Research Article Remaining Problems in Interpretation of...

Research ArticleRemaining Problems in Interpretation of the CosmicMicrowave Background

Hans-Joumlrg Fahr and Michael Sokaliwska

Argelander Institut fur Astronomie Universitat Bonn Auf dem Hugel 71 53121 Bonn Germany

Correspondence should be addressed to Michael Sokaliwska msokaliwskagmxde

Received 25 June 2014 Accepted 7 April 2015

Academic Editor Avishai Dekel

Copyright copy 2015 H-J Fahr and M SokaliwskaThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

By three independent hints it will be demonstrated that still at present there is a substantial lack of theoretical understandingof the CMB phenomenon One point as we show is that at the phase of the recombination era one cannot assume completethermodynamic equilibrium conditions but has to face both deviations in the velocity distributions of leptons and baryons froma Maxwell-Boltzmann distribution and automatically correlated deviations of photons from a Planck law Another point is thatat the conventional understanding of the CMB evolution in an expanding universe one has to face growing CMB temperatureswith growing look-back times We show however here that the expected CMB temperature increases would be prohibitive to starformation in galaxies at redshifts higher than 119911 = 2 where nevertheless the cosmologically most relevant supernovae have beenobservedThe third point in our present study has to do with the assumption of a constant vacuum energy density which is requiredby the present ΛCDM-cosmology Our studies here rather lead to the conclusion that cosmic vacuum energy density scales withthe inverse square of the cosmic expansion scale 119877 = 119877(119905) Thus we come to the conclusion that with the interpretation of thepresent-day high quality CMB data still needs to be considered carefully

1 Introduction

The cosmic background radiation (CMB) has been continu-ously full-sky monitored since 1989 beginning with COBEcontinued by WMAP [1] and now recently by PLANCK[2] Though with these series of successful and continuousmeasurements our knowledge of the structure of the CMBhas tremendously grown representing nowadays this cos-mologically highly relevant phenomenon in an enormousquality of spectral and spatial resolution these data howevergood in quality do not speak for themselves They ratherneed to be interpreted on the basis of a theoretical contextunderstanding of the CMB origin The latter however hasnot grown in quality as CMB data have This paper wantsto show some aspects of modern cosmological research innew lights Thereby it may also serve readers with somehesitation towards present-day cosmology and give themsome encouragement One needs to be convinced that ascientific discipline like cosmology is built on safe conceptualand physical grounds before one can appreciate the mostrecent messages from modern precision cosmology One

only can appreciate cosmological numbers like a Hubbleconstant of 1198670 = 73 kmsMpc and an age of the universe of1205910 = 137GYr [1] as eminent findings of the present epochwhen one accepts a universe that presently expands in anaccelerated form due to being driven by vacuum pressureThis puts the question what are the basic prerequisites ofmodern cosmology

At first it is the assumption that all relevant facts deter-mining the global structures of the universe and their internaldynamics have been found at present times This puts thequestion what part of the world may presently be screenedout by our world horizon which nevertheless influencesthe cosmological reality inside If as generally believed thecosmic microwave background (CMB) sky is such a horizonthen everything deeper in the cosmological past must beinvented as a cosmologic ingredient that never becomes anobservational fact On the other hand when inside thathorizon only something not of global but of local relevanceis seen then the extrapolation from what is seen to the wholeuniverse is scientifically questionable

Hindawi Publishing CorporationPhysics Research InternationalVolume 2015 Article ID 503106 15 pageshttpdxdoiorg1011552015503106

2 Physics Research International

In this paper we start out from a critical look on theproperties of cosmic microwave background (CMB) radi-ation the oldest picture of the universe and investigatebasic assumptions made when taking this background as thealmanac of basic cosmological facts Neither the exact initialthermodynamical equilibrium state of this CMB radiation isguaranteed nor its behaviour during the epochs of cosmicexpansion is predictable without strong assumptions on anunperturbed homologous expansion of the universe Theclaim connectedwith this assumption that theCMB radiationmust have been much hotter in the past may even bringcosmologists in unexpected explanatory needs to explain starformation in the early universe as will be shown

2 Does Planck Stay Planck If It Ever Was

21TheCosmicMicrowave BackgroundTested byCosmicTher-mometers It is generally well known that we are surroundedby the so-called cosmic microwave background (CMB)radiation This highly homogeneous and isotropic black-body radiation [1 5ndash7] is understood as relict of the earlycosmic recombination era when due to removal of electricallycharged particles by electron-proton recombinations the uni-verse for the times furtheron became transparent for photonsSince that time cosmic photons persistent from the times ofmatter-antimatter annihilations thus are propagating freelyon light geodetic trajectories through the spacetime geometryof the expanding universe up to the present days

Assuming that at the times before recombination matterand photons coexisted in perfect thermodynamical equilib-rium despite the expansion of the cosmic volume (we shallcome back to this problematic point in the next section)then this allows one to expect that these cosmic photonsinitially had a spectral distribution according to a perfectblack-body radiator that is a Planckıan spectrum It is thengenerally concluded that a perfectly homogeneous Planckıanradiation in an expanding universe stays rigorously Planckıanover all times that follow At this point one however one hasto emphasize that this conclusion can only be drawn if (a)the initial spectrum really is perfectly Planckian and if (b)the universe is perfectly homogeneous and expands in thehighest symmetrical form possible that is the one describedby the so-called Robertson-Walker spacetime geometry

Then it can be demonstrated (eg see [7]) that thePlanckıan character of the CMB spectral photon densityinitially given by

119889119899119903(120582) =

21205824

119889120582

exp [ℎ119888119870119879119903120582] minus 1

(1)

where 119889119899119903(120582) denotes the spectral photon density at the time

of recombination per wavelength interval 119889120582 at wavelength120582 and 119879

119903is the temperature of the Planck radiation at this

time is conserved for all ongoing periods of the expandinguniverse

Readers should however keep in mind that this isonly guaranteed if the universe has isotropic curvatureand expands in a homologous Robertson-Walker symmetricmanner (see eg [8]) Due to this fact it then turns out

that the initially Planckıan spectral photon density changeswith time so that for all cosmic future it maintains itsPlanckıan character however associated to a cosmologicallyreduced temperature 119879 lt 119879

119903 On one hand at a later time

119905 photons appear cosmologically redshifted to a wavelength1205821015840

= 120582(119877119877119903) and on the other hand they are redistributed

to a space volume increased by a factor (119877119877119903)3 Taking both

effects together shows that at a later time 119905 gt 119905119903the resulting

spectrum is given by

119889119899 (120582) = (

119877119903

119877

)

212058210158404

1198891205821015840

exp [ℎ1198881198701198791199031205821015840] minus 1

=

21205824

119889120582

exp [ℎ119888119870120582119879119903(119877119877119903)] minus 1

(2)

which with the help of Wienrsquos displacement law 119879 sdot 120582 =

const reveals that at later times it again is a Planck spectrumhowever with temperature 119879 = 119879

119903sdot (119877119903119877) This already

indicates that the present-day CMB should be associated to atemperature1198790 given by1198790 = 119879

119903sdot(1198771199031198770)where the quantities

indexed with ldquo0rdquo are those associated to the universe at thepresent time 119905 = 1199050 Depending on cosmic densities at therecombination phase the temperature 119879

119903should have been

between 3500K and 4500K (see [9])This indicates that withthe present-day CMB value of 1198790 = 2735K [1] a ratio ofcosmic expansion scales of

27353500

ge (

119877119903

1198770) = (

1198790119879119903

) ge

27354500 (3)

is disputableThe abovementioned theory of a homologous cosmic

expansion then also allows to derive an expression for thecosmic CMB temperature as a function of the cosmic photonredshift 119911 = (1205820 minus 120582

119890)120582119890at which astronomers are seeing

distant galactic objects Here 1205820 is the wavelength whichis observed at present that is at us while the associatedwavelength 120582

119890is emitted at the distant object With the

validity of the cosmological redshift relation in a Robertson-Walker universe

120582119890

1205820=

119877119890

1198770 (4)

where 119877119890and 1198770 denote the cosmic scale parameters at the

time 119905119890when the photon was emitted from the distant galaxy

and at the present time 1199050 Thus one obtains by definition

119911 =

1205820 minus 120582119890

120582119890

=

1198770119877119890

minus 1 (5)

This relation taken together with Wienrsquos law of spectralshift 120582max sdot 119879CMB = 03 in this context is expressed by

120582max (119911 = 0) sdot 119879CMB (119911 = 0) = 120582max (119911) sdot 119879CMB (119911) (6)

then finally allows to write the CMB temperature as thefollowing function of redshift

119879CMB (119911) = 119879CMB (119911 = 0) sdot120582max (119911 = 0)120582max (119911)

= 1198790CMB sdot (119911 + 1)

(7)

Physics Research International 3

22 Particle Distribution Functions in Expanding SpacetimesUsually it is assumed that at the recombination era photonsand matter that is electrons and protons in this phase ofthe cosmic evolution are dynamically tightly bound to eachother and undergo strong mutual interactions via Coulombcollisions and Compton collisions These conditions arethought to then evidently guarantee a pure thermodynam-ical equilibrium state implying that particles are Maxwelldistributed and photons have a Planckian blackbody dis-tribution It is however by far not so evident that theseassumptions really are fulfilled This is because photonsand particles are reacting to the cosmological expansionvery differently photons generally are cooling cosmologicallybeing redshifted while particles in first order are not directlyfeeling the expansion unless they feel it adiabatically bymediation through numerous Coulomb collisions which arerelevant here in a fully ionized plasma before recombinationlike they do in a box with subsonic expansion of its walls ButCoulomb collisions have a specific property which is highlyproblematic in this context

This is because Coulomb collision cross sections arestrongly dependent on the particle velocity V namely beingproportional to (1V4) (see [10]) This evidently causes thathigh-velocity particles are much less collision-dominatedcompared to low-velocity ones they are even collision-freeat supercritical velocities V ge V

119888 So while the low-velocity

branch of the distribution may still cool adiabatically andthus feels cosmic expansion in an adiabatic form the high-velocity branch in contrast behaves collision-free and hencechanges in a different form This violates the concept of ajoint equilibrium temperature and of a resulting Maxwellianvelocity distribution function and means that there may bea critical evolutionary phase of the universe due to differentforms of cooling in the low- and high-velocity branches of theparticle velocity distribution function which do not permitthe persistence of a Maxwellian equilibrium distribution tolater cosmic times

In the following part of the paper we demonstrate thateven if a Maxwellian distribution would still prevail at thebeginning of the collision-free expansion phase that is thepostrecombination phase era it would not persist in the uni-verse during the ongoing of the collision-free expansion Forthat purpose let us first consider a collision-free populationin an expanding Robertson-Walker universe It is clear thatdue to the cosmological principle and connected with it thehomogeneity requirement the velocity distribution functionof the particles must be isotropic that is independent on thelocal place and thus of the following general form

119891 (V 119905) = 119899 (119905) sdot 119891 (V 119905) (8)

where 119899(119905) denotes the cosmologically varying density onlydepending on the worldtime 119905 and 119891(V 119905) is the normalizedtime-dependent isotropic velocity distribution function withthe property int119891(V 119905)1198893V = 1

If we assume that particles moving freely with theirvelocity V into the V-associated direction over a distance119897 are restituting at this new place despite the differentialHubble flow and the explicit time-dependence of 119891 a locally

prevailing covariant but perhaps form-invariant distributionfunction 119891

1015840

(V1015840 1199051015840) then the associated functions 119891(V1015840 119905) and119891(V 119905) must be related to each other in a very specific wayTo define this relation needs some special care since particlesthat are freelymoving in an homologously expandingHubbleuniverse do in this case at their motions not conservetheir associated phasespace volumes 119889

6120601 = 119889

3V1198893119909 since

no Lagrangian exists and thus no Hamiltonian canonicalrelations for their dynamical coordinates V and 119909 are validHence Liouvillersquos theorem then requires that the conjugateddifferential phase space densities are identical that is

1198911015840

(V1015840 119905) 1198893V101584011988931199091015840

= 119891 (V 119905) 1198893V1198893119909 (9)

At the placewhere they arrive after passage over a distance119897 the particle population has a relative Hubble drift given byV119867

= 119897 sdot 119867 coaligned with V where 119867 = 119867(119905) means thetime-dependent Hubble parameterThus the original particlevelocity V is locally turned to V1015840 = V minus 119897 sdot 119867 All dimensionsof the space volume within a time Δ119905 are cosmologicallyexpanded so that 119889119909

1015840

= 119889119909(1 + 119867Δ119905) holds Completereintegration into the locally valid distribution function thenimplies with linearizably small quantities Δ119905 ≃ 119897V and ΔV =

minus119897 sdot 119867 that one can express the above requirement in thefollowing form

1198911015840

(V1015840 1199051015840) 1198893V101584011988931199091015840

= (119891 (V 119905) +120597119891

120597119905

Δ119905 +

120597119891

120597VΔV)

times(1+ ΔVV

)

2(1+119867Δ119905)

31198893V1198893

119909

= 119891 (V 119905) 1198893V1198893119909

(10)

This then means for terms of first order that

120597119891

120597119905

Δ119905 +

120597119891

120597VΔV+ 2ΔV

V119891+ 3119867Δ119905119891 = 0 (11)

and thus

120597119891

120597119905

119897

Vminus 119897119867

120597119891

120597Vminus 2 119897119867

V119891+ 3119867 119897

V119891 = 0 (12)

or the following requirement

120597119891

120597119905

= V119867120597119891

120597Vminus119867119891 (13)

Looking first here for interesting velocity moments ofthe function 119891 fulfilling the above partial differential equa-tion by multiplying this equation with (a) 4120587V2119889V and (b)(41205873)119898V4119889V and integrating over velocity space then leadsto

119886 119899 = 1198990 exp(minus2119867(119905 minus 1199050))and119887 119875 = 1198750 exp(minus4119867(119905 minus 1199050))


which then immediately makes evident that with the abovesolutions one finds that

119875

119899120574= (

1198750

119899120574

0) exp (minus (4minus 2120574)119867 (119905 minus 1199050))

= (

1198750

119899120574

0) exp(minus(

23)119867 (119905 minus 1199050))

(14)

is not constant meaning that no adiabatic behaviour of theexpanding gas occurs and that the gas entropy 119878 also is notconstant but decreasing and given by

119878 = 119878 (119905) = 1198780 ln119875

119899120574= minus

23119867(119905 minus 1199050) (15)

It is perhaps historically interesting to see that assumingHamilton canonical relations to be valid the Liouville the-orem would then instead of (9) simply require 119891

1015840

(V1015840 119905) =

119891(V 119905) and hence would lead to the following form of aVlasow equation

120597119891

120597119905

minus V119867120597119891

120597V= 0 (16)

In that case the first velocity moment is found with

120597119899

120597119905

= int 4120587V3119867120597119891

120597V119889V

= 4120587119867int

120597

120597V(V3119891)minus 12120587119867int V2119891119889V

(17)

yielding

120597119899

120597119905

= minus 3119899119867 (18)

which agrees with 119899 sim 119877minus3 Looking also for the higher

moment 119875 then leads to

120597119875

120597119905

=

41205873

int V5119867120597119891

120597V119889V = minus 5119867119875 (19)

which now in this case shows that

119875

119899120574=

1198750

119899120574

0exp (minus (5minus 3120574)119867 (119905 minus 1199050)) = const (20)

That means in this case an adiabatic expansion is foundhowever based on wrong assumptions

Now going back to the correct Vlasow equation (13) onecan then check whether or not this equation allows thatan initial Maxwellian velocity distribution function persistsduring the ongoing collision-free expansion Here we find for119891 sim 119899119879

minus32 exp[minus119898V22119870119879] 119899 and 119879 being time-dependentthat one has

120597119891

120597119905

= 119891[

119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

]

120597119891

120597V= minus119891

119898V119870119879

(21)

leading to the following Vlasow requirement (see (13))

119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

= minus119867(

119898V2

119870119879

+ 1) (22)

In order to fulfill the above equation obviously the termswith V2 have to cancel each other since 119899 and 119879 are velocitymoments of119891 hence independent on VThis is evidently onlysatisfied if the change of the temperature with cosmic time isgiven by

119879 = 1198790 exp (minus2119867(119905 minus 1199050)) (23)

This dependence in fact is obtained when inspecting theearlier found solutions for the moments 119899 and 119875 (see (13) and(14)) because these solutions exactly give

119879 =

119875

119870119899

=

11987501198701198990

exp (minus (4minus 2)119867 (119905 minus 1199050))

= 1198790 exp (minus2119867(119905 minus 1199050))

(24)

With that the above requirement (22) then only reducesto

119889 ln 119899

119889119905

minus

32

119879

119879

= minus119867 (25)

which then leads to

minus 2119867minus

32(minus2119867) = minus119867 (26)

making it evident that this requirement is not fulfilled andthus meaning that consequently a Maxwellian distributioncannot be maintained even not at a collision-free expansion

This finally leads to the statement that a correctly derivedVlasow equation for the cosmic gas particles leads to acollision-free expansion behaviour that neither runs adi-abatic nor does it conserve the Maxwellian form of thedistribution function119891 Under these auspices it can howeveralso easily be demonstrated (see [11]) that collisional interac-tion of cosmic photons with cosmic particles via Comptoncollisions in case of non-Maxwellian particle distributionsdoes unavoidably lead to deviations from the Planckianblackbody spectrum This makes it hard to be convinced bya pure Planck spectrum of the CMB photons at the time 119905recaround the cosmic matter recombination

Let us therefore now look into other basic concepts ofcosmology to see whether perhaps also there problems canbe identified which should caution cosmologists

23 Can the Cosmological CMB Cooling Be Confirmed Inthe following part of the paper we now want to investigatewhether or not the cosmological cooling of theCMBphotonsfreely propagating in the expanding Robertson-Walker spacetime geometry can be confirmed by observationsThe accessto this problem is given by the connection that in an expand-ing universe at earlier cosmic times theCMB radiation shouldhave been hotter according to cosmological expectationsfor example as derived in [7] Hence the decisive question


is whether it can be confirmed that the galaxies at largerredshifts that is those seen at times in the distant pastreally give indications that they in fact are embedded ina correspondingly hotter CMB radiation environment Forthat purpose one generally uses appropriate so-called CMBradiation thermometers like interstellar CN- CH- or CO-molecular species (see [3 12 13] or [4])

Assuming that molecular interstellar gas phases withinthese galaxies are in optically thin contact to the CMB thatactually surrounds these galaxies allows one to assume thatsuchmolecular species are populated in their electronic levelsaccording to a quasistationary equilibrium state populationIn this respect especially interesting are molecular specieswith an energy splitting of vibrational or rotational excitationlevels 119894 119895 that correspond to mean energies of the surround-ing CMB photons that is 119864

119894minus 119864119895

= ℎ]CMB Under suchconditions the relative level populations 119899

119894 119899119895essentially are

given by the associated Boltzmann factor

119899119894

119899119895

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879CMB] (27)

where 119892119894119895

are the state multiplicities In the years of therecent past interstellar CO-molecules have been proven tobe best suited in this respect as highly appropriate CMBthermometers This was demonstrated by Srianand et al [3]and Noterdaeme et al [4]

The carbon monoxide molecule CO splits into differentrotational excitation levels according to different rotationalquantum numbers 119869 According to these numbers a splittingof CO lines occurs with transitions characterized by Δ119869 = 1In this respect the transition 119869 = 1 rarr 119869 = 0 leads to abasic emission line at 12058210 = 26mm (ie ]0 = 1156GHz)The CO-molecule is biatomic with a rotation around anaxis perpendicular to the atomic interconnection line Thequantum energies 119864rot(119869) are given by

119864rot (119869) =ℎ2

81205872119868

119869 (119869 + 1) = 1198782(119869)

2119868= 119868

1205962(119869)

2 (28)

where 119868 is the moment of inertia of the CO-rotator and isgiven by

119868 (CO) = 1198862 119898C119898O119898C + 119898O

(29)

Here 119886 is the interconnection distance and 119898C 119898O arethe masses of the carbon and oxygen atom respectively 119878(119869)is the angular momentum of the state with quantum number119869 and 120596(119869) is the associated angular rotation frequency Theemission wavelengths from the excited states of the CO-A-Xbands (119869 ge 2) thus are given by

120582119895ge2 = 1205820 [

12minus

1119869 (119869 + 1)

] (30)

Usually it is hardly possible to detect these CO-finestructure emissions from distant galaxies directly due totheir weaknesses and due to the strong perturbations andcontaminations in this frequency range by the infrared (ie

ge115 GHz) Instead the relative population of these rotationalfine structure levels can much better be observed in absorp-tion appearing in the optical range To actually use such aconstellation to determine the relative populations of CO finestructure levels one needs a broadband continuum emitter inthe cosmic background behind a gas-containing galaxy in theforeground As in case of the object investigated by Srianandet al [3] the foreground galaxy is at a redshift of 119911abs =

241837 illuminated by a background quasar SDSS J14391204+ 1117405 Then the CO fine structure lines appear inabsorption at wavelengths between 4900 A and 5200 A andby fitting them with Voigt-profiles the relative populations(119899(119869119894)119899(119869119895)) of these fine structure levels can be determined

Assuming now optically thin conditions of the absorbinggas with respect to CMB photons one can assume that in aphotostationary equilibrium these relative populations areconnected with the abovementioned Boltzmann factor as

119899 (119869119894)

119899 (119869119895)

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879lowast

CMB] (31)

where now 119879lowast

CMB is the CMB Planck temperature at cosmicredshift 119911abs = 241837 On the basis of the abovementionedassumptions Srianand et al [3] depending on the specifictransitions which they fit find CMB excitation temperaturesof 119879lowastCMB(0 1) = 911 plusmn 123K 119879lowastCMB(1 2) = 919 plusmn 121Kand 119879

lowast

CMB(0 2) = 916 plusmn 077K while according to standardcosmology (see (7)) at a redshift 119911abs = 241837 one shouldhave a CMB temperature of119879lowastCMB = (1+119911abs)119879

0CMB = 9315K

where 1198790CMB = 2725K is the present-day CMB temperature

(see [14])Though this clearly points to the fact that CMB tempera-

tures 119879lowastCMB at higher redshifts are indicated to be higher thanthe present-day temperature 119879

0CMB it also demonstrates that

the cosmologically expected value should have been a fewpercent higher than these fitted valuesThis however cannotquestion the applicability of the above described method ingeneral though some basic caveats have to be mentionedhere

First of all observers with similar observations are oftenrunning into optically thick CO absorption conditions whichwill render the fitting procedure more difficult Noterdaemeet al [4] for instance can show that the fitted CMB temper-ature differs with the CO-column density of the foregroundabsorber (see Figure 1) The determination of these columndensities in itself is a highly nontrivial endeavour and onlycan be carried out assuming some fixed correlations betweenCO- and H

2-column densities the latter being much better

measurableThe second caveat in this context is connected with the

assumption that relative populations of fine structure levelsare purely determined by a photon excitation equilibriumwith the surrounding CMB photons If in addition anybinary collisions with other molecules or any photons otherthan CMB photons are interfering into these populationprocesses then of course the fitted 119879

lowast

CMB values have to betaken with correspondingly great caution Especially in theinfrared range delivering the relevant photons for excitationsor deexcitations the CMB spectrum is strongly contaminated


10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

4690 4691 4692 4693 4395 4396 4397 4398

4585 4586 4587 4588 4309 4310 4311 4312

4487 4488 4489 4490 4229 4230 4231

Observed wavelength (Aring) Observed wavelength (Aring)

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

P3R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

14

12

10

8

6

4

2

0140 141 142 143

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340

Tex(CO) = 86+11minus10 K

LogN(CO) (cmminus2)

2120590

CO A

X(3ndash0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4ndash0)

CO A

X(2ndash0)

CO A

X(5ndash0)

zabs = 2038

Figure 1 CO absorption profiles observed at SDSS-J1-70542 + 354340 (119911abs = 2038) figure taken from Srianand et al [3]

by galactic dust emissions [1 15 16] Facing then the possi-bility that galaxies at higher redshifts are more pronouncedin galactic dust emissions compared to our present galax-ies nearby then makes CMB temperature determinationsperhaps questionable Nevertheless the results obtained byNoterdaeme et al [4] when determining CO-excitationtemperatures at foreground galaxies with different redshiftsperhaps for most readers do convincingly demonstrate that alinear correlation of the CMB temperature with redshift canbe confirmed (see Figure 2) as expected

24 Problems with a Hot CMB in the Past Though fromthe results displayed in the above Figure 2 it seems as if thecosmological CMB cooling with time can be surprisingly wellconfirmed one nevertheless should not too carelessly takethat as an observational fact We remind the reader first tothe theoretical prerequisites of a cosmologic CMB coolingreflected in a decrease of the Planck temperature 119879CMB of thisradiation a Planckian spectrum only stays a Planckian if

(a) it was Planckian already at the beginning that is atthe recombination phase and if

(b) since that time a completely homologous cosmicexpansion took place till today

Point (a) is questionable because the thermodynamicequilibrium state between baryons and photons in the earlyphase of fast cosmic expansionmay quite well be disturbed orincomplete (see [7 11] Section 22 of this paper) Point (b) is

14

12

10

8

6

4

2

TCM

B(K

)

00 05 10 15 20 25 30

T0CMB(1 + z)

T0CMB(1 + z)(1minus120573)

⧬

z

Figure 2 CMB temperatures as function of redshift 119911 derived fromCO-excitation temperatures figure taken from Noterdaeme et al[4]

questionable since at present times we find a highly struc-tured inhomogeneous cosmic matter distribution whichdoes not originate from a homogeneous matter cosmos witha pure unperturbed Robertson-Walker cosmic expansion

The present universe actually is highly structured bygalaxies galaxy clusters superclusters and walls [17 18]Although perhaps the matter distribution was quite homo-geneous at the epoch of the last scattering of cosmic photonswhen the CMB photons were in close contact to the cosmic


matter during the evolutionary times after that matter dis-tribution has evidently become very inhomogeneous by thegravitational growth of seed structures Thus fitting a per-fectly symmetrical Robertson-Walker spacetime geometry toa universe with a lumpy matter distribution appears highlyquestionable [19] This is an eminent general relativisticproblem as discussed by Buchert [20] Buchert [21] Buchert[22] Buchert [23] Buchert [24] and Wiltshire [25] If due tothat structuring processes in the cosmic past and the asso-ciated geometrical perturbations of the Robertson-Walkergeometry we would look back into direction-dependentdifferent expansion histories of the universe this would pointtowards associated CMB fluctuations (see [7])

Thus it should be kept in mind that a CMB Planckspectrum is only seen with the same temperature fromall directions of the sky if in all these directions thesame expansion dynamics of the universe took place IfCMB photons arriving from different directions of the skyhave seen different expansion histories then their Plancktemperatures would of course be different and anisotropicdestroying completely the Planckian character of the CMBThis situation evidently comes up in case an anisotropicand nonhomologous cosmic expansion takes place like thatenvisioned and described in theories by Buchert [23] Buchert[26] Buchert [27] Buchert [24] or Wiltshire [25] Let uscheck this situation by a simple-minded approach here inthe two-phase universe consisting of void and wall regionsas described byWiltshire [25] void expansions turn out to bedifferent from wall expansions and when looking out fromthe surface border of a wall region in the one hemisphereone would see the void expansion dynamics whereas in theopposite hemisphere one sees the wall expansion dynamicsThus CMB photons arriving from the two opposite sides aredifferently cosmologically redshifted and thus in no case doconstitute one common Planckian spectrum with one jointtemperature 119879CMB but rather a bipolar feature of the localCMB-horizon

In fact if one hemisphere expands different from theopposite hemisphere then as a reaction also different CMBPlanck temperatures would have to be ascribed to the CMBphotons arriving from these opposite hemispherical direc-tions If for instance the present values of the characteristicscale in the two opposite hemispheres are1198771 and1198772 then thiswould lead to a hemispheric CMB temperature difference ofΔ11987912 given by (see [7])

Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)

and would give an alternative to the present-day CMB-dipoleexplanation

25 Hot CMB Impedes Gas Fragmentation Stars are formeddue to gravitational fragmentation of parts of a condensedinterstellar molecular cloud For the occurrence of an initialhydrostatic contraction of a self-gravitating primordial stellargas cloud the radiation environmental conditions have to beappropriate Cloud contraction namely can only continueas long as the contracting cloud can get rid of its increased

gravitational binding energy by thermal radiation from theborder of the cloud into open space Hence in the followingwe show that in this respect the cloud-surrounding CMBradiation can take a critical control on that contractionprocess occurring or not occurring

Here we simply start from the gravitational bindingenergy of a homogeneous gas cloud given by

119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)

where 119866 is the gravitation constant 120588 is the mass density ofthe gas 119877 is the radius of the cloud and 119872 is the total gasmass of the cloud

A contraction of the cloud during the hydrostatic collapsephase (see [28]) is only possible if the associated change ininternal binding energy 119864

119861can effectively be radiated off to

space from the outer surface of the cloud that is if

119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)

where 120590sb denote the Stefan-Boltzmann constant and 119879119888the

thermal radiation temperature of the cloud respectivelyThis already makes evident that further contraction of

the cloud is impeded if the surrounding CMB temperatureexceeds the cloud temperature that is if 119879CMB gt 119879

119888 because

then the only possibility is 119889119877119889119905 ge 0 that is expansionIn order to calculate the radiation temperature 119879

119888of the

contracting cloud one can determine an average value of theshrinking rate during this hydrostatic collapse phase by useof the following expression

⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)

where 120591ff is the so-called free-fall time period of the cloudmass (see [29]) Thus from the above contraction conditiontogether with this shrinking rate one thus obtains the follow-ing requirement for ongoing shrinking

35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)

which allows to find the following value for the cloudtemperature

1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)

To give an idea for the magnitude of this cloud temper-ature 119879

119888we here assume that the typical cloud mass can be

adopted with 119872 = 10119872⊙and that for mass fragmentation

of that size to occur primordial molecular cloud conditionswith an H2minusdensity of the order of 1205882119898 = 105 cmminus3 must beadopted With these values one then calculates a temperatureof

119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)

This result must be interpreted as saying that as soon asin the past of cosmic evolution the CMB temperatures 119879CMB


were becoming greater than this above value 119879119888 then stellar

mass fragmentations of masses of the order of 119872 ≃ 10119872⊙

were not possible anymore This would mean that galaxies atsupercritical distances correlated with redshifts 119911 ge 119911

119888should

not be able to produce stars with stellar masses larger than10119872⊙ This critical redshift can be easily calculated from the

linear cooling relation 119879119888= 119879

0CMB sdot (1 + 119911

119888) and interestingly

enough delivers 119911119888= 119879119888119879

0CMB minus 1 = 209 This means that

galaxies at distances beyond such redshifts that is with 119911 ge

119911119888= 209 should not be able to produce stars with stellar

masses greater than 10119872⊙

If on the other hand it iswell known amongst astronomersthat galaxies with redshifts 119911 ge 119911

119888are known to show distant

supernovae events [30] even serving as valuable cosmic lightunit-candles and distance tracers while such events just areassociated with the collapse of 10119872

⊙-stars then cosmology

obviously is running into a substantial problem

3 Conclusions

This paper hopefully has at least made evident that the ldquoso-calledrdquo modern precision cosmology will perhaps not lead usdirectly into a complete understanding of the world and theevolution of the universe Too many basic concepts inherentto the application of general relativity on describing thewholeuniverse are still not settled on safe grounds as we have pin-pointed in the foregoing sections of this paper

We have shown in Section 2 of this paper that thecosmic microwave background radiation (CMB) only thencan reasonably well be understood as a relict of the Big-Bang if (a) it was already a purely Planckian radiation at thebeginning of the recombination era and if (b) the universefrom that time onwards did expand rigorously isotropic andhomologous according to a Robertson-Walker symmetricalexpansion As we have shown that however both pointsare highly questionable since (a) matter and radiation arecooling differently in the expanding cosmos so that thetransition to the collisionless expansion induces a degen-eration from thermodynamical equilibrium conditions withparticle distribution functions deviating from Maxwelliansand radiation distributions deviating from Planckians (seeSection 22) Furthermore since (b) the cosmic expansioncannot have continued up to the present days in a purelyRobertson-Walker-like style otherwise no cosmic structuresand material hierarchies could have formed

It is hard to say anything quantitative at thismomentwhatneeds to be concluded from these results in Section 22 Factis that during and after the phase of matter recombination inthe universe Maxwellian velocity distributions for electronsand protons do not survive as Maxwellians but are degen-erating into non-Maxwellian nonequilibrium distributionsimplying the drastic consequence that baryon densities arenot falling off as (11198773

) but as (11198772) The interaction of

the originating CMB photons with nonequilibrium electronsby Compton collisions will then in consequence also changethe resulting CMB spectrum to become a non-Planckianspectrum as shown by Fahr and Loch [11] Essentially theeffect is that from Wienrsquos branch CMB photons are removedwhich instead reappear in the Rayleigh-Jeans branch

The critical frequency limit is at around 103 GHz witheffective radiation temperatures being reduced at higherfrequencies with respect to those at lower frequencies Theexact degree of these changes depends on many things forexample like the cosmologic expansion dynamics duringthe recombination phase and the matter density during thisphase However the consequence is that the effective CMBradiation temperature measured at frequencies higher than103 GHz are lower than CMB temperatures at frequenciesbelow 103 GHzOur estimate for conventionally assumed cos-mologicmodel ingredients (Omegas) would be by about 1119870

UnfortunatelyCMBmeasurements at frequencies beyond103 GHZ are practically absent up to now and do not allowto identify these differences If in upcoming time periodson the basis of upcoming better measurements in the Wienrsquosbranch of the CMB no such differences will be found thenthe conclusions should not be drawn that the theoreticalderivations of such changes presented here in ourmanuscriptmust be wrong but rather that the explanation of the CMB asa relict radiation of the recombination era may be wrong

Though indeed as we discuss in Section 23 there areindications given by cosmic radiation thermometers likeCN- CO- or CH-molecules that the CMB radiation hasbeen hotter in the past of cosmic evolution we also pointout however that alternative explanations of these moleculeexcitation data like by collisional excitations and by infraredexcitations through dust emissions should not be overlookedand that the conventionally claimed redshift-relatedly hotterCMB in the past (ie following the relation 119879CMB sim (1 +

119911)1198790CMB) in fact brings astrophysicists rather into severeproblems in understanding the origin of massive stars indistant galaxies seen at large redshifts 119911 ge 119911

119888(see Section 25)

For those readers interested in more hints why theconventional cosmology could be in error we are presentingother related controversial points in the Appendices

Appendices

A Behaviour of Cosmic Masses andInfluence on Cosmology

All massive objects in space have inertia that is reactwith resistance to forces acting upon them Physicists andcosmologists as well do know this as a basic fact but nearlynone of them puts the question why this must be so Evencelestial bodies at greatest cosmic distances appear to moveas if they are equipped with inertia and only resistantly reactto cosmic forces It nearly seems as if nothing real exists thatis not resistant to accelerating forces While this already is amystery in itself it is even more mysterious what dictates themeasure of this inertia One attempt to clarify this mysterygoes back to Newtonrsquos concept of absolute space and themotions of objects with respect to this space According toI Newton inertial reactions proportional to objectsrsquo massesalways appear when the motion of these objects is to bechanged However this concept of absolute space is alreadyobsolete since the beginning of the last century Insteadmodern relativity theory only talks about inertial systems


(IRF) being in a constant nonaccelerated motion Amongstthese all IRF systems are alike and equally suited to describephysics Inertia thus must be something more basic whichwas touched by ideas of Mach [31] and Sciama [32] In thefollowing we shall follow these pioneering ideas a little more

A1 Linear Masses and Scimarsquos Approach to Machrsquos IdeaVelocity and acceleration of an object can only be definedwith respect to reference points like for example anotherobject or the origin of a Cartesian coordinate system Anacceleration with respect to the empty universe without anyreference points does however not seem to make physicalsense because in that case no change of location can bedefined A reasonable concept insteadwould require to defineaccelerations with respect to other masses or bodies in theuniverse But which masses should be serving as referencepoints All Perhaps weighted in some specific way Oronly some selected ones And how should a resistance atthe objectrsquos acceleration with respect to all masses in theuniverse be quantifiableThis first thinking already show thatthe question of inertia very directly brings one into deepestcalamitiesThefirstmore constructive thinking in this respectwere started with the Austrian physicist Mach in 1883 (see[31]) For him any rational concept must ascribe inertia to thephenomenon appearing as resistance with respect to acceler-ations relative to all other bodies in the universeThus inertiacannot be taken as a genuine quantity of every body ratherinertiamust be a ldquorelationalrdquo quantity imprinted to each bodyby the existence and constellation of all other bodies in theuniverse a so-called inertial interdependence between allbodies This principle of mutual interdependence has beencalled Machrsquos inertial principle and the fathers of relativityand cosmology always were deeply moved by this principlethough they all never managed to construct a cosmologictheory which did fulfill Machrsquos principle (see [33])

The English physicist Sciama [32] however tried to takeserious the Machian principle and looked for a Machianformulation of inertial masses With the help of an enlargedgravity theory expanded inMaxwellian analogy to scalar andvectorial gravity potentials he tried to show that inertia ofsingle objects depends on all other masses in the universeup to its greatest distances Sciamarsquos ideas go back to earlierconcepts of Thomson [34] and Searle [35] and start withintroducing a scalar potential Φ and a vector potential for the complete description of the cosmic gravitational fieldThe vector potential thereby describes the gravitationalaction of mass currents 119895

119898 and each moving object in the

universe immediately is subject to the field of these cosmicmass currents 119895

119898= 120588( 119903)V

120588( 119903)which are intimately connected

with the objectrsquos own motion Here 120588( 119903) and V120588denote the

mass density and its bulk motion respectivelyIn a homogeneous matter universe with mass density 120588

the scalar field potentialΦ for a test particle at rest is given by

Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)

where the upper integration border is the mass horizon 119903infin

=

119888119867 Furthermore the vector potential in a homogeneous

and homologously expanding Hubble universe vanishessince

= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)

In contrast for a moving particle the scalar potential Φis the same as for the test particle at rest however the vectorpotential now is given by

= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)

The gravitoelectromagnetic fields 119892and

119892seen by the

moving particle thus are

119892= minus gradΦminus

1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)

Assuming an additional body with mass 119872 at a distance119903 from the test particle where 119903 may be taken as collinear toV then leads to the following total gravitational force actingon the test particle

= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)

Considering Newtonrsquos second law describing the gravi-tational attraction between two masses it then requires for = 0 that the following relation is valid

119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)

and thus indicates that the apparent inertial mass of the testparticle is proportional toΦ and thus ultimately is associatedwith the very distant cosmic masses The inertial mass 119898

119895of

this test object 119895 at a cosmic place 119903119895 when replacing density

by distributed masses119898119894 is represented by the expression

119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)

where again here 120588 is the cosmic mass density and 119903 is thedistance to the cosmic mass source with the volume 119889119881 Theabove summation runs over all other objects ldquo119894rdquo in cosmicspace besides that with index ldquo119895rdquo Hereby it turns out thataccording to Sciamas theory the required inertial masses 119898

119895

are related to all other cosmic masses 119898119894and their inverse

distances (1119903119894) Hence this formulation fulfills Machrsquos basic

idea that is its ideological requestTo enable this argumentation a Maxwellian analogy of

gravity to electromagnetism was adopted This howeverseems justified through papers like those by Fahr [36]and Fahr and Sokaliwska [37] were it is shown that theanomalous gravity needed for stably rotating disk galaxiesand needed for conformal invariance of gravity fields withrespect to special relativistic transformations do require thegravitational actions of mass currents


A2 Centrifugal Masses It furthermore appears that massconstellations in the universe do also play the decisive role atcentrifugal forces acting on rotating bodies Accelerations arenot only manifested when the velocity of the object changesin the direction parallel to its motion (linear acceleration)but also if the velocity changes its direction (directionalacceleration) without changing its magnitude (eg in caseof orbital motions of planets) Under these latter conditionsthe inertia at rotational motions leading to centrifugal forcescan be tested The question here is what determines themagnitude of such centrifugal forces Newton with hisfamous thought experiment of a water-filled rotating bucket(see eg [9 38]) had intended to prove that what counts interms of centrifugal forces is the motion or rotation relativeto the absolute space According to Mach [31] centrifugalforces rather should however be a reaction with respect tophysically relevant massive reference points in the universeThus also inertia with respect to centrifugal forces is aMachian phenomenon and again is a relational quantityconnected with the constellation of other cosmic masses

According to Mach the reaction of the water in therotating bucket in forming a parabolic surface is an inertialreaction to centrifugal forces due to the rotation with respectto the whole universe marked by cosmic mass points theso-called cosmic rest frame (see eg [33]) Since the earthrotates centrifugal forces act and the earthrsquos ocean producesa centrifugal bulge at the equator with a differential heightof about 10m The question what determines the exactmagnitude of these centrifugal forces is generally answeredthe rotation period of the earth But this answer just nowcontains the real basic question namely the rotation period120591Ωwith respect to what To the moon To the sun To the

center of the galaxy It easily turns out that it onlymakes senseto talk about rotation with respect to the stellar firmamentthat is the fixstar horizon Why however just these mostdistant stars at a rotation should determine the inertialreaction of the earthrsquos ocean While this again would provethe Machian constellation of the universe it neverthelessis hard to give any good reason for that Perhaps the onlyelucidating answerwas given byThirring [39] who started outfrom the principle of the relativity of rotations requiring thatidentical physical phenomena should be described irrelevantfrom what reference the description is formulated

Thus whether one describes the earth as rotating withrespect to the universe at rest or the universe as counterro-tating with respect to the earth at rest should lead to identicalphenomena that is identical forces To test this expectationThirring [39] gave a general-relativistic description of therotating universe and calculated geometrical perturbationforces induced by this rotating universe

Within a Newtonian approximation of general relativityhe could show that a rotating universe at the surface of theearth leads to metrical perturbation forces which are similarto centrifugal forces of the rotating earth For a rigorousidentity of both systems (a) rotating earth and (b) rotatinguniverse a special requirement must however be fulfilled(see Figure 3)

To carry out his calculations he needed to simplify themass constellation in the universe In his case the whole

120596

M

R

minus120596

M

Figure 3 The universe as a mass-shell

universe was represented by an infinitely thin rotatingspherical mass shell with radius 119877

119880and a homogeneous

mass deposition 119872119880

representing the whole mass of theuniverse Fahr and Zoennchen [9] have shown that thisstrongly artificial assumption can be easily relaxed to auniverse represented as an extended system of spherical massshells and still leads to Thirringrsquos findings namely that a fullequivalence of the systems (a) earth rotating and (b) universerotating only exists if the ratio (119872

119880119877119880) is a constant

where119872119880is the total mass of the universe within the cosmic

mass horizon 119877119880

= 1198881198670 increasing proportional with theincreasing age of the universe Here 119888 is the light velocity and1198670 denotes the present Hubble constant

However this request would have very interesting con-sequences for an expanding Hubble universe with 119877

119880being

time-dependent and increasing with worldtime 119905 as 119877119880(119905) =

119888119867(119905) It would namely mean that the equivalence ofrotations in an expanding universe can only be and stay validif the mass of the universe increases with time such that(11987211988001198771198800) = 119872

119880(119905)119877119880(119905) stays constant that is if the

mass 119872119880increases linearly with 119877

119880 This is an exciting and

also wonderful result at the same time because on one handit is absolutely surprising to have a hint for an increasingworld mass and on the other hand it fulfills Machrsquos ideaof inertia in a perfect way The above request related toevery single mass in the universe would namely requirethat its mass varies if all ambient cosmic masses increasetheir cosmic distances unless these masses change linearlywith their changing distances The same result would alsocome out from the analysis presented by Sciama [32] (seeAppendix A) which lead to a mass 119898

119895of the object 119895 given

by all other masses119898119895by the expression

119898119895sim sum

119894

119898119894

119903119894

(A8)

If now in addition with Thirringrsquos relation in a homolo-gously expanding universe one can adopt that

119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)


We obtain the result that both Machian effects on inertiataken together just guarantee a constant inertia of every singleobject in the universe

B What Is the Mass of the Universe

Following Mach [31] inertial masses of cosmic particlesare not of a particle-genuine character but have relationalcharacter and are determined by the constellation of othercosmic masses in the universe (see reviews by Barbour andPfister [33] Barbour [40] Wesson et al [41] and Jammerand Bain [42]) Einstein at his first attempts to develop hisfield equations of general relativity was deeply impressed byMachrsquos principle but later in his career he abandoned it [43]Up to the present days it is debated whether or not EinsteinrsquosGR theory can be called a ldquoMachianrdquo or a ldquonon-Machianrdquotheory At least some attempts have been made to develop anadequate form of a ldquorelationalrdquo that is Machian mechanics[8 44ndash47] In particular the requested scale-dependenceof cosmic masses is unclear though perhaps suggested bysymmetry requirements or general relativistic action princi-ple arguments given by arguments discussed by Hoyle [48]Hoyle [49] Hoyle [50] Hoyle et al [51] and Hoyle et al [52]along the line of the general relativistic action principle

As we have shown above Thirringrsquos considerations of thenature of centrifugal forces were based on the concept of themass of the universe 119872

119880 To better understand Thirringrsquos

result that this mass 119872119880should vary with the radius 119877

119880

of the universe one should have a clear understanding ofhow this world mass might conceptually be defined insteadsimply treating it as a mere number Most rational would beto conceive 119872

119880as a space-like summation of all masses in

the universe that is an expression representing the space-like sum over all cosmicmasses present in the universe at thesame event of time In a uniform universe this number119872

119880is

independent on the selected reference point This means119872119880

represents the space-like sum of all masses simultaneouslysurrounding this point within its associated mass horizon Ifat the time 119905 a cosmic mass density 120588(119905) prevails then thewhole mass integral up to the greatest distances has to becarried out using this density 120588(119905) disregarded the fact thatmore distant region are seen at earlier cosmic times

Fahr and Heyl [53] in order to calculate this space-like sum considered an arbitrary spacepoint surrounded bycosmicmass shells all characterized by an actual mass density120588 = 1205880 The situation is similar to a point in the center of astar being surrounded by mass shells with a stellar density inthe stellar case variable with central distance For the cosmiccase the spacegeometry of this space-like mass system isthen given by the inner Schwarzschild metric Under theseauspices the quantity 119872

119880as shown by Fahr and Heyl [53] is

given by the following expression

1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)

where the function in the numerator of the integrand is givenby the following metrical expression

exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)

The space-like metric in this cosmic case is given by aninner Schwarzschildmetric however with thematter densitygiven by the actual cosmic density 120588

0and taking into account

the fact that cosmic matter in a homologously expandinguniverse equipped with the Hubble dynamics leads to arelativistic mass increase taken into account by a cosmicLorentz factor 120574(119903) = (1 minus (1198670119903119888)

2)minus1 Assuming that within

the integration border Hubble motions are subrelativistic onemay evaluate the above expression with 120574(119903) = 1

Then the above expression for119872119880shows that real-valued

mass contributions are collected up to a critical outer radiuswhich one may call the local Schwarzschild infinity 119903 = 119877

119880

defined by that point-associated Schwarzschild mass horizonwhich is given by (see [53])

119877119880=

1120587

radic1198882

21198661205880 (B3)

This result is very interesting sincemeaning that thismasshorizon distance119877

119880appears related to the actual cosmicmass

density by the expression

1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)

and as evident from carrying out the integration in (B2) leadsto a point-associated mass119872

119880of the universe given by

119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)

This not only points to the surprising fact that with theuse of the above concept for119872

119880Thirringrsquos relation in (B1) is

in fact fulfilled but also proves that Machrsquos idea on the basisof this newly introduced definition of the mass 119872

119880of the

universe can be put on a solid basis

C A Physically Logic Conception ofEmpty Spacetime

The correct treatment of empty space in cosmology needs ananswer to the following fundamental problemWhat should apriori be expectable from empty space and how to formulateuncontroversial conditions for it and its physical behaviourThe main point to pay attention to is perhaps that the basicmechanical principle which was pretty clear at Newtonrsquosepoch of classical mechanics namely ldquoactio = reactiordquo shouldsomehow also still be valid at times of modern cosmologySo if at all the energy of empty space causes something tohappen then that ldquosomethingrdquo should somehow react back


to the energy of empty space Thus an action without anybackreaction contains a conceptual error that is a miscon-ception That means if empty space causes something tochange in terms of spacegeometry because it represents someenergy that serves as a source of spacetime geometry perhapssince space itself is energy-loaded then with some evidencethis vacuum-influenced spacegeometry should change theenergy-loading of space (see [54ndash56]) There is howevera direct hint that modern precision cosmology [1] doesnot respect this principle This is because the modern Λ-cosmology describes a universe carrying out an acceleratedexpansion due to the action of vacuum pressure while thevacuum energy density nevertheless is taken to be constant(eg see [57]) How could a remedy of this flaw thus look like

The cosmological concept of vacuum has a long and evennot yet finished history (see eg [41 54 58ndash64]) Due to itsenergy content this vacuum influences spacetime geometrybut it is not yet clear in which way specifically Normalbaryonic or darkionic matter (ie constituted by baryons ordark matter particles darkions resp) general-relativisticallyact through their associated energy-momentum tensors 119879119887

120583]

and 119879119889

120583] (eg see [65]) Consequently it has been tried to alsodescribe the GRT action of an energy-loaded vacuum intro-ducing an associated hydrodynamical energy-momentumtensor 119879

vac120583] in close analogy to that of matter The prob-

lem however now is that in these hydrodynamical energy-momentum tensors the contributing substances are treated asfluids described by their scalar pressures and their mass den-sitiesThequestion then evidently ariseswhat is vacuumpres-sure 119901vac and what is vacuum mass density 120588vac Not goingdeeper into this point at the moment one nevertheless thencan give the tensor 119879vac

120583] in the following form (see eg [38])

119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)

where 119880120582are the components of the vacuum fluid 4-velocity

vector and 119892120583] is themetrical tensor If now as done in the so-

called Λ-cosmology (see [1]) vacuum energy density is con-sidered to be constant then the following relation betweenmass density and pressure of the vacuum fluid can be derived120588vac119888

2= minus119901vac (eg see [38 57]) Under these prerequisites

the vacuum fluid tensor 119879vac120583] attains the simple form

119879vac120583] = 120588vac119888

2119892120583] (C2)

The above term for 119879vac120583] for a constant vacuum energy

density 120588vac can be combined with the famous integrationconstantΛ that was introduced by Einstein [66] into his GRTfield equations and then formally leads to something like anldquoeffective cosmological constantrdquo

Λ eff =

81205871198661198882 120588vac minusΛ (C3)

Under this convention then the following interestingchance opens up namely to fix the unknown and unde-fined value of Einsteinrsquos integration constant Λ so that theabsolutely empty space despite its vacuum energy density

120588vac = 120588vac0 does not gravitate at all or curve spacetimebecause this completely empty space is just described by avanishing effective constant (ie pure vacuumdoes not curvespacetime)

Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)

Very interesting implications connected with that vieware discussed by Overduin and Fahr [38] Fahr [54] or FahrandHeyl [55] It for instance implies that a completely emptyspace does not accelerate its expansion but can stagnate andleave cosmic test photons without permanently increasingredshifts and that on the other hand a matter-filled universewith a vacuum energy density different from 120588vac0 leads to aneffective value of Λ eff which now in general does not need tobe constant It nevertheless remains a hard problem to deter-mine this function Λ eff for a matter-filled universe in whicha matter-polarized vacuum (see [56]) different from the vac-uum of the empty space prevails In the following we brieflydiscuss general options one has to describe this vacuum

If vacuum is addressed as done in modern cosmologyas a purely spacetime- or volume-related quantity it never-theless is by far not evident that ldquovacuum energy densityrdquoshould thus be a constant quantity simply because the unitof space volume is not a cosmologically relevant quantityIt may perhaps be much more reasonable to envision thatthe amount of vacuum energy of a homologously comovingproper volume 119863119881 is something that does not change itsmagnitude at cosmological expansions because this propervolume is a cosmologically relevant quantity This new viewthen however would mean that the cosmologically constantquantity instead of vacuumenergy density 120598vac is the vacuumenergy within a proper volume given by

119889119890vac = 120598vacradicminus11989231198893119881 (C5)

where 1198923 is the determinant of the 3D-space metricIn case of a Robertson-Walker geometry this is given by

1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)

Here 119870 is the curvature parameter and 119877 = 119877(119905) is thetime-dependent scale of the universeThe differential 3-spacevolume element in normalized polar coordinates is given by1198893119881 = 119889119903 119889120599 119889120593 and thus leads to

119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)

If119889119890vac now is taken as a cosmologically constant quantitythen it evidently requires that vacuum energy density has tochange like

120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)


The invariance of the vacuum energy per comovingproper volume 119889119890vac is a reasonable requirement if thisenergy content does not do work on the dynamics of the cos-mic geometry especially by physically or causally influencingthe evolution of the scale factor 119877(119905) of the universe

If on the other hand work is done by vacuum energyinfluencing the dynamics of the cosmic spacetime (either byinflation or deflation) as is always the case for a nonvanishingenergy-momentum tensor then automatically thermody-namic requirements need to be fulfilled for example relatingvacuum energy density and vacuum pressure in a homoge-nous universe by the most simple standard thermodynamicrelation (see [65])

119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)

This equation is fulfilled by a functional relation of theform

119901vac = minus

3 minus ]3

120598vac (C10)

for a scale-dependent vacuum energy density in the form120598vac sim 119877

minus] Then it is evident that the above thermodynamiccondition besides for the trivial case ] = 3 when vacuumdoes not act (since 119901vac(] = 3) = 0 ie pressure-lessvacuum) is as well fulfilled by other values of ] as forinstance by ] = 0 that is a constant vacuum energy density120598vac sim 119877

0= const

The exponent ] is however more rigorously restrictedif under more general cosmic conditions the above ther-modynamic expression (C9) needs to be enlarged by aterm describing the work done by the expanding volumeagainst the internal gravitational binding in this volume Formesoscalic gas dynamics (aerodynamics meteorology etc)this term is generally of no importance however for cosmicscales there is definitely a need for this term Under cosmicperspectives this term for binding energy is an essentialquantity as for instance evident from star formation theoryand has been quantitatively formulated by Fahr and Heyl[55] and Fahr and Heyl [67] With this term the enlargedthermodynamic equation (C9) then attains the followingcompleted form

119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)

where the last term on the RHS accounts for internalgravitational binding energy of the vacuum With this termthe above thermodynamic equation can also tentatively besolved by the 119901vac = minus((3 minus ])3)120598vac which then leads to

minus3 (3 minus ]) 119901vac3 minus ]

1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)

As evident however now the above relation is onlyfulfilled by ] = 2 prescribing that the corresponding cosmicvacuum energy density must vary and only vary like

120598vac sim 119877minus2 (C13)

expressing the fact that under these general conditionsvacuum energy density should fall off with 119877

minus2 instead ofbeing constant

If we then take all these results together we see that notonly the mass density in the Robertson-Walker cosmos butalso the vacuum energy density should scale with 119877

minus2 Thefirst to conclude from this is that the vacuum pressure 119901vacunder this condition should behave like prescribed by thethermodynamic equation (C9)

119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)

and thus under the new auspices given now yield

119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)

meaning that now the following polytropic relation holds

119901vac = minus

13120598vac (C16)

meaning that again a negative pressure for the vacuum isfound but smaller than in the case of constant vacuum energydensity

With the additional points presented in the Appendicesit may all the more become evident that modern cosmologyhas to undergo a substantial reformation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C L Bennett R S Hill GHinshaw et al ldquoFirst-yearWilkinsonMicrowave Anisotropy Probe (WMAP) observations fore-ground emissionrdquo Astrophysical Journal Supplement Series vol148 no 1 pp 97ndash117 2003

[2] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XVI Cosmological parametersrdquo httparxivorgabs13035076

[3] R Srianand N Gupta P Petitjean P Noterdaeme and D JSaikia ldquoDetection of the 2175 A extinction feature and 21-cmabsorption in two Mg II systems at z sim 13rdquo Monthly Notices ofthe Royal Astronomical Society Letters vol 391 no 1 pp L69ndashL73 2008

[4] P Noterdaeme P Petitjean R Srianand C Ledoux and SLopez ldquoThe evolution of the cosmic microwave backgroundtemperature measurements of T

119862119872119861at high redshift from

carbon monoxide excitationrdquo Astronomy amp Astrophysics vol526 no 11 article L7 2011


[5] A A Penzias and R W Wilson ldquoA Measurement of excessantenna temperature at 4080Mcsrdquo The Astrophysical Journalvol 142 pp 419ndash421 1965

[6] G Smoot ldquoTheoretical and observational aspects of the CMBrdquoin Cosmology 2000 2000

[7] H J Fahr and J H Zonnchen ldquoThe lsquowriting on the cosmic wallrsquois there a straightforward explanation of the cosmic microwavebackgroundrdquoAnnalen der Physik vol 18 no 10-11 pp 699ndash7212009

[8] H FM Goenner ldquoMachrsquos principle and theories of gravitationrdquoinMachrsquos Principle From Newtonrsquos Bucket to Quantum GravityJ B Barbour and H Pfister Eds p 442 Birkhauser 1995

[9] H J Fahr and J H Zoennchen ldquoCosmological implications ofthe Machian principlerdquo Naturwissenschaften vol 93 no 12 pp577ndash587 2006

[10] L SpitzerPhysics of Fully IonizedGases Interscience Publishers1956

[11] H J Fahr and R Loch ldquoPhoton stigmata from the recombi-nation phase superimposed on the cosmological backgroundradiationrdquo Astronomy amp Astrophysics vol 246 pp 1ndash9 1991

[12] J N Bahcall and R A Wolf ldquoFine-structure transitionsrdquo TheAstrophysical Journal vol 152 p 701 1968

[13] DMMeyer andM Jura ldquoA precisemeasurement of the cosmicmicrowave background temperature from optical observationsof interstellar CNrdquo The Astrophysical Journal vol 297 pp 119ndash132 1985

[14] J C Mather D J Fixsen R A Shafer C Mosier and DT Wilkinson ldquoCalibrator design for the COBE Far InfraredAbsolute Spectrophotometer (FIRAS)rdquo Astrophysical JournalLetters vol 512 no 2 pp 511ndash520 1999

[15] M G Hauser R G Arendt T Kelsall et al ldquoThe COBE diffuseinfrared background experiment search for the cosmic infraredbackground I Limits and detectionsrdquo The Astrophysical Jour-nal vol 508 no 1 pp 25ndash43 1998

[16] R C Henry ldquoDiffuse background radiationrdquoThe AstrophysicalJournal vol 516 pp L49ndashL52 1999

[17] M J Geller and J P Huchra ldquoMapping the universerdquo Sciencevol 246 no 4932 pp 897ndash903 1989

[18] R S Ellis ldquoGalaxy formationrdquo in Proceedings of the 19th TexasSymposium on Relativistic Astrophysics J Paul T Montmerleand E Aubourg Eds Paris France December 1998

[19] K K S Wu O Lahav and M J Rees ldquoThe large-scalesmoothness of the UniverserdquoNature vol 397 no 6716 pp 225ndash230 1999

[20] T Buchert ldquoCosmogony of generic structuresrdquo in Publicationsof the Beijing Astronomical Observatory vol 1 pp 59ndash70 1995

[21] T Buchert ldquoAveraging hypotheses in Newtonian cosmologyrdquo inMapping Measuring and Modelling the Universe P Coles VMartinez andM-J Pons-Borderia Eds vol 94 ofAstronomicalSociety of the Pacific Conference Series p 349 AstronomicalSociety of the Pacific 1996

[22] T Buchert ldquoAveraging inhomogeneous cosmologiesmdasha dia-loguerdquo in Research in Particle-AstrophysicsmdashProceedings RBender T Buchert P Schneider and F von Feilitzsch Edspp 71ndash82 Max-Planck-Institut fur Astrophysik Garching Ger-many 1997

[23] T Buchert ldquoOn average properties of inhomogeneous cosmolo-giesrdquo General Relativity and Gravitation vol 33 pp 1381ndash13902001 httparxivorgabsgr-qc0001056

[24] T Buchert ldquoDark energy from structure a status reportrdquoGeneral Relativity and Gravitation vol 40 no 2-3 pp 467ndash5272008

[25] D L Wiltshire ldquoCosmic clocks cosmic variance and cosmicaveragesrdquo New Journal of Physics vol 9 article 377 2007

[26] T Buchert ldquoOn average properties of inhomogeneous fluids ingeneral relativity perfect fluid cosmologiesrdquo General Relativityand Gravitation vol 33 no 8 pp 1381ndash1405 2001

[27] T Buchert ldquoA cosmic equation of state for the inhomogeneousuniverse can a global far-from-equilibrium state explain darkenergyrdquo Classical and Quantum Gravity vol 22 no 19 ppL113ndashL119 2005

[28] V S Safronov Evolution of the Protoplanetary Cloud andFormation of the Earth and Planets 1972

[29] J H Jeans ldquoThe planetesimal hypothesisrdquoTheObservatory vol52 pp 172ndash173 1929

[30] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsof omega and lambda from 42 high-redshift supernovaerdquo TheAstrophysical Journal vol 517 pp 565ndash586 1999

[31] E Mach Die Mechanik in ihrer Entiwcklung Eine historitschkritische Darstellung FA Brockhaus 1883

[32] D W Sciama ldquoOn the origin of inertiardquoMonthly Notices of theRoyal Astronomical Society vol 113 no 1 pp 34ndash42 1953

[33] J B Barbour and H Pfister Eds Machrsquos Principle FromNewtonrsquos Bucket to Quantum Gravity Birkhauser 1995

[34] J Thomson ldquoOn the electric and magnetic effects produced bythemotion of electrified bodiesrdquo PhilosophicalMagazine vol 11pp 229ndash249 1981

[35] G Searle ldquoOn the steady motion of an electrified ellipsoidrdquoPhilosophical Magazine vol 44 pp 329ndash341 1897

[36] H J Fahr ldquoThe Maxwellian alternative to the dark matterproblem in galaxiesrdquo Astronomy amp Astrophysics vol 236 pp86ndash94 1990

[37] H-J Fahr and M Sokaliwska ldquoThe influence of gravitationalbinding energy on cosmic expansion dynamics new perspec-tives for cosmologyrdquo Astrophysics and Space Science vol 339no 2 pp 379ndash387 2012

[38] J Overduin andH-J Fahr ldquoMatter spacetime and the vacuumrdquoNaturwissenschaften vol 88 no 12 pp 491ndash503 2001

[39] H Thirring ldquoUber die Wirkung rotierender ferner Massenin der Einsteinschen Gravitationstheorierdquo PhysikalischeZeitschrift vol 19 pp 33ndash39 1918

[40] J B Barbour ldquo General relativity as a perfectlyMachian theoryrdquoinMachrsquos Principle From Newtonrsquos Bucket to Quantum GravityJ B Barbour and H Pfister Eds p 214 Birkhauser 1995

[41] P S Wesson J Ponce de Leon H Liu et al ldquoA theory of spacetime andmatterrdquo International Journal ofModern Physics A vol11 no 18 pp 3247ndash3255 1996

[42] M Jammer and J Bain ldquoConcepts of mass in contemporaryphysics and philosophyrdquoPhysics Today vol 53 no 12 pp 67ndash682000

[43] G J Holtonl Einstein and the Search for Reality vol 6 1970[44] H Dehnen andH Honl ldquoFinite universe andMachrsquos principlerdquo

Nature vol 196 no 4852 pp 362ndash363 1962[45] W Hofmann ldquoMotion and inertiardquo in Machrsquos Principle From

Newtonrsquos Bucket to Quantum Gravity J B Barbour and HPfister Eds p 128 Birkhauser 1995

[46] H Reissner ldquoOn the relativity of accelerations in mechanicsrdquo inMachrsquos Principle From Newtonrsquos Bucket to Quantum Gravity JB Barbour and H Pfister Eds p 134 Birkhauser 1995

[47] D Lynden-Bell and J Katz ldquoClassical mechanics withoutabsolute spacerdquoPhysical ReviewD vol 52 no 12 pp 7322ndash73241995


[48] F Hoyle ldquoA new model for the expanding universerdquo MonthlyNotices of the Royal Astronomical Society vol 108 p 372 1948

[49] F Hoyle ldquoOn the relation of the large numbers problem to thenature of massrdquo Astrophysics and Space Science vol 168 no 1pp 59ndash88 1990

[50] F Hoyle ldquoMathematical theory of the origin of matterrdquo Astro-physics and Space Science vol 198 no 2 pp 195ndash230 1992

[51] F Hoyle G Burbidge and J V Narlikar ldquoA quasi-steady statecosmological model with creation of matterrdquoThe AstrophysicalJournal vol 410 no 2 pp 437ndash457 1993

[52] F Hoyle G Burbidge and J V Narlikar ldquoAstrophysical deduc-tions from the quasi-steady-state cosmologyrdquo Monthly Noticesof the Royal Astronomical Society vol 267 no 4 pp 1007ndash10191994

[53] H J Fahr and M Heyl ldquoConcerning the instantaneous massand the extent of an expanding universerdquo AstronomischeNachrichten vol 327 no 7 pp 733ndash736 2006

[54] H J Fahr ldquoThe cosmology of empty space how heavy is thevacuumrdquo Philosophy of Natural Sciences vol 33 pp 339ndash3532004

[55] H J Fahr andMHeyl ldquoAbout universes with scale-related totalmasses and their abolition of presently outstanding cosmolog-ical problemsrdquo Astronomische Nachrichten vol 328 no 2 pp192ndash199 2007

[56] H J Fahr and M Sokaliwska ldquoRevised concepts for cosmicvacuum energy and binding energy Innovative cosmologyrdquo inAspects of Todays Cosmology pp 95ndash120 2011

[57] P J Peebles and B Ratra ldquoThe cosmological constant and darkenergyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash6062003

[58] H-J Blome and W Priester ldquoVacuum energy in a Friedmann-Lemaıtre cosmosrdquoNaturwissenschaften vol 71 no 10 pp 528ndash531 1984

[59] H J Fahr ldquoThemodern concept of vacuum and its relevance forthe cosmological models of the universerdquo Philosophy of NaturalSciences vol 17 pp 48ndash60 1989

[60] I B Zeldovich ldquoVacuum theorymdasha possible solution to thesingularity problem of cosmologyrdquo Soviet Physics Uspekhi vol133 pp 479ndash503 1981

[61] ND Birrell andP CWDavies ldquoBook-reviewmdashquantumfieldsin curved spacerdquo Science vol 217 p 50 1982

[62] S K Lamoreaux ldquoSystematic correction for lsquodemonstrationof the Casimir force in the 06 to 6 micrometer rangersquordquohttparxivorgabs10074276

[63] J D Barrow Guide to Performing Relative Quantitation ofGene Expression Using Real-Time Quantitative PCR AppliedBiosystems Foster City Calif USA 2000

[64] H J Blome J Hoell andW Priester ldquoKosmologierdquo in Lehrbuchder Experimentalphysik vol 8 pp 439ndash582 Walter de GruyterBerlin Germany 2002

[65] HGoennerEinfuehrung in die Spezielle undAllgemeine Relativ-itaetstheorie Spektrum Akademischer Heidelberg Germany1996

[66] A Einstein ldquoKosmologische Betrachtungen zur allgemeinenRelativitatstheorierdquo in Sitzungsberichte der Koniglich Preussis-chen Akademie der Wissenschaften pp 142ndash152 Nabu Press1917

[67] H Fahr and M Heyl ldquoCosmic vacuum energy decay andcreation of cosmic matterrdquo Naturwissenschaften vol 94 pp709ndash724 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


In this paper we start out from a critical look on theproperties of cosmic microwave background (CMB) radi-ation the oldest picture of the universe and investigatebasic assumptions made when taking this background as thealmanac of basic cosmological facts Neither the exact initialthermodynamical equilibrium state of this CMB radiation isguaranteed nor its behaviour during the epochs of cosmicexpansion is predictable without strong assumptions on anunperturbed homologous expansion of the universe Theclaim connectedwith this assumption that theCMB radiationmust have been much hotter in the past may even bringcosmologists in unexpected explanatory needs to explain starformation in the early universe as will be shown

2 Does Planck Stay Planck If It Ever Was

21TheCosmicMicrowave BackgroundTested byCosmicTher-mometers It is generally well known that we are surroundedby the so-called cosmic microwave background (CMB)radiation This highly homogeneous and isotropic black-body radiation [1 5ndash7] is understood as relict of the earlycosmic recombination era when due to removal of electricallycharged particles by electron-proton recombinations the uni-verse for the times furtheron became transparent for photonsSince that time cosmic photons persistent from the times ofmatter-antimatter annihilations thus are propagating freelyon light geodetic trajectories through the spacetime geometryof the expanding universe up to the present days

Assuming that at the times before recombination matterand photons coexisted in perfect thermodynamical equilib-rium despite the expansion of the cosmic volume (we shallcome back to this problematic point in the next section)then this allows one to expect that these cosmic photonsinitially had a spectral distribution according to a perfectblack-body radiator that is a Planckıan spectrum It is thengenerally concluded that a perfectly homogeneous Planckıanradiation in an expanding universe stays rigorously Planckıanover all times that follow At this point one however one hasto emphasize that this conclusion can only be drawn if (a)the initial spectrum really is perfectly Planckian and if (b)the universe is perfectly homogeneous and expands in thehighest symmetrical form possible that is the one describedby the so-called Robertson-Walker spacetime geometry

Then it can be demonstrated (eg see [7]) that thePlanckıan character of the CMB spectral photon densityinitially given by

119889119899119903(120582) =

21205824

119889120582

exp [ℎ119888119870119879119903120582] minus 1

(1)

where 119889119899119903(120582) denotes the spectral photon density at the time

of recombination per wavelength interval 119889120582 at wavelength120582 and 119879

119903is the temperature of the Planck radiation at this

time is conserved for all ongoing periods of the expandinguniverse

Readers should however keep in mind that this isonly guaranteed if the universe has isotropic curvatureand expands in a homologous Robertson-Walker symmetricmanner (see eg [8]) Due to this fact it then turns out

that the initially Planckıan spectral photon density changeswith time so that for all cosmic future it maintains itsPlanckıan character however associated to a cosmologicallyreduced temperature 119879 lt 119879

119903 On one hand at a later time

119905 photons appear cosmologically redshifted to a wavelength1205821015840

= 120582(119877119877119903) and on the other hand they are redistributed

to a space volume increased by a factor (119877119877119903)3 Taking both

effects together shows that at a later time 119905 gt 119905119903the resulting

spectrum is given by

119889119899 (120582) = (

119877119903

119877

)

212058210158404

1198891205821015840

exp [ℎ1198881198701198791199031205821015840] minus 1

=

21205824

119889120582

exp [ℎ119888119870120582119879119903(119877119877119903)] minus 1

(2)

which with the help of Wienrsquos displacement law 119879 sdot 120582 =

const reveals that at later times it again is a Planck spectrumhowever with temperature 119879 = 119879

119903sdot (119877119903119877) This already

indicates that the present-day CMB should be associated to atemperature1198790 given by1198790 = 119879

119903sdot(1198771199031198770)where the quantities

indexed with ldquo0rdquo are those associated to the universe at thepresent time 119905 = 1199050 Depending on cosmic densities at therecombination phase the temperature 119879

119903should have been

between 3500K and 4500K (see [9])This indicates that withthe present-day CMB value of 1198790 = 2735K [1] a ratio ofcosmic expansion scales of

27353500

ge (

119877119903

1198770) = (

1198790119879119903

) ge

27354500 (3)

is disputableThe abovementioned theory of a homologous cosmic

expansion then also allows to derive an expression for thecosmic CMB temperature as a function of the cosmic photonredshift 119911 = (1205820 minus 120582

119890)120582119890at which astronomers are seeing

distant galactic objects Here 1205820 is the wavelength whichis observed at present that is at us while the associatedwavelength 120582

119890is emitted at the distant object With the

validity of the cosmological redshift relation in a Robertson-Walker universe

120582119890

1205820=

119877119890

1198770 (4)

where 119877119890and 1198770 denote the cosmic scale parameters at the

time 119905119890when the photon was emitted from the distant galaxy

and at the present time 1199050 Thus one obtains by definition

119911 =

1205820 minus 120582119890

120582119890

=

1198770119877119890

minus 1 (5)

This relation taken together with Wienrsquos law of spectralshift 120582max sdot 119879CMB = 03 in this context is expressed by

120582max (119911 = 0) sdot 119879CMB (119911 = 0) = 120582max (119911) sdot 119879CMB (119911) (6)

then finally allows to write the CMB temperature as thefollowing function of redshift

119879CMB (119911) = 119879CMB (119911 = 0) sdot120582max (119911 = 0)120582max (119911)

= 1198790CMB sdot (119911 + 1)

(7)







119891 (V 119905) = 119899 (119905) sdot 119891 (V 119905) (8)




1015840


6120601 = 119889

3V1198893119909 since


1198911015840

(V1015840 119905) 1198893V101584011988931199091015840

= 119891 (V 119905) 1198893V1198893119909 (9)



1015840



1198911015840

(V1015840 1199051015840) 1198893V101584011988931199091015840

= (119891 (V 119905) +120597119891

120597119905

Δ119905 +

120597119891

120597VΔV)

times(1+ ΔVV

)

2(1+119867Δ119905)

31198893V1198893

119909

= 119891 (V 119905) 1198893V1198893119909

(10)


120597119891

120597119905

Δ119905 +

120597119891

120597VΔV+ 2ΔV

V119891+ 3119867Δ119905119891 = 0 (11)

and thus

120597119891

120597119905

119897

Vminus 119897119867

120597119891

120597Vminus 2 119897119867

V119891+ 3119867 119897

V119891 = 0 (12)


120597119891

120597119905

= V119867120597119891

120597Vminus119867119891 (13)





119875

119899120574= (

1198750

119899120574


= (

1198750

119899120574

0) exp(minus(

23)119867 (119905 minus 1199050))

(14)


119878 = 119878 (119905) = 1198780 ln119875

119899120574= minus

23119867(119905 minus 1199050) (15)


1015840

(V1015840 119905) =


120597119891

120597119905

minus V119867120597119891

120597V= 0 (16)


120597119899

120597119905

= int 4120587V3119867120597119891

120597V119889V

= 4120587119867int

120597

120597V(V3119891)minus 12120587119867int V2119891119889V

(17)

yielding

120597119899

120597119905

= minus 3119899119867 (18)



120597119875

120597119905

=

41205873

int V5119867120597119891

120597V119889V = minus 5119867119875 (19)


119875

119899120574=

1198750

119899120574





120597119891

120597119905

= 119891[

119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

]

120597119891

120597V= minus119891

119898V119870119879

(21)


119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

= minus119867(

119898V2

119870119879

+ 1) (22)


119879 = 1198790 exp (minus2119867(119905 minus 1199050)) (23)


119879 =

119875

119870119899

=

11987501198701198990


= 1198790 exp (minus2119867(119905 minus 1199050))

(24)


119889 ln 119899

119889119905

minus

32

119879

119879

= minus119867 (25)

which then leads to

minus 2119867minus

32(minus2119867) = minus119867 (26)








119894minus 119864119895




119899119894

119899119895

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879CMB] (27)

where 119892119894119895



119864rot (119869) =ℎ2

81205872119868

119869 (119869 + 1) = 1198782(119869)

2119868= 119868

1205962(119869)

2 (28)


119868 (CO) = 1198862 119898C119898O119898C + 119898O

(29)


120582119895ge2 = 1205820 [

12minus

1119869 (119869 + 1)

] (30)





119899 (119869119894)

119899 (119869119895)

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879lowast

CMB] (31)



lowast


0CMB = 9315K










lowast



10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

4690 4691 4692 4693 4395 4396 4397 4398

4585 4586 4587 4588 4309 4310 4311 4312

4487 4488 4489 4490 4229 4230 4231


R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

P3R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

14

12

10

8

6

4

2

0140 141 142 143

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340


LogN(CO) (cmminus2)

2120590

CO A

X(3ndash0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4ndash0)

CO A

X(2ndash0)

CO A

X(5ndash0)

zabs = 2038







14

12

10

8

6

4

2

TCM

B(K

)

00 05 10 15 20 25 30

T0CMB(1 + z)

T0CMB(1 + z)(1minus120573)

⧬

z








Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)





119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)





119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)




119888 because


119888of the


⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)


35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)


1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)





119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)






119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









119891 (V 119905) = 119899 (119905) sdot 119891 (V 119905) (8)




1015840


6120601 = 119889

3V1198893119909 since


1198911015840

(V1015840 119905) 1198893V101584011988931199091015840

= 119891 (V 119905) 1198893V1198893119909 (9)



1015840



1198911015840

(V1015840 1199051015840) 1198893V101584011988931199091015840

= (119891 (V 119905) +120597119891

120597119905

Δ119905 +

120597119891

120597VΔV)

times(1+ ΔVV

)

2(1+119867Δ119905)

31198893V1198893

119909

= 119891 (V 119905) 1198893V1198893119909

(10)


120597119891

120597119905

Δ119905 +

120597119891

120597VΔV+ 2ΔV

V119891+ 3119867Δ119905119891 = 0 (11)

and thus

120597119891

120597119905

119897

Vminus 119897119867

120597119891

120597Vminus 2 119897119867

V119891+ 3119867 119897

V119891 = 0 (12)


120597119891

120597119905

= V119867120597119891

120597Vminus119867119891 (13)





119875

119899120574= (

1198750

119899120574


= (

1198750

119899120574

0) exp(minus(

23)119867 (119905 minus 1199050))

(14)


119878 = 119878 (119905) = 1198780 ln119875

119899120574= minus

23119867(119905 minus 1199050) (15)


1015840

(V1015840 119905) =


120597119891

120597119905

minus V119867120597119891

120597V= 0 (16)


120597119899

120597119905

= int 4120587V3119867120597119891

120597V119889V

= 4120587119867int

120597

120597V(V3119891)minus 12120587119867int V2119891119889V

(17)

yielding

120597119899

120597119905

= minus 3119899119867 (18)



120597119875

120597119905

=

41205873

int V5119867120597119891

120597V119889V = minus 5119867119875 (19)


119875

119899120574=

1198750

119899120574





120597119891

120597119905

= 119891[

119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

]

120597119891

120597V= minus119891

119898V119870119879

(21)


119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

= minus119867(

119898V2

119870119879

+ 1) (22)


119879 = 1198790 exp (minus2119867(119905 minus 1199050)) (23)


119879 =

119875

119870119899

=

11987501198701198990


= 1198790 exp (minus2119867(119905 minus 1199050))

(24)


119889 ln 119899

119889119905

minus

32

119879

119879

= minus119867 (25)

which then leads to

minus 2119867minus

32(minus2119867) = minus119867 (26)








119894minus 119864119895




119899119894

119899119895

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879CMB] (27)

where 119892119894119895



119864rot (119869) =ℎ2

81205872119868

119869 (119869 + 1) = 1198782(119869)

2119868= 119868

1205962(119869)

2 (28)


119868 (CO) = 1198862 119898C119898O119898C + 119898O

(29)


120582119895ge2 = 1205820 [

12minus

1119869 (119869 + 1)

] (30)





119899 (119869119894)

119899 (119869119895)

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879lowast

CMB] (31)



lowast


0CMB = 9315K










lowast



10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

4690 4691 4692 4693 4395 4396 4397 4398

4585 4586 4587 4588 4309 4310 4311 4312

4487 4488 4489 4490 4229 4230 4231


R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

P3R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

14

12

10

8

6

4

2

0140 141 142 143

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340


LogN(CO) (cmminus2)

2120590

CO A

X(3ndash0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4ndash0)

CO A

X(2ndash0)

CO A

X(5ndash0)

zabs = 2038







14

12

10

8

6

4

2

TCM

B(K

)

00 05 10 15 20 25 30

T0CMB(1 + z)

T0CMB(1 + z)(1minus120573)

⧬

z








Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)





119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)





119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)




119888 because


119888of the


⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)


35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)


1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)





119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)






119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119875

119899120574= (

1198750

119899120574


= (

1198750

119899120574

0) exp(minus(

23)119867 (119905 minus 1199050))

(14)


119878 = 119878 (119905) = 1198780 ln119875

119899120574= minus

23119867(119905 minus 1199050) (15)


1015840

(V1015840 119905) =


120597119891

120597119905

minus V119867120597119891

120597V= 0 (16)


120597119899

120597119905

= int 4120587V3119867120597119891

120597V119889V

= 4120587119867int

120597

120597V(V3119891)minus 12120587119867int V2119891119889V

(17)

yielding

120597119899

120597119905

= minus 3119899119867 (18)



120597119875

120597119905

=

41205873

int V5119867120597119891

120597V119889V = minus 5119867119875 (19)


119875

119899120574=

1198750

119899120574





120597119891

120597119905

= 119891[

119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

]

120597119891

120597V= minus119891

119898V119870119879

(21)


119889 ln 119899

119889119905

minus

32

119879

119879

+

119898V2

2119870119879

119879

119879

= minus119867(

119898V2

119870119879

+ 1) (22)


119879 = 1198790 exp (minus2119867(119905 minus 1199050)) (23)


119879 =

119875

119870119899

=

11987501198701198990


= 1198790 exp (minus2119867(119905 minus 1199050))

(24)


119889 ln 119899

119889119905

minus

32

119879

119879

= minus119867 (25)

which then leads to

minus 2119867minus

32(minus2119867) = minus119867 (26)








119894minus 119864119895




119899119894

119899119895

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879CMB] (27)

where 119892119894119895



119864rot (119869) =ℎ2

81205872119868

119869 (119869 + 1) = 1198782(119869)

2119868= 119868

1205962(119869)

2 (28)


119868 (CO) = 1198862 119898C119898O119898C + 119898O

(29)


120582119895ge2 = 1205820 [

12minus

1119869 (119869 + 1)

] (30)





119899 (119869119894)

119899 (119869119895)

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879lowast

CMB] (31)



lowast


0CMB = 9315K










lowast



10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

4690 4691 4692 4693 4395 4396 4397 4398

4585 4586 4587 4588 4309 4310 4311 4312

4487 4488 4489 4490 4229 4230 4231


R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

P3R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

14

12

10

8

6

4

2

0140 141 142 143

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340


LogN(CO) (cmminus2)

2120590

CO A

X(3ndash0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4ndash0)

CO A

X(2ndash0)

CO A

X(5ndash0)

zabs = 2038







14

12

10

8

6

4

2

TCM

B(K

)

00 05 10 15 20 25 30

T0CMB(1 + z)

T0CMB(1 + z)(1minus120573)

⧬

z








Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)





119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)





119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)




119888 because


119888of the


⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)


35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)


1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)





119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)






119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119894minus 119864119895




119899119894

119899119895

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879CMB] (27)

where 119892119894119895



119864rot (119869) =ℎ2

81205872119868

119869 (119869 + 1) = 1198782(119869)

2119868= 119868

1205962(119869)

2 (28)


119868 (CO) = 1198862 119898C119898O119898C + 119898O

(29)


120582119895ge2 = 1205820 [

12minus

1119869 (119869 + 1)

] (30)





119899 (119869119894)

119899 (119869119895)

sim

119892119894

119892119895

exp[minus

ℎ (119864119894minus 119864119895)

119870119879lowast

CMB] (31)



lowast


0CMB = 9315K










lowast



10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

4690 4691 4692 4693 4395 4396 4397 4398

4585 4586 4587 4588 4309 4310 4311 4312

4487 4488 4489 4490 4229 4230 4231


R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

P3R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

14

12

10

8

6

4

2

0140 141 142 143

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340


LogN(CO) (cmminus2)

2120590

CO A

X(3ndash0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4ndash0)

CO A

X(2ndash0)

CO A

X(5ndash0)

zabs = 2038







14

12

10

8

6

4

2

TCM

B(K

)

00 05 10 15 20 25 30

T0CMB(1 + z)

T0CMB(1 + z)(1minus120573)

⧬

z








Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)





119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)





119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)




119888 because


119888of the


⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)


35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)


1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)





119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)






119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

10

05

00Nor

mal

ized

flux

4690 4691 4692 4693 4395 4396 4397 4398

4585 4586 4587 4588 4309 4310 4311 4312

4487 4488 4489 4490 4229 4230 4231


R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

P3R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

R3 R2 R1 R0 Q1

Q2

Q3 P2 P3 R3 R2 R1 R0 Q1

Q2

Q3 P2 P3

14

12

10

8

6

4

2

0140 141 142 143

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340


LogN(CO) (cmminus2)

2120590

CO A

X(3ndash0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4ndash0)

CO A

X(2ndash0)

CO A

X(5ndash0)

zabs = 2038







14

12

10

8

6

4

2

TCM

B(K

)

00 05 10 15 20 25 30

T0CMB(1 + z)

T0CMB(1 + z)(1minus120573)

⧬

z








Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)





119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)





119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)




119888 because


119888of the


⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)


35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)


1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)





119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)






119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







Δ11987912 = 119879119903[

119877119903

1198771minus

119877119903

1198772] (32)





119864119861=

1615

1205872119866120588

21198775=

35119866

1198722

119877

(33)





119889119864119861

119889119905

= minus

35119866

1198722

1198772119889119877

119889119905

= 41205871198772120590sb (119879

4119888minus119879

4CMB) (34)




119888 because


119888of the


⟨

119889119877

119889119905

⟩ = minus

119877

120591ff= minus119877radic4120587119866120588 (35)


35119866

1198722

1198772 119877radic4120587119866120588 = 41205871198772

120590sb1198794119888

(36)


1198794119888=

320120587120590sb

119866

1198722

1198773 radic4120587119866120588 =

radic41205875120590sb

11986632

11987212058832

(37)





119879119888= (5220)14 119870 ≃ 85 sdot 119870 (38)






119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119888should



0CMB sdot (1 + 119911


enough delivers 119911119888= 119879119888119879










3 Conclusions












Appendices









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









119898= 120588( 119903)V





Φ = minusint

119881

120588

119903

119889119881 = minus120588int

119903infin

4120587119903 119889119903 = minus 21205871205881199032infin (A1)


=



= minus

1119888

int

119881

119895

119903

119889119881 = 0 (A2)


= minusint

119881

119895

119903

119889119881 = minus

1119888

int

119881

120588 (V + 119903119867)

119903

119889119881 =

Φ

119888

V (A3)


119892seen by the



1119888

120597

120597119905

= minus

Φ

1198882120597V120597119905

119892= rot = 0

(A4)


= minus

119872

1199032 (

119903

119903

) minus

Φ

1198882120597V120597119905

(A5)


119872

1199032 = minus

Φ

1198882120597V120597119905

(A6)


119895of



119898119895sim int120588

119889119881

119903

= sum

119894

119898119894

119903119894

(A7)


119895












120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










120596

M

R

minus120596

M






119880119877119880) is a constant





119880being









119898119895sim sum

119894

119898119894

119903119894

(A8)


119898119895sim sum

119894

119898119894

119903119894

= int120588

119889119881

119903

= 120588int

119877119906 41205871199032119889119903

119903

=

32119872119906

119877119906

= const

(A9)








119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










119880




119880is






1198721198801198882= 41205871205880119888

2int

119877119880

0

exp (120582 (119903) 2) 1199032119889119903

radic1 minus (1198670119903119888)2

(B1)


exp (120582 (119903))

=

1

1 minus (81205871198661199031198882) 1205880 int119903

0 (1199092119889119909radic1 minus (1198670119909119888)

2)

(B2)








119880


119877119880=

1120587

radic1198882

21198661205880 (B3)




1205880 (119877119880) =1198882

21205872119866119877

2119880

(B4)



119872119880= 16152119888

2

120587119866

119877119880≃

1198882

119866

119877119880 (B5)




119880of the







120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






120583]

and 119879119889





119879vac120583] = (120588vac119888

2+119901vac)119880120583119880] minus119901vac119892120583] (C1)






119879vac120583] = 120588vac119888

2119892120583] (C2)



Λ eff =

81205871198661198882 120588vac minusΛ (C3)



Λ eff =

81205871198661198882 (120588vac minus120588vac0) = 0 (C4)





1198923 = 119892111198922211989233 = minus

11987761199034sin2120599

1 minus 1198701199032 (C6)


119889119890vac = 120598vacradic

11987761199034sin2120599

1 minus 1198701199032 119889119903119889120599119889120593

= 120598vac1198773 119903

2 sin 120599

radic1 minus 1198701199032119889119903119889120599119889120593

(C7)


120598vac = 120588vac1198882sim 119877 (119905)

minus3 (C8)




119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C9)


119901vac = minus

3 minus ]3

120598vac (C10)



0= const


119889120598vac1198773

119889119877

= minus119901vac119889

119889119877

1198773

minus

81205872119866

151198884119889

119889119877

[(120598vac + 3119901vac)21198775]

(C11)



1198772= minus 3119901vac119877

2

minus

81205872119866

1511988846 minus 3]3 minus ]

119889119901vac1198775

119889119903

(C12)


120598vac sim 119877minus2 (C13)





119889

119889119877

(120598vac1198773) = minus119901vac

119889

119889119877

1198773 (C14)


119889

119889119877

(120598vac011987720

1198772119877

3) = 120598vac119877

2= minus 3119901vac119877

2 (C15)


119901vac = minus

13120598vac (C16)





References













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Remaining Problems in Interpretation of...

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