Research Article Remaining Problems in Interpretation of the Cosmic … · 2017. 12. 4. · Research Article Remaining Problems in Interpretation of the Cosmic Microwave Background

Research ArticleRemaining Problems in Interpretation of the CosmicMicrowave Background

Hans-Jörg Fahr and Michael Sokaliwska

Argelander Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany

Correspondence should be addressed to Michael Sokaliwska; [email protected]

Received 25 June 2014; Accepted 7 April 2015

Academic Editor: Avishai Dekel

Copyright © 2015 H.-J. Fahr and M. Sokaliwska.This 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 𝑧 = 2 where nevertheless the cosmologically most relevant supernovae have beenobserved.The 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 𝑅 = 𝑅(𝑡). 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 COBE,continued 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 𝐻0 = 73 km/s/Mpc and an age of the universe of𝜏0 = 13.7GYr [1] as eminent findings of the present epoch,when one accepts a universe that presently expands in anaccelerated form due to being driven by vacuum pressure.This 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 horizon,then 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 pageshttp://dx.doi.org/10.1155/2015/503106

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?

2.1.TheCosmicMicrowave 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, 5–7] 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 photons.Since 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 Planckı́an spectrum. It is thengenerally concluded that a perfectly homogeneous Planckı́anradiation in an expanding universe stays rigorously Planckı́anover 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 Planckiàn 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 (e.g., see [7]) that thePlanckı́an character of the CMB spectral photon densityinitially given by

𝑑𝑛𝑟(𝜆) =

2𝜆4

𝑑𝜆

exp [ℎ𝑐/𝐾𝑇𝑟𝜆] − 1

, (1)

where 𝑑𝑛𝑟(𝜆) denotes the spectral photon density at the time

of recombination per wavelength interval 𝑑𝜆 at wavelength𝜆 and 𝑇

𝑟is 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, e.g., [8]). Due to this fact it then turns out

that the initially Planckı́an spectral photon density changeswith time so that for all cosmic future it maintains itsPlanckı̀an character, however, associated to a cosmologicallyreduced temperature 𝑇 < 𝑇

𝑟. On one hand at a later time

𝑡 photons appear cosmologically redshifted to a wavelength𝜆

= 𝜆(𝑅/𝑅𝑟), and on the other hand they are redistributed

to a space volume increased by a factor (𝑅/𝑅𝑟)3. Taking both

effects together shows that at a later time 𝑡 > 𝑡𝑟the resulting

spectrum is given by

𝑑𝑛 (𝜆) = (

𝑅𝑟

𝑅

)

2𝜆4

𝑑𝜆

exp [ℎ𝑐/𝐾𝑇𝑟𝜆] − 1

=

2𝜆4

𝑑𝜆

exp [ℎ𝑐/𝐾𝜆𝑇𝑟(𝑅/𝑅𝑟)] − 1

(2)

which with the help of Wien’s displacement law 𝑇 ⋅ 𝜆 =const reveals that at later times it again is a Planck spectrum,however, with temperature 𝑇 = 𝑇

𝑟⋅ (𝑅𝑟/𝑅). This already

indicates that the present-day CMB should be associated to atemperature𝑇0 given by𝑇0 = 𝑇𝑟 ⋅(𝑅𝑟/𝑅0)where the quantitiesindexed with “0” are those associated to the universe at thepresent time 𝑡 = 𝑡0. Depending on cosmic densities at therecombination phase the temperature 𝑇

𝑟should have been

between 3500K and 4500K (see [9]).This indicates that withthe present-day CMB value of 𝑇0 = 2.735K [1] a ratio ofcosmic expansion scales of

2.7353500

≥ (

𝑅𝑟

𝑅0) = (

𝑇0𝑇𝑟

) ≥

2.7354500 (3)

is disputable.The 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 𝑧 = (𝜆0 − 𝜆𝑒)/𝜆𝑒 at which astronomers are seeingdistant galactic objects. Here 𝜆0 is the wavelength whichis observed at present, that is, at us, while the associatedwavelength 𝜆

𝑒is emitted at the distant object. With the

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

𝜆𝑒

𝜆0=

𝑅𝑒

𝑅0, (4)

where 𝑅𝑒and 𝑅0 denote the cosmic scale parameters at the

time 𝑡𝑒when the photon was emitted from the distant galaxy

and at the present time 𝑡0. Thus one obtains by definition

𝑧 =

𝜆0 − 𝜆𝑒𝜆𝑒

=

𝑅0𝑅𝑒

− 1. (5)

This relation taken together with Wien’s law of spectralshift 𝜆max ⋅ 𝑇CMB = 0.3 in this context is expressed by

𝜆max (𝑧 = 0) ⋅ 𝑇CMB (𝑧 = 0) = 𝜆max (𝑧) ⋅ 𝑇CMB (𝑧) (6)

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

𝑇CMB (𝑧) = 𝑇CMB (𝑧 = 0) ⋅𝜆max (𝑧 = 0)𝜆max (𝑧)

= 𝑇0CMB ⋅ (𝑧 + 1) .

(7)

Physics Research International 3

2.2. Particle Distribution Functions in Expanding Spacetimes.Usually 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 recombination,like 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 (1/V4) (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 ≥ V

𝑐. 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:

𝑓 (V, 𝑡) = 𝑛 (𝑡) ⋅ 𝑓 (V, 𝑡) , (8)

where 𝑛(𝑡) denotes the cosmologically varying density onlydepending on the worldtime 𝑡 and 𝑓(V, 𝑡) is the normalized,time-dependent isotropic velocity distribution function withthe property: ∫𝑓(V, 𝑡)𝑑3V = 1.

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

prevailing covariant, but perhaps form-invariant distributionfunction 𝑓(V, 𝑡), then the associated functions 𝑓(V, 𝑡) and𝑓(V, 𝑡) must be related to each other in a very specific way.To 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 𝑑6𝜙 = 𝑑3V𝑑3𝑥, sinceno Lagrangian exists and thus no Hamiltonian canonicalrelations for their dynamical coordinates V and 𝑥 are valid.Hence Liouville’s theorem then requires that the conjugateddifferential phase space densities are identical; that is,

𝑓

(V, 𝑡) 𝑑3V𝑑3𝑥 = 𝑓 (V, 𝑡) 𝑑3V𝑑3𝑥. (9)

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

= 𝑙 ⋅ 𝐻 coaligned with V⃗, where 𝐻 = 𝐻(𝑡) means thetime-dependent Hubble parameter.Thus the original particlevelocity V is locally turned to V = V − 𝑙 ⋅ 𝐻. All dimensionsof the space volume within a time Δ𝑡 are cosmologicallyexpanded, so that 𝑑𝑥 = 𝑑𝑥(1 + 𝐻Δ𝑡) holds. Completereintegration into the locally valid distribution function thenimplies, with linearizably small quantities Δ𝑡 ≃ 𝑙/V and ΔV =−𝑙 ⋅ 𝐻, that one can express the above requirement in thefollowing form:

𝑓

(V, 𝑡) 𝑑3V𝑑3𝑥

= (𝑓 (V, 𝑡) +𝜕𝑓

𝜕𝑡

Δ𝑡 +

𝜕𝑓

𝜕VΔV)

×(1+ ΔVV

)

2(1+𝐻Δ𝑡)3 𝑑3V𝑑3𝑥

= 𝑓 (V, 𝑡) 𝑑3V𝑑3𝑥.

(10)

This then means for terms of first order that

𝜕𝑓

𝜕𝑡

Δ𝑡 +

𝜕𝑓

𝜕VΔV+ 2ΔV

V𝑓+ 3𝐻Δ𝑡𝑓 = 0 (11)

and thus

𝜕𝑓

𝜕𝑡

𝑙

V− 𝑙𝐻

𝜕𝑓

𝜕V− 2 𝑙𝐻

V𝑓+ 3𝐻 𝑙

V𝑓 = 0 (12)

or the following requirement:

𝜕𝑓

𝜕𝑡

= V𝐻𝜕𝑓

𝜕V−𝐻𝑓. (13)

Looking first here for interesting velocity moments ofthe function 𝑓 fulfilling the above partial differential equa-tion by multiplying this equation with (a) 4𝜋V2𝑑V and (b)(4𝜋/3)𝑚V4𝑑V and integrating over velocity space then leadsto

𝑎 : 𝑛 = 𝑛0 exp(−2𝐻(𝑡 − 𝑡0)),and𝑏 : 𝑃 = 𝑃0 exp(−4𝐻(𝑡 − 𝑡0))


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

𝑃

𝑛𝛾= (

𝑃0

𝑛𝛾

0) exp (− (4− 2𝛾)𝐻 (𝑡 − 𝑡0))

= (

𝑃0

𝑛𝛾

0) exp(−(2

3)𝐻 (𝑡 − 𝑡0))

(14)

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

𝑆 = 𝑆 (𝑡) = 𝑆0 ln𝑃

𝑛𝛾= −

23𝐻(𝑡 − 𝑡0) . (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 𝑓(V, 𝑡) =𝑓(V, 𝑡) and hence would lead to the following form of aVlasow equation:

𝜕𝑓

𝜕𝑡

− V𝐻𝜕𝑓

𝜕V= 0. (16)

In that case the first velocity moment is found with

𝜕𝑛

𝜕𝑡

= ∫ 4𝜋V3𝐻𝜕𝑓

𝜕V𝑑V

= 4𝜋𝐻∫ 𝜕𝜕V

(V3𝑓)− 12𝜋𝐻∫ V2𝑓𝑑V(17)

yielding

𝜕𝑛

𝜕𝑡

= − 3𝑛𝐻 (18)

which agrees with 𝑛 ∼ 𝑅−3. Looking also for the highermoment 𝑃 then leads to

𝜕𝑃

𝜕𝑡

=

4𝜋3

∫ V5𝐻𝜕𝑓

𝜕V𝑑V = − 5𝐻𝑃 (19)

which now in this case shows that

𝑃

𝑛𝛾=

𝑃0

𝑛𝛾

0exp (− (5− 3𝛾)𝐻 (𝑡 − 𝑡0)) = const!!!. (20)

That means in this case an adiabatic expansion is found,however, 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 for𝑓 ∼ 𝑛𝑇

−3/2 exp[−𝑚V2/2𝐾𝑇], 𝑛 and 𝑇 being time-dependent,that one has

𝜕𝑓

𝜕𝑡

= 𝑓[

𝑑 ln 𝑛𝑑𝑡

−

32

̇𝑇

𝑇

+

𝑚V2

2𝐾𝑇̇

𝑇

𝑇

] ,

𝜕𝑓

𝜕V= −𝑓

𝑚V𝐾𝑇

(21)

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


−

32

̇𝑇

𝑇

+

𝑚V2

2𝐾𝑇̇

𝑇

𝑇

= −𝐻(

𝑚V2

𝐾𝑇

+ 1) . (22)

In order to fulfill the above equation obviously the termswith V2 have to cancel each other, since 𝑛 and 𝑇 are velocitymoments of𝑓, hence independent on V.This is evidently onlysatisfied, if the change of the temperature with cosmic time isgiven by

𝑇 = 𝑇0 exp (−2𝐻(𝑡 − 𝑡0)) . (23)

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

𝑇 =

𝑃

𝐾𝑛

=

𝑃0𝐾𝑛0

exp (− (4− 2)𝐻 (𝑡 − 𝑡0))

= 𝑇0 exp (−2𝐻(𝑡 − 𝑡0)) .(24)

With that the above requirement (22) then only reducesto


−

32

̇𝑇

𝑇

= −𝐻 (25)

which then leads to

− 2𝐻− 32(−2𝐻) = −𝐻 (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 function𝑓. Under these auspices it can, however,also 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 𝑡recaround 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.

2.3. Can the Cosmological CMB Cooling Be Confirmed? Inthe following part of the paper we now want to investigatewhether or not the cosmological cooling of theCMBphotons,freely propagating in the expanding Robertson-Walker spacetime geometry, can be confirmed by observations.The 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 expectations,for 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 past,really 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 population.In this respect especially interesting are molecular specieswith an energy splitting of vibrational or rotational excitationlevels 𝑖, 𝑗 that correspond to mean energies of the surround-ing CMB photons; that is, 𝐸

𝑖− 𝐸𝑗

= ℎ]CMB. Under suchconditions the relative level populations 𝑛

𝑖, 𝑛𝑗essentially are

given by the associated Boltzmann factor

𝑛𝑖

𝑛𝑗

∼

𝑔𝑖

𝑔𝑗

exp[−ℎ (𝐸𝑖− 𝐸𝑗)

𝐾𝑇CMB] , (27)

where 𝑔𝑖,𝑗

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 𝐽. According to these numbers a splittingof CO lines occurs with transitions characterized by Δ𝐽 = 1.In this respect the transition 𝐽 = 1 → 𝐽 = 0 leads to abasic emission line at 𝜆1,0 = 2.6mm (i.e., ]0 = 115,6GHz).The CO-molecule is biatomic with a rotation around anaxis perpendicular to the atomic interconnection line. Thequantum energies 𝐸rot(𝐽) are given by

𝐸rot (𝐽) =ℎ2

8𝜋2𝐼𝐽 (𝐽 + 1) = 𝑆

2(𝐽)

2𝐼= 𝐼

𝜔2(𝐽)

2, (28)

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

𝐼 (CO) = 𝑎2𝑚C𝑚O

𝑚C + 𝑚O. (29)

Here 𝑎 is the interconnection distance, and 𝑚C, 𝑚O arethe masses of the carbon and oxygen atom, respectively. 𝑆(𝐽)is the angular momentum of the state with quantum number𝐽, and 𝜔(𝐽) is the associated angular rotation frequency. Theemission wavelengths from the excited states of the CO-A-Xbands (𝐽 ≥ 2) thus are given by

𝜆𝑗≥2 = 𝜆0 [

12−

1𝐽 (𝐽 + 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 (i.e.,

≥115 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 𝑧abs =2.41837 illuminated by a background quasar SDSS J143912.04+ 111740.5. Then the CO fine structure lines appear inabsorption at wavelengths between 4900 Å and 5200 Å and,by fitting them with Voigt-profiles, the relative populations(𝑛(𝐽𝑖)/𝑛(𝐽𝑗)) 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

𝑛 (𝐽𝑖)

𝑛 (𝐽𝑗)

∼

𝑔𝑖

𝑔𝑗

exp[−ℎ (𝐸𝑖− 𝐸𝑗)

𝐾𝑇∗

CMB] , (31)

where now 𝑇∗CMB is the CMB Planck temperature at cosmicredshift 𝑧abs = 2.41837. On the basis of the abovementionedassumptions Srianand et al. [3], depending on the specifictransitions which they fit, find CMB excitation temperaturesof 𝑇∗CMB(0, 1) = 9.11 ± 1.23K; 𝑇

∗

CMB(1, 2) = 9.19 ± 1.21K;and 𝑇∗CMB(0, 2) = 9.16 ± 0.77K, while according to standardcosmology (see (7)) at a redshift 𝑧abs = 2.41837 one shouldhave a CMB temperature of𝑇∗CMB = (1+𝑧abs)𝑇

0CMB = 9.315K,

where 𝑇0CMB = 2.725K is the present-day CMB temperature(see [14]).

Though this clearly points to the fact that CMB tempera-tures 𝑇∗CMB at higher redshifts are indicated to be higher thanthe present-day temperature 𝑇0CMB, it also demonstrates thatthe cosmologically expected value should have been a fewpercent higher than these fitted values.This, 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

measurable.The 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 𝑇∗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


1.0

0.5

0.0Nor

mal

ized

flux

1.0

0.5

0.0Nor

mal

ized

flux

1.0

0.5

0.0Nor

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 (Å) Observed wavelength (Å)

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

014.0 14.1 14.2 14.3

Tex(C

O)

(K)

SDSS-J1-70542 +

+

354340

Tex(CO) = 8.6+1.1−1.0 K

LogN(CO) (cm−2)

2𝜎

CO A

X(3–0)

CO A

X(1-0)

CO A

X(0-0)

CO A

X(4–0)

CO A

X(2–0)

CO A

X(5–0)

zabs = 2.038

Figure 1: CO absorption profiles observed at SDSS-J1-70542 + 354340 (𝑧abs = 2.038); 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.

2.4. 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 𝑇CMB 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 2.2 of this paper). Point (b) is

14

12

10

8

6

4

2

TCM

B(K

)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T0CMB(1 + z)

T0CMB(1 + z)(1−𝛽)

⧬

z

Figure 2: CMB temperatures as function of redshift 𝑧 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 anisotropic,destroying completely the Planckian character of the CMB.This 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 regions,as 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 dynamics.Thus CMB photons arriving from the two opposite sides aredifferently cosmologically redshifted and thus in no case doconstitute one common Planckian spectrum with one jointtemperature 𝑇CMB, 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 are𝑅1 and𝑅2, then thiswould lead to a hemispheric CMB temperature difference ofΔ𝑇1,2 given by (see [7])

Δ𝑇1,2 = 𝑇𝑟 [𝑅𝑟

𝑅1−

𝑅𝑟

𝑅2] (32)

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

2.5. 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

𝐸𝐵=

1615

𝜋2𝐺𝜌

2𝑅5=

35𝐺

𝑀2

𝑅

, (33)

where 𝐺 is the gravitation constant, 𝜌 is the mass density ofthe gas, 𝑅 is the radius of the cloud, and 𝑀 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 𝐸

𝐵can effectively be radiated off to

space from the outer surface of the cloud, that is, if

𝑑𝐸𝐵

𝑑𝑡

= −

35𝐺

𝑀2

𝑅2𝑑𝑅

𝑑𝑡

= 4𝜋𝑅2𝜎sb (𝑇4𝑐−𝑇

4CMB) , (34)

where 𝜎sb denote the Stefan-Boltzmann constant and 𝑇𝑐 thethermal radiation temperature of the cloud, respectively.

This already makes evident that further contraction ofthe cloud is impeded, if the surrounding CMB temperatureexceeds the cloud temperature, that is, if 𝑇CMB > 𝑇𝑐, becausethen the only possibility is 𝑑𝑅/𝑑𝑡 ≥ 0!, that is, expansion!In order to calculate the radiation temperature 𝑇

𝑐of the

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

⟨

𝑑𝑅

𝑑𝑡

⟩ = −

𝑅

𝜏ff= −𝑅√4𝜋𝐺𝜌, (35)

where 𝜏ff 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:

35𝐺

𝑀2

𝑅2 𝑅√4𝜋𝐺𝜌 = 4𝜋𝑅

2𝜎sb𝑇

4𝑐

(36)

which allows to find the following value for the cloudtemperature:

𝑇4𝑐=

320𝜋𝜎sb

𝐺

𝑀2

𝑅3 √4𝜋𝐺𝜌 =

√4𝜋5𝜎sb

𝐺3/2

𝑀𝜌3/2

. (37)

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

𝑐we here assume that the typical cloud mass can be

adopted with 𝑀 = 10𝑀⊙and that, for mass fragmentation

of that size to occur, primordial molecular cloud conditionswith an H2−density of the order of 𝜌/2𝑚 = 10

5 cm−3 must beadopted. With these values one then calculates a temperatureof

𝑇𝑐= (5220)1/4 𝐾 ≃ 8.5 ⋅ 𝐾. (38)

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


were becoming greater than this above value 𝑇𝑐, then stellar

mass fragmentations of masses of the order of 𝑀 ≃ 10𝑀⊙

were not possible anymore. This would mean that galaxies atsupercritical distances correlated with redshifts 𝑧 ≥ 𝑧

𝑐should

not be able to produce stars with stellar masses larger than10𝑀⊙. This critical redshift can be easily calculated from the

linear cooling relation 𝑇𝑐= 𝑇

0CMB ⋅ (1 + 𝑧𝑐) and interestingly

enough delivers 𝑧𝑐= 𝑇𝑐/𝑇

0CMB − 1 = 2.09. This means that

galaxies at distances beyond such redshifts, that is, with 𝑧 ≥𝑧𝑐= 2.09, should not be able to produce stars with stellar

masses greater than 10𝑀⊙.

If on the other hand it iswell known amongst astronomersthat galaxies with redshifts 𝑧 ≥ 𝑧

𝑐are 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 10𝑀

⊙-stars, then cosmology

obviously is running into a substantial problem.

3. Conclusions

This paper hopefully has at least made evident that the “so-called” 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 2.2). 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 2.2. 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 distributions,implying the drastic consequence that baryon densities arenot falling off as (1/𝑅3) but as (1/𝑅2). The interaction ofthe 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 Wien’s 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 GHz.Our estimate for conventionally assumed cos-mologicmodel ingredients (Omegas!) would be by about 1𝐾!

UnfortunatelyCMBmeasurements at frequencies beyond103 GHZ are practically absent up to now and do not allowto identify these differences. If in upcoming time periods,on the basis of upcoming better measurements in the Wien’sbranch 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 2.3, 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 (i.e., following the relation 𝑇CMB ∼ (1 +𝑧)𝑇0,CMB) in fact brings astrophysicists rather into severeproblems in understanding the origin of massive stars indistant galaxies seen at large redshifts 𝑧 ≥ 𝑧

𝑐(see Section 2.5).

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 move,as 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 Newton’s concept of absolute space and themotions of objects with respect to this space. According toI. Newton, inertial reactions proportional to objects’ 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.

A.1. Linear Masses and Scima’s Approach to Mach’s Idea.Velocity 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 object’s acceleration with respect to all masses in theuniverse be quantifiable?This first thinking already show thatthe question of inertia very directly brings one into deepestcalamities.Thefirstmore 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 universe.Thus inertiacannot be taken as a genuine quantity of every body, ratherinertiamust be a “relational” 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 Mach’s inertial principle, and the fathers of relativityand cosmology always were deeply moved by this principle,though they all never managed to construct a cosmologictheory which did fulfill Mach’s 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. Sciama’s 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 field.The vector potential �⃗� thereby describes the gravitationalaction of mass currents ⃗𝑗

𝑚, and each moving object in the

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

𝑚= 𝜌( ⃗𝑟)V⃗

𝜌( ⃗𝑟)which are intimately connected

with the object’s own motion. Here 𝜌( ⃗𝑟) and V⃗𝜌denote the

mass density and its bulk motion, respectively.In a homogeneous matter universe with mass density 𝜌

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

Φ = −∫

𝑉

𝜌

𝑟

𝑑𝑉 = −𝜌∫

𝑟∞

4𝜋𝑟 𝑑𝑟 = − 2𝜋𝜌𝑟2∞, (A.1)

where the upper integration border is the mass horizon 𝑟∞

=

𝑐/𝐻. Furthermore the vector potential �⃗� in a homogeneous

and homologously expanding Hubble universe vanishes,since

�⃗� = −

1𝑐

∫

𝑉

⃗𝑗

𝑟

𝑑𝑉 = 0. (A.2)

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

�⃗� = −∫

𝑉

⃗𝑗

𝑟

𝑑𝑉 = −

1𝑐

∫

𝑉

𝜌 (V⃗ + ⃗𝑟𝐻)𝑟

𝑑𝑉 =

Φ

𝑐

V⃗. (A.3)

The gravitoelectromagnetic fields �⃗�𝑔and �⃗�

𝑔seen by the

moving particle thus are

�⃗�𝑔= − gradΦ− 1

𝑐

𝜕�⃗�

𝜕𝑡

= −

Φ

𝑐2𝜕V⃗𝜕𝑡

,

�⃗�𝑔= rot�⃗� = 0.

(A.4)

Assuming an additional body with mass 𝑀 at a distance⃗𝑟 from the test particle, where ⃗𝑟 may be taken as collinear toV⃗, then leads to the following total gravitational force actingon the test particle:

�⃗� = −

𝑀

𝑟2 (

⃗𝑟

𝑟

) −

Φ

𝑐2𝜕V⃗𝜕𝑡

. (A.5)

Considering Newton’s second law, describing the gravi-tational attraction between two masses, it then requires for�⃗� = 0 that the following relation is valid:

𝑀

𝑟2 = −

Φ

𝑐2𝜕V𝜕𝑡

(A.6)

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 𝑚

𝑗of

this test object 𝑗 at a cosmic place 𝑟𝑗, when replacing density

by distributed masses𝑚𝑖, is represented by the expression

𝑚𝑗∼ ∫𝜌

𝑑𝑉

𝑟

= ∑

𝑖

𝑚𝑖

𝑟𝑖

, (A.7)

where again here 𝜌 is the cosmic mass density and 𝑟 is thedistance to the cosmic mass source with the volume 𝑑𝑉. Theabove summation runs over all other objects “𝑖” in cosmicspace besides that with index “𝑗.” Hereby it turns out thataccording to Sciamaś theory the required inertial masses 𝑚

𝑗

are related to all other cosmic masses 𝑚𝑖and their inverse

distances (1/𝑟𝑖). Hence this formulation fulfills Mach’s basic

idea, that is, its ideological request.To enable this argumentation a Maxwellian analogy of

gravity to electromagnetism was adopted. This, however,seems 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.


A.2. 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 (e.g., 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, e.g., [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 universe.Thus 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, e.g., [33]). Since the earthrotates, centrifugal forces act, and the earth’s 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 answered:the rotation period of the earth! But this answer just nowcontains the real basic question: namely, the rotation period𝜏Ωwith 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 firmament,that is, the fixstar horizon. Why however just these mostdistant stars at a rotation should determine the inertialreaction of the earth’s 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

𝜔

M

R

−𝜔

M

?

Figure 3: The universe as a mass-shell.

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

𝑈and a homogeneous

mass deposition 𝑀𝑈

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 Thirring’s findings; namely, that a fullequivalence of the systems, (a) earth rotating and (b) universerotating, only exists, if the ratio (𝑀

𝑈/𝑅𝑈) is a constant,

where𝑀𝑈is the total mass of the universe within the cosmic

mass horizon 𝑅𝑈

= 𝑐/𝐻0 increasing proportional with theincreasing age of the universe. Here 𝑐 is the light velocity, and𝐻0 denotes the present Hubble constant.

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

𝑈being

time-dependent and increasing with worldtime 𝑡 as 𝑅𝑈(𝑡) =

𝑐/𝐻(𝑡). It would, namely, mean that the equivalence ofrotations in an expanding universe can only be and stay valid,if the mass of the universe increases with time such that(𝑀𝑈,0/𝑅𝑈,0) = 𝑀𝑈(𝑡)/𝑅𝑈(𝑡) stays constant, that is, if the

mass 𝑀𝑈increases linearly with 𝑅

𝑈. 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 Mach’s 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 𝑚

𝑗of the object 𝑗 given

by all other masses𝑚𝑗by the expression

𝑚𝑗∼ ∑

𝑖

𝑚𝑖

𝑟𝑖

. (A.8)

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

𝑚𝑗∼ ∑

𝑖

𝑚𝑖

𝑟𝑖

= ∫𝜌

𝑑𝑉

𝑟

= 𝜌∫

𝑅𝑢 4𝜋𝑟2𝑑𝑟

𝑟

=

32𝑀𝑢

𝑅𝑢

= const.

(A.9)


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 byMach’s principle, but later in his career he abandoned it [43].Up to the present days it is debated whether or not Einstein’sGR theory can be called a “Machian” or a “non-Machian”theory. At least some attempts have been made to develop anadequate form of a “relational,” that is, Machian, mechanics[8, 44–47]. 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 Thirring’s considerations of thenature of centrifugal forces were based on the concept of themass of the universe 𝑀

𝑈. To better understand Thirring’s

result that this mass 𝑀𝑈should vary with the radius 𝑅

𝑈

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 𝑀

𝑈as 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 number𝑀

𝑈is

independent on the selected reference point. This means𝑀𝑈

represents the space-like sum of all masses simultaneouslysurrounding this point within its associated mass horizon. Ifat the time 𝑡 a cosmic mass density 𝜌(𝑡) prevails, then thewhole mass integral up to the greatest distances has to becarried out using this density 𝜌(𝑡), 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 density𝜌 = 𝜌0. 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 𝑀

𝑈as shown by Fahr and Heyl [53] is

given by the following expression:

𝑀𝑈𝑐2= 4𝜋𝜌0𝑐

2∫

𝑅𝑈

0

exp (𝜆 (𝑟) /2) 𝑟2𝑑𝑟

√1 − (𝐻0𝑟/𝑐)2

, (B.1)

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

exp (𝜆 (𝑟))

=

1

1 − (8𝜋𝐺/𝑟𝑐2) 𝜌0 ∫𝑟

0 (𝑥2𝑑𝑥/√1 − (𝐻0𝑥/𝑐)

2)

.(B.2)

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

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 𝛾(𝑟) = (1 − (𝐻0𝑟/𝑐)

2)−1. Assuming that within

the integration border Hubble motions are subrelativistic onemay evaluate the above expression with 𝛾(𝑟) = 1.

Then the above expression for𝑀𝑈shows that real-valued

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

𝑈

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

𝑅𝑈=

1𝜋

√𝑐2

2𝐺𝜌0. (B.3)

This result is very interesting sincemeaning that thismasshorizon distance𝑅

𝑈appears related to the actual cosmicmass

density by the expression

𝜌0 (𝑅𝑈) =𝑐2

2𝜋2𝐺𝑅2𝑈

(B.4)

and as evident from carrying out the integration in (B.2) leadsto a point-associated mass𝑀

𝑈of the universe given by

𝑀𝑈= 1.6152𝑐

2

𝜋𝐺

𝑅𝑈≃

𝑐2

𝐺

𝑅𝑈. (B.5)

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

𝑈Thirring’s relation in (B.1) is

in fact fulfilled but also proves that Mach’s idea on the basisof this newly introduced definition of the mass 𝑀

𝑈of 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 problem:What should apriori be expectable from empty space and how to formulateuncontroversial conditions for it and its physical behaviour?The main point to pay attention to is perhaps that the basicmechanical principle which was pretty clear at Newton’sepoch of classical mechanics, namely, “actio = reactio”, shouldsomehow also still be valid at times of modern cosmology.So if at all the energy of empty space causes something tohappen, then that “something” 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 [54–56]). There is, however,a 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.(e.g., 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, e.g., [41, 54, 58–64]). Due to itsenergy content, this vacuum influences spacetime geometry,but it is not yet clear in which way specifically. Normalbaryonic or darkionic matter (i.e., constituted by baryons ordark matter particles, darkions, resp.) general-relativisticallyact through their associated energy-momentum tensors 𝑇𝑏

𝜇]

and 𝑇𝑑𝜇] (e.g., see [65]). Consequently it has been tried to also

describe the GRT action of an energy-loaded vacuum intro-ducing an associated hydrodynamical energy-momentumtensor 𝑇vac

𝜇] 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-sities.Thequestion then evidently ariseswhat is vacuumpres-sure 𝑝vac and what is vacuum mass density 𝜌vac? Not goingdeeper into this point at the moment, one nevertheless thencan give the tensor 𝑇vac

𝜇] in the following form (see, e.g., [38]):

𝑇vac𝜇] = (𝜌vac𝑐

2+𝑝vac)𝑈𝜇𝑈] −𝑝vac𝑔𝜇], (C.1)

where 𝑈𝜆are the components of the vacuum fluid 4-velocity

vector and 𝑔𝜇] 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 derived𝜌vac𝑐

2= −𝑝vac (e.g., see [38, 57]). Under these prerequisites

the vacuum fluid tensor 𝑇vac𝜇] attains the simple form

𝑇vac𝜇] = 𝜌vac𝑐

2𝑔𝜇]. (C.2)

The above term for 𝑇vac𝜇] for a constant vacuum energy

density 𝜌vac can be combined with the famous integrationconstantΛ that was introduced by Einstein [66] into his GRTfield equations and then formally leads to something like an“effective cosmological constant”:

Λ eff =8𝜋𝐺𝑐2 𝜌vac −Λ. (C.3)

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

𝜌vac = 𝜌vac,0 does not gravitate at all or curve spacetime,because this completely empty space is just described by avanishing effective constant (i.e., pure vacuumdoes not curvespacetime):

Λ eff =8𝜋𝐺𝑐2 (𝜌vac −𝜌vac,0) = 0. (C.4)

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 𝜌vac,0 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 cosmology,as a purely spacetime- or volume-related quantity, it never-theless is by far not evident that “vacuum energy density”should thus be a constant quantity, simply because the unitof space volume is not a cosmologically relevant quantity.It may perhaps be much more reasonable to envision thatthe amount of vacuum energy of a homologously comovingproper volume 𝐷𝑉 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 𝜖vac, is the vacuumenergy within a proper volume given by

𝑑𝑒vac = 𝜖vac√−𝑔3𝑑3𝑉, (C.5)

where 𝑔3 is the determinant of the 3D-space metric.In case of a Robertson-Walker geometry this is given by

𝑔3 = 𝑔11𝑔22𝑔33 = −𝑅6𝑟4sin2𝜗

1 − 𝐾𝑟2. (C.6)

Here 𝐾 is the curvature parameter and 𝑅 = 𝑅(𝑡) is thetime-dependent scale of the universe.The differential 3-spacevolume element in normalized polar coordinates is given by𝑑3𝑉 = 𝑑𝑟 𝑑𝜗 𝑑𝜑 and thus leads to

𝑑𝑒vac = 𝜖vac√

𝑅6𝑟4sin2𝜗

1 − 𝐾𝑟2𝑑𝑟𝑑𝜗𝑑𝜑

= 𝜖vac𝑅3 𝑟

2 sin 𝜗√1 − 𝐾𝑟2

𝑑𝑟𝑑𝜗𝑑𝜑.

(C.7)

If𝑑𝑒vac now is taken as a cosmologically constant quantity,then it evidently requires that vacuum energy density has tochange like

𝜖vac = 𝜌vac𝑐2∼ 𝑅 (𝑡)

−3. (C.8)


The invariance of the vacuum energy per comovingproper volume, 𝑑𝑒vac, 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 𝑅(𝑡) 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]):

𝑑

𝑑𝑅

(𝜖vac𝑅3) = −𝑝vac

𝑑

𝑑𝑅

𝑅3. (C.9)

This equation is fulfilled by a functional relation of theform

𝑝vac = −3 − ]3

𝜖vac (C.10)

for a scale-dependent vacuum energy density in the form𝜖vac ∼ 𝑅

−]. Then it is evident that the above thermodynamiccondition, besides for the trivial case ] = 3 when vacuumdoes not act (since: 𝑝vac(] = 3) = 0!, i.e., pressure-lessvacuum), is as well fulfilled by other values of ], as, forinstance, by ] = 0, that is, a constant vacuum energy density𝜖vac ∼ 𝑅

0= const.

The exponent ] is, however, more rigorously restricted,if under more general cosmic conditions the above ther-modynamic expression (C.9) 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 theory,and has been quantitatively formulated by Fahr and Heyl[55] and Fahr and Heyl [67]. With this term, the enlargedthermodynamic equation (C.9) then attains the followingcompleted form:

𝑑𝜖vac𝑅3

𝑑𝑅

= −𝑝vac𝑑

𝑑𝑅

𝑅3

−

8𝜋2𝐺15𝑐4

𝑑

𝑑𝑅

[(𝜖vac + 3𝑝vac)2𝑅5] ,

(C.11)

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 𝑝vac = −((3 − ])/3)𝜖vac which then leads to

−3 (3 − ]) 𝑝vac3 − ]

𝑅2= − 3𝑝vac𝑅

2

−

8𝜋2𝐺15𝑐4

6 − 3]3 − ]

𝑑𝑝vac𝑅5

𝑑𝑟

.

(C.12)

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

𝜖vac ∼ 𝑅−2 (C.13)

expressing the fact that under these general conditionsvacuum energy density should fall off with 𝑅−2, 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 𝑅−2. Thefirst to conclude from this is that the vacuum pressure 𝑝vacunder this condition should behave like prescribed by thethermodynamic equation (C.9):

𝑑

𝑑𝑅

(𝜖vac𝑅3) = −𝑝vac

𝑑

𝑑𝑅

𝑅3 (C.14)

and thus under the new auspices given now yield

𝑑

𝑑𝑅

(𝜖vac,0𝑅20

𝑅2𝑅

3) = 𝜖vac𝑅

2= − 3𝑝vac𝑅

2 (C.15)

meaning that now the following polytropic relation holds

𝑝vac = −13𝜖vac (C.16)

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.

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