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Thermodynamics of a Dust Universe, Energy density, Temperature, Pressure and Entropy for Co

Thermodynamics of a Dust Universe, Energy density, Temperature, Pressure and Entropy for Co
Thermodynamics of a Dust Universe, Energy density, Temperature, Pressure and Entropy for Co

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Thermodynamics of a Dust Universe Energy density,Temperature,Pressure and Entropy for Cosmic Microwave Background James G.Gilson j.g.gilson@https://www.doczj.com/doc/e37160673.html, ?January 24,2007Abstract This paper continues the building of the cosmological theory that was introduced in two earlier papers under the title A Dust Universe Solution to the Dark Energy Problem .The model introduced in this theory has existence before time zero so that it is not necessary to interpret it as of big-bang origin.The location of the Cosmic Microwave Background,within the theoretical structure gives a closing of the fundamentals of the model in terms of the de?nitions of Temperature,Entropy and other Thermodynamic aspects.Thus opening up a research tool in cosmology in exact agreement with experiment that can compete with the so-called Standard Big Bang Model as a mathematical-physical description of our universe based rigorously on Einstein’s general relativity.It is suggested that the singularity at time zero involves a population inversion in the statistical mechanics sense and so justi?es the use of negative temperature for the CMB at negative times.This also has the satisfactory consequence that the Universe’s evolution involves entropy steadily increasing over all time from minus in?nity through the singularity to plus in?nity.Keywords:Dust Universe,Dark Energy,Friedman Equations,Entropy,Population Inversion,Negative Temperature PACS Nos.:98.80.-k,98.80.Es,98.80.Jk,98.80.Qc 1Introduction

The work to be described in this paper is an extension and general discussion of the signi?cance and physical interpretations of the papers A Dust Universe Solution to the Dark Energy Problem [23]and Existence of Negative Gravity Ma-terial.Identi?cation of Dark Energy [24].The conclusions arrived at in those papers was that the dark energy substance is physical material with a positive density,as is usual,but with a negative gravity,-G,characteristic and which is twice as abundant as has usually been considered to be the case.References to equations in those papers will be prefaced with the letter A and B respectively.The work in A ,B ,the discussion here and the extensions here have origins in the studies of Einstein’s general relativity in the Friedman equations context to be found in references ([16],[22],[21],[20],[19],[18],[4],[23])and similarly mo-tivated work in references ([10],[9],[8],[7],[5])and ([12],[13],[14],[15],[7]).Other

2DEFECT IN STANDARD MODEL2 useful sources of information have been references([17],[3])with the measure-ment essentials coming from references([1],[2],[11]).Further references will be mentioned as necessary.After writing A and B,I found that Abbe Georges Lema??tre[25]had produced much the same model in1927but had presented it with a greatly di?erent emphasis and interpretation while also missing impor-tant aspects of the signi?cance of the model that have emerged from the version and structure that I had found.He seems to have been only aware of the pos-itive time solution and then only in a restricted form in his1927paper.See remarks on page66of McVitties’s book[28].Abbe Georges Lema??tre is often referred to as the father of the big-bang but it seems to me that this reputation arose out of his omission of the negative time solution in his considerations of this early model.The explicit solution for r(t),equation(3.8)here,is also not mentioned in the1927paper.Up to date,there are generally few references to the explicit solution,(3.8),and sparse developments of its form.In particular, I have found a substantial application of this formula by the authors Ronald J.Adler et al[26]in a paper called Finite Cosmology and a CMB Cold Spot. Their work has common ground with the present paper.However,they fail to realise the power of the scale factor(3.8)and use it in conjunction with a time patching scheme making use of the related looking formula(1.1),

a(t)~sinh1/2(t/(2R d))(1.1) where R d is the de Sitter radius,to represent the radiation dominated era just as the formula(2.2)is used in the standard model to represent this era.This patching will be discussed in detail in the next section.I have not found who?rst wrote down the formula(3.8).The scale factor(3.8)and its universe although known about for many years,seems to have been almost totally eclipsed and overshadowed by the vast e?ort put into developing over the last eighty years of what is now called The Standard Model.

2Defect in Standard model

The standard cosmological model is generally accepted as giving a good and plausible description of the evolution of the universe from the initially assumed big bang through to the present time and into predicting the future of the universe.At the same time,it can be used to give a convincing case for how the material structures of which the universe is composed now has evolved from simpler and more fundamental materials that were likely predominant at earlier times.This is seen as a building and complexifying process with time which is assumed to be driven by temperature change from the very high temperature in the vicinity of the big bang the low temperate of the present day.The physical processes that are involved can be identi?ed from general earthbound physical theory and technology that associates the need for speci?c ranges of temperature with the possibility of known processes occurring.Thus the dependence of the temperature T(t)on time,t,is the main key to the structural evolution of the universe.

At this moment in time,we have only one convincing theory of gravity, Einstein’s General Relativity and its consequence the Friedman equations,as a description for the process of cosmological evolution with epoch.It is generally assumed that the standard cosmological model is rigorously a solution to the Einstein?eld equations.I do not wish to be dogmatic but I believe the truth of this assumption is questionable.To examine this issue let us consider the mathematical structures that constitutes the standard model.Nowadays,the standard model is said to describe four main phases in the birth and evolution

2DEFECT IN STANDARD MODEL3 of the universe.They are(1)in?ation and the big bang,(2)a radiation domi-nated phase,(3)a matter dominated phase and(4)accelerated expansion into the distant future.Essentially three distinct rigorous solutions to Einstein’s ?eld equations as represented by their distinct scale factors,r A(t),r B(t),r C(t) and distinct temperature time relations T A(t),T B(t),T C(t)can be found to ac-commodate the type of physical process assumed to be taking place within each epoch interval A,B and C.The three scale factors or radii involved are usually represented as,

r A(t)~exp(Ht).(2.1)

This?rst A stage involves Guth’s in?ation determined by a very large value for Einstein’s cosmological constant,ΛA and the big-bang event.

r B(t)~t1/2.(2.2)

This second stage B is the radiation dominated epoch,with some appropriate value for Einstein’s cosmological constant,ΛB.

r C(t)~t2/3.(2.3)

This third stage C which includes the present time,t0=t?,is the mass dom-inated epoch with accelerating expansion into the future determined by the de?nite measured value of Einstein’s cosmological constant,ΛC=Λ.I have emphasised that the three solutions A,B,C above are rigorously each a solution of Einstein’s?eld equations.Further,full versions can be found with detailed coe?cients so that the~sign can be replaced with the=sign.However,these solution are called the Standard Model when they are patched together in time sequence.Mathematically,this patching process can be carried through for both the scale factors and the time temperature relations to?nd a single form covering the time range0→+∞.It seems to me that this?nal form is not strictly speaking a rigorous solution to Einstein’s?eld equations and for the following reasons.Einstein’s?eld equations are causal in the sense that paths calculated from theory are determined for all time by initial conditions and this applies to the points on the boundary of the expanding universe given by the radial variable r(t).Thus for rigorous solutions r(t)cannot change its form in time passage without some external intervention outside the guiding in?uence of general relativity.There is no statistical aspect of relativity in its accepted unmodi?ed form that allow for choice between random selection of options at any stage of an evolution process.Statistics is involved in the thermal aspects of modern cosmology but this process the CMB is e?ectively locked away in a background subspace and can only in?uence paths through a constant thermal mass contribution in spite of its density changing with temperature and epoch. If such a change of motion had occurred in actual history,it could be regarded as due to the actual randomness of natural events or as an act of God but this later explanation is certainly outside relativity.Such a change would be on a par with the big-bang concept itself which is generally admitted not to be understood or likely to be explainable by present theory.From the theoretical point of view of the standard model,such changes have been allowed by mod-ern cosmologists in order to conform to human intuitions and preconceptions of how things should be.They may,of course,be right with their intuitive pictures of event sequences but still this is not science and not necessarily proof that the time patched solutions are rigorous solutions to the?eld equations. This argument against the standard model is strongly reinforced by the fact that solutions of Einstein’s?eld equation involving the cosmological constant requires that this quantity really remains constant over all the time for which

3SUMMARY OF MATHEMATICAL STRUCTURE OF MODEL4 the theory or solution is being used.We have seen that the standard model involves at least two de?nite changes to its parameter values for r(t)and T(t) andΛ.There is another important aspect of this issue and that is that the model I am proposing is rigorously a solution in isolation.It does not need patching up or joining to other solution to give a full time solution.Further it can supply all the facilities and options that are o?ered by the standard model for material synthesis resulting from temperature and pressure conditions that change with epoch and more.Showing that this is the case is what this paper is all about.

3Summary of Mathematical Structure of Model The main theoretical basis for the work to be discussed here are the two Fried-man equations that derive from general relativity with the curvature parameter k=0and a positively valuedΛ.For ease of reference these equations and the main results obtained so far from the dark energy model are listed next,

8πGρ(t)r2/3=˙r2?|Λ|r2c2/3(3.1)?8πGP(t)r/c2=2¨r+˙r2/r?|Λ|rc2.(3.2) r(t)=(RΛ/c)2/3C1/3sinh2/3(±3ct/(2RΛ))(3.3)

b=(RΛ/c)2/3C1/3(3.4)

C=8πGρ(t)r3/3(3.5)

RΛ=|3/Λ|1/2(3.6)

θ±(t)=±3ct/(2RΛ)(3.7)

r(t)=b sinh2/3(θ±(t))(3.8)

v(t)=±(bc/RΛ)sinh?1/3(θ±(t))cosh(θ±(t))(3.9)

a(t)=b(c/(RΛ))2sinh2/3(θ±(t))(3?coth2(θ±(t)))/2(3.10)

H(t)=(c/RΛ)coth(±3ct/(2RΛ))(3.11)

P(t)=(?c2/(8πG))(2¨r(t)/r(t)+H2(t)?3(c/RΛ)2)(3.12)

P(t)≡0?t(3.13)

PΛ=(?3c2/(8πG))(c/RΛ)2(3.14)

P G=(3c2/(8πG))(c/RΛ)2(3.15)

ρΛ=(3/(8πG))(c/RΛ)2(3.16)

ρ?Λ=(3/(4πG))(c/RΛ)2=2ρΛ.(3.17)

ρ(t)=(3/(8πG))(c/RΛ)2(sinh?2(3ct/(2RΛ))(3.18)

=3M U/(4πr3(t))=M U/V U(t).(3.19)ρG(t)=(G+ρ(t)+G?ρ?Λ)/G(3.20)

G+=+G(3.21)

G?=?G,(3.22)

4THERMODYNAMICS OF THE CMB5

¨r(t)=?4πr(t)GρG(t)/3(3.23)

?ρ(t))/3=a(t)(3.24)

¨r(t)=4πrG(ρ?

Λ

?ρ(t))/(3r2)(3.25)

=4πr3G(ρ?

Λ

=M?ΛG/r2?M U G/r2(3.26)

M?Λ=4πr3ρ?Λ/3(3.27)

M U=4πr3ρ(t)/3(3.28) P G(t)/(c2ρG(t))=1/(coth2(3ct/(2RΛ))?3)=ωG(t)(3.29)

ωΛ=PΛ/(c2ρΛ)=?1.(3.30) where M?Λis the total dark energy mass within the universe and M U is the total non-dark energy mass within the universe.Equation(3.23)or equation(3.24) is exactly the classical Newtonian result for the acceleration at the boundary of a spherical distribution positive G mass density.The function r(t)(3.8)is the radius or scale factor1for this model([3]).After writing A and B,I found the connection with Lema??tre’s work.

An interesting and signi?cant result that follows easily from this theory is the identi?cation of the Newtonian gravitational Coulomb like potential that is equivalent to Einstein’s general relativity with his positive cosmological con-stantΛ>0or expressed otherwise,the Newtonian Coulomb potential limit of relativity with Einstein’s dark energy.All that is needed is to replace the Newtonian equivalent density for a point source potential at epoch-time,t,with the general relativity gravity weighted density,ρG(t),equation(3.20),

ρG(t)=ρ(t)?ρ?Λ.

This procedure gives the result,

¨r=?MG/r2+rΛc2/3

in place of the Newtonian result

¨r=?MG/r2.

The inverse square is simply modi?ed with the addition of a term linear in r and proportional to Einstein’sΛ.

4Thermodynamics of the CMB

The thermodynamics of dust with its characteristically zero pressure as de-scribed here by the quantity P(t)equation(3.12)may seem to present this the-ory with some conceptual di?culties,if it is to include the cosmical microwave background as part of its structure.However,this turns out not to be the case as will be explained.The model is geometrically and dynamically completely de?ned,it does have pressure as part of its structure but the thermodynamical signi?cance of this pressure which is identically zero over the whole life span of this model and any relation it may have with a temperature is not obvious.The zero dust pressure does decompose into positive and negative signed components and as usual the dark energy material contributes the negative pressure.The dark energy component is identi?ed?rmly with truly negatively characterised gravitating material but takes the form of a positive mass density,ρ?Λwhich is twice the dark energy density,ρΛ,identi?ed by Einstein.There is no reason to

4THERMODYNAMICS OF THE CMB6 believe that the CMB should be directly associated with the dark energy vac-uum constituent and also be negative G characterised and so to add the CMB into the structure the obvious choice is to make it part of the conserved mass of the universe which has been denoted earlier by M U,equation(3.28).This placement for the CMB is reinforced by the fact that blackbody radiation is not an absolute constant as is the dark energy density but rather depends on temperature which itself is usually assumed to vary with epoch time.I therefore make the strong assumption,

M U=M?+MΓ,(4.1)

where M U,the total conserved non-dark energy mass of the universe,M?is the conserved mass that is neither CMB nor non-dark energy mass and MΓ, is the conserved CMB mass of the universe and all are taken to be absolute constants.In this paper,I shall restrict the discussion to the assumption(4.1).

I refer to this assumption as strong as there are other option possibilities all of which can keep M U costant over all time because the model depends essentially on the assumption that Rindler’s constant C=2M U G is an absolute constant and this is what makes the integration of the Friedman equations yield the model.Clearly,C can be kept constant if M U varies with time with whatever side e?ects that may have.The main consequence of the assumption(4.1)is that total non-dark energy density,M U and both?andΓcomponent densities acquire an epoch time dependence as a result of the equations,

M U=ρ(t)V U(t),(4.2)

M?=ρ?(t)V U(t),(4.3)

MΓ=ρΓ(t)V U(t),(4.4)

where the volume of the universe at time t is given by V U(t)=4πr3(t)/3.Taking the cosmic microwave background radiation to conform to the usual blackbody radiation description,the mass density function for the CMB will have the form

ρΓ(t)=aT4(t)/c2,(4.5)

a=π2k4/(15 3c3),(4.6)

=4σ/c,(4.7)

whereσis the Stephan-Boltzmann constant,

σ=π2k4/(60 3c2),(4.8)

=5.670400×10?8W m?2K?4.(4.9)

The total mass associated with the CMB is given by(4.4).That is

MΓ=ρΓ(t)V U(t),(4.10) =(aT4(t)/c2)4πr3(t)/3(4.11)

=(aT4(t)/(3c2))4πb3sinh2(θ±(t))(4.12)

=(aT4(t)/(3c2))4π(RΛ/c)22M U G sinh2(θ±(t))(4.13)

=(8πaT4(t)/(3c4))(RΛ)2M U G sinh2(θ±(t))(4.14)

It follows from(4.14)that the temperature,T(t)as a function of epoch can be expressed as,

(8πaT4(t)/(3c4))=MΓ/((RΛ)2M U G sinh2(θ±(t)))(4.15) T4(t)=MΓ(3c4)/(8πa(RΛ)2M U G sinh2(θ±(t)))(4.16)

T(t)=±(MΓ3c4/(8πa(RΛ)2M U G sinh2(θ±(t))))1/4(4.17)

T(t)=±((MΓ3c2/(4πa))(r(t)3))1/4.(4.18)

4THERMODYNAMICS OF THE CMB7 The fourth power equation in T has four solutions two of which are real and two of which are complex and so the latter two can be neglected.The positive solution is obviously important but the negative solution also turns out to play an important role in this theory.I shall return to this issue.

The well established fact that at the present time,t?,the temperature of the CMB is

T(t?)=T?=2.728K(4.19) can be used to simplify the formula(4.18)because with(4.19)it implies that 0

T(t)/T(t?)=±(sinh2(θ±(t?)))1/4/(sinh2(θ±(t)))1/4(4.21)

T(t)=±T? sinh(θ±(t?))

?T V=(4a/3)V U(t)T3(t),(4.24) PΓ(t)= ??F

5RATIO OF RADIATION MASS TO DELTA MASS8 ranges that suit various quantum particle physical structure generation pro-cesses.They can be found in this theory with the single temperature all-time formula,T(t),given by equation(4.22).

Approximate list of physical processes associated

with temperature ranges implied by T(t): Preliminary remark:Processes are associated with times in this list using times often quoted in the standard model[29].The association of process with temperature in the standard model is of very low-level accuracy so I have made no attempt to get accurate connection of time with process.The list is just to show qualitatively that the processes described in the standard model can also be described in the model that I am proposing with at least equal detail.

The zero and positive valued times listed above from the standard model, t A→t H,interspersed with theoretical values(0,t1,t c,t?,t2)from the present theory,in numerical time order,are given in the following list.The present theory contains time symmetrical event t→?t that are not listed.

Main Events in History of the Universe

t=0The singularity when the radial speed is ambiguously,±∞

t A Planck epoch range of super-fast in?ation

t B Post in?ation when there is a hot soup of electrons and quarks

t C Rapid cooling when quarks convert into protons and neutrons

t D Super hot fog of charged electrons and protons impede passage of photons t E Electrons protons and neutrons form atoms and photons have free passage t1The time when the radial boundary speed of the universe has descended through?nite values from∞to the value c

t F Hydrogen and helium coalesce under gravity to eventually become galaxies t c The time when the universes motion changes from deceleration to accelera-tion

t G=t?The present time,t?,conditions as they are now

t H Conditions109years into the future

t2The time when the radial boundary speed of the universe ascends from below to attain the value c again

t∞=∞

5Ratio of Radiation Mass to Delta Mass

The accelerating universe astronomical observational workers[1]give measured values of the three?s,and wΛto be

(5.1)

?M,0=8πGρ0/(3H20)=0.25+0.07

?0.06

?Λ,0=Λc2/(3H20)=0.75+0.06

(5.2)

?0.07

?k,0=?kc2/(r20H20)=0,?k=0,(5.3)

ωΛ=PΛ/(c2ρΛ)=?1±≈0.3.(5.4)

5RATIO OF RADIATION MASS TO DELTA MASS9 According to the strong assumption(4.1)M U,the total conserved mass of the universe,M?the conserved?mass of the universe and MΓ,the conserved CMB mass of the universe are all absolute constants.Thus the ratio of CMB mass to the?mass,rΓ,?=MΓ/M?,will be an absolute constant with epoch change together with the same ratio in terms of the corresponding time dependent densities,ρΓ(t)andρ?(t).The value of this ratio is,

rΓ,?=ρΓ(t)/ρ?(t)≈0.00019151≈2×10?4(5.5)

?M(t)=??(t)+?Γ(t)(5.6)

rΓ,?=?Γ(t)/??(t)≈2×10?4(5.7)

?M(t)≈??(t)(1+2×10?4)(5.8)

??(t)≈?M(t)(1?2×10?4)(5.9)

??(t)≈?M(t)?0.00005,?M,0=?M(t?)=0.25(5.10)

?Γ(t)≈?M(t)×2×10?4,(5.11)

where(5.6)is the?equivalent of the strong assumption,(4.1).Hence(5.7) through to(5.11).

If we compare(5.10)with(5.1),it is clear that the mass weight,MΓ,of CMB contribution,?Γ(t),to the conserved mass contribution M U is way inside the error limits for?M,0.That is to say that as far as mass is concerned, the CMB could be neglected in relation to the??contribution as it makes no signi?cant numerical contribution to the basic structure of the theory or the integration process used to derive it.However,it does introduce detail internally to the theory and it is clearly vital to the quantum synthesis of structures story attached to the theory.It reduces homogeneity in an important and useful way and,as will be shown,it induces a partitioning of the pressure P(t)in the forms for the CMB pressure PΓand the pressure P?from the?mass component. This will be addressed next.

The CMB pressure from equation(4.25)is

PΓ(t)=(a/3)T4(t),(5.12)

=EΓ/(3V U(t)),(5.13)

=MΓc2/(3V U(t))=ρΓ(t)c2/3.(5.14)

From(3.13)and following equations the pressure of the dust universe is given by

P(t)=P G+PΛ≡0?t.(5.15)

where both P G and PΛare oppositely signed but numerically equal absolute constants.In the previous section,the CMB mass was incorporating into the theory by partitioning the total mass M U which has been given a?xed numerical value from experiment into the two parts M?and MΓwhile keeping the same constant value for M U.This de?nes the?mass,M?,but otherwise makes no di?erence to the structure of the theory or its correctness as rigorous solution to Einstein’s?eld equations.However,as the value of the pressure,P(t),is determined by the value within the theory of M U,the non-dark energy part of the universe mass,the partitioning procedure implies a partitioning of the non-dark energy part of the pressure P(t)which is P G(5.15)corresponding to the partitioning of mass,(4.1).

There is an unfortunate anomaly in Friedman cosmology in the perception of pressure and in the way that it is de?ned.The pressure term in the Friedman equations is assumed to arise basically from the gravitational attraction of the material within the universe.Thus at the universe boundary massive objects

6ENTROPY OF THE CMB10 will be attracted to within the universe by gravitation and this is regarded as the source of the positive pressure term,P,in the Friedman equations.Thus, e?ectively,P,is identi?ed with the pressure that an in?ated balloon in equilib-rium would produce in reaction to the air pressure,?P,within it at equilibrium. It follows that,for consistency,if there is radiation pressure present within the universe it should be placed as a contribution to the P(t)term,within the the-ory,with a negative sign.That is we should write for the induced partitioning of P(t)

P(t)=P G+PΛ≡0?t(5.16)

P G=P?(t)?PΓ(t)(5.17) as the dark energy pressure,PΛ,is not involved being not part of an e?ect from the conserved mass of the universe.The pressure,P G,associated with the gravitational attraction of all the mass within the universe at any time t,as has been shown earlier,is an absolute constant while PΓ(t)depends on time.Thus the non-CMB part of the pressure,P?(t)must also depend on time.Equation, (5.17)can be written as

1=P?(t)/P G?PΓ(t)/P G(5.18)

r PΓG(t)=PΓ(t)/P G(5.19)

r P?G(t)=r PΓG(t)+1.(5.20) The ratios r PΓG(t)and r P?G(t)above give the weight of CMB pressure and the weight of non-CMB to the total non-dark pressure respectively Let us now consider the relation between the CMB pressure,PΓ(t),and the ?mass density,

PΓ(t)=(a/3)T4(t),(5.21)

=EΓ/(3V U(t)),(5.22)

=MΓc2/(3V U(t))=ρΓ(t)c2/3(5.23)

≈ρ?(t)c2×2×10?4/3.(5.24) Equations(5.16)and(5.17)together,expressed as

PΓ(t)≡P?(t)+PΛ?t,(5.25) gives the important interpretational result that the CMB is in mechanical equi-librium with the?andΛmasses combined.This is an alternative characteri-sation of the dust universe property.

6Entropy of the CMB

We found at equation(4.18)that there is a choice available for de?ning physical temperature as a positive or a negative quantity.As we all know,the usual choice in terrestrial physics is the positive one.This choice is made because S is a monotonic increasing function of E,

1

>0.(6.1)

?E

However,there are also physical theory situations where negative values of temperature are encountered and accommodated within conventional thermal

6ENTROPY OF THE CMB11 physics.Negative temperatures occur for example in nuclear spin systems.Orig-inal work on this aspect is associated with the researchers Purcell and Pound ([31],[27]).The second of the previous two references gives much detail about such systems and explanations of their thermodynamics The condition under which negative temperature occurs is often considered to be due to the existence of a sub-system with?nite maximum energy that is in weak interaction with the rest of the system so that its own temperature can be de?ned independently from the rest.The sub-system can then reach equilibrium without being in equi-librium with the whole system.In statistical mechanics,the process involved is sometimes referred to as population inversion with the temperature passing from positive in?nity to negative in?nity or conversely at the inversion time. This subsystem property is largely the situation in the cosmological model be-ing presented here where the sub-system is the CMB which has a constant?xed total energy EΓand so its interaction with the whole universe system which changes markedly with time t must be judged as weak.The property is exactly ful?lled in the present model if,at the singularity time,population inversion actually takes place with a?nite negative temperature just before time t=0 and reaching?∞at the singularity then jumping to+∞immediately after the singularity to then descend through?nite positive values to zero as epoch time advances.Such a singularity event depends theoretically on the temperature for negative time being chosen also to be negative and as we have seen we have this option from the theory(4.17).The in?nite temperature jump at t=0 can,as is well known,be avoided if one works with the parameter1/T instead of T.This is equivalent to regarding inverse temperature as more physically signi?cant than temperature itself.My feeling is that the idea of a singularity population inversion taking place at t=0is a likely scenario.However,I have to admit that the choice of negative temperature for t<0is speculative in spite of it apparent very good?t with the mathematics of the present model.In particular,this assumption does lead to the important conclusion that,if true, the entropy associated with the CMB steadily increases from t=?∞→+∞and,of course,that is what we would like to be the case.Firm decisions about what happens at the singularity at this time in theory development can only be speculative.Quantum Cosmology when it arrives may hopefully change the situation.

Starting at equation(6.2)is a list of the values of the ratio r PΓG(t)of CMB pressure to the non-dark energy component component P G of the pressure P(t) that arises from the?mass,r PΓ?(t),for fourteen important values of epoch time,t.This model is time symmetric about the time t=0,the time of the singularity.Thus this model has existence before the time t=0.This is evident from the form of the radius or scale factor equation,(3.8)provided it is understood that the square is taken before the cube root is taken in the index of the sinh function.

r PΓG(±0)≈∞(6.2)

r PΓG(±t1/10)≈6.94625(6.3)

r PΓG(±t1)≈0.06493(6.4)

r PΓG(±t c)≈0.00012(6.5)

r PΓG(±t?)≈0.00002(6.6)

r PΓG(±t2)≈2.03×10?6(6.7)

r PΓG(±∞)=0(6.8) The ratio r PΓG(t)is a measure of how the CMB mass pressure relates to the P G mass pressure.Towards the singularity,t=0,it approaches∞.The?mass pressure ratio to P G at the times above is given by adding1to the corresponding

7CONCLUSIONS12 values above.Thus towards t=0it also approaches∞.Thus from the pressure point of view at the singularity there is mechanical equilibrium between CNB mass and the?mass andΛmass combined with the contribution from theΛmass being negligible(5.25).At the singularity temperature and pressure are both in?nite.This is the equivalent of the idea from big-bang theory that near the singularity the universe is radiation dominated.

7Conclusions

The cosmological model developed in A and B and further ampli?ed here to show how the thermodynamics of the CMB is included in its structure is rigor-ously a solution to the full set of Einstein’s?eld equations of general relativity via the Friedman equations.The model also satis?es exactly the recent measure-ments by the astronomical dark energy workers.The model need not necessarily be considered to be of the big-bang type because it has a history extending from t=?∞to t=+∞.However,if the reader cannot accept or just disagrees with the existence of the solution before t=0being joined to the solution after t=0,he or she can simply disregard the negative time phase and stick with the big-bang idea.The negative time phase can then be dismissed as an unphysical solution to Einstein’s?eld equations.I shall now give a brief account of the time evolution of the model over the full time range t=?∞to t=+∞and mention a few of the new puzzles thrown up by it.The reader would?nd it helpful to look at the graphs for velocity and acceleration over this range as I outline the history of events as the model evolves with epoch.See the?le darkenergy.pdf [32],presentation to the2006PIRT Conference and the list Main Events in History of the Universe in section4.

The theory implies the following sequence of steps starting at t=?∞.A de?nite quantity of positively gravitational mass,M u,at density zero uniformly distributed over the whole of an in?nitely extended spherical three dimensional Euclidean region of hyperspace is collapsing at a very high speed,v?c,towards a de?nite centre.This is the initial situation.Apart from the fact that this mentions the kinematics of a spherical boundary moving towards its centre and as the density of the contents of this collapsing sphere is essentially zero almost nothing physical is happening.Clearly this initial situation is somewhat like the big-bang.However,I think it can be given a plausible explanation with a slight extension of the theory which does not damage its correctness under relativity.I shall come back to this point at the end of this section.The hyperspace is itself ?lled with a uniform constant unchanging density of negatively characterised gravitating mass,dark energy from Einstein’s positiveΛ.Thus the repulsive gravitational e?ect of this material within the descending sphere will cause the incoming high velocity of the boundary to steadily be reduced until it reaches from above the value c at the negative time,?t2.All this?rst phase does present some physical mystery because the boundary motion is superluminal. However,if this early stage motion can be seen as just kinematic that might reduce any detraction by this aspect.Similar problems occur in the standard model.We note that at and above the positive time,t2,we have no information about what physical processes are taking place except that the mass density of the universe is very low indeed which is also obviously the case below and up to the negative time?t2.The next main stage above?t2and below?t1 is acceptable as within known physics and probably involves physical processes like those occurring above t1and below t2.However,it is not necessarily true that the processes in this range are the time-reversed processes in the time symmetric range,displayed in the Events in History https://www.doczj.com/doc/e37160673.html,stly,in the negative time range from?t1to the singularity at0we are into a mystery superluminal

REFERENCES13 range again but this will likely contain events like those in the positive time range0to t1but with the same caveat as for the previous range.The sequence of events for the positive time range can be read o?from the Main Events in History list which takes us to the end of history at t∞with a very low density universe sphere with its boundary rushing to in?nity with v?c.

Returning to the point raised earlier about understanding the initial state, we see that the?nal state at+∞is the same as the initial state at?∞with re-versed radial speed of the boundary.This can be given a reasonable explanation by making a reinterpretation of the3-dimension hyper-space into which the uni-verse is contracting and then expanding after the singularity.Suppose that the original3D-hyperspace is replaced with a?xed3D-surface of a4D-sphere of very large radius,a well know geometrical trick.Thus the?xed hyper-space becomes a closed3D-space.The expanding Universe can then be regarded as expanding from a point to cover the4D-sphere surface to some maximum hyper-area and then to contract back over the adjacent hyper-surface to become a point sphere again.This is most easily picture by considering the lower dimensional situation, a circle on an ordinary3D-sphere surface expanding from a point on the surface to approaching a great circle at great positive speed away from its start point to when,after passing the great circle,descends initially at great negative speed to become an antipodal point.This is just like the kinematic process in the model is seen to be happening through t=±∞.This hyper-spherical interpretation is not part of the model as it stands but it could be incorporated in the structure without damage to the rigorousness of the model as a solution of the Einstein ?eld equations.We can in fact just use this idea to interpret what is happen-ing at in?nity and then let the radius of the4D-hypersphere go to in?nity.Its surface then becomes indistinguishable from the original hyperspace.

The main conclusions are that this model for the evolution of the universe has all the advantages that are found in the cosmological standard model together with new perspectives on the nature of dark energy and the amount of it that is present in the background together with a new and clearer representations for temperature,pressure and entropy.Importantly,it does not su?er from the serious defect of the standard model of not being de?nitely a solution of Einstein’s?eld equations.

Acknowledgements

I am greatly indebted to Professors Clive Kilmister and Wolfgang Rindler for help,encouragement and inspiration over many years.

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Essential Statistical Thermodynamics

Appendix B Essential Statistical Thermodynamics Karl K. Irikura Physical and Chemical Properties Division, National Institute of Standards and Technology, Gaithersburg, MD 20899 Some computational methods, particularly ab initio techniques, produce detailed molecular information but no thermodynamic information directly. Further calculations are needed to generate familiar, ideal-gas quantities such as the standard molar entropy (S E ), heat capacity (C p E ),and enthalpy change [H E (T )-H E (0)]. This Appendix details the necessary procedures, including worked examples. Thermochemical calculations can be extended to transition states of chemical reactions. Procedures are provided for converting such information into rate constants. Tables are also provided for unit conversions and physical constants. Statistical thermodynamics calculations are necessary to compute properties as functions of temperature. In some computations, such as ab initio electronic calculations of molecular energy, the raw results do not even correspond to properties at absolute zero temperature and must always be corrected. All the corrections are based upon molecular spectroscopy, with temperature-dependence implicit in the molecular partition function,Q . The partition function is used not only for theoretical predictions, but also to generate most published thermochemical tables. Many data compilations include descriptions of calculational procedures (1-3). Corrections Unique to Ab Initio Predictions By convention, energies from ab initio calculations are reported in hartrees, the atomic unit of energy (1 hartree = 2625.5 kJ/mol = 627.51 kcal/mol = 219474.6 cm -1) (4). These energies are negative, with the defined zero of energy being the fully-dissociated limit (free electrons and bare nuclei). Ab initio models also invoke the approximation that the atomic nuclei are stationary, with the electrons swarming about them. This is a good approximation because nuclei are much heavier than electrons. Consequently, the Irikura, K. K. In "Computational Thermochemistry: Prediction and Estimation of Molecular Thermodynamics" (ACS Symposium Series 677); Irikura, K. K. and Frurip, D. J., Eds.; American Chemical Society: Washington, DC, 1998. Errata corrected through January 19, 2001

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新视野大学英语第三版第二册课文语法讲解 Unit4

新视野三版读写B2U4Text A College sweethearts 1I smile at my two lovely daughters and they seem so much more mature than we,their parents,when we were college sweethearts.Linda,who's21,had a boyfriend in her freshman year she thought she would marry,but they're not together anymore.Melissa,who's19,hasn't had a steady boyfriend yet.My daughters wonder when they will meet"The One",their great love.They think their father and I had a classic fairy-tale romance heading for marriage from the outset.Perhaps,they're right but it didn't seem so at the time.In a way, love just happens when you least expect it.Who would have thought that Butch and I would end up getting married to each other?He became my boyfriend because of my shallow agenda:I wanted a cute boyfriend! 2We met through my college roommate at the university cafeteria.That fateful night,I was merely curious,but for him I think it was love at first sight."You have beautiful eyes",he said as he gazed at my face.He kept staring at me all night long.I really wasn't that interested for two reasons.First,he looked like he was a really wild boy,maybe even dangerous.Second,although he was very cute,he seemed a little weird. 3Riding on his bicycle,he'd ride past my dorm as if"by accident"and pretend to be surprised to see me.I liked the attention but was cautious about his wild,dynamic personality.He had a charming way with words which would charm any girl.Fear came over me when I started to fall in love.His exciting"bad boy image"was just too tempting to resist.What was it that attracted me?I always had an excellent reputation.My concentration was solely on my studies to get superior grades.But for what?College is supposed to be a time of great learning and also some fun.I had nearly achieved a great education,and graduation was just one semester away.But I hadn't had any fun;my life was stale with no component of fun!I needed a boyfriend.Not just any boyfriend.He had to be cute.My goal that semester became: Be ambitious and grab the cutest boyfriend I can find. 4I worried what he'd think of me.True,we lived in a time when a dramatic shift in sexual attitudes was taking place,but I was a traditional girl who wasn't ready for the new ways that seemed common on campus.Butch looked superb!I was not immune to his personality,but I was scared.The night when he announced to the world that I was his girlfriend,I went along

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