# Remaining Problems in Interpretation of the Cosmic Microwave Background

Authors: Hans-Jörg Fahr; Michael Sokaliwska
Journal: Physics Research International
 (2015)

---

## Abstract

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


---

## Body

## 1. Introduction

The cosmic background radiation (CMB) has been continuously 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 continuous measurements our knowledge of the structure of the CMB has tremendously grown, representing nowadays this cosmologically highly relevant phenomenon in an enormous quality of spectral and spatial resolution; these data, however good in quality, do not speak for themselves. They rather need to be interpreted on the basis of a theoretical context understanding of the CMB origin. The latter, however, has not grown in quality as CMB data have. This paper wants to show some aspects of modern cosmological research in new lights. Thereby it may also serve readers with some hesitation towards present-day cosmology and give them some encouragement. One needs to be convinced that a scientific discipline like cosmology is built on safe conceptual and physical grounds, before one can appreciate the most recent messages from modern precision cosmology. One only can appreciate cosmological numbers like a Hubble constant of H
0
=
73 km/s/Mpc and an age of the universe of τ
0
=
13.7 GYr [1] as eminent findings of the present epoch, when one accepts a universe that presently expands in an accelerated form due to being driven by vacuum pressure. This puts the question what are the basic prerequisites of modern cosmology?At first it is the assumption that all relevant facts determining the global structures of the universe and their internal dynamics have been found at present times. This puts the question what part of the world may presently be screened out by our world horizon, which nevertheless influences the cosmological reality inside? If, as generally believed, the cosmic microwave background (CMB) sky is such a horizon, then everything deeper in the cosmological past must be invented as a cosmologic ingredient that never becomes an observational fact. On the other hand, when inside that horizon only something not of global but of local relevance is seen, then the extrapolation from what is seen to the whole universe is scientifically questionable.In this paper we start out from a critical look on the properties of cosmic microwave background (CMB) radiation, the oldest picture of the universe, and investigate basic assumptions made when taking this background as the almanac of basic cosmological facts. Neither the exact initial thermodynamical equilibrium state of this CMB radiation is guaranteed, nor its behaviour during the epochs of cosmic expansion is predictable without strong assumptions on an unperturbed homologous expansion of the universe. The claim connected with this assumption that the CMB radiation must have been much hotter in the past may even bring cosmologists in unexpected explanatory needs to explain star formation in the early universe as will be shown.


## 2. Does Planck Stay Planck, If It Ever Was?

### 2.1. The Cosmic Microwave Background Tested by Cosmic Thermometers

It is generally well known that we are surrounded by 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 early cosmic recombination era when due to removal of electrically charged particles by electron-proton recombinations the universe for the times furtheron became transparent for photons. Since that time cosmic photons, persistent from the times of matter-antimatter annihilations, thus are propagating freely on light geodetic trajectories through the spacetime geometry of the expanding universe up to the present days.Assuming that at the times before recombination matter and photons coexisted in perfect thermodynamical equilibrium, despite the expansion of the cosmic volume (we shall come back to this problematic point in the next section), then this allows one to expect that these cosmic photons initially had a spectral distribution according to a perfect black-body radiator, that is, a Planckían spectrum. It is then generally concluded that a perfectly homogeneous Planckían radiation in an expanding universe stays rigorously Planckían over all times that follow. At this point one, however, one has to 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 the highest symmetrical form possible, that is, the one described by the so-called Robertson-Walker spacetime geometry.Then it can be demonstrated (e.g., see [7]) that the Planckían character of the CMB spectral photon density initially given by(1)
d
n
r
λ
=
2
λ
4
d
λ
exp
⁡
h
c
/
K
T
r
λ
-
1
,where d
n
r
(
λ
) denotes the spectral photon density at the time of recombination per wavelength interval d
λ at wavelength λ and T
r is the temperature of the Planck radiation at this time, is conserved for all ongoing periods of the expanding universe.Readers should, however, keep in mind that this is only guaranteed, if the universe has isotropic curvature and expands in a homologous, Robertson-Walker symmetric manner. (see, e.g., [8]). Due to this fact it then turns out that the initially Planckían spectral photon density changes with time so that for all cosmic future it maintains its Planckìan character, however, associated to a cosmologically reduced temperature T
<
T
r. On one hand at a later time t photons appear cosmologically redshifted to a wavelength λ
′
=
λ
(
R
/
R
r
), and on the other hand they are redistributed to a space volume increased by a factor (
R
/
R
r
)
3. Taking both effects together shows that at a later time t
>
t
r the resulting spectrum is given by(2)
d
n
λ
=
R
r
R
2
λ
′
4
d
λ
′
exp
⁡
h
c
/
K
T
r
λ
′
-
1
=
2
λ
4
d
λ
exp
⁡
h
c
/
K
λ
T
r
R
/
R
r
-
1which with the help of Wien’s displacement law T
·
λ
=
c
o
n
s
t reveals that at later times it again is a Planck spectrum, however, with temperature T
=
T
r
·
(
R
r
/
R
). This already indicates that the present-day CMB should be associated to a temperature T
0 given by T
0
=
T
r
·
(
R
r
/
R
0
) where the quantities indexed with “0” are those associated to the universe at the present time t
=
t
0. Depending on cosmic densities at the recombination phase the temperature T
r should have been between 3500 K and 4500 K (see [9]). This indicates that with the present-day CMB value of T
0
=
2.735 K [1] a ratio of cosmic expansion scales of(3)
2.735
3500
≥
R
r
R
0
=
T
0
T
r
≥
2.735
4500is disputable.The abovementioned theory of a homologous cosmic expansion then also allows to derive an expression for the cosmic CMB temperature as a function of the cosmic photon redshiftz
=
(
λ
0
-
λ
e
)
/
λ
e at which astronomers are seeing distant galactic objects. Here λ
0 is the wavelength which is observed at present, that is, at us, while the associated wavelength λ
e is emitted at the distant object. With the validity of the cosmological redshift relation in a Robertson-Walker universe,(4)
λ
e
λ
0
=
R
e
R
0
,where R
e and R
0 denote the cosmic scale parameters at the time t
e when the photon was emitted from the distant galaxy and at the present time t
0. Thus one obtains by definition(5)
z
=
λ
0
-
λ
e
λ
e
=
R
0
R
e
-
1
.This relation taken together with Wien’s law of spectral shiftλ
m
a
x
·
T
C
M
B
=
0.3 in this context is expressed by(6)
λ
m
a
x
z
=
0
·
T
C
M
B
z
=
0
=
λ
m
a
x
z
·
T
C
M
B
zthen finally allows to write the CMB temperature as the following function of redshift:(7)
T
C
M
B
z
=
T
C
M
B
z
=
0
·
λ
m
a
x
z
=
0
λ
m
a
x
z
=
T
C
M
B
0
·
z
+
1
.


### 2.2. Particle Distribution Functions in Expanding Spacetimes

Usually it is assumed that at the recombination era photons and matter, that is, electrons and protons in this phase of the cosmic evolution, are dynamically tightly bound to each other and undergo strong mutual interactions via Coulomb collisions and Compton collisions. These conditions are thought to then evidently guarantee a pure thermodynamical equilibrium state, implying that particles are Maxwell distributed and photons have a Planckian blackbody distribution. It is, however, by far not so evident that these assumptions really are fulfilled. This is because photons and particles are reacting to the cosmological expansion very differently; photons generally are cooling cosmologically being redshifted, while particles in first order are not directly feeling the expansion, unless they feel it adiabatically by mediation through numerous Coulomb collisions, which are relevant here in a fully ionized plasma before recombination, like they do in a box with subsonic expansion of its walls. But Coulomb collisions have a specific property which is highly problematic in this context.This is because Coulomb collision cross sections are strongly dependent on the particle velocityv, namely, being proportional to (
1
/
v
4
) (see [10]). This evidently causes that high-velocity particles are much less collision-dominated compared to low-velocity ones; they are even collision-free at supercritical velocities v
≥
v
c. So while the low-velocity branch of the distribution may still cool adiabatically and thus feels cosmic expansion in an adiabatic form, the high-velocity branch in contrast behaves collision-free and hence changes in a different form. This violates the concept of a joint equilibrium temperature and of a resulting Maxwellian velocity distribution function and means that there may be a critical evolutionary phase of the universe, due to different forms of cooling in the low- and high-velocity branches of the particle velocity distribution function, which do not permit the persistence of a Maxwellian equilibrium distribution to later cosmic times.In the following part of the paper we demonstrate that even if a Maxwellian distribution would still prevail at the beginning of the collision-free expansion phase, that is, the postrecombination phase era, it would not persist in the universe during the ongoing of the collision-free expansion. For that purpose let us first consider a collision-free population in an expanding Robertson-Walker universe. It is clear that due to the cosmological principle and, connected with it, the homogeneity requirement, the velocity distribution function of the particles must be isotropic, that is, independent on the local place, and thus of the following general form:(8)
f
v
,
t
=
n
t
·
f
¯
v
,
t
,where n
(
t
) denotes the cosmologically varying density only depending on the worldtime t and f
¯
(
v
,
t
) is the normalized, time-dependent isotropic velocity distribution function with the property: ∫
f
¯
(
v
,
t
)
d
3
v
=
1.If we assume that particles, moving freely with their velocityv into the v
→-associated direction over a distance l, are restituting at this new place, despite the differential Hubble flow and the explicit time-dependence of f, a locally prevailing covariant, but perhaps form-invariant distribution function f
′
(
v
′
,
t
′
), then the associated functions f
(
v
′
,
t
) and f
(
v
,
t
) must be related to each other in a very specific way. To define this relation needs some special care, since particles that are freely moving in an homologously expanding Hubble universe do in this case at their motions not conserve their associated phasespace volumes d
6
ϕ
=
d
3
v
d
3
x, since no Lagrangian exists and thus no Hamiltonian canonical relations for their dynamical coordinates v and x are valid. Hence Liouville’s theorem then requires that the conjugated differential phase space densities are identical; that is,(9)
f
′
v
′
,
t
d
3
v
′
d
3
x
′
=
f
v
,
t
d
3
v
d
3
x
.At the place where they arrive after passage over a distancel the particle population has a relative Hubble drift given by v
H
=
l
·
H coaligned with v
→, where H
=
H
(
t
) means the time-dependent Hubble parameter. Thus the original particle velocity v is locally turned to v
′
=
v
-
l
·
H. All dimensions of the space volume within a time Δ
t are cosmologically expanded, so that d
x
′
=
d
x
(
1
+
H
Δ
t
) holds. Complete reintegration into the locally valid distribution function then implies, with linearizably small quantities Δ
t
≃
l
/
v and Δ
v
=
-
l
·
H, that one can express the above requirement in the following form:(10)
f
′
v
′
,
t
′
d
3
v
′
d
3
x
′
=
f
v
,
t
+
∂
f
∂
t
Δ
t
+
∂
f
∂
v
Δ
v
×
1
+
Δ
v
v
2
1
+
H
Δ
t
3
d
3
v
d
3
x
=
f
v
,
t
d
3
v
d
3
x
.This then means for terms of first order that(11)
∂
f
∂
t
Δ
t
+
∂
f
∂
v
Δ
v
+
2
Δ
v
v
f
+
3
H
Δ
t
f
=
0and thus(12)
∂
f
∂
t
l
v
-
l
H
∂
f
∂
v
-
2
l
H
v
f
+
3
H
l
v
f
=
0or the following requirement:(13)
∂
f
∂
t
=
v
H
∂
f
∂
v
-
H
f
.Looking first here for interesting velocity moments of the functionf fulfilling the above partial differential equation by multiplying this equation with (a) 4
π
v
2
d
v and (b) (
4
π
/
3
)
m
v
4
d
v and integrating over velocity space then leads toa
:
n
=
n
0
exp
⁡
-
2
H
t
-
t
0,
and
b
:
P
=
P
0
exp
⁡
-
4
H
t
-
t
0which then immediately makes evident that with the above solutions one finds that(14)
P
n
γ
=
P
0
n
0
γ
exp
⁡
-
4
-
2
γ
H
t
-
t
0
=
P
0
n
0
γ
exp
⁡
-
2
3
H
t
-
t
0is not constant, meaning that no adiabatic behaviour of the expanding gas occurs and that the gas entropy S also is not constant but decreasing and given by(15)
S
=
S
t
=
S
0
ln
⁡
P
n
γ
=
-
2
3
H
t
-
t
0
.It is perhaps historically interesting to see that assuming Hamilton canonical relations to be valid the Liouville theorem would then instead of (9) simply require f
′
(
v
′
,
t
)
=
f
(
v
,
t
) and hence would lead to the following form of a Vlasow equation:(16)
∂
f
∂
t
-
v
H
∂
f
∂
v
=
0
.In that case the first velocity moment is found with(17)
∂
n
∂
t
=
∫
4
π
v
3
H
∂
f
∂
v
d
v
=
4
π
H
∫
∂
∂
v
v
3
f
-
12
π
H
∫
v
2
f
d
vyielding(18)
∂
n
∂
t
=
-
3
n
Hwhich agrees withn
~
R
-
3. Looking also for the higher moment P then leads to(19)
∂
P
∂
t
=
4
π
3
∫
v
5
H
∂
f
∂
v
d
v
=
-
5
H
Pwhich now in this case shows that(20)
P
n
γ
=
P
0
n
0
γ
exp
⁡
-
5
-
3
γ
H
t
-
t
0
=
const
!
!
!
.That means in this case an adiabatic expansion is found, however, based on wrong assumptions!Now going back to the correct Vlasow equation (13) one can then check whether or not this equation allows that an initial Maxwellian velocity distribution function persists during the ongoing collision-free expansion. Here we find for f
~
n
T
-
3
/
2
exp
⁡
[
-
m
v
2
/
2
K
T
], n and T being time-dependent, that one has(21)
∂
f
∂
t
=
f
d
ln
⁡
n
d
t
-
3
2
T
˙
T
+
m
v
2
2
K
T
T
˙
T
,
∂
f
∂
v
=
-
f
m
v
K
Tleading to the following Vlasow requirement (see (13)):(22)
d
ln
⁡
n
d
t
-
3
2
T
˙
T
+
m
v
2
2
K
T
T
˙
T
=
-
H
m
v
2
K
T
+
1
.In order to fulfill the above equation obviously the terms withv
2 have to cancel each other, since n and T are velocity moments of f, hence independent on v. This is evidently only satisfied, if the change of the temperature with cosmic time is given by(23)
T
=
T
0
exp
⁡
-
2
H
t
-
t
0
.This dependence in fact is obtained when inspecting the earlier found solutions for the momentsn and P (see (13) and (14)), because these solutions exactly give(24)
T
=
P
K
n
=
P
0
K
n
0
exp
⁡
-
4
-
2
H
t
-
t
0
=
T
0
exp
⁡
-
2
H
t
-
t
0
.With that the above requirement (22) then only reduces to(25)
d
ln
⁡
n
d
t
-
3
2
T
˙
T
=
-
Hwhich then leads to(26)
-
2
H
-
3
2
-
2
H
=
-
Hmaking it evident that this requirement is not fulfilled and thus meaning that consequently a Maxwellian distribution cannot be maintained, even not at a collision-free expansion.This finally leads to the statement that a correctly derived Vlasow equation for the cosmic gas particles leads to a collision-free expansion behaviour that neither runs adiabatic nor does it conserve the Maxwellian form of the distribution functionf. Under these auspices it can, however, also easily be demonstrated (see [11]) that collisional interaction of cosmic photons with cosmic particles via Compton collisions in case of non-Maxwellian particle distributions does unavoidably lead to deviations from the Planckian blackbody spectrum. This makes it hard to be convinced by a pure Planck spectrum of the CMB photons at the time t
r
e
c around the cosmic matter recombination.Let us therefore now look into other basic concepts of cosmology to see whether perhaps also there problems can be identified which should caution cosmologists.


### 2.3. Can the Cosmological CMB Cooling Be Confirmed?

In the following part of the paper we now want to investigate whether or not the cosmological cooling of the CMB photons, freely propagating in the expanding Robertson-Walker space time geometry, can be confirmed by observations. The access to this problem is given by the connection that in an expanding universe at earlier cosmic times the CMB radiation should have 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 larger redshifts, that is, those seen at times in the distant past, really give indications that they in fact are embedded in a correspondingly hotter CMB radiation environment. For that purpose one generally uses appropriate, so-called CMB radiation thermometers like interstellar CN-, CH-, or CO-molecular species (see [3, 12, 13], or [4]).Assuming that molecular interstellar gas phases within these galaxies are in optically thin contact to the CMB that actually surrounds these galaxies allows one to assume that such molecular species are populated in their electronic levels according to a quasistationary equilibrium state population. In this respect especially interesting are molecular species with an energy splitting of vibrational or rotational excitation levelsi
,
j that correspond to mean energies of the surrounding CMB photons; that is, E
i
-
E
j
=
h
ν
C
M
B. Under such conditions the relative level populations n
i
,
n
j essentially are given by the associated Boltzmann factor(27)
n
i
n
j
~
g
i
g
j
exp
⁡
-
h
E
i
-
E
j
K
T
C
M
B
,where g
i
,
j are the state multiplicities. In the years of the recent past interstellar CO-molecules have been proven to be best suited in this respect as highly appropriate CMB thermometers. This was demonstrated by Srianand et al. [3] and Noterdaeme et al. [4].The carbon monoxide molecule CO splits into different rotational excitation levels according to different rotational quantum numbersJ. According to these numbers a splitting of CO lines occurs with transitions characterized by Δ
J
=
1. In this respect the transition J
=
1

→

J
=
0 leads to a basic emission line at λ
1,0
=
2.6 mm (i.e., ν
0
=
115
,

6 GHz). The CO-molecule is biatomic with a rotation around an axis perpendicular to the atomic interconnection line. The quantum energies E
r
o
t
(
J
) are given by (28)
E
r
o
t
J
=
h
2
8
π
2
I
J
J
+
1
=
S
2
J
2
I
=
I
ω
2
J
2
,where I is the moment of inertia of the CO-rotator and is given by(29)
I
C
O
=
a
2
m
C
m
O
m
C
+
m
O
.Herea is the interconnection distance, and m
C, m
O are the masses of the carbon and oxygen atom, respectively. S
(
J
) is the angular momentum of the state with quantum number J, and ω
(
J
) is the associated angular rotation frequency. The emission wavelengths from the excited states of the CO-A-X bands (J
≥
2) thus are given by(30)
λ
j
≥
2
=
λ
0
1
2
-
1
J
J
+
1
.Usually it is hardly possible to detect these CO-fine structure emissions from distant galaxies directly, due to their weaknesses and due to the strong perturbations and contaminations in this frequency range by the infrared (i.e., ≥115 GHz). Instead the relative population of these rotational fine structure levels can much better be observed in absorption appearing in the optical range. To actually use such a constellation to determine the relative populations of CO fine structure levels one needs a broadband continuum emitter in the cosmic background behind a gas-containing galaxy in the foreground. As in case of the object investigated by Srianand et al. [3] the foreground galaxy is at a redshift of z
a
b
s
=
2.41837 illuminated by a background quasar SDSS J143912.04 + 111740.5. Then the CO fine structure lines appear in absorption at wavelengths between 4900 Å and 5200 Å and, by fitting them with Voigt-profiles, the relative populations (
n
(
J
i
)
/
n
(
J
j
)
) of these fine structure levels can be determined. Assuming now optically thin conditions of the absorbing gas with respect to CMB photons, one can assume that in a photostationary equilibrium these relative populations are connected with the abovementioned Boltzmann factor as(31)
n
J
i
n
J
j
~
g
i
g
j
exp
⁡
-
h
E
i
-
E
j
K
T
C
M
B
∗
,where now T
C
M
B
∗ is the CMB Planck temperature at cosmic redshift z
a
b
s
=
2.41837. On the basis of the abovementioned assumptions Srianand et al. [3], depending on the specific transitions which they fit, find CMB excitation temperatures of T
C
M
B
∗
(
0,1
)
=
9.11
±
1.23 K; T
C
M
B
∗
(
1,2
)
=
9.19
±
1.21 K; and T
C
M
B
∗
(
0,2
)
=
9.16
±
0.77 K, while according to standard cosmology (see (7)) at a redshift z
a
b
s
=
2.41837 one should have a CMB temperature of T
C
M
B
∗
=
(
1
+
z
a
b
s
)
T
C
M
B
0
=
9.315 K, where T
C
M
B
0
=
2.725 K is the present-day CMB temperature (see [14]).Though this clearly points to the fact that CMB temperaturesT
C
M
B
∗ at higher redshifts are indicated to be higher than the present-day temperature T
C
M
B
0, it also demonstrates that the cosmologically expected value should have been a few percent higher than these fitted values. This, however, cannot question the applicability of the above described method in general, though some basic caveats have to be mentioned here.First of all, observers with similar observations are often running into optically thick CO absorption conditions which will render the fitting procedure more difficult. Noterdaeme et al. [4], for instance, can show that the fitted CMB temperature differs with the CO-column density of the foreground absorber (see Figure 1). The determination of these column densities in itself is a highly nontrivial endeavour and only can be carried out assuming some fixed correlations between CO- and H2-column densities, the latter being much better measurable.Figure 1
CO absorption profiles observed at SDSS-J1-70542 + 354340 (z
a
b
s
=
2.038); figure taken from Srianand et al. [3].The second caveat in this context is connected with the assumption that relative populations of fine structure levels are purely determined by a photon excitation equilibrium with the surrounding CMB photons. If in addition any binary collisions with other molecules or any photons other than CMB photons are interfering into these population processes, then of course the fittedT
C
M
B
∗ values have to be taken with correspondingly great caution. Especially in the infrared range delivering the relevant photons for excitations or deexcitations the CMB spectrum is strongly contaminated by galactic dust emissions [1, 15, 16]. Facing then the possibility that galaxies at higher redshifts are more pronounced in galactic dust emissions compared to our present galaxies nearby then makes CMB temperature determinations perhaps questionable. Nevertheless the results obtained by Noterdaeme et al. [4] when determining CO-excitation temperatures at foreground galaxies with different redshifts perhaps for most readers do convincingly demonstrate that a linear correlation of the CMB temperature with redshift can be confirmed (see Figure 2) as expected.Figure 2
CMB temperatures as function of redshiftz derived from CO-excitation temperatures; figure taken from Noterdaeme et al. [4].


### 2.4. Problems with a Hot CMB in the Past

Though from the results displayed in the above Figure2 it seems as if the cosmological CMB cooling with time can be surprisingly well confirmed, one nevertheless should not too carelessly take that as an observational fact. We remind the reader first to the theoretical prerequisites of a cosmologic CMB cooling reflected in a decrease of the Planck temperature T
C
M
B of this radiation: a Planckian spectrum only stays a Planckian, if(a)
it was Planckian already at the beginning, that is, at the recombination phase, and if
(b)
since that time a completely homologous cosmic expansion took place till today.Point (a) is questionable because the thermodynamic equilibrium state between baryons and photons in the early phase of fast cosmic expansion may quite well be disturbed or incomplete (see [7, 11], Section 2.2 of this paper). Point (b) is questionable, since at present times we find a highly structured, inhomogeneous cosmic matter distribution which does not originate from a homogeneous matter cosmos with a pure, unperturbed Robertson-Walker cosmic expansion.The present universe actually is highly structured by galaxies, galaxy clusters, superclusters, and walls [17, 18]. Although perhaps the matter distribution was quite homogeneous at the epoch of the last scattering of cosmic photons when the CMB photons were in close contact to the cosmic matter, during the evolutionary times after that matter distribution has evidently become very inhomogeneous by the gravitational growth of seed structures. Thus fitting a perfectly symmetrical Robertson-Walker spacetime geometry to a universe with a lumpy matter distribution appears highly questionable [19]. This is an eminent general relativistic problem as discussed by Buchert [20], Buchert [21], Buchert [22], Buchert [23], Buchert [24], and Wiltshire [25]. If due to that structuring processes in the cosmic past and the associated geometrical perturbations of the Robertson-Walker geometry we would look back into direction-dependent different expansion histories of the universe, this would point towards associated CMB fluctuations (see [7]).Thus it should be kept in mind that a CMB Planck spectrum is only seen with the same temperature from all directions of the sky, if in all these directions the same expansion dynamics of the universe took place. If CMB photons arriving from different directions of the sky have seen different expansion histories, then their Planck temperatures would of course be different and anisotropic, destroying completely the Planckian character of the CMB. This situation evidently comes up in case an anisotropic and nonhomologous cosmic expansion takes place like that envisioned and described in theories by Buchert [23], Buchert [26], Buchert [27], Buchert [24], or Wiltshire [25]. Let us check this situation by a simple-minded approach here: in the two-phase universe consisting of void and wall regions, as described by Wiltshire [25], void expansions turn out to be different from wall expansions, and, when looking out from the surface border of a wall region, in the one hemisphere one would see the void expansion dynamics, whereas in the opposite hemisphere one sees the wall expansion dynamics. Thus CMB photons arriving from the two opposite sides are differently cosmologically redshifted and thus in no case do constitute one common Planckian spectrum with one joint temperature T
C
M
B, but rather a bipolar feature of the local CMB-horizon.In fact if one hemisphere expands different from the opposite hemisphere, then as a reaction also different CMB Planck temperatures would have to be ascribed to the CMB photons arriving from these opposite hemispherical directions. If, for instance, the present values of the characteristic scale in the two opposite hemispheres areR
1 and R
2, then this would lead to a hemispheric CMB temperature difference of Δ
T
1,2 given by (see [7])(32)
Δ
T
1,2
=
T
r
R
r
R
1
-
R
r
R
2and would give an alternative to the present-day CMB-dipole explanation.


### 2.5. Hot CMB Impedes Gas Fragmentation

Stars are formed due to gravitational fragmentation of parts of a condensed interstellar molecular cloud. For the occurrence of an initial hydrostatic contraction of a self-gravitating primordial stellar gas cloud the radiation environmental conditions have to be appropriate. Cloud contraction, namely, can only continue as long as the contracting cloud can get rid of its increased gravitational binding energy by thermal radiation from the border of the cloud into open space. Hence in the following we show that in this respect the cloud-surrounding CMB radiation can take a critical control on that contraction process occurring or not occurring.Here we simply start from the gravitational binding energy of a homogeneous gas cloud given by(33)
E
B
=
16
15
π
2
G
ρ
2
R
5
=
3
5
G
M
2
R
,where G is the gravitation constant, ρ is the mass density of the gas, R is the radius of the cloud, and M is the total gas mass of the cloud.A contraction of the cloud during the hydrostatic collapse phase (see [28]) is only possible, if the associated change in internal binding energy E
B can effectively be radiated off to space from the outer surface of the cloud, that is, if(34)
d
E
B
d
t
=
-
3
5
G
M
2
R
2
d
R
d
t
=
4
π
R
2
σ
s
b
T
c
4
-
T
C
M
B
4
,where σ
s
b denote the Stefan-Boltzmann constant and T
c the thermal radiation temperature of the cloud, respectively.This already makes evident that further contraction of the cloud is impeded, if the surrounding CMB temperature exceeds the cloud temperature, that is, ifT
C
M
B
>
T
c, because then the only possibility is d
R
/
d
t
≥
0
!, that is, expansion! In order to calculate the radiation temperature T
c of the contracting cloud, one can determine an average value of the shrinking rate during this hydrostatic collapse phase by use of the following expression:(35)
d
R
d
t
=
-
R
τ
f
f
=
-
R
4
π
G
ρ
,where τ
f
f is the so-called free-fall time period of the cloud mass (see [29]). Thus from the above contraction condition together with this shrinking rate one thus obtains the following requirement for ongoing shrinking:(36)
3
5
G
M
2
R
2
R
4
π
G
ρ
=
4
π
R
2
σ
s
b
T
c
4which allows to find the following value for the cloud temperature:(37)
T
c
4
=
3
20
π
σ
s
b
G
M
2
R
3
4
π
G
ρ
=
4
π
5
σ
s
b
G
3
/
2
M
ρ
3
/
2
.To give an idea for the magnitude of this cloud temperatureT
c we here assume that the typical cloud mass can be adopted with M
=
10
M
⊙ and that, for mass fragmentation of that size to occur, primordial molecular cloud conditions with an H
2
-density of the order of ρ
/
2
m
=
1
0
5 cm−3 must be adopted. With these values one then calculates a temperature of(38)
T
c
=
5220
1
/
4
K
≃
8.5
·
K
.This result must be interpreted as saying that as soon as in the past of cosmic evolution the CMB temperaturesT
C
M
B were becoming greater than this above value T
c, then stellar mass fragmentations of masses of the order of M
≃
10
M
⊙ were not possible anymore. This would mean that galaxies at supercritical distances correlated with redshifts z
≥
z
c should not be able to produce stars with stellar masses larger than 10
M
⊙. This critical redshift can be easily calculated from the linear cooling relation T
c
=
T
C
M
B
0
·
(
1
+
z
c
) and interestingly enough delivers z
c
=
T
c
/
T
C
M
B
0
-
1
=
2.09. This means that galaxies at distances beyond such redshifts, that is, with z
≥
z
c
=
2.09, should not be able to produce stars with stellar masses greater than 10
M
⊙.If on the other hand it is well known amongst astronomers that galaxies with redshiftsz
≥
z
c are known to show distant supernovae events [30], even serving as valuable cosmic light unit-candles and distance tracers, while such events just are associated with the collapse of 10
M
⊙-stars, then cosmology obviously is running into a substantial problem.


## 2.1. The Cosmic Microwave Background Tested by Cosmic Thermometers

It is generally well known that we are surrounded by 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 early cosmic recombination era when due to removal of electrically charged particles by electron-proton recombinations the universe for the times furtheron became transparent for photons. Since that time cosmic photons, persistent from the times of matter-antimatter annihilations, thus are propagating freely on light geodetic trajectories through the spacetime geometry of the expanding universe up to the present days.Assuming that at the times before recombination matter and photons coexisted in perfect thermodynamical equilibrium, despite the expansion of the cosmic volume (we shall come back to this problematic point in the next section), then this allows one to expect that these cosmic photons initially had a spectral distribution according to a perfect black-body radiator, that is, a Planckían spectrum. It is then generally concluded that a perfectly homogeneous Planckían radiation in an expanding universe stays rigorously Planckían over all times that follow. At this point one, however, one has to 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 the highest symmetrical form possible, that is, the one described by the so-called Robertson-Walker spacetime geometry.Then it can be demonstrated (e.g., see [7]) that the Planckían character of the CMB spectral photon density initially given by(1)
d
n
r
λ
=
2
λ
4
d
λ
exp
⁡
h
c
/
K
T
r
λ
-
1
,where d
n
r
(
λ
) denotes the spectral photon density at the time of recombination per wavelength interval d
λ at wavelength λ and T
r is the temperature of the Planck radiation at this time, is conserved for all ongoing periods of the expanding universe.Readers should, however, keep in mind that this is only guaranteed, if the universe has isotropic curvature and expands in a homologous, Robertson-Walker symmetric manner. (see, e.g., [8]). Due to this fact it then turns out that the initially Planckían spectral photon density changes with time so that for all cosmic future it maintains its Planckìan character, however, associated to a cosmologically reduced temperature T
<
T
r. On one hand at a later time t photons appear cosmologically redshifted to a wavelength λ
′
=
λ
(
R
/
R
r
), and on the other hand they are redistributed to a space volume increased by a factor (
R
/
R
r
)
3. Taking both effects together shows that at a later time t
>
t
r the resulting spectrum is given by(2)
d
n
λ
=
R
r
R
2
λ
′
4
d
λ
′
exp
⁡
h
c
/
K
T
r
λ
′
-
1
=
2
λ
4
d
λ
exp
⁡
h
c
/
K
λ
T
r
R
/
R
r
-
1which with the help of Wien’s displacement law T
·
λ
=
c
o
n
s
t reveals that at later times it again is a Planck spectrum, however, with temperature T
=
T
r
·
(
R
r
/
R
). This already indicates that the present-day CMB should be associated to a temperature T
0 given by T
0
=
T
r
·
(
R
r
/
R
0
) where the quantities indexed with “0” are those associated to the universe at the present time t
=
t
0. Depending on cosmic densities at the recombination phase the temperature T
r should have been between 3500 K and 4500 K (see [9]). This indicates that with the present-day CMB value of T
0
=
2.735 K [1] a ratio of cosmic expansion scales of(3)
2.735
3500
≥
R
r
R
0
=
T
0
T
r
≥
2.735
4500is disputable.The abovementioned theory of a homologous cosmic expansion then also allows to derive an expression for the cosmic CMB temperature as a function of the cosmic photon redshiftz
=
(
λ
0
-
λ
e
)
/
λ
e at which astronomers are seeing distant galactic objects. Here λ
0 is the wavelength which is observed at present, that is, at us, while the associated wavelength λ
e is emitted at the distant object. With the validity of the cosmological redshift relation in a Robertson-Walker universe,(4)
λ
e
λ
0
=
R
e
R
0
,where R
e and R
0 denote the cosmic scale parameters at the time t
e when the photon was emitted from the distant galaxy and at the present time t
0. Thus one obtains by definition(5)
z
=
λ
0
-
λ
e
λ
e
=
R
0
R
e
-
1
.This relation taken together with Wien’s law of spectral shiftλ
m
a
x
·
T
C
M
B
=
0.3 in this context is expressed by(6)
λ
m
a
x
z
=
0
·
T
C
M
B
z
=
0
=
λ
m
a
x
z
·
T
C
M
B
zthen finally allows to write the CMB temperature as the following function of redshift:(7)
T
C
M
B
z
=
T
C
M
B
z
=
0
·
λ
m
a
x
z
=
0
λ
m
a
x
z
=
T
C
M
B
0
·
z
+
1
.


## 2.2. Particle Distribution Functions in Expanding Spacetimes

Usually it is assumed that at the recombination era photons and matter, that is, electrons and protons in this phase of the cosmic evolution, are dynamically tightly bound to each other and undergo strong mutual interactions via Coulomb collisions and Compton collisions. These conditions are thought to then evidently guarantee a pure thermodynamical equilibrium state, implying that particles are Maxwell distributed and photons have a Planckian blackbody distribution. It is, however, by far not so evident that these assumptions really are fulfilled. This is because photons and particles are reacting to the cosmological expansion very differently; photons generally are cooling cosmologically being redshifted, while particles in first order are not directly feeling the expansion, unless they feel it adiabatically by mediation through numerous Coulomb collisions, which are relevant here in a fully ionized plasma before recombination, like they do in a box with subsonic expansion of its walls. But Coulomb collisions have a specific property which is highly problematic in this context.This is because Coulomb collision cross sections are strongly dependent on the particle velocityv, namely, being proportional to (
1
/
v
4
) (see [10]). This evidently causes that high-velocity particles are much less collision-dominated compared to low-velocity ones; they are even collision-free at supercritical velocities v
≥
v
c. So while the low-velocity branch of the distribution may still cool adiabatically and thus feels cosmic expansion in an adiabatic form, the high-velocity branch in contrast behaves collision-free and hence changes in a different form. This violates the concept of a joint equilibrium temperature and of a resulting Maxwellian velocity distribution function and means that there may be a critical evolutionary phase of the universe, due to different forms of cooling in the low- and high-velocity branches of the particle velocity distribution function, which do not permit the persistence of a Maxwellian equilibrium distribution to later cosmic times.In the following part of the paper we demonstrate that even if a Maxwellian distribution would still prevail at the beginning of the collision-free expansion phase, that is, the postrecombination phase era, it would not persist in the universe during the ongoing of the collision-free expansion. For that purpose let us first consider a collision-free population in an expanding Robertson-Walker universe. It is clear that due to the cosmological principle and, connected with it, the homogeneity requirement, the velocity distribution function of the particles must be isotropic, that is, independent on the local place, and thus of the following general form:(8)
f
v
,
t
=
n
t
·
f
¯
v
,
t
,where n
(
t
) denotes the cosmologically varying density only depending on the worldtime t and f
¯
(
v
,
t
) is the normalized, time-dependent isotropic velocity distribution function with the property: ∫
f
¯
(
v
,
t
)
d
3
v
=
1.If we assume that particles, moving freely with their velocityv into the v
→-associated direction over a distance l, are restituting at this new place, despite the differential Hubble flow and the explicit time-dependence of f, a locally prevailing covariant, but perhaps form-invariant distribution function f
′
(
v
′
,
t
′
), then the associated functions f
(
v
′
,
t
) and f
(
v
,
t
) must be related to each other in a very specific way. To define this relation needs some special care, since particles that are freely moving in an homologously expanding Hubble universe do in this case at their motions not conserve their associated phasespace volumes d
6
ϕ
=
d
3
v
d
3
x, since no Lagrangian exists and thus no Hamiltonian canonical relations for their dynamical coordinates v and x are valid. Hence Liouville’s theorem then requires that the conjugated differential phase space densities are identical; that is,(9)
f
′
v
′
,
t
d
3
v
′
d
3
x
′
=
f
v
,
t
d
3
v
d
3
x
.At the place where they arrive after passage over a distancel the particle population has a relative Hubble drift given by v
H
=
l
·
H coaligned with v
→, where H
=
H
(
t
) means the time-dependent Hubble parameter. Thus the original particle velocity v is locally turned to v
′
=
v
-
l
·
H. All dimensions of the space volume within a time Δ
t are cosmologically expanded, so that d
x
′
=
d
x
(
1
+
H
Δ
t
) holds. Complete reintegration into the locally valid distribution function then implies, with linearizably small quantities Δ
t
≃
l
/
v and Δ
v
=
-
l
·
H, that one can express the above requirement in the following form:(10)
f
′
v
′
,
t
′
d
3
v
′
d
3
x
′
=
f
v
,
t
+
∂
f
∂
t
Δ
t
+
∂
f
∂
v
Δ
v
×
1
+
Δ
v
v
2
1
+
H
Δ
t
3
d
3
v
d
3
x
=
f
v
,
t
d
3
v
d
3
x
.This then means for terms of first order that(11)
∂
f
∂
t
Δ
t
+
∂
f
∂
v
Δ
v
+
2
Δ
v
v
f
+
3
H
Δ
t
f
=
0and thus(12)
∂
f
∂
t
l
v
-
l
H
∂
f
∂
v
-
2
l
H
v
f
+
3
H
l
v
f
=
0or the following requirement:(13)
∂
f
∂
t
=
v
H
∂
f
∂
v
-
H
f
.Looking first here for interesting velocity moments of the functionf fulfilling the above partial differential equation by multiplying this equation with (a) 4
π
v
2
d
v and (b) (
4
π
/
3
)
m
v
4
d
v and integrating over velocity space then leads toa
:
n
=
n
0
exp
⁡
-
2
H
t
-
t
0,
and
b
:
P
=
P
0
exp
⁡
-
4
H
t
-
t
0which then immediately makes evident that with the above solutions one finds that(14)
P
n
γ
=
P
0
n
0
γ
exp
⁡
-
4
-
2
γ
H
t
-
t
0
=
P
0
n
0
γ
exp
⁡
-
2
3
H
t
-
t
0is not constant, meaning that no adiabatic behaviour of the expanding gas occurs and that the gas entropy S also is not constant but decreasing and given by(15)
S
=
S
t
=
S
0
ln
⁡
P
n
γ
=
-
2
3
H
t
-
t
0
.It is perhaps historically interesting to see that assuming Hamilton canonical relations to be valid the Liouville theorem would then instead of (9) simply require f
′
(
v
′
,
t
)
=
f
(
v
,
t
) and hence would lead to the following form of a Vlasow equation:(16)
∂
f
∂
t
-
v
H
∂
f
∂
v
=
0
.In that case the first velocity moment is found with(17)
∂
n
∂
t
=
∫
4
π
v
3
H
∂
f
∂
v
d
v
=
4
π
H
∫
∂
∂
v
v
3
f
-
12
π
H
∫
v
2
f
d
vyielding(18)
∂
n
∂
t
=
-
3
n
Hwhich agrees withn
~
R
-
3. Looking also for the higher moment P then leads to(19)
∂
P
∂
t
=
4
π
3
∫
v
5
H
∂
f
∂
v
d
v
=
-
5
H
Pwhich now in this case shows that(20)
P
n
γ
=
P
0
n
0
γ
exp
⁡
-
5
-
3
γ
H
t
-
t
0
=
const
!
!
!
.That means in this case an adiabatic expansion is found, however, based on wrong assumptions!Now going back to the correct Vlasow equation (13) one can then check whether or not this equation allows that an initial Maxwellian velocity distribution function persists during the ongoing collision-free expansion. Here we find for f
~
n
T
-
3
/
2
exp
⁡
[
-
m
v
2
/
2
K
T
], n and T being time-dependent, that one has(21)
∂
f
∂
t
=
f
d
ln
⁡
n
d
t
-
3
2
T
˙
T
+
m
v
2
2
K
T
T
˙
T
,
∂
f
∂
v
=
-
f
m
v
K
Tleading to the following Vlasow requirement (see (13)):(22)
d
ln
⁡
n
d
t
-
3
2
T
˙
T
+
m
v
2
2
K
T
T
˙
T
=
-
H
m
v
2
K
T
+
1
.In order to fulfill the above equation obviously the terms withv
2 have to cancel each other, since n and T are velocity moments of f, hence independent on v. This is evidently only satisfied, if the change of the temperature with cosmic time is given by(23)
T
=
T
0
exp
⁡
-
2
H
t
-
t
0
.This dependence in fact is obtained when inspecting the earlier found solutions for the momentsn and P (see (13) and (14)), because these solutions exactly give(24)
T
=
P
K
n
=
P
0
K
n
0
exp
⁡
-
4
-
2
H
t
-
t
0
=
T
0
exp
⁡
-
2
H
t
-
t
0
.With that the above requirement (22) then only reduces to(25)
d
ln
⁡
n
d
t
-
3
2
T
˙
T
=
-
Hwhich then leads to(26)
-
2
H
-
3
2
-
2
H
=
-
Hmaking it evident that this requirement is not fulfilled and thus meaning that consequently a Maxwellian distribution cannot be maintained, even not at a collision-free expansion.This finally leads to the statement that a correctly derived Vlasow equation for the cosmic gas particles leads to a collision-free expansion behaviour that neither runs adiabatic nor does it conserve the Maxwellian form of the distribution functionf. Under these auspices it can, however, also easily be demonstrated (see [11]) that collisional interaction of cosmic photons with cosmic particles via Compton collisions in case of non-Maxwellian particle distributions does unavoidably lead to deviations from the Planckian blackbody spectrum. This makes it hard to be convinced by a pure Planck spectrum of the CMB photons at the time t
r
e
c around the cosmic matter recombination.Let us therefore now look into other basic concepts of cosmology to see whether perhaps also there problems can be identified which should caution cosmologists.


## 2.3. Can the Cosmological CMB Cooling Be Confirmed?

In the following part of the paper we now want to investigate whether or not the cosmological cooling of the CMB photons, freely propagating in the expanding Robertson-Walker space time geometry, can be confirmed by observations. The access to this problem is given by the connection that in an expanding universe at earlier cosmic times the CMB radiation should have 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 larger redshifts, that is, those seen at times in the distant past, really give indications that they in fact are embedded in a correspondingly hotter CMB radiation environment. For that purpose one generally uses appropriate, so-called CMB radiation thermometers like interstellar CN-, CH-, or CO-molecular species (see [3, 12, 13], or [4]).Assuming that molecular interstellar gas phases within these galaxies are in optically thin contact to the CMB that actually surrounds these galaxies allows one to assume that such molecular species are populated in their electronic levels according to a quasistationary equilibrium state population. In this respect especially interesting are molecular species with an energy splitting of vibrational or rotational excitation levelsi
,
j that correspond to mean energies of the surrounding CMB photons; that is, E
i
-
E
j
=
h
ν
C
M
B. Under such conditions the relative level populations n
i
,
n
j essentially are given by the associated Boltzmann factor(27)
n
i
n
j
~
g
i
g
j
exp
⁡
-
h
E
i
-
E
j
K
T
C
M
B
,where g
i
,
j are the state multiplicities. In the years of the recent past interstellar CO-molecules have been proven to be best suited in this respect as highly appropriate CMB thermometers. This was demonstrated by Srianand et al. [3] and Noterdaeme et al. [4].The carbon monoxide molecule CO splits into different rotational excitation levels according to different rotational quantum numbersJ. According to these numbers a splitting of CO lines occurs with transitions characterized by Δ
J
=
1. In this respect the transition J
=
1

→

J
=
0 leads to a basic emission line at λ
1,0
=
2.6 mm (i.e., ν
0
=
115
,

6 GHz). The CO-molecule is biatomic with a rotation around an axis perpendicular to the atomic interconnection line. The quantum energies E
r
o
t
(
J
) are given by (28)
E
r
o
t
J
=
h
2
8
π
2
I
J
J
+
1
=
S
2
J
2
I
=
I
ω
2
J
2
,where I is the moment of inertia of the CO-rotator and is given by(29)
I
C
O
=
a
2
m
C
m
O
m
C
+
m
O
.Herea is the interconnection distance, and m
C, m
O are the masses of the carbon and oxygen atom, respectively. S
(
J
) is the angular momentum of the state with quantum number J, and ω
(
J
) is the associated angular rotation frequency. The emission wavelengths from the excited states of the CO-A-X bands (J
≥
2) thus are given by(30)
λ
j
≥
2
=
λ
0
1
2
-
1
J
J
+
1
.Usually it is hardly possible to detect these CO-fine structure emissions from distant galaxies directly, due to their weaknesses and due to the strong perturbations and contaminations in this frequency range by the infrared (i.e., ≥115 GHz). Instead the relative population of these rotational fine structure levels can much better be observed in absorption appearing in the optical range. To actually use such a constellation to determine the relative populations of CO fine structure levels one needs a broadband continuum emitter in the cosmic background behind a gas-containing galaxy in the foreground. As in case of the object investigated by Srianand et al. [3] the foreground galaxy is at a redshift of z
a
b
s
=
2.41837 illuminated by a background quasar SDSS J143912.04 + 111740.5. Then the CO fine structure lines appear in absorption at wavelengths between 4900 Å and 5200 Å and, by fitting them with Voigt-profiles, the relative populations (
n
(
J
i
)
/
n
(
J
j
)
) of these fine structure levels can be determined. Assuming now optically thin conditions of the absorbing gas with respect to CMB photons, one can assume that in a photostationary equilibrium these relative populations are connected with the abovementioned Boltzmann factor as(31)
n
J
i
n
J
j
~
g
i
g
j
exp
⁡
-
h
E
i
-
E
j
K
T
C
M
B
∗
,where now T
C
M
B
∗ is the CMB Planck temperature at cosmic redshift z
a
b
s
=
2.41837. On the basis of the abovementioned assumptions Srianand et al. [3], depending on the specific transitions which they fit, find CMB excitation temperatures of T
C
M
B
∗
(
0,1
)
=
9.11
±
1.23 K; T
C
M
B
∗
(
1,2
)
=
9.19
±
1.21 K; and T
C
M
B
∗
(
0,2
)
=
9.16
±
0.77 K, while according to standard cosmology (see (7)) at a redshift z
a
b
s
=
2.41837 one should have a CMB temperature of T
C
M
B
∗
=
(
1
+
z
a
b
s
)
T
C
M
B
0
=
9.315 K, where T
C
M
B
0
=
2.725 K is the present-day CMB temperature (see [14]).Though this clearly points to the fact that CMB temperaturesT
C
M
B
∗ at higher redshifts are indicated to be higher than the present-day temperature T
C
M
B
0, it also demonstrates that the cosmologically expected value should have been a few percent higher than these fitted values. This, however, cannot question the applicability of the above described method in general, though some basic caveats have to be mentioned here.First of all, observers with similar observations are often running into optically thick CO absorption conditions which will render the fitting procedure more difficult. Noterdaeme et al. [4], for instance, can show that the fitted CMB temperature differs with the CO-column density of the foreground absorber (see Figure 1). The determination of these column densities in itself is a highly nontrivial endeavour and only can be carried out assuming some fixed correlations between CO- and H2-column densities, the latter being much better measurable.Figure 1
CO absorption profiles observed at SDSS-J1-70542 + 354340 (z
a
b
s
=
2.038); figure taken from Srianand et al. [3].The second caveat in this context is connected with the assumption that relative populations of fine structure levels are purely determined by a photon excitation equilibrium with the surrounding CMB photons. If in addition any binary collisions with other molecules or any photons other than CMB photons are interfering into these population processes, then of course the fittedT
C
M
B
∗ values have to be taken with correspondingly great caution. Especially in the infrared range delivering the relevant photons for excitations or deexcitations the CMB spectrum is strongly contaminated by galactic dust emissions [1, 15, 16]. Facing then the possibility that galaxies at higher redshifts are more pronounced in galactic dust emissions compared to our present galaxies nearby then makes CMB temperature determinations perhaps questionable. Nevertheless the results obtained by Noterdaeme et al. [4] when determining CO-excitation temperatures at foreground galaxies with different redshifts perhaps for most readers do convincingly demonstrate that a linear correlation of the CMB temperature with redshift can be confirmed (see Figure 2) as expected.Figure 2
CMB temperatures as function of redshiftz derived from CO-excitation temperatures; figure taken from Noterdaeme et al. [4].


## 2.4. Problems with a Hot CMB in the Past

Though from the results displayed in the above Figure2 it seems as if the cosmological CMB cooling with time can be surprisingly well confirmed, one nevertheless should not too carelessly take that as an observational fact. We remind the reader first to the theoretical prerequisites of a cosmologic CMB cooling reflected in a decrease of the Planck temperature T
C
M
B of this radiation: a Planckian spectrum only stays a Planckian, if(a)
it was Planckian already at the beginning, that is, at the recombination phase, and if
(b)
since that time a completely homologous cosmic expansion took place till today.Point (a) is questionable because the thermodynamic equilibrium state between baryons and photons in the early phase of fast cosmic expansion may quite well be disturbed or incomplete (see [7, 11], Section 2.2 of this paper). Point (b) is questionable, since at present times we find a highly structured, inhomogeneous cosmic matter distribution which does not originate from a homogeneous matter cosmos with a pure, unperturbed Robertson-Walker cosmic expansion.The present universe actually is highly structured by galaxies, galaxy clusters, superclusters, and walls [17, 18]. Although perhaps the matter distribution was quite homogeneous at the epoch of the last scattering of cosmic photons when the CMB photons were in close contact to the cosmic matter, during the evolutionary times after that matter distribution has evidently become very inhomogeneous by the gravitational growth of seed structures. Thus fitting a perfectly symmetrical Robertson-Walker spacetime geometry to a universe with a lumpy matter distribution appears highly questionable [19]. This is an eminent general relativistic problem as discussed by Buchert [20], Buchert [21], Buchert [22], Buchert [23], Buchert [24], and Wiltshire [25]. If due to that structuring processes in the cosmic past and the associated geometrical perturbations of the Robertson-Walker geometry we would look back into direction-dependent different expansion histories of the universe, this would point towards associated CMB fluctuations (see [7]).Thus it should be kept in mind that a CMB Planck spectrum is only seen with the same temperature from all directions of the sky, if in all these directions the same expansion dynamics of the universe took place. If CMB photons arriving from different directions of the sky have seen different expansion histories, then their Planck temperatures would of course be different and anisotropic, destroying completely the Planckian character of the CMB. This situation evidently comes up in case an anisotropic and nonhomologous cosmic expansion takes place like that envisioned and described in theories by Buchert [23], Buchert [26], Buchert [27], Buchert [24], or Wiltshire [25]. Let us check this situation by a simple-minded approach here: in the two-phase universe consisting of void and wall regions, as described by Wiltshire [25], void expansions turn out to be different from wall expansions, and, when looking out from the surface border of a wall region, in the one hemisphere one would see the void expansion dynamics, whereas in the opposite hemisphere one sees the wall expansion dynamics. Thus CMB photons arriving from the two opposite sides are differently cosmologically redshifted and thus in no case do constitute one common Planckian spectrum with one joint temperature T
C
M
B, but rather a bipolar feature of the local CMB-horizon.In fact if one hemisphere expands different from the opposite hemisphere, then as a reaction also different CMB Planck temperatures would have to be ascribed to the CMB photons arriving from these opposite hemispherical directions. If, for instance, the present values of the characteristic scale in the two opposite hemispheres areR
1 and R
2, then this would lead to a hemispheric CMB temperature difference of Δ
T
1,2 given by (see [7])(32)
Δ
T
1,2
=
T
r
R
r
R
1
-
R
r
R
2and would give an alternative to the present-day CMB-dipole explanation.


## 2.5. Hot CMB Impedes Gas Fragmentation

Stars are formed due to gravitational fragmentation of parts of a condensed interstellar molecular cloud. For the occurrence of an initial hydrostatic contraction of a self-gravitating primordial stellar gas cloud the radiation environmental conditions have to be appropriate. Cloud contraction, namely, can only continue as long as the contracting cloud can get rid of its increased gravitational binding energy by thermal radiation from the border of the cloud into open space. Hence in the following we show that in this respect the cloud-surrounding CMB radiation can take a critical control on that contraction process occurring or not occurring.Here we simply start from the gravitational binding energy of a homogeneous gas cloud given by(33)
E
B
=
16
15
π
2
G
ρ
2
R
5
=
3
5
G
M
2
R
,where G is the gravitation constant, ρ is the mass density of the gas, R is the radius of the cloud, and M is the total gas mass of the cloud.A contraction of the cloud during the hydrostatic collapse phase (see [28]) is only possible, if the associated change in internal binding energy E
B can effectively be radiated off to space from the outer surface of the cloud, that is, if(34)
d
E
B
d
t
=
-
3
5
G
M
2
R
2
d
R
d
t
=
4
π
R
2
σ
s
b
T
c
4
-
T
C
M
B
4
,where σ
s
b denote the Stefan-Boltzmann constant and T
c the thermal radiation temperature of the cloud, respectively.This already makes evident that further contraction of the cloud is impeded, if the surrounding CMB temperature exceeds the cloud temperature, that is, ifT
C
M
B
>
T
c, because then the only possibility is d
R
/
d
t
≥
0
!, that is, expansion! In order to calculate the radiation temperature T
c of the contracting cloud, one can determine an average value of the shrinking rate during this hydrostatic collapse phase by use of the following expression:(35)
d
R
d
t
=
-
R
τ
f
f
=
-
R
4
π
G
ρ
,where τ
f
f is the so-called free-fall time period of the cloud mass (see [29]). Thus from the above contraction condition together with this shrinking rate one thus obtains the following requirement for ongoing shrinking:(36)
3
5
G
M
2
R
2
R
4
π
G
ρ
=
4
π
R
2
σ
s
b
T
c
4which allows to find the following value for the cloud temperature:(37)
T
c
4
=
3
20
π
σ
s
b
G
M
2
R
3
4
π
G
ρ
=
4
π
5
σ
s
b
G
3
/
2
M
ρ
3
/
2
.To give an idea for the magnitude of this cloud temperatureT
c we here assume that the typical cloud mass can be adopted with M
=
10
M
⊙ and that, for mass fragmentation of that size to occur, primordial molecular cloud conditions with an H
2
-density of the order of ρ
/
2
m
=
1
0
5 cm−3 must be adopted. With these values one then calculates a temperature of(38)
T
c
=
5220
1
/
4
K
≃
8.5
·
K
.This result must be interpreted as saying that as soon as in the past of cosmic evolution the CMB temperaturesT
C
M
B were becoming greater than this above value T
c, then stellar mass fragmentations of masses of the order of M
≃
10
M
⊙ were not possible anymore. This would mean that galaxies at supercritical distances correlated with redshifts z
≥
z
c should not be able to produce stars with stellar masses larger than 10
M
⊙. This critical redshift can be easily calculated from the linear cooling relation T
c
=
T
C
M
B
0
·
(
1
+
z
c
) and interestingly enough delivers z
c
=
T
c
/
T
C
M
B
0
-
1
=
2.09. This means that galaxies at distances beyond such redshifts, that is, with z
≥
z
c
=
2.09, should not be able to produce stars with stellar masses greater than 10
M
⊙.If on the other hand it is well known amongst astronomers that galaxies with redshiftsz
≥
z
c are known to show distant supernovae events [30], even serving as valuable cosmic light unit-candles and distance tracers, while such events just are associated with the collapse of 10
M
⊙-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 us directly into a complete understanding of the world and the evolution of the universe. Too many basic concepts inherent to the application of general relativity on describing the whole universe are still not settled on safe grounds, as we have pin-pointed in the foregoing sections of this paper.We have shown in Section2 of this paper that the cosmic microwave background radiation (CMB) only then can reasonably well be understood as a relict of the Big-Bang, if (a) it was already a purely Planckian radiation at the beginning of the recombination era, and if (b) the universe from that time onwards did expand rigorously isotropic and homologous according to a Robertson-Walker symmetrical expansion. As we have shown that, however, both points are highly questionable, since (a) matter and radiation are cooling differently in the expanding cosmos, so that the transition to the collisionless expansion induces a degeneration from thermodynamical equilibrium conditions with particle distribution functions deviating from Maxwellians and radiation distributions deviating from Planckians (see Section 2.2). Furthermore since (b) the cosmic expansion cannot have continued up to the present days in a purely Robertson-Walker-like style, otherwise no cosmic structures and material hierarchies could have formed.It is hard to say anything quantitative at this moment what needs to be concluded from these results in Section2.2. Fact is that during and after the phase of matter recombination in the universe Maxwellian velocity distributions for electrons and protons do not survive as Maxwellians but are degenerating into non-Maxwellian, nonequilibrium distributions, implying the drastic consequence that baryon densities are not falling off as (
1
/
R
3
) but as (
1
/
R
2
). The interaction of the originating CMB photons with nonequilibrium electrons by Compton collisions will then in consequence also change the resulting CMB spectrum to become a non-Planckian spectrum, as shown by Fahr and Loch [11]. Essentially the effect is that from Wien’s branch CMB photons are removed which instead reappear in the Rayleigh-Jeans branch.The critical frequency limit is at around 103 GHz, with effective radiation temperatures being reduced at higher frequencies with respect to those at lower frequencies. The exact degree of these changes depends on many things, for example, like the cosmologic expansion dynamics during the recombination phase and the matter density during this phase. However, the consequence is that the effective CMB radiation temperature measured at frequencies higher than 103 GHz are lower than CMB temperatures at frequencies below 103 GHz. Our estimate for conventionally assumed cosmologic model ingredients (Omegas!) would be by about 1
K!Unfortunately CMB measurements at frequencies beyond 103 GHZ are practically absent up to now and do not allow to identify these differences. If in upcoming time periods, on the basis of upcoming better measurements in the Wien’s branch of the CMB, no such differences will be found, then the conclusions should not be drawn that the theoretical derivations of such changes presented here in our manuscript must be wrong, but rather that the explanation of the CMB as a relict radiation of the recombination era may be wrong.Though indeed, as we discuss in Section2.3, there are indications given by cosmic radiation thermometers like CN-, CO-, or CH-molecules that the CMB radiation has been hotter in the past of cosmic evolution, we also point out, however, that alternative explanations of these molecule excitation data like by collisional excitations and by infrared excitations through dust emissions should not be overlooked and that the conventionally claimed, redshift-relatedly hotter CMB in the past (i.e., following the relation T
C
M
B
~
(
1
+
z
)
T
0
,
C
M
B) in fact brings astrophysicists rather into severe problems in understanding the origin of massive stars in distant galaxies seen at large redshifts z
≥
z
c (see Section 2.5).For those readers interested in more hints why the conventional cosmology could be in error, we are presenting other related controversial points in the Appendices.



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*Source: 503106-2015-06-09.xml*