The present size of the universe 1198860= 0302119897

Λ= 132 sdot

1026m is precisely of the order of the scale 119886

2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand

1199050sim 1199052sim 119905Λ Therefore we live just at the transition between

the matter era and the dark energy era (see bullets in Figures11 12 13 14 15 and 16) (In the context of the standardΛCDMmodel the way to state this observation is to say that thepresent ratio Ω

Λ0Ω1198980

= 322 between dark energy andmatter is of order unity Since the matter density changes

10 12 14 16 18 2010

20

30

40

50

(late inflation)Dark energy era

Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ

Figure 11 Evolution of the scale factor 119886 with the time 119905 inlogarithmic scales This figure clearly shows the transition betweenthe matter era (EdS) and the dark energy era (de Sitter) In thedark energy era the radius increases exponentially rapidly on atimescale of the order of the cosmological time 119905

Λ= 146sdot10

18 sThiscorresponds to a phase of late inflation The universe is deceleratingfor 119886 lt 119886

1015840

119888and accelerating for 119886 gt 119886

1015840

119888with 119886

1015840

119888= 711 sdot 10

25m Thetransition between thematter era and the dark energy era takes placeat 1198862= 895 sdot 10

25m Coincidentally the present universe turns outto be close to the transition point (119886

0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2

Figure 12 Evolution of the scale factor 119886 with the time 119905 in linearscales

as 119886minus3 the ratio between matter and dark energy is of orderunity only during a brief period of time in the evolutionof the universe It turns out that we live precisely in thisperiod This coincidence is intriguing and often referred toas the ldquocosmic coincidence problemrdquo [58] Several theorieshave been proposed to explain why Ω

Λ0Ω1198980

sim 1 [59ndash61]However this may be just a pure coincidence without deeperreason Life (and researchers inquiring about cosmology)may have emerged at this special moment of the history of theuniverse sim14 Gyrs after the Big Bang precisely at the epochwhere Ω

ΛΩ119898

sim 1 Life cannot have emerged much earlierbecause galaxies stars and planets are not created and it


10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]

Figure 13 Evolution of the density 120588 with the time 119905 in logarithmicscales During the late inflation the density remains approximatelyconstant with the cosmological value 120588min = 120588

Λrepresenting a lower

bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ

Figure 14 Evolution of the density 120588with the time 119905 in linear scalesGeneral relativity (cosmological constant) limits the decay of thedensity to the cosmological value 120588

Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0

Figure 15 Evolution of the temperature of radiation119879with the time119905 in linear scales

0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2

Figure 16 Evolution of the pressure of radiation 119901119892with the time 119905

in linear scales

may not persistmuch later because planets stars and galaxieswill dieTherefore this coincidence is not really a ldquoproblemrdquo)The present density of the universe is 120588

0= 131120588

Λ= 920 sdot

10minus24 gm3 and the age of the universe is 119905

0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10

17s sim 14Gyrs The present values of theequation of state parameter and deceleration parameter are1199080= minusΩ

Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]

Numerically 1199080= minus0763 and 119902

0= minus0645

55 The Whole Evolution of the Universe In the standardΛCDM model the radiation baryonic matter dark matterand dark energy are treated as four independent speciescharacterized by different equations of state The equation ofstate is 119901 = 120588119888

23 for the radiation 119901 = 0 for the baryonic

matter and dark matter and 119901 = minus1205881198882 for the dark energy

One then solves the equation of continuity (15) for these fourspecies in order to obtain the relation between their densityand the scale factor that is 120588rad = 120588rad0(1198861198860)

4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0

Finally weadd their individual densities 120588 = 120588rad + 120588

119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)

and solve the Friedmann equation (17) This leads to theequation

119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)

with Ωrad + Ω119861+ ΩDM + Ω

Λ= 1 This standard model

provides a good description of the universe after the Planckera However it exhibits a primordial singularity at 119905 = 0 (BigBang) Furthermore it does not describe the phase of earlyinflation This phase is usually described by another theoryfor example a scalar field theory and it is then connected tothe standard model

In the present model only three species coexist baryonicmatter (see (66a)) darkmatter (see (66b)) and a ldquogeneralized


radiationrdquo unifying vacuum energy radiation and darkenergy (see (65)) As a result the equation describing thewhole evolution of the universe is (In the standard modelthe radiation density diverges as 119886minus4 when 119886 rarr 0 whilethe matter density diverges as 119886

minus3 which is subdominantTherefore we can safely extrapolate the matter density to119886 = 0 (its contribution is negligible anyway) In the presentmodel the generalized radiation density (65) does not divergeanymore when 119886 rarr 0 since 119886

1= 0 Therefore it becomes

necessary to state explicitly that matter appears at sufficientlylate time in order to avoid a spurious divergence of thematterdensity when 119886 rarr 0 To that purpose it is sufficient to add aHeaviside function119867(119886 minus 119886

1) in the matter density and write

Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)

with Ω119861+ Ωrad + ΩDM + Ω

Λ= 1 For 119886

1= 0 we

obtain the same equation as in the standard ΛCDM modelwhere the contributions of radiation baryonic matter darkmatter and dark energy are added individually [1] Howeveras we have previously mentioned the ΛCDM model doesnot describe the phase of early inflation and it presents asingularity at 119905 = 0 (Big Bang) For 119886

11198860= 197 sdot 10

minus32we obtain a generalized model which does not present aprimordial singularity and which displays a phase of earlyinflation In this model the universe always exists in thepast but for 119905 lt 0 it has a very small radius smallerthan the Planck length (As already stated in Section 3 ourmodel certainly breaks down in this period since it doesnot take quantum fluctuations into account The Planckera may not be described in terms of an equation of state119901(120588) or even in terms of the Einstein equations as we haveassumed It probably requires the development of a theoryof quantum gravity that does not exist for the moment Aninteresting description of the early inflation has been givenby Monerat et al [62] in terms of a quantized model basedon a simplified Wheeler-DeWitt equation In that model aquantum tunneling process explains the birth of the universewith a well-defined size after tunneling Therefore otherinflationary scenarios are possible in addition to the onebased on the generalized equation of state (63)) At 119905 = 0it undergoes an inflationary expansion in a very short lapseof time of the order of 233 Planck times For 119905 gt 233119905

119875

this model gives the same results as the standard ΛCDMmodel the universe first undergoes an algebraic expansion inthe radiation and matter eras then an exponential expansion(second inflation) in the dark energy era A nice feature ofthis model is its simplicity since it incorporates a phase ofearly inflation in a very simple manner We just have to add aterm +(119886

11198860)4 in the standard equation (99) Therefore the

modification implied by (65) to describe the early inflationis very natural On the other hand this model gives thesame results in the late universe as the standard model sothis does not bring any modification to the usual equation(99) Therefore our simple model completes the standard

ΛCDMmodel by incorporating the phase of early inflation ina natural manner This is an interest of this description sincethe standard model gives results that agree with observationsat late times

The evolution of the scale factor with time is obtained bysolving the first-order differential equation (100) This yields

int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)

where 119886119894= 1198861= 0 in the standard model and 119886

119894= 119897119875in the

generalized model The age of the universe is

1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)

Of course for the determination of the age of the universewe can neglect the contribution of vacuum energy in theearly universe and take 119886

1= 0 (strictly speaking the age of

the universe is infinite since it has no origin however wedefine the age of the universe from the time 119905 = 0 at which119886 = 119897

119875) We obtain the standard result 119905

0= 103 119867

minus1

0=

453 sdot 1017 s = 144Gyr (The Hubble constant is usually

written as 1198670= 2268 ℎ

710minus18 sminus1 where the dimensionless

parameter ℎ7is about 10of unity [1] For simplicity we have

taken ℎ7= 1 in the numerical applicationsThe current value

is ℎ7

= 105 plusmn 005 If we take ℎ7

= 105 the age of theuniverse is 119905

0= 137Gyr) Actually we find the same result

if we neglect radiation and use the analytical expression (87)instead of (102)

In Figures 17ndash19 we have represented the evolution of theradius density and temperature of the universe as a functionof timeThe universe exhibits two types of inflations an earlyinflation due to the Planck density 120588

119875= 516 sdot 10

99 gm3(vacuum energy) and a late inflation due to the cosmologicaldensity 120588

Λ= 702 sdot 10

minus24 gm3 (dark energy) There exists astriking ldquosymmetryrdquo between the early and the late evolutionof the universe and the cosmological constant in the lateuniverse plays the same role as the Planck constant in the earlyuniverse In particular Figure 18 shows that the density variesbetween two bounds 120588max = 120588

119875and 120588min = 120588

Λthat are fixed

by fundamental constants These values differ by a factor ofthe order 10122 The early universe is governed by quantummechanics (ℎ) and the late universe by general relativity (Λ)

Remark 1 We have described dark matter as a presurelessfluid with 119901 = 0 There are some indications that dark mattermay be described by an isothermal equation of state 119901 = 120572120588119888

2

[63] In that case the term Ωlowast

DM0(1198861198860)3 in (100) should

be replaced by Ωlowast

DM0(1198861198860)3(1+120572) Another generalization


20 30

0

20

40

Radiation era

(dark energy)Matter era

Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20

minus10 0 10log [t (s)]

log [a

(m)]

simlPsimtP

sima1

Figure 17 Evolution of the scale factor 119886 as a function of time inlogarithmic scalesThe early universe undergoes a phase of inflationdue to the vacuum energy before entering into the radiation eraDuring the early inflation the scale factor increases by 29 ordersof magnitude in less than 10

minus42 s This is followed by the matter eraand the dark energy era responsible for the accelerated expansion ofthe universe The universe exhibits two types of inflation an earlyinflation corresponding to the Planck density 120588

119875(vacuum energy)

due to quantum mechanics (Planck constant) and a late inflationcorresponding to the cosmological density 120588

Λ(dark energy) due to

general relativity (cosmological constant)The evolution of the earlyand late universe is remarkably symmetric We have represented indashed line the standard model leading to a primordial singularity(Big Bang) The dotted line corresponds to the model of Section 53where the radiation is neglected This analytical model provides agood description of the late universe We have also represented thelocation of the present universe (bullet) It happens to be just at thetransition between the matter era and the dark energy era

may be introduced if dark matter is made of Bose-Einsteincondensates (BECs) instead of being pressureless In thatcase the term Ω

lowast

DM0(1198861198860)3 in (100) should be replaced

by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)

3] as proposed in [64 65]

It is shown in these papers that the presence of BECscan substantially accelerate the formation of the large-scalestructures of the universe

6 Conclusion

There are different manners to describe the evolution of theuniverse

In the standardΛCDMmodel [1] the radiation baryonicmatter dark matter and dark energy are treated as fourindependent species characterized by different equations ofstate The equation of state is 119901 = 120588119888

23 for the radiation

119901 = 0 for the baryonic matter and dark matter and 119901 =

minus1205881198882 for the dark energy One then solves the equation of

continuity (15) for each species in order to obtain the relationbetween their density and the scale factor Finally we addtheir densities 120588 = 120588rad + 120588

119861+ 120588DM + 120588

Λand solve the

Friedmann equation (17) This leads to (99) This standardmodel provides a good description of the universe after thePlanck era However it exhibits a primordial singularity (BigBang) and does not display a phase of early inflation

100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation

Late inflation(dark energy)

(vacuum energy)

a0minus50

minus60 minus50 minus40 minus30 minus20 minus10

log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]

Figure 18 Evolution of the density 120588 as a function of time inlogarithmic scales The density goes from a maximum value 120588max =

120588119875determined by the Planck constant (quantum mechanics) to

a minimum value 120588min = 120588Λdetermined by the cosmological

constant (general relativity) These two bounds which are fixed byfundamental constants of physics are responsible for the early andlate inflation of the universe In between the density decreases as119905minus2

0

50

Early inflation

Radiation eraMatter era


(vacuum energy)

a0

20 30minus60 minus50 minus40 minus30 minus20 minus10 0 10log [t (s)]

simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K

Figure 19 Evolution of the temperature 119879 of the ldquogeneralizedrdquoradiation as a function of time in logarithmic scales Before the earlyinflation the universe is extremely cold (119879 lt 10

minus173 K) During theearly inflation the temperature increases by 204 orders ofmagnitudein less than 10

minus42 s During the radiation era the temperaturedecreases algebraically The present value of the temperature is 119879 ≃

27K (bullet) During the late inflation the temperature decreasesexponentially rapidly

In recent papers [34ndash36] we have proposed a differentdescription We have described the early universe (vacuumenergy + radiation) and the late universe (matter + darkenergy) by two symmetric equations of state The transitionbetween the vacuum energy era and the radiation era in theearly universe has been described by a single equation ofstate of the form 119901 = minus(43)(120588

2120588119875)1198882+ 12058811988823 Similarly the

transition between the matter era and the dark energy erain the late universe has been described by a single equationof state 119901 = minus120588

Λ1198882 These equations of state may be viewed


as two polytropic equations of state with index 119899 = 1 and119899 = minus1 respectivelyThis description leads to very symmetricmathematical formulae in the early and late universe thecosmological density in the late universe plays a role similarto that of the Planck density in the early universe In orderto obtain a single equation describing the whole evolutionof the universe we have just added the density of these twophases 120588 = 120588early + 120588late in the Friedmann equation (17) Thisleads to (100) with 119886

11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875

the evolution is the same as in the ΛCDM model Howeverthis model incorporates a phase of early inflation in a naturalmanner and yields a model without primordial singularity(although we should not extrapolate it in the infinite pastdue to our neglect of quantum fluctuations) This may becalled ldquoaioniotic universerdquo since it has no origin nor end Asmentioned in the Introduction this model may be viewed asa generalization of the Chaplygin gas model

Finally in this paper we have adopted a different pointof view We have described the vacuum energy + radiation+ dark energy as a single species called ldquogeneralized radi-ationrdquo characterized by a quadratic equation of state 119901

119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588

Λ1198882 From this equation of state

we have obtained a generalization of the Stefan-Boltzmannlaw (see (69)) The baryonic matter and dark matter aretreated as different species with a pressureless equation ofstate 119901

119861= 119901DM = 0 We have solved the equation of

continuity (15) to obtain the relation between the densityand the scale factor for these three species In order toobtain a single equation describing the whole evolution of theuniverse we have just added their densities 120588 = 120588

119892+120588119861+120588DM

in the Friedmann equation (17) This again leads to (100)as in our previous approach However the point of view isdifferent The present approach suggests that vacuum energyand dark energy may correspond to a form of ldquogeneralizedradiationrdquo This may open new perspectives in cosmologyOf course more work is needed to justify this quadraticequation of state However the present paper shows that thisequation of state is able to reproduce the main properties ofour universe

Appendices

A The Intermediate UniverseTransition between the Radiation Era andthe Matter Era

In this Appendix we consider the epoch where the evolutionof the universe is dominated by radiation and matter Thetotal density can be written as

120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)

The transition between the radiation era and the matterera takes place when 120588rad = 120588

119898leading to 119886 = 119886

4=

(Ωrad0Ω1198980)1198860 Numerically 1198864= 358 sdot 10

minus41198860= 108 sdot

10minus4119897Λ hence

1198864= 473 sdot 10

22m (A2)

This corresponds to a density 1205884= 2(Ω

4

1198980Ω3

rad0)1205880 and atemperature 119879

4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ

= 952 sdot 10minus14 gm3 and 119879

4= 279 sdot 10

31198790=

869 sdot 103 K

The Friedmann equation (100) simplifies into

119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)

The evolution of the scale factor with time is obtained bysolving the first-order differential equation (A3) This yields

int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)

The integral can be performed analytically leading to

1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)

In the radiation era (119886 ≪ 1198864) we recover (79) and in the

matter era (119886 ≫ 1198864) we recover (96) The transition between

the radiation era and the matter era takes place at a time11986701199054= (13)(4 minus 2radic2)(Ωrad0)

32(Ω1198980

)2 Numerically 119905

4=

543 sdot 10minus6119867minus1

0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s

B Analytical Solution ofthe Friedmann Equations for a ToyUniverse without Matter

For the sake of completeness we give the solution of theFriedmann equations for a universe described by the pureequation of state (63) This solution is interesting mathemat-ically because it is analytical and it exhibits two phases ofinflation separated by a radiation era However it is unphysi-cal in the intermediate phase because it ignores the contribu-tion of baryonic and dark matter The numerical solution of(100) studied in the core of the paper is more physical

In the early universe the equation of state (63) reducesto (20) The density is given as a function of the scale factorby (21) The evolution of the scale factor is given by (31)This solution is satisfactory physically since matter can beneglected in the early universe

In the late universe the equation of state (63) reduces to(51) The density is given as a function of the scale factor by(52) If we neglect the contribution of matter the solution ofthe Friedmann equation (17) with the density (52) is

119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)

For 119905 rarr 0 it reduces to (58) and for 119905 rarr +infin we obtain(55) with a prefactor 1radic2 Furthermore we can associate to


the equation of state (51) a scalar field as in Section 37 Thepotential of this scalar field is

119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)

For 119905 rarr 0 using (58) we get

120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)

For 119905 rarr +infin using (55) with a prefactor 1radic2 we get

120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)

More generally using (B1) the evolution of the scalar field isgiven by

120595 = minussinhminus1 [ 1

sinh (radic(321205873) (119905119905Λ))] (B5)

For the quadratic equation of state (63) the densityis related to the scale factor by (64) If we neglect thecontribution of matter it is possible to solve the Friedmannequation (17) with the density-radius relation (64) analyti-cally Introducing 119877 = 119886119886

1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)

which can be integrated into

1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)

where the constant 119862 is determined such that 119886 = 119897119875at

119905 = 0 This solution is interesting mathematically becauseit describes analytically a phase of early inflation and aphase of late inflation connected by a power-law evolutioncorresponding to the radiation However since (B7) doesnot take into account the contribution of baryonic matterand dark matter the intermediate phase is unphysical thesolution (B7) describes a universe without matter The cor-rect solution is obtained by solving the complete Friedmannequation (100) as in Section 55

References

[1] J Binney and S Tremaine Galactic Dynamics Princeton Uni-versity Press 2008

[2] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 pp347ndash356 1981

[3] A D Linde ldquoA new inflationary universe scenario a possiblesolution of the horizon flatness homogeneity isotropy andprimordial monopole problemsrdquo Physics Letters B vol 108 no6 pp 389ndash393 1982

[4] A Albrecht P J Steinhardt M S Turner and FWilczek ldquoReh-eating an inflationary universerdquo Physical Review Letters vol 48no 20 pp 1437ndash1440 1982

[5] A LindeParticle Physics and InflationaryCosmology HarwoodChur Switzerland 1990

[6] S Weinberg Gravitation and Cosmology John Wiley amp Sons1972

[7] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 pp 1753ndash1935 2006

[8] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[9] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ and Λ from 42 High-Redshift SupernovaerdquoThe Astrophys-ical Journal vol 517 no 2 pp 565ndash586 1999

[10] P de Bernardis P A RAde J J Bock et al ldquoAflatUniverse fromhigh-resolution maps of the cosmic microwave backgroundradiationrdquo Nature vol 404 pp 955ndash959 2000

[11] S Hanany P Ade A Balbi et al ldquoMAXIMA-1 a measurementof the cosmic microwave background anisotropy on angularscales of 101015840-5∘rdquo The Astrophysical Journal Letters vol 545 no1 pp L5ndashL9 2000

[12] B Ratra and J Peebles ldquoCosmological consequences of a rollinghomogeneous scalar fieldrdquo Physical Review D vol 37 no 12 pp3406ndash3427 1988

[13] A A Starobinsky ldquoHow to determine an effective potentialfor a variable cosmological termrdquo Journal of Experimental andTheoretical Physics Letters vol 68 no 10 pp 757ndash763 1998

[14] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy component with general equation of staterdquoPhysical Review Letters vol 80 pp 1582ndash1585 1998

[15] Ph Brax and JMartin ldquoQuintessence and supergravityrdquo PhysicsLetters B vol 468 no 1-2 pp 40ndash45 1999

[16] A Albrecht and C Skordis ldquoPhenomenology of a realisticaccelerating universe using only Planck-scale physicsrdquo PhysicalReview Letters vol 84 pp 2076ndash2079 2000

[17] T Barreiro E J Copeland and N J Nunes ldquoQuintessence aris-ing from exponential potentialsrdquo Physical Review D vol 61 no12 Article ID 127301 4 pages 2000

[18] L A Urena-Lopez and T Matos ldquoNew cosmological trackersolution for quintessencerdquo Physical ReviewD vol 62 Article ID081302 4 pages 2000

[19] P Brax and J Martin ldquoRobustness of quintessencerdquo PhysicalReview D vol 61 no 10 Article ID 103502 14 pages 2000

[20] T D Saini S Raychaudhury V Sahni and A A StarobinskyldquoReconstructing the cosmic equation of state from supernovadistancesrdquo Physical Review Letters vol 85 no 6 pp 1162ndash11652000

[21] V Sahni andAA Starobinsky ldquoThe case for a positive cosmolo-gical Λ-termrdquo International Journal of Modern Physics D vol 9no 4 pp 373ndash443 2000

[22] V Sahni ldquoThe cosmological constant problem and quintessen-cerdquo Classical and Quantum Gravity vol 19 no 13 pp 3435ndash3448 2002


[23] M Pavlov C RubanoM Sazhin and P Scudellaro ldquoAnalysis oftwo quintessence models with Type Ia supernova datardquo Astro-physical Journal Letters vol 566 no 2 pp 619ndash622 2002

[24] V Sahni T D Saini A A Starobinsky and U Alam ldquoStatefin-dermdasha new geometrical diagnostic of dark energyrdquo Journal ofExperimental and Theoretical Physics Letters vol 77 no 5 pp201ndash206 2003

[25] A Kamenshchik U Moschella and V Pasquier ldquoAn alternativeto quintessencerdquo Physics Letters B vol 511 no 2ndash4 pp 265ndash2682001

[26] N Bilic G B Tuper and R Viollier ldquoUnification of darkmatterand dark energy the inhomogeneous Chaplygin gasrdquo PhysicsLetters B vol 535 pp 17ndash21 2002

[27] J S Fabris S V Goncalves and P E de Souza ldquoLetter densityperturbations in a universe dominated by the Chaplygin gasrdquoGeneral Relativity andGravitation vol 34 no 1 pp 53ndash63 2002

[28] M C Bento O Bertolami and A A Sen ldquoGeneralized Chaply-gin gas accelerated expansion and dark-energy-matter unifi-cationrdquo Physical Review D vol 66 no 4 Article ID 043507 5pages 2002

[29] H B Benaoum ldquoAccelerated universe frommodifiedChaplygingas and Tachyonic fluidrdquo submitted httparxivorgabshep-th0205140

[30] V Gorini A Kamenshchik and U Moschella ldquoCan the Chap-lygin gas be a plausible model for dark energyrdquo Physical ReviewD vol 67 no 6 Article ID 063509 2003

[31] M C Bento O Bertolami andA A Sen ldquoRevival of the unifieddark energyndashdarkmattermodelrdquoPhysical ReviewD vol 70 no8 Article ID 083519 7 pages 2004

[32] U Debnath A Banerjee and S Chakraborty ldquoRole of modifiedChaplygin gas in accelerated universerdquo Classical and QuantumGravity vol 21 no 23 pp 5609ndash5617 2004

[33] K Karami S Ghaari and J Fehri ldquoInteracting polytropic gasmodel of phantom dark energy in non-flat universerdquoThe Euro-pean Physical Journal C vol 64 no 1 pp 85ndash88 2009

[34] P H Chavanis ldquoModels of universe with a polytropic equationof state I The early universerdquo submitted httparxivorgabs12080797

[35] P H Chavanis ldquoModels of universe with a polytropic equationof state II The late universerdquo submitted httparxivorgabs12080801

[36] P H Chavanis ldquoModels of universe with a polytropic equationof state IIIThe phantom universerdquo submitted httparxivorgabs12081185

[37] H B Sandvik M Tegmark M Zaldarriaga and I Waga ldquoTheend of unified dark matterrdquo Physical Review D vol 69 no 12Article ID 123524 7 pages 2004

[38] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[39] T PadmanabhanTheoretical Astrophysics vol 3 ofGalaxies andCosmology Cambridge University Press 2002

[40] R R Caldwell ldquoA phantom menace Cosmological conse-quences of a dark energy component with super-negativeequation of staterdquo Physics Letters B vol 545 pp 23ndash29 2002

[41] R R Caldwell M Kamionkowski and N NWeinberg ldquoPhan-tom energy dark energy with 119908 lt minus1 causes a cosmic dooms-dayrdquo Physical Review Letters vol 91 no 7 Article ID 071301 4pages 2003

[42] B McInnes ldquoThe dSCFT correspondence and the big smashrdquoJournal of High Energy Physics vol 2002 no 8 p 29 2002

[43] S M Carroll M Hoffman and M Trodden ldquoCan the darkenergy equation-of-state parameter w be less than -1rdquo PhysicalReview D vol 68 no 2 Article ID 023509 11 pages 2003

[44] P Singh M Sami and N Dadhich ldquoCosmological dynamics ofa phantom fieldrdquo Physical Review D vol 68 Article ID 0235227 pages 2003

[45] J M Cline S Jeon and G D Moore ldquoThe phantom menacedconstraints on low-energy effective ghostsrdquo Physical Review Dvol 70 no 4 Article ID 043543 4 pages 2004

[46] M Sami and A Toporensky ldquoPhantom field and the fate of theuniverserdquoModern Physics Letters A vol 19 no 20 pp 1509ndash15172004

[47] S Nesseris and L Perivolaropoulos ldquoFate of bound systemsin phantom and quintessence cosmologiesrdquo Physical Review Dvol 70 no 12 Article ID 123529 9 pages 2004

[48] E Babichev V Dokuchaev and Y Eroshenko ldquoBlack hole massdecreasing due to phantom energy accretionrdquo Physical ReviewLetters vol 93 no 2 Article ID 021102 4 pages 2004

[49] P F Gonzalez-Dıaz and C L Siguenza ldquoThe fate of black holesin an accelerating universerdquo Physics Letters B vol 589 pp 78ndash82 2004

[50] P F Gonzalez-Dıaz and C L Siguenza ldquoPhantom thermody-namicsrdquo Nuclear Physics B vol 697 pp 363ndash386 2004

[51] S Nojiri and S D Odintsov ldquoFinal state and thermodynamicsof a dark energy universerdquo Physical Review D vol 70 no 10Article ID 103522 15 pages 2004

[52] S Nojiri S D Odintsov and S Tsujikawa ldquoProperties ofsingularities in the (phantom) dark energy universerdquo PhysicalReview D vol 71 no 6 Article ID 063004 16 pages 2005

[53] H Stefancic ldquoExpansion around the vacuum equation of statesudden future singularities and asymptotic behaviorrdquo PhysicalReview D vol 71 no 8 Article ID 084024 9 pages 2005

[54] M Bouhmadi-Lopez P F Gonzalez-Dıaz and P Martın-Mor-uno ldquoWorse than a big riprdquo Physics Letters B vol 659 pp 1ndash52008

[55] H Garcıa-Compean G Garcıa-Jimenez O Obregon and CRamırez ldquoCrossing the phantom divide in an interacting gen-eralized Chaplygin gasrdquo Journal of Cosmology and AstroparticlePhysics vol 2008 no 7 p 16 2008

[56] L Fernandez-Jambrina ldquow-cosmological singularitiesrdquo Physi-cal Review D vol 82 Article ID 124004 5 pages 2010

[57] P H Frampton K J Ludwick and R J Scherrer ldquoThe little riprdquoPhysical Review D vol 84 Article ID 063003 5 pages 2011

[58] P Steinhardt ldquoCosmological challenges for the 21st centuryrdquo inCritical Problems in Physics V L Fitch and D R Marlow EdsPrinceton University Press Princeton NJ USA 1997

[59] I Zlatev L Wang and P J Steinhardt ldquoQuintessence cosmiccoincidence and the cosmological constantrdquo Physical ReviewLetters vol 82 no 5 pp 896ndash899 1999

[60] L Amendola and D Tocchini-Valentini ldquoStationary dark ener-gy the present universe as a global attractorrdquo Physical ReviewD vol 64 no 4 Article ID 043509 5 pages 2001

[61] L P Chimento A S Jakubi and D Pavon ldquoDark energy dis-sipation and the coincidence problemrdquo Physical Review D vol67 no 8 Article ID 087302 3 pages 2003

[62] GAMonerat GOliveira-Neto E V Correa Silva et al ldquoDyna-mics of the early universe and the initial conditions for inflationin a model with radiation and a Chaplygin gasrdquo Physical ReviewD vol 76 no 2 Article ID 024017 11 pages 2007


[63] C M Muller ldquoCosmological bounds on the equation of state ofdarkmatterrdquo Physical ReviewD vol 71 no 4 Article ID 0473024 pages 2005

[64] T Harko ldquoEvolution of cosmological perturbations in BosendashEinstein condensate dark matterrdquo Monthly Notices of the RoyalAstronomical Society vol 413 pp 3095ndash3104 2011

[65] P H Chavanis ldquoGrowth of perturbations in an expanding uni-verse with Bose-Einstein condensate darkmatterrdquoAstronomyampAstrophysics vol 537 article A127 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Computational Methods in Physics

Journal of


Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013


Condensed Matter PhysicsAdvances in

OpticsInternational Journal of



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Volume 2013

ISRN High Energy Physics



Advances in

Astronomy


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ISRN Condensed Matter Physics



AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom


High Energy PhysicsAdvances in


Soft MatterJournal of


Statistical MechanicsInternational Journal of


PhotonicsJournal of

ISRN Optics



ThermodynamicsJournal of


As a result of the vacuum energy the universe expandsexponentially rapidly on a timescale of the order of the Plancktime 119905

119875= 539 sdot 10

minus44 s This phase of early inflation isfollowed by the radiation era described by an equation ofstate 119901 = 120588119888

23 In the vacuum energy era the density has

a constant value 120588 ≃ 120588119875 while it decreases as 120588 prop 119886

minus4 duringthe radiation era

The phase of acceleration in the late universe is usuallyascribed to the cosmological constant Λ which is equivalentto a constant energy density120588

Λ= Λ(8120587119866) = 702sdot10

minus24 gm3called the dark energy This acceleration can be modeled byan equation of state 119901 = minus120588119888

2 implying a constant energydensity identified with the cosmological density 120588

Λ As a

result of the dark energy the universe expands exponentiallyrapidly on a timescale of the order of the cosmological time119905Λ

= 146 sdot 1018 s (de Sitter solution) This leads to a

phase of late inflation Inspired by the analogy with the earlyinflation some authors have represented the dark energy bya scalar field called quintessence [12ndash24] As an alternativeto the quintessence other authors have proposed to modelthe acceleration of the universe by an exotic fluid with anequation of state of the form119901119888

2= 119861120588minus119860120588

119886 with119860 ge 0 and119886 ge minus1 called the Chaplygin gas when (119861 119886) = (0 1) and thegeneralized Chaplygin gas otherwise [25ndash32] This equationof state may be rewritten as

119901 = (120572120588 + 1198961205881+1119899

) 1198882 (1)

with 119896 le 0 in order to emphasize its analogy with thepolytropic equation of state [33ndash36] Indeed it may be viewedas the sum of a linear equation of state 119901 = 120572120588119888

2 describingdust matter (120572 = 0) or radiation (120572 = 13) and a polytropicequation of state 119901 = 119896120588

1+11198991198882 We shall call (1) a generalized

polytropic equation of state At late times this equation ofstate with 119899 lt 0 leads to a constant energy density implyingan exponential growth of the scale factor that is similar to theeffect of the cosmological constant At earlier times and for120572 = 0 this equation of state returns the results of the dustmatter model (119901 ≃ 0) Therefore this hydrodynamic modelprovides a unification of matter (120588 prop 119886

minus3) and dark energy(120588 = 120588

Λ) in the late universe [25ndash36]

In [34ndash36] we remarked that a constant pressure 119901 =

minus120588Λ1198882 has the same virtues as the Chaplygin gas model

(Actually this is a particular case of the generalized Chap-lygin gas model corresponding to 119861 = 0 and 119886 = 0) Thisequation of state may be viewed as a generalized polytropicequation of state (1) with polytropic index 119899 = minus1 polytropicconstant 119896 = minus120588

Λ and linear coefficient 120572 = 0 At late

times this equation of state leads to a constant energy densityequal to 120588

Λimplying an exponential growth of the scale factor

(late inflation) At earlier times this equation of state returnsthe results of the dust matter model (119901 ≃ 0) Thereforeas any Chaplygin gas model with 120572 = 0 and 119899 lt 0 thisequation of state provides a unification of matter (120588 prop 119886

minus3)and dark energy (120588 = 120588

Λ) in the late universe Moreover

this equation of state gives the same results as the standardΛCDM model where matter and dark energy are treatedas independent species This is interesting since the ΛCDM

model is consistent with observations and provides a gooddescription of the late universe (By contrast the Chaplygingas model is not consistent with observations unless 119886 isclose to zero [37] Therefore observations tend to select theconstant pressure equation of state 119901 = minus120588

Λ1198882 (119886 = 0 119899 = minus1)

among the whole family of Chaplygin gas models of the form119901 = minus119860120588

119886) In [34ndash36] we also proposed to describe thetransition between the vacuum energy era and the radiationera in the early universe by a unique equation of state of theform 119901119888

2= minus4120588

23120588119875+ 1205883 This equation of state may be

viewed as a generalized polytropic equation of state (1) withpolytropic index 119899 = 1 polytropic constant 119896 = minus4(3120588

119875)

and linear coefficient 120572 = 13 At early times this equation ofstate leads to a constant energy density equal to 120588

119875implying

an exponential growth of the scale factor (early inflation) Atlater times and for 120572 = 13 this equation of state returnsthe results of the radiation model (119901 = 120588119888

23) Therefore

this hydrodynamic model provides a unification of vacuumenergy (120588 = 120588

119875) and radiation (120588 prop 119886

minus4) in the early universeThis equation of state may be viewed as an extension of theChaplygin (or polytropic) gas model to the early universeIn this approach the late universe is described by negativepolytropic indices 119899 lt 0 (eg 119899 = minus1) and the early universeby positive polytropic indices 119899 gt 0 (eg 119899 = +1) [34ndash36]Indeed for 119899 lt 0 the polytropic equation of state dominatesin the late universe where the density is low and for 119899 gt 0 thepolytropic equation of state dominates in the early universewhere the density is high

In the viewpoint just exposed one tries to ldquounifyrdquo vacuumenergy + radiation on the one hand and dust matter +dark energy on the other hand (It is oftentimes arguedthat the dark energy (cosmological constant) corresponds tothe vacuum energy This leads to the so-called cosmologicalproblem [38] since the cosmological density 120588

Λ= 702 sdot

10minus24 gm3 and the Planck density 120588

119875= 516sdot10

99 gm3 differby about 122 orders ofmagnitudeWe think that it is amistaketo identify the dark energy with the vacuum energy In thispaper we shall regard the vacuum energy and the dark energyas two distinct entities We shall call vacuum energy theenergy associatedwith the Planck density and dark energy theenergy associated with the cosmological densityThe vacuumenergy is responsible for the inflation in the early universe andthe dark energy for the inflation in the late universe In thisviewpoint the vacuum energy is due to quantum mechanicsand the dark energy is an effect of general relativity Thecosmological constant Λ is interpreted as a fundamentalconstant of nature applying to the cosmophysics in the sameway the Planck constant ℎ applies to the microphysics) Inthis paper we adopt a different point of view We proposethat vacuum energy radiation and dark energy may be themanifestation of a unique form of ldquogeneralized radiationrdquoWepropose to describe it by a quadratic equation of state

119901 = minus41205882

3120588119875

1198882+1

31205881198882minus4

3120588Λ1198882 (2)

involving the Planck density and the cosmological density Inthe early universe we recover the equation of state 119901119888

2=

minus412058823120588119875+ 1205883 unifying vacuum energy and radiation In










Λin the





1198891199042= 11988821198891199052minus 119886(119905)

2

1198891199032

1 minus 1198961199032+ 1199032(1198891205792+ sin21205791198891206012) (3)


0= 119888119867

0



119886(119905)1198860 It satisfies 119886






119895is

1198790

0= 1205881198882 119879

1

1= 1198792

2= 1198793

3= minus119901 (4)


119877119894

119895minus1

2119892119894

119895119877 minus Λ119892

119894

119895= minus

8120587119866

1198882119879119894

119895(5)




8120587119866120588 + Λ = 31198862+ 1198961198882

1198862

8120587119866

1198882119901 minus Λ = minus

2119886 119886 + 1198862+ 1198961198882

1198862

(6)



119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (7)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) +

Λ

3 (8)

1198672= (

119886

119886)

2

=8120587119866

3120588 minus

1198961198882

1198862+Λ

3 (9)





119886119889120588

119889119886= minus3 (120588 +

119901

1198882) (10)

119889

119889119886(1205881198863) = minus3

119901

11988821198862 (11)

1198863 119889119901

119889119905=

119889

119889119905[1198863(119901 + 120588119888

2)] (12)








119889119901

119889119879=

1

119879(1205881198882+ 119901) (13)



119878 = 0 where 119878 =1198863

119879(119901 + 120588119888

2) (14)




119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (15)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) (16)

1198672= (

119886

119886)

2

=8120587119866

3120588 (17)


119902 (119905) = minus119886119886

1198862 (18)



119902 (119905) =1 + 3119908 (119905)

2 (19)





119901 =1

31205881198882minus41205882

3120588119875

1198882 (20)

where 120588119875= 11988851198662ℎ = 516 sdot 10


119875)



119875 we get 119901 rarr minus120588

1198751198882


120588 =120588119875

(1198861198861)4

+ 1 (21)



2 and 120588 = 120588119875) and


1198751198864

11198864) It pro-


1

120588=

1

120588rad+

1

120588119875

(22)

At 119886 = 1198861we have 120588rad = 120588

119875so that 120588 = 120588

1198752



119908 =1

3minus

4120588

3120588119875

119902 = 1 minus 2120588

120588119875

(23)


119904= 1199011015840(120588) is given by

1198882

119904

1198882=1

3minus8

3

120588

120588119875

(24)





minus73 to 13



120588 = 120588max = 120588119875 (25)


119875=





12119905minus1


Planck time 119905119875= 1(119866120588

119875)12

= (ℎ1198661198885)12 Numerically119867 =



119886 (119905) sim 119897119875119890(81205873)

12119905119905119875 (26)





= 119888119905119875

=

(119866ℎ1198883)12




119875


tion

120588rad sim1205881198751198864

1

1198864 (27)



4 implies that1205881198751198864

1= 120588rad0119886

4


and 1198860

= 1198881198670




= 441 sdot 1017 s)


2

08120587119866 = 920 sdot




1198861= 261 sdot 10

minus6m (28)



29 and11988611198860= 197 sdot 10



yields

119886

1198861

sim (32120587

3)

14

(119905

119905119875

)

12

(29)

We then have120588

120588119875

sim3

32120587(119905119875

119905)

2

(30)



radic(1198861198861)4

+ 1 minus ln(1 + radic(119886119886

1)4

+ 1

(1198861198861)2

)

= 2(8120587

3)

12119905

119905119875

+ 119862

(31)


119875at 119905 = 0 Setting 120598 = 119897

1198751198861 we get


1205982) (32)



119886 (119905) sim 1198861119890(81205873)

12119905119905119875+119863

(33)


119888(ie


119888(ie 120588 lt 120588

119888) where

119886119888= 1198861and 120588119888= 1205881198752The time 119905




119875


119888


(119905119888= 1199051) Numerically 120588

119888= 1205881= 05 120588

119875= 258 sdot 10

99 gm3119886119888= 1198861= 261 sdot 10

minus6m and 119905119888= 1199051= 233 119905

119875= 125 sdot 10

minus42 sAt 119905 = 119905

119894= 0 119886

119894= 119897119875and 120588

119894≃ 120588119875 Numerically 119886

119894= 119897119875=


119894≃ 120588119875= 516 sdot 10

99 gm3


119879 = 119879119875(15

1205872)

14

(1 minus120588

120588119875

)

74

(120588

120588119875

)

14

(34)


119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (35)


119875 it


119879 = 119879119875(15

1205872)

14

(120588

120588119875

)

14

(36)



51198661198962

119861)12

= 142 sdot 1032 K



119879

119879119875

≃ (15

1205872)

14

(1 minus120588

120588119875

)

74

(37)

According to (35) 119879 sim 119879119875(15120587

2)14

(1198861198861)7 for 119886 ≪ 119886

1and

119879 sim 119879119875(15120587

2)14

(1198861119886) for 119886 ≫ 119886

1


119879

119879119875

sim (15

1205872)

14

(119897119875

1198861

)

7

1198907(81205873)

12119905119905119875 (38)



119879

119879119875

sim (45

321205873)

14

(119905119875

119905)

12

(39)


119890=

1205881198758 119886119890= 714

1198861 119879119890= (78)

74(158120587

2)14


at a time 119905119890= (332120587)


119875

Numerically 120588119890= 0125 120588

119875= 644 sdot 10

98 gm3 119886119890= 163119886

1=


119890= 0523 119879

119875= 740sdot10

31 K and 119905119890= 236 119905

119875=

127 sdot 10minus42 s At 119905 = 119905

119894= 0 119879

119894≃ 119879119875(15120587

2)14

(1198971198751198861)7


minus205119879119875


119905 = 119905119888= 1199051 119879119888= (14)(15120587

2)14

119879119875 Numerically 119879

119888= 1198791=

0278 119879119875= 393 sdot 10


119901 =1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882 (40)





119894≃ minus1205881198751198882 Numeri-



119908= 314

1198861 120588119908= 1205881198754 119879119908= (916)(5120587

2)14

119879119875



119862]119905119875 Numerically 119886

119908= 132119886

1= 343sdot10

minus6m120588119908= 025120588

119875=

129 sdot 1099 gm3 119879

119908= 0475 119879

119875= 674 sdot 10

31 K and 119905119908

=

234119905119875= 126sdot10




114 gms2 At 119905 = 119905119888= 1199051 we have 119901

119888= 1199011=


119888= 1199011= minus773 sdot 10


given by

119878 =4

3(1205872

15)

14

1205881198751198882

119879119875

1198863

1 (41)


1087


119875= 516 sdot 10


1198751198882= minus464sdot10


119875= 162 sdot 10






120588119875= 516 sdot 10

99 gm3 119879119894= 391 sdot 10

minus205119879119875= 554 sdot 10

minus173 K






119894= 0

and ends at 1199051= 233 119905

119875= 125sdot10


119894= 119897119875=


1= 261sdot10


119894= 391sdot10

minus205119879119875= 554sdot10

minus173 K to1198791= 0278 119879

119875=



99 gm3and 119901

119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 to 1205881= 05 120588

119875=

258 sdot 1099 gm3 and 119901

1= minus0167120588

1198751198882= minus773 sdot 10


= 234119905119875


119908=

025120588119875= 129 sdot 10

99 gm3 119886119908= 132 119886

1= 343 sdot 10

minus6m and119879119908= 0475 119879

119875= 674sdot10





119875=

740 sdot 1031 K and 119901

119890= 208 sdot 10

minus21205881198751198882= 966 sdot 10

114 gms2at 119905 = 119905

119890= 236 119905

119875= 127 sdot 10


119890= 0125 120588

119875= 644 sdot 10

98 gm3 and the radius119886119890= 163 119886

1= 424sdot10


119888= 1199051= 233 119905

119875= 125 sdot 10







119891with 119905

119891= 10minus33 s For 119905 gt


119894 the





0

20

Inflation

Radiation era

minus20


simlPsimtP

sima1

log [t (s)]

log [a

(m)]


0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1


119875= 162 sdot 10




119888minus 119879)12) but



0

50

100


simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]




0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P


119875=

516 sdot 1099 gm3


119875gt 0 at 119905 = 0 instead of 119886 = 0



0= 105 sdot 10



1


0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics













Λin the





1198891199042= 11988821198891199052minus 119886(119905)

2

1198891199032

1 minus 1198961199032+ 1199032(1198891205792+ sin21205791198891206012) (3)


0= 119888119867

0



119886(119905)1198860 It satisfies 119886






119895is

1198790

0= 1205881198882 119879

1

1= 1198792

2= 1198793

3= minus119901 (4)


119877119894

119895minus1

2119892119894

119895119877 minus Λ119892

119894

119895= minus

8120587119866

1198882119879119894

119895(5)




8120587119866120588 + Λ = 31198862+ 1198961198882

1198862

8120587119866

1198882119901 minus Λ = minus

2119886 119886 + 1198862+ 1198961198882

1198862

(6)



119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (7)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) +

Λ

3 (8)

1198672= (

119886

119886)

2

=8120587119866

3120588 minus

1198961198882

1198862+Λ

3 (9)





119886119889120588

119889119886= minus3 (120588 +

119901

1198882) (10)

119889

119889119886(1205881198863) = minus3

119901

11988821198862 (11)

1198863 119889119901

119889119905=

119889

119889119905[1198863(119901 + 120588119888

2)] (12)








119889119901

119889119879=

1

119879(1205881198882+ 119901) (13)



119878 = 0 where 119878 =1198863

119879(119901 + 120588119888

2) (14)




119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (15)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) (16)

1198672= (

119886

119886)

2

=8120587119866

3120588 (17)


119902 (119905) = minus119886119886

1198862 (18)



119902 (119905) =1 + 3119908 (119905)

2 (19)





119901 =1

31205881198882minus41205882

3120588119875

1198882 (20)

where 120588119875= 11988851198662ℎ = 516 sdot 10


119875)



119875 we get 119901 rarr minus120588

1198751198882


120588 =120588119875

(1198861198861)4

+ 1 (21)



2 and 120588 = 120588119875) and


1198751198864

11198864) It pro-


1

120588=

1

120588rad+

1

120588119875

(22)

At 119886 = 1198861we have 120588rad = 120588

119875so that 120588 = 120588

1198752



119908 =1

3minus

4120588

3120588119875

119902 = 1 minus 2120588

120588119875

(23)


119904= 1199011015840(120588) is given by

1198882

119904

1198882=1

3minus8

3

120588

120588119875

(24)





minus73 to 13



120588 = 120588max = 120588119875 (25)


119875=





12119905minus1


Planck time 119905119875= 1(119866120588

119875)12

= (ℎ1198661198885)12 Numerically119867 =



119886 (119905) sim 119897119875119890(81205873)

12119905119905119875 (26)





= 119888119905119875

=

(119866ℎ1198883)12




119875


tion

120588rad sim1205881198751198864

1

1198864 (27)



4 implies that1205881198751198864

1= 120588rad0119886

4


and 1198860

= 1198881198670




= 441 sdot 1017 s)


2

08120587119866 = 920 sdot




1198861= 261 sdot 10

minus6m (28)



29 and11988611198860= 197 sdot 10



yields

119886

1198861

sim (32120587

3)

14

(119905

119905119875

)

12

(29)

We then have120588

120588119875

sim3

32120587(119905119875

119905)

2

(30)



radic(1198861198861)4

+ 1 minus ln(1 + radic(119886119886

1)4

+ 1

(1198861198861)2

)

= 2(8120587

3)

12119905

119905119875

+ 119862

(31)


119875at 119905 = 0 Setting 120598 = 119897

1198751198861 we get


1205982) (32)



119886 (119905) sim 1198861119890(81205873)

12119905119905119875+119863

(33)


119888(ie


119888(ie 120588 lt 120588

119888) where

119886119888= 1198861and 120588119888= 1205881198752The time 119905




119875


119888


(119905119888= 1199051) Numerically 120588

119888= 1205881= 05 120588

119875= 258 sdot 10

99 gm3119886119888= 1198861= 261 sdot 10

minus6m and 119905119888= 1199051= 233 119905

119875= 125 sdot 10

minus42 sAt 119905 = 119905

119894= 0 119886

119894= 119897119875and 120588

119894≃ 120588119875 Numerically 119886

119894= 119897119875=


119894≃ 120588119875= 516 sdot 10

99 gm3


119879 = 119879119875(15

1205872)

14

(1 minus120588

120588119875

)

74

(120588

120588119875

)

14

(34)


119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (35)


119875 it


119879 = 119879119875(15

1205872)

14

(120588

120588119875

)

14

(36)



51198661198962

119861)12

= 142 sdot 1032 K



119879

119879119875

≃ (15

1205872)

14

(1 minus120588

120588119875

)

74

(37)

According to (35) 119879 sim 119879119875(15120587

2)14

(1198861198861)7 for 119886 ≪ 119886

1and

119879 sim 119879119875(15120587

2)14

(1198861119886) for 119886 ≫ 119886

1


119879

119879119875

sim (15

1205872)

14

(119897119875

1198861

)

7

1198907(81205873)

12119905119905119875 (38)



119879

119879119875

sim (45

321205873)

14

(119905119875

119905)

12

(39)


119890=

1205881198758 119886119890= 714

1198861 119879119890= (78)

74(158120587

2)14


at a time 119905119890= (332120587)


119875

Numerically 120588119890= 0125 120588

119875= 644 sdot 10

98 gm3 119886119890= 163119886

1=


119890= 0523 119879

119875= 740sdot10

31 K and 119905119890= 236 119905

119875=

127 sdot 10minus42 s At 119905 = 119905

119894= 0 119879

119894≃ 119879119875(15120587

2)14

(1198971198751198861)7


minus205119879119875


119905 = 119905119888= 1199051 119879119888= (14)(15120587

2)14

119879119875 Numerically 119879

119888= 1198791=

0278 119879119875= 393 sdot 10


119901 =1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882 (40)





119894≃ minus1205881198751198882 Numeri-



119908= 314

1198861 120588119908= 1205881198754 119879119908= (916)(5120587

2)14

119879119875



119862]119905119875 Numerically 119886

119908= 132119886

1= 343sdot10

minus6m120588119908= 025120588

119875=

129 sdot 1099 gm3 119879

119908= 0475 119879

119875= 674 sdot 10

31 K and 119905119908

=

234119905119875= 126sdot10




114 gms2 At 119905 = 119905119888= 1199051 we have 119901

119888= 1199011=


119888= 1199011= minus773 sdot 10


given by

119878 =4

3(1205872

15)

14

1205881198751198882

119879119875

1198863

1 (41)


1087


119875= 516 sdot 10


1198751198882= minus464sdot10


119875= 162 sdot 10






120588119875= 516 sdot 10

99 gm3 119879119894= 391 sdot 10

minus205119879119875= 554 sdot 10

minus173 K






119894= 0

and ends at 1199051= 233 119905

119875= 125sdot10


119894= 119897119875=


1= 261sdot10


119894= 391sdot10

minus205119879119875= 554sdot10

minus173 K to1198791= 0278 119879

119875=



99 gm3and 119901

119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 to 1205881= 05 120588

119875=

258 sdot 1099 gm3 and 119901

1= minus0167120588

1198751198882= minus773 sdot 10


= 234119905119875


119908=

025120588119875= 129 sdot 10

99 gm3 119886119908= 132 119886

1= 343 sdot 10

minus6m and119879119908= 0475 119879

119875= 674sdot10





119875=

740 sdot 1031 K and 119901

119890= 208 sdot 10

minus21205881198751198882= 966 sdot 10

114 gms2at 119905 = 119905

119890= 236 119905

119875= 127 sdot 10


119890= 0125 120588

119875= 644 sdot 10

98 gm3 and the radius119886119890= 163 119886

1= 424sdot10


119888= 1199051= 233 119905

119875= 125 sdot 10







119891with 119905

119891= 10minus33 s For 119905 gt


119894 the





0

20

Inflation

Radiation era

minus20


simlPsimtP

sima1

log [t (s)]

log [a

(m)]


0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1


119875= 162 sdot 10




119888minus 119879)12) but



0

50

100


simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]




0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P


119875=

516 sdot 1099 gm3


119875gt 0 at 119905 = 0 instead of 119886 = 0



0= 105 sdot 10



1


0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics









119889119901

119889119879=

1

119879(1205881198882+ 119901) (13)



119878 = 0 where 119878 =1198863

119879(119901 + 120588119888

2) (14)




119889120588

119889119905+ 3

119886

119886(120588 +

119901

1198882) = 0 (15)

119886

119886= minus

4120587119866

3(120588 +

3119901

1198882) (16)

1198672= (

119886

119886)

2

=8120587119866

3120588 (17)


119902 (119905) = minus119886119886

1198862 (18)



119902 (119905) =1 + 3119908 (119905)

2 (19)





119901 =1

31205881198882minus41205882

3120588119875

1198882 (20)

where 120588119875= 11988851198662ℎ = 516 sdot 10


119875)



119875 we get 119901 rarr minus120588

1198751198882


120588 =120588119875

(1198861198861)4

+ 1 (21)



2 and 120588 = 120588119875) and


1198751198864

11198864) It pro-


1

120588=

1

120588rad+

1

120588119875

(22)

At 119886 = 1198861we have 120588rad = 120588

119875so that 120588 = 120588

1198752



119908 =1

3minus

4120588

3120588119875

119902 = 1 minus 2120588

120588119875

(23)


119904= 1199011015840(120588) is given by

1198882

119904

1198882=1

3minus8

3

120588

120588119875

(24)





minus73 to 13



120588 = 120588max = 120588119875 (25)


119875=





12119905minus1


Planck time 119905119875= 1(119866120588

119875)12

= (ℎ1198661198885)12 Numerically119867 =



119886 (119905) sim 119897119875119890(81205873)

12119905119905119875 (26)





= 119888119905119875

=

(119866ℎ1198883)12




119875


tion

120588rad sim1205881198751198864

1

1198864 (27)



4 implies that1205881198751198864

1= 120588rad0119886

4


and 1198860

= 1198881198670




= 441 sdot 1017 s)


2

08120587119866 = 920 sdot




1198861= 261 sdot 10

minus6m (28)



29 and11988611198860= 197 sdot 10



yields

119886

1198861

sim (32120587

3)

14

(119905

119905119875

)

12

(29)

We then have120588

120588119875

sim3

32120587(119905119875

119905)

2

(30)



radic(1198861198861)4

+ 1 minus ln(1 + radic(119886119886

1)4

+ 1

(1198861198861)2

)

= 2(8120587

3)

12119905

119905119875

+ 119862

(31)


119875at 119905 = 0 Setting 120598 = 119897

1198751198861 we get


1205982) (32)



119886 (119905) sim 1198861119890(81205873)

12119905119905119875+119863

(33)


119888(ie


119888(ie 120588 lt 120588

119888) where

119886119888= 1198861and 120588119888= 1205881198752The time 119905




119875


119888


(119905119888= 1199051) Numerically 120588

119888= 1205881= 05 120588

119875= 258 sdot 10

99 gm3119886119888= 1198861= 261 sdot 10

minus6m and 119905119888= 1199051= 233 119905

119875= 125 sdot 10

minus42 sAt 119905 = 119905

119894= 0 119886

119894= 119897119875and 120588

119894≃ 120588119875 Numerically 119886

119894= 119897119875=


119894≃ 120588119875= 516 sdot 10

99 gm3


119879 = 119879119875(15

1205872)

14

(1 minus120588

120588119875

)

74

(120588

120588119875

)

14

(34)


119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (35)


119875 it


119879 = 119879119875(15

1205872)

14

(120588

120588119875

)

14

(36)



51198661198962

119861)12

= 142 sdot 1032 K



119879

119879119875

≃ (15

1205872)

14

(1 minus120588

120588119875

)

74

(37)

According to (35) 119879 sim 119879119875(15120587

2)14

(1198861198861)7 for 119886 ≪ 119886

1and

119879 sim 119879119875(15120587

2)14

(1198861119886) for 119886 ≫ 119886

1


119879

119879119875

sim (15

1205872)

14

(119897119875

1198861

)

7

1198907(81205873)

12119905119905119875 (38)



119879

119879119875

sim (45

321205873)

14

(119905119875

119905)

12

(39)


119890=

1205881198758 119886119890= 714

1198861 119879119890= (78)

74(158120587

2)14


at a time 119905119890= (332120587)


119875

Numerically 120588119890= 0125 120588

119875= 644 sdot 10

98 gm3 119886119890= 163119886

1=


119890= 0523 119879

119875= 740sdot10

31 K and 119905119890= 236 119905

119875=

127 sdot 10minus42 s At 119905 = 119905

119894= 0 119879

119894≃ 119879119875(15120587

2)14

(1198971198751198861)7


minus205119879119875


119905 = 119905119888= 1199051 119879119888= (14)(15120587

2)14

119879119875 Numerically 119879

119888= 1198791=

0278 119879119875= 393 sdot 10


119901 =1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882 (40)





119894≃ minus1205881198751198882 Numeri-



119908= 314

1198861 120588119908= 1205881198754 119879119908= (916)(5120587

2)14

119879119875



119862]119905119875 Numerically 119886

119908= 132119886

1= 343sdot10

minus6m120588119908= 025120588

119875=

129 sdot 1099 gm3 119879

119908= 0475 119879

119875= 674 sdot 10

31 K and 119905119908

=

234119905119875= 126sdot10




114 gms2 At 119905 = 119905119888= 1199051 we have 119901

119888= 1199011=


119888= 1199011= minus773 sdot 10


given by

119878 =4

3(1205872

15)

14

1205881198751198882

119879119875

1198863

1 (41)


1087


119875= 516 sdot 10


1198751198882= minus464sdot10


119875= 162 sdot 10






120588119875= 516 sdot 10

99 gm3 119879119894= 391 sdot 10

minus205119879119875= 554 sdot 10

minus173 K






119894= 0

and ends at 1199051= 233 119905

119875= 125sdot10


119894= 119897119875=


1= 261sdot10


119894= 391sdot10

minus205119879119875= 554sdot10

minus173 K to1198791= 0278 119879

119875=



99 gm3and 119901

119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 to 1205881= 05 120588

119875=

258 sdot 1099 gm3 and 119901

1= minus0167120588

1198751198882= minus773 sdot 10


= 234119905119875


119908=

025120588119875= 129 sdot 10

99 gm3 119886119908= 132 119886

1= 343 sdot 10

minus6m and119879119908= 0475 119879

119875= 674sdot10





119875=

740 sdot 1031 K and 119901

119890= 208 sdot 10

minus21205881198751198882= 966 sdot 10

114 gms2at 119905 = 119905

119890= 236 119905

119875= 127 sdot 10


119890= 0125 120588

119875= 644 sdot 10

98 gm3 and the radius119886119890= 163 119886

1= 424sdot10


119888= 1199051= 233 119905

119875= 125 sdot 10







119891with 119905

119891= 10minus33 s For 119905 gt


119894 the





0

20

Inflation

Radiation era

minus20


simlPsimtP

sima1

log [t (s)]

log [a

(m)]


0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1


119875= 162 sdot 10




119888minus 119879)12) but



0

50

100


simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]




0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P


119875=

516 sdot 1099 gm3


119875gt 0 at 119905 = 0 instead of 119886 = 0



0= 105 sdot 10



1


0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics








= 119888119905119875

=

(119866ℎ1198883)12




119875


tion

120588rad sim1205881198751198864

1

1198864 (27)



4 implies that1205881198751198864

1= 120588rad0119886

4


and 1198860

= 1198881198670




= 441 sdot 1017 s)


2

08120587119866 = 920 sdot




1198861= 261 sdot 10

minus6m (28)



29 and11988611198860= 197 sdot 10



yields

119886

1198861

sim (32120587

3)

14

(119905

119905119875

)

12

(29)

We then have120588

120588119875

sim3

32120587(119905119875

119905)

2

(30)



radic(1198861198861)4

+ 1 minus ln(1 + radic(119886119886

1)4

+ 1

(1198861198861)2

)

= 2(8120587

3)

12119905

119905119875

+ 119862

(31)


119875at 119905 = 0 Setting 120598 = 119897

1198751198861 we get


1205982) (32)



119886 (119905) sim 1198861119890(81205873)

12119905119905119875+119863

(33)


119888(ie


119888(ie 120588 lt 120588

119888) where

119886119888= 1198861and 120588119888= 1205881198752The time 119905




119875


119888


(119905119888= 1199051) Numerically 120588

119888= 1205881= 05 120588

119875= 258 sdot 10

99 gm3119886119888= 1198861= 261 sdot 10

minus6m and 119905119888= 1199051= 233 119905

119875= 125 sdot 10

minus42 sAt 119905 = 119905

119894= 0 119886

119894= 119897119875and 120588

119894≃ 120588119875 Numerically 119886

119894= 119897119875=


119894≃ 120588119875= 516 sdot 10

99 gm3


119879 = 119879119875(15

1205872)

14

(1 minus120588

120588119875

)

74

(120588

120588119875

)

14

(34)


119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (35)


119875 it


119879 = 119879119875(15

1205872)

14

(120588

120588119875

)

14

(36)



51198661198962

119861)12

= 142 sdot 1032 K



119879

119879119875

≃ (15

1205872)

14

(1 minus120588

120588119875

)

74

(37)

According to (35) 119879 sim 119879119875(15120587

2)14

(1198861198861)7 for 119886 ≪ 119886

1and

119879 sim 119879119875(15120587

2)14

(1198861119886) for 119886 ≫ 119886

1


119879

119879119875

sim (15

1205872)

14

(119897119875

1198861

)

7

1198907(81205873)

12119905119905119875 (38)



119879

119879119875

sim (45

321205873)

14

(119905119875

119905)

12

(39)


119890=

1205881198758 119886119890= 714

1198861 119879119890= (78)

74(158120587

2)14


at a time 119905119890= (332120587)


119875

Numerically 120588119890= 0125 120588

119875= 644 sdot 10

98 gm3 119886119890= 163119886

1=


119890= 0523 119879

119875= 740sdot10

31 K and 119905119890= 236 119905

119875=

127 sdot 10minus42 s At 119905 = 119905

119894= 0 119879

119894≃ 119879119875(15120587

2)14

(1198971198751198861)7


minus205119879119875


119905 = 119905119888= 1199051 119879119888= (14)(15120587

2)14

119879119875 Numerically 119879

119888= 1198791=

0278 119879119875= 393 sdot 10


119901 =1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882 (40)





119894≃ minus1205881198751198882 Numeri-



119908= 314

1198861 120588119908= 1205881198754 119879119908= (916)(5120587

2)14

119879119875



119862]119905119875 Numerically 119886

119908= 132119886

1= 343sdot10

minus6m120588119908= 025120588

119875=

129 sdot 1099 gm3 119879

119908= 0475 119879

119875= 674 sdot 10

31 K and 119905119908

=

234119905119875= 126sdot10




114 gms2 At 119905 = 119905119888= 1199051 we have 119901

119888= 1199011=


119888= 1199011= minus773 sdot 10


given by

119878 =4

3(1205872

15)

14

1205881198751198882

119879119875

1198863

1 (41)


1087


119875= 516 sdot 10


1198751198882= minus464sdot10


119875= 162 sdot 10






120588119875= 516 sdot 10

99 gm3 119879119894= 391 sdot 10

minus205119879119875= 554 sdot 10

minus173 K






119894= 0

and ends at 1199051= 233 119905

119875= 125sdot10


119894= 119897119875=


1= 261sdot10


119894= 391sdot10

minus205119879119875= 554sdot10

minus173 K to1198791= 0278 119879

119875=



99 gm3and 119901

119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 to 1205881= 05 120588

119875=

258 sdot 1099 gm3 and 119901

1= minus0167120588

1198751198882= minus773 sdot 10


= 234119905119875


119908=

025120588119875= 129 sdot 10

99 gm3 119886119908= 132 119886

1= 343 sdot 10

minus6m and119879119908= 0475 119879

119875= 674sdot10





119875=

740 sdot 1031 K and 119901

119890= 208 sdot 10

minus21205881198751198882= 966 sdot 10

114 gms2at 119905 = 119905

119890= 236 119905

119875= 127 sdot 10


119890= 0125 120588

119875= 644 sdot 10

98 gm3 and the radius119886119890= 163 119886

1= 424sdot10


119888= 1199051= 233 119905

119875= 125 sdot 10







119891with 119905

119891= 10minus33 s For 119905 gt


119894 the





0

20

Inflation

Radiation era

minus20


simlPsimtP

sima1

log [t (s)]

log [a

(m)]


0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1


119875= 162 sdot 10




119888minus 119879)12) but



0

50

100


simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]




0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P


119875=

516 sdot 1099 gm3


119875gt 0 at 119905 = 0 instead of 119886 = 0



0= 105 sdot 10



1


0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics






119879

119879119875

sim (45

321205873)

14

(119905119875

119905)

12

(39)


119890=

1205881198758 119886119890= 714

1198861 119879119890= (78)

74(158120587

2)14


at a time 119905119890= (332120587)


119875

Numerically 120588119890= 0125 120588

119875= 644 sdot 10

98 gm3 119886119890= 163119886

1=


119890= 0523 119879

119875= 740sdot10

31 K and 119905119890= 236 119905

119875=

127 sdot 10minus42 s At 119905 = 119905

119894= 0 119879

119894≃ 119879119875(15120587

2)14

(1198971198751198861)7


minus205119879119875


119905 = 119905119888= 1199051 119879119888= (14)(15120587

2)14

119879119875 Numerically 119879

119888= 1198791=

0278 119879119875= 393 sdot 10


119901 =1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882 (40)





119894≃ minus1205881198751198882 Numeri-



119908= 314

1198861 120588119908= 1205881198754 119879119908= (916)(5120587

2)14

119879119875



119862]119905119875 Numerically 119886

119908= 132119886

1= 343sdot10

minus6m120588119908= 025120588

119875=

129 sdot 1099 gm3 119879

119908= 0475 119879

119875= 674 sdot 10

31 K and 119905119908

=

234119905119875= 126sdot10




114 gms2 At 119905 = 119905119888= 1199051 we have 119901

119888= 1199011=


119888= 1199011= minus773 sdot 10


given by

119878 =4

3(1205872

15)

14

1205881198751198882

119879119875

1198863

1 (41)


1087


119875= 516 sdot 10


1198751198882= minus464sdot10


119875= 162 sdot 10






120588119875= 516 sdot 10

99 gm3 119879119894= 391 sdot 10

minus205119879119875= 554 sdot 10

minus173 K






119894= 0

and ends at 1199051= 233 119905

119875= 125sdot10


119894= 119897119875=


1= 261sdot10


119894= 391sdot10

minus205119879119875= 554sdot10

minus173 K to1198791= 0278 119879

119875=



99 gm3and 119901

119894≃ minus1205881198751198882= minus464 sdot 10

116 gms2 to 1205881= 05 120588

119875=

258 sdot 1099 gm3 and 119901

1= minus0167120588

1198751198882= minus773 sdot 10


= 234119905119875


119908=

025120588119875= 129 sdot 10

99 gm3 119886119908= 132 119886

1= 343 sdot 10

minus6m and119879119908= 0475 119879

119875= 674sdot10





119875=

740 sdot 1031 K and 119901

119890= 208 sdot 10

minus21205881198751198882= 966 sdot 10

114 gms2at 119905 = 119905

119890= 236 119905

119875= 127 sdot 10


119890= 0125 120588

119875= 644 sdot 10

98 gm3 and the radius119886119890= 163 119886

1= 424sdot10


119888= 1199051= 233 119905

119875= 125 sdot 10







119891with 119905

119891= 10minus33 s For 119905 gt


119894 the





0

20

Inflation

Radiation era

minus20


simlPsimtP

sima1

log [t (s)]

log [a

(m)]


0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1


119875= 162 sdot 10




119888minus 119879)12) but



0

50

100


simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]




0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P


119875=

516 sdot 1099 gm3


119875gt 0 at 119905 = 0 instead of 119886 = 0



0= 105 sdot 10



1


0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics





0

20

Inflation

Radiation era

minus20


simlPsimtP

sima1

log [t (s)]

log [a

(m)]


0 10 20 30 40 50 60 70

0

5

10

15

minus10

aa

1

ttP

lP t1a1


119875= 162 sdot 10




119888minus 119879)12) but



0

50

100


simtP

log [t (s)]

sim120588P

log [120588

(gm

3)]




0

02

04

06

08

1

0 10 20 30 40 50 60 70minus10

t1

ttP

120588120588

P

sim120588P


119875=

516 sdot 1099 gm3


119875gt 0 at 119905 = 0 instead of 119886 = 0



0= 105 sdot 10



1


0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics





0

minus60minus200

minus150

minus100


log [T

(K)] minus50

simtP

simTP

sim(lPa1)7TP





0

01

02

03

04

05

t1

simTP

sim(lPa1)7TP

0 10 20 30 40 50 60 70minus10

ttP

TT

P


119875= 142 sdot 10


We introduce 119886 = 1198861198860

1and = 119905119905

0

119875 Using 119897

119875= 119888119905119875and

1205881198751198864

1= 120588rad0119886

4

0 we obtain

119897119875

1198970119875

=119905119875

1199050119875

= (1198861

11988601

)

2

= ℎ12

(42)


120598

1205980

=1198971198751198861

119897011987511988601

=1198861

11988601

= ℎ14

(43)

0

02

t1

minus02

minus04

minus06

minus08

minus1

0 10 20 30 40 50 60 70minus10

ttP

p120588

Pc2

sim120588Pc2


119875119888248 = 966 sdot



=


where 1205980= 1198970

1198751198860

1= 620sdot10


radic1198864

ℎ+ 1 minus ln(

1 + radic1198864ℎ + 1

1198862ℎ12)

= 2(8120587

3)

12

ℎ12+ 119862 (120598

0ℎ14

)

(44)



120601 + 3119867 120601 +119889119881

119889120601= 0 (45)


1205881198882=1

21206012+ 119881 (120601) 119901 =

1

21206012minus 119881 (120601) (46)


80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics





80 100

0

5

10

15

Vacuum

Radiation

energy

20 40 60minus20 0t(tP)0

h rarr 0

a(a 1) 0



119881 (120595) =1

31205881198751198882 cosh2120595 + 2

cosh4120595(120595 ge 0)

(119886

1198861

)

2

= sinh120595 120595 = (8120587119866

1198882)

12

120601

(47)


120595 sim (119897119875

1198861

)

2

119890radic321205873119905119905

119875 997888rarr 0 (48)


120595 ≃ ln 119905

119905119875

+1

2ln 8120587

3+ 2 ln (2) 997888rarr +infin (49)



) = 2(8120587

3)

12119905

119905119875

+ 119862 (50)




119901 =1

31205881198882minus4

3120588Λ1198882 (51)

0 05 1 15 2 25 30

02

04

06

08

1

120595

V120588

Pc2


0

1

2

3

4

5

6

7

120595

0 10 20 30 40 50 60 70minus10

ttP


where 120588Λ= Λ8120587119866 = 702 sdot 10








120588 = 120588Λ[(

1198863

119886)

4

+ 1] (52)



23 and 120588rad sim 120588

Λ(1198863119886)4)


Λ) It provides




120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics






120588 = 120588rad + 120588Λ (53)

At 119886 = 1198863we have 120588rad = 120588

Λso that 120588 = 2120588

Λ



120588 = 120588min = 120588Λ (54)


sity 120588Λ= Λ8120587119866 = 702 sdot10




Λ)12

=

(8120587Λ)12


Λ= 119888119905Λ=

(81205871198882Λ)12



12119905minus1



119886 (119905) prop 119890(81205873)

12119905119905Λ (55)


Λ= 146 sdot 10





tion

120588rad sim120588Λ1198864

3

1198864 (56)




3= 120588rad0119886

4

0 Using 120588

Λ= ΩΛ0

1205880and 120588rad0 = Ωrad01205880 with

ΩΛ0


[1] we obtain 11988631198860= (Ωrad0ΩΛ0)

14 hence

1198863= 136 sdot 10

25m (57)


0= 309 sdot 10

minus2119897Λwhere



3 the Friedmann


119886

1198863

sim (32120587

3)

14

(119905

119905Λ

)

12

(58)

We then have

120588

120588Λ

sim3

32120587(119905Λ

119905)

2

(59)


119879 = 119879119875(15

1205872)

14

(120588 minus 120588Λ

120588119875

)

14

(60)


119879 = 119879119875(15

1205872)

14

(120588Λ

120588119875

)

141198863

119886 (61)


Λ it reduces




= (ℎ11988851198661198962

119861)12




119875(4532120587

3)14

(120588Λ120588119875)14

(119905Λ119905)12 and


minus(81205873)12119905119905Λ


119901 = [1

3(1198863

119886)

4

minus 1] 120588Λ1198882 (62)





= 1198863314 1205881015840119908

= 4120588Λ 1198791015840119908

= 119879119875(45

1205872)14

(120588Λ120588119875)14 Numerically 1198861015840

119908= 0760119886

3= 103 sdot 10

25m1205881015840

119908= 4120588Λ= 281 sdot 10

minus23 gm3 1198791015840119908= 281 sdot 10

minus31119879119875= 399K

5 The General Model


119901119892= minus

41205882

119892

3120588119875

1198882+1

31205881198921198882minus4

3120588Λ1198882 (63)



Λ≪ 120588119892≪


119892rarr 120588Λ119901119892rarr minus120588

Λ1198882 (dark





120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics







120588119892=

120588119875

(1198861198861)4

+ 1+ 120588Λ (64)

where 1198861= 261 sdot 10


119875≫ 120588Λso that 119901

1198921198882+

120588119892≃ (43120588


119892rarr 120588119875


≪ 120588119892≪ 120588119875 120588119892sim 120588119875(1198861198861)minus4



1198751198864

1=

120588Λ1198864



1= 120588rad0119886

4


form

120588119892=

120588rad0

(1198861198860)4

+ (11988611198860)4+ 120588Λ (65)

where 11988611198860= 197 sdot 10


119861= 0 and


120588119861=

1205881198610

(1198861198860)3 (66a)

120588DM =120588DM0

(1198861198860)3 (66b)



120588 =120588rad0

(1198861198860)4

+ (11988611198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ (67)


11986101205880 120588DM0 = ΩDM01205880 and

120588Λ= ΩΛ0

1205880 we obtain

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(68)


= 00455ΩDM0 = 01915 Ω

1198980= 0237 andΩ

Λ0= 0763 [1]


119879 = 119879119875(15

1205872)

14

(1 minus120588119892

120588119875

)

74

(120588119892

120588119875

minus120588Λ

120588119875

)

14

(69)

Using (64) we get

119879 = 119879119875(15

1205872)

14 (1198861198861)7

[(1198861198861)4

+ 1]2 (70)


119879 = 1198790

(1198861198860)7

[(1198861198860)4

+ (11988611198860)4

]2 (71)

with 1198790= 119879119875(15120587

2)14

(11988611198860) Numerically 119879

0= 311K

Since 1198861



119879 sim 1198790

1198860

119886 (72)


119901119892=1

3

(1198861198861)4

minus 3

[(1198861198861)4

+ 1]21205881198751198882minus 120588Λ1198882 (73)


120588 =120588rad0

(1198861198860)4

+ (11988611198860)4 (74)


119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4 (75)



11198860

= (120588rad0120588119875)14 1205881

= 1205881198752

1198791= (14)(15120587

2)14

119879119875 Numerically 119886

1= 197 sdot 10

minus321198860=


1= 05 120588

119875= 258 sdot 10

99 gm3 1198791= 0278 119879

119875=


1= 233119905

119875= 125 sdot






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics






constant

120588 ≃ 120588119875 (76)


(81205873)12

119905minus1



1) we have

120588 sim120588rad0119886

4

0

1198864 (77)


119867

1198670

= radicΩrad0

(1198861198860)4

(78)

and yields

119886

1198860

sim Ω14

rad0radic21198670119905

120588

1205880

sim1

(21198670119905)2

119879

1198790

sim1198860

119886sim

1

Ω14

rad0radic21198670119905

(79)


120588 =1205881198980

(1198861198860)3+ 120588Λ (80)



119867

1198670

= radicΩ1198980

(1198861198860)3+ ΩΛ0

(81)

withΩ1198980

+ ΩΛ0


119886

1198860

= (Ω1198980

ΩΛ0

)

13


1198670119905) (82)


120588

1205880

=ΩΛ0


1198670119905) (83)


119879 = 119879119875(15

1205872)

14

(120588119892

120588119875

minus120588Λ

120588119875

)

14

119879 = 1198790

1198860

119886

(84)


119879 =1198790

(Ω1198980

ΩΛ0


Λ01198670119905)

(85)


119901119892=1

3

120588rad01198882

(1198861198860)4minus 120588Λ1198882 (86)


1199050=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 [(ΩΛ0

Ω1198980

)

12

] (87)

Numerically 1199050= 103 119867

minus1

0= 0310119905

Λ= 453 sdot 10

17 s sim


0


with 120588 ≃ 120588119898+ 120588Λand 119901 ≃ minus120588

Λ1198882] Using 120588

1198980= Ω1198980

1205880and

120588Λ= ΩΛ0

1205880with Ω

1198980= 0237 (matter) and Ω

Λ0= 0763


119888and 119905 = 119905

1015840

119888with

1198861015840

119888

1198860

= (Ω1198980

2ΩΛ0

)

13

1199051015840

119888=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 ( 1

radic2)

(88)

At that point 120588 = 1205881015840

119888= 3120588Λand 119879 = 119879

1015840

119888= 119879011988601198861015840

119888 Numeri-

cally 1198861015840119888= 0538119886

0= 0162119897

Λ= 711 sdot 10

25m 1199051015840119888= 0504119867

minus1

0=

0152119905Λ

= 222 sdot 1017 s 1205881015840

119888= 3120588Λ


1198791015840

119888= 186119879


1015840

119888


119888


119898= 120588Λ This leads to 119886 = 119886

2and 119905 = 119905

2

with

1198862

1198860

= (Ω1198980

ΩΛ0

)

13

1199052=

1

1198670

2

3

1

radicΩΛ0

sinhminus1 (1) (89)


2= 119879011988601198862 Numeri-

cally 1198862= 0677119886

0= 0204119897

Λ= 895sdot10

25m 1199052= 0674119867

minus1

0=

0203119905Λ= 297 sdot 10

17 s 1205882= 2120588Λ= 140 sdot 10



119888= (12)

131198862= 0794119886

2

Of course the value

1198862= 895 sdot 10

25m (90)



120588 ≃ 120588Λ (91)


1198670radicΩΛ0


119886

1198860

sim (Ω1198980

4ΩΛ0

)

13

119890radicΩΛ01198670119905 (92)



119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics






119879

1198790

sim (4ΩΛ0

Ω1198980

)

13

119890minusradicΩΛ01198670119905 (93)


120588 sim1205881198980

1198863

0

1198863 (94)


119867

1198670

= radicΩ1198980

(1198861198860)3 (95)

and we have

119886

1198860

= Ω13

1198980(3

21198670119905)

23

120588

1205880

=1

((32)1198670119905)2 (96)

119879

1198790

sim1

Ω13

1198980((32)119867

0119905)23

(97)









place at 1198862= 0677119886

0= 0204119897

Λ= 895 sdot 10

25m 1205882= 2120588Λ=

140 sdot 10minus23 gm3 119879

2= 148119879

0= 459K and 119905

2= 0674119867

minus1

0=

0203119905Λ



1015840

119888=

0504119867minus1

0= 0152119905

Λ= 222 sdot 10


119888= 0538119886

0= 0162119897

Λ= 0794119886

2= 711 sdot 10

25m a density1205881015840

119888= 3120588Λ= 211sdot10


0=





Λ= 132 sdot


2= 0204119897

Λ=

895 sdot 1025m (119886

0= 148119886

2) We have 119886

0sim 1198862

sim 119897Λand



Λ0Ω1198980


10 12 14 16 18 2010

20

30

40

50


Matter era

log [a

(m)]

log [t (s)]

a998400c a0

a21Λ


Λ= 146sdot10


1015840


1015840

119888with 119886

1015840

119888= 711 sdot 10



0sim 1198862)

0 02 04 06 080

1

2

3

4

5

6

a998400c

a0a2

1Λ

ttΛ

aa 2



Λ0Ω1198980


ΛΩ119898



10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics





10 12 14 16 18 20

Dark energy

Matter era

era

log [t (s)]

a998400ca0

a21Λ

minus10

minus15

minus20

minus25120588Λ

log [120588

(gm

3)]



bound

0 02 04 06 080

1

2

3

4

5

6

7

8

9

10

a998400c

a0

a2

1Λ

ttΛ

120588120588

Λ


Λ= 702 sdot 10

minus24 gm3

0 02 04 06 080

1

2

3

4

5

6

TT

2

a998400c a2

1Λ

ttΛ

a0


0 02 04 06 08

0

1

a998400c a0a2 1Λ

ttΛ

minus1

pg120588

Λc2


in linear scales


0= 131120588

Λ= 920 sdot


0= 103 119867

minus1

0=

0310119905Λ= 454 sdot 10


Λ0and 119902

0= (1 minus 3Ω

Λ0)2 [since 119901 ≃ minus120588

Λ1198882]


0= minus0645





4 120588119861

=

1205881198610(1198861198860)3 120588DM = 120588DM0(1198861198860)

3 and 120588Λ= 120588Λ0


119861+ 120588DM + 120588

Λ that

is

120588 =120588rad0

(1198861198860)4+

1205881198610

(1198861198860)3+

120588DM0

(1198861198860)3+ 120588Λ0 (98)


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198610

(1198861198860)3+

ΩDM0

(1198861198860)3+ ΩΛ0

(99)











Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics










Ωlowast

1198980= Ω1198980

119867(119886 minus 1198861))

119867

1198670

= radicΩrad0

(1198861198860)4

+ (11988611198860)4+

Ωlowast

1198610

(1198861198860)3+

Ωlowast

DM0

(1198861198860)3+ ΩΛ0

(100)


Λ= 1 For 119886

1= 0 we


11198860= 197 sdot 10


119875






int

1198861198860

1198861198941198860

119889119909(119909radicΩrad0

1199094 + (11988611198860)4+Ωlowast

1198610

1199093+Ωlowast

DM0

1199093+ ΩΛ0

)

minus1

= 1198670119905

(101)


119894= 119897119875in the


1199050=

1

1198670

times int

1

0

119889119909

119909radic(Ωrad01199094) + (Ω

11986101199093) + (ΩDM0119909

3)+ ΩΛ0

(102)





0= 103 119867

minus1

0=


written as 1198670= 2268 ℎ




is ℎ7






119875= 516 sdot 10


Λ= 702 sdot 10


119875and 120588min = 120588

Λthat are fixed



2


DM0(1198861198860)3 in (100) should




20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics





20 30

0

20

40

Radiation era


Early inflation

Late inflation

(vacuum energy)

a0 t0

minus60minus60

minus50 minus40

minus40

minus30 minus20

minus20


log [a

(m)]

simlPsimtP

sima1








lowast


by Ωlowast

DM0[(1198861198860)3∓ (119886BEC1198860)



6 Conclusion







119861+ 120588DM + 120588

Λand solve the


100 20 30 40 50 60

0

50

100

150

Radiation era

Matter era

Early inflation


(vacuum energy)

a0minus50


log [t (s)]

simtP

simtΛ

120588Λ

120588P

log [120588

(gm

3)]





0

50

Early inflation



(vacuum energy)

a0


simtP

minus200

minus150

minus100

minus50

log [T

(K)]

simTP

sim(lPa1)7TP

sim27 K






2120588119875)1198882+ 12058811988823 Similarly the





11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics






11198860

= 197 sdot 10minus32 For 119905 gt 233119905

119875



119892=

minus(43)(1205882

119892120588119875)1198882+ 12058811989211988823 minus 120588





119892+120588119861+120588DM


Appendices



120588 =120588rad0

(1198861198860)4+

1205881198980

(1198861198860)3 (A1)


119898leading to 119886 = 119886

4=


minus41198860= 108 sdot


1198864= 473 sdot 10

22m (A2)


4

1198980Ω3


4= 119879011988601198864 Numerically 120588

4= 103 sdot 10

101205880=

136 sdot 1010120588Λ


4= 279 sdot 10

31198790=

869 sdot 103 K


119867

1198670

= radicΩrad0

(1198861198860)4+

Ω1198980

(1198861198860)3 (A3)


int

1198861198860

0

119889119909

119909radic(Ωrad01199094) + (Ω

11989801199093)

= 1198670119905 (A4)


1198670119905 =

4

3

(Ωrad0)32

(Ω1198980

)2

minus2

3

1

(Ω1198980

)12

(2Ωrad0

Ω1198980

minus119886

1198860

)radicΩrad0

Ω1198980

+119886

1198860

(A5)




32(Ω1198980


4=


0= 164 sdot 10

minus6119905Λ= 239 sdot 10

12 s





119886

1198863

= sinh12 [(321205873

)

12119905

119905Λ

] (B1)




119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics






119881 (120595) =1

3120588Λ1198882(cosh2120595 + 2) (120595 le 0)

(1198863

119886)

2

= minus sinh120595 120595 = (8120587119866

1198882)

12

120601

(B2)


120595 ≃ ln( 119905

119905Λ

) + (1

2) ln(3120587

2) + ln(4

3) (B3)


120595 sim minus2119890minus(321205873)

12119905119905Λ (B4)



sinh (radic(321205873) (119905119905Λ))] (B5)


1and 120582 = 120588

Λ120588119875≪ 1 we obtain

intradic1 + 1198774

119877radic1 + 1205821198774119889119877 = (

8120587

3)

12119905

119905119875

+ 119862 (B6)


1

radic120582ln [1 + 2120582119877

4+ 2radic120582 (1 + 1198774 + 1205821198778)]

minus ln[2 + 1198774+ 2radic1 + 1198774 + 1205821198778

1198774] = 4(

8120587

3)

12119905

119905119875

+ 119862

(B7)



References






































































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics



















































FluidsJournal of




Journal of




Volume 2013







Volume 2013




Advances in

Astronomy


GravityJournal of





Volume 2013

ISRN Thermodynamics









PhotonicsJournal of

ISRN Optics




research article a cosmological model based on a quadratic equation of state unifying vacuum

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