Physics Research International

Volume 2015, Article ID 503106, 15 pages

http://dx.doi.org/10.1155/2015/503106

## Remaining Problems in Interpretation of the Cosmic Microwave Background

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

Received 25 June 2014; Accepted 7 April 2015

Academic Editor: Avishai Dekel

Copyright © 2015 Hans-Jörg Fahr and Michael Sokaliwska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 than 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 CDM-cosmology. Our studies here rather lead to the conclusion that cosmic vacuum energy density scales with the inverse square of the cosmic expansion scale . Thus we come to the conclusion that with the interpretation of the present-day high quality CMB data still needs to be considered carefully.

#### 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 km/s/Mpc and an age of the universe of 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 bywhere denotes the spectral photon density at the time of recombination per wavelength interval at wavelength and 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 . On one hand at a later time photons appear cosmologically redshifted to a wavelength , and on the other hand they are redistributed to a space volume increased by a factor . Taking both effects together shows that at a later time the resulting spectrum is given bywhich with the help of Wien’s displacement law reveals that at later times it again is a Planck spectrum, however, with temperature . This already indicates that the present-day CMB should be associated to a temperature given by where the quantities indexed with “0” are those associated to the universe at the present time . Depending on cosmic densities at the recombination phase the temperature should have been between 3500 K and 4500 K (see [9]). This indicates that with the present-day CMB value of K [1] a ratio of cosmic expansion scales ofis 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 redshift at which astronomers are seeing distant galactic objects. Here is the wavelength which is observed at present, that is, at us, while the associated wavelength is emitted at the distant object. With the validity of the cosmological redshift relation in a Robertson-Walker universe,where and denote the cosmic scale parameters at the time when the photon was emitted from the distant galaxy and at the present time . Thus one obtains by definition

This relation taken together with Wien’s law of spectral shift in this context is expressed bythen finally allows to write the CMB temperature as the following function of redshift:

##### 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 velocity , namely, being proportional to (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 . 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:where denotes the cosmologically varying density only depending on the worldtime and is the normalized, time-dependent isotropic velocity distribution function with the property: .

If we assume that particles, moving freely with their velocity into the -associated direction over a distance , are restituting at this new place, despite the differential Hubble flow and the explicit time-dependence of , a locally prevailing covariant, but perhaps form-invariant distribution function , then the associated functions and 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 , since no Lagrangian exists and thus no Hamiltonian canonical relations for their dynamical coordinates and are valid. Hence Liouville’s theorem then requires that the conjugated differential phase space densities are identical; that is,

At the place where they arrive after passage over a distance the particle population has a relative Hubble drift given by coaligned with , where means the time-dependent Hubble parameter. Thus the original particle velocity is locally turned to . All dimensions of the space volume within a time are cosmologically expanded, so that holds. Complete reintegration into the locally valid distribution function then implies, with linearizably small quantities and , that one can express the above requirement in the following form:

This then means for terms of first order thatand thusor the following requirement:

Looking first here for interesting velocity moments of the function fulfilling the above partial differential equation by multiplying this equation with (a) and (b) and integrating over velocity space then leads to , and which then immediately makes evident that with the above solutions one finds thatis not constant, meaning that no adiabatic behaviour of the expanding gas occurs and that the gas entropy also is not constant but decreasing and given by

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 and hence would lead to the following form of a Vlasow equation:

In that case the first velocity moment is found withyieldingwhich agrees with. Looking also for the higher moment then leads towhich now in this case shows that

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 , and being time-dependent, that one hasleading to the following Vlasow requirement (see (13)):

In order to fulfill the above equation obviously the terms with have to cancel each other, since and are velocity moments of , hence independent on . This is evidently only satisfied, if the change of the temperature with cosmic time is given by

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

With that the above requirement (22) then only reduces towhich then leads tomaking 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 function . 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 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 levels that correspond to mean energies of the surrounding CMB photons; that is, . Under such conditions the relative level populations essentially are given by the associated Boltzmann factorwhere 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 numbers . According to these numbers a splitting of CO lines occurs with transitions characterized by . In this respect the transition leads to a basic emission line at mm (i.e., GHz). The CO-molecule is biatomic with a rotation around an axis perpendicular to the atomic interconnection line. The quantum energies are given by where is the moment of inertia of the CO-rotator and is given by

Here is the interconnection distance, and , are the masses of the carbon and oxygen atom, respectively. is the angular momentum of the state with quantum number , and is the associated angular rotation frequency. The emission wavelengths from the excited states of the CO-A-X bands () thus are given by

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 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 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 aswhere now is the CMB Planck temperature at cosmic redshift . On the basis of the abovementioned assumptions Srianand et al. [3], depending on the specific transitions which they fit, find CMB excitation temperatures of K; K; and K, while according to standard cosmology (see (7)) at a redshift one should have a CMB temperature of K, where K is the present-day CMB temperature (see [14]).

Though this clearly points to the fact that CMB temperatures at higher redshifts are indicated to be higher than the present-day temperature , 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 H_{2}-column densities, the latter being much better measurable.