SPECULATIONS ON A CONTINUUM COSMOLOGICAL MODEL
DESCRIBED IN FRIEDMANN-LEMAITRE SPACETIME
Conceptual Considerations


Moishe Garfinkle
PHILADELPHIA COLLEGE OF TEXTILES AND SCIENCE
Philadelphia, PA 19144

(215) 235-5042

CONTENTS

1 INTRODUCTION
2 BACKGROUND
3 STANDARD COSMOLOGICAL MODEL
        SCM Closure Problem
        SCM Horizon Problem
        SCM Symmetry Problem
        SCM Smoothness Problem
4  CONTINUUM COSMOLOGICAL MODEL
        World Form
        Friedmann-Lemaître Spacetime
        Spectral Shift
        Singularity
        Pseudo-Stellar Fireball
        Planck Interface
        Spacetime Curvature
5 SYNTHESIS
        Fermiosynthesis
        Leptosynthesis
        Baryosynthesis
        Baryon Symmetry Violation
        Critique of Evidence for Baryon Symmetry Violation
6  INTERACTIONS
        Gravitational Potentials
        Potential Form
        Baryonic Clustering
        Freeze-Out
        Large-Scale Structure
        Nucleosynthesis
        Decoupling
        Dicke Coincidence
        Black Holes, White Holes, and Other Anomolies
        Cosmic Background Radiation
7  CCM CONCERNS
        Closure
        Horizon
        Symmetry
        Smoothness
        Homogeniety
8  VERIFICATION
        Lepton Experiment
        Baryon Experiment
9  DISCUSSION
        Cosmogony and Eschatology
10 SUMMARY


1 INTRODUCTION



The evolution of cosmological models is almost as fascinating a study as that of cosmological evolution itself, and speculation on cosmology are of course as old as man's imagination. However, greater strides have been made in the last half-century to place cosmology on a firm analytical foundation than all of the conjectures presented since ancient Assyria. This progress has occurred because of our ability to look both inwards and outwards has increased so greatly in this period: both downward to the depths of the sub-nuclear universe and upward to the furthest extent of the astronomical universe; for only in the beginning did these universes coincide. Cosmology simply could not have been studied with any rigor until the processes underlying both particle physics and astrophysics could be understood with some degree of certainty and their findings correlated with some degree of confidence.

Modern analytical cosmology was fathered by Friemann and independently by Lemaître during the 1920s. They derived a series of simple solutions to Enstein's General Relativity field equations involving highly spherically-symmetrical spacetime geodesics that were time-dependent and could be interpreted as descriptions of an expanding universe. Such a dynamic universe must have expanded from a smaller universe, in direct contrast to the prevalent static models developed by Einstein and de Sitter. The Friedmann-Lemaître evolutionary model could not predict explictly the curvature of the expansion however, permitting three possible solutions: either hyperbolic, flat or spherical expansion, all of which are unbound but only the latter is closed.

In terms of Riemannian space these solutions predict monotonic spacetime divergence of hyperbolic and flat universes. The spherical universe alone provides a cusp beyond which spacetime reconverges to an unfortunate conclusion: either a final crunch or perhaps a rebound, resulting in either an unending cycle of birth and death or a gradual running down of the universe with each cycle.

Modern empirical cosmology was sired by Hubble, who correlated the spectrometric red-shifts of distant celestial objects with the radial velocity of such objects, albeit poorly, but with enough verve to pronounce the correlation a fact. Hubble proposed a linear correlation between red-shift and receding velocity whose slope H has since been denoted Hubble's Constant. However this correlation did not necessarily point to the Friedmann-Lemaître model as opposed to the Einstein-de Sitter model, either of which could accommodate this finding inasmuch as spectrometric red-shifts need not be related to spacetime expansion (gravitational effect) but to radial motion in a static framework (Doppler effect).

However, in the light of the excellent empirical observations of Humason and reconsideration of the theoretical conclusions of Wirtz the Hubble observations were generally interpreted to be considerably more consistant with a dynamic model which would encompass both gravitational and Doppler effects than a static model which would accommodate only the latter. When both Einstein and de Sitter also accepted the dynamic interpretation of the red-shift expansion achieved acceptability. Accordingly Hubble's Constant became a time-dependent variable related to spacetime expansion, the determination of whose present value Ho still involves much difficulty. Hubble never fully reconciled himself to an expansion model because such a model must presuppose a past smaller universe, and therefore logically a beginning. In agreement Eddington professed that "the notion of a beginning of the present order of nature is repugnant to me", while, in contrast to Hubble, Eddington paradoxically supported the Friedmann-Lemaître model as the model most closely supported by observations.

The three solutions to the Friedmann-Lemaître model can be quantized by normalizing the Riemannian expansion factor K with K=-1 for hyperbolic space, K=0 for flat space, and K=+1 for spherical space. Because spacetime curvature depends on the density of the enclosed space D relative to a critical density for closure Dc, the present density ratio Wo can be defined as Wo=Do/Dc. Hence K=-1 for Wo<1, K=0 for Wo=1,and K=+1 for Wo>1.

Within this framework H>0 for a hyperbolic universe, H=0 for a flat universe, and H<0 for a spherical universe. However there is simply too much uncertainty in the red-shift velocity observations to fix the variance in H, let alone its current value Ho. The present value of Ho ranges from 40 to 100 km/s-Mpc, depending on the investigator, each reporting an error of no greater than ±10 km/s-Mpc. This unseeming uncertainty arises from the method of measurement, the objects observed, and particularly the reliability of standard candles required for absolute distance determinations at great displacements. The Cepheid variables, the most reliable candles, can be calibrated to less than 1% of the present Hubble radius: c/Ho.


2 BACKGROUND



Only after it was generally accepted by astronomers that the present universe evolved from a smaller precursor was the concept of expansion seriously considered by cosmologists. During the 1930s Tolman suggested that intergalactic radiation would cool in an expanding universe, though maintaining its Planck distribution, permitting its temperature to be determined.  During the late 1940s two cosmologies ostensibly consistant with the Friedmann-Lemaître model were introduced that have attained sufficient recognition to have been studied seriously by cosmologists.

In large measure the hot big-bang as the origin from which the universe expanded was placed on a quantitative basis by the seminal work of Gamov concerning the relative abundance of the elements in the universe. If a big bang is a fact then remnent cosmic background radiation would fill the universe, and if its temperature could be measured it would reveal quantitative information as to the age of the universe.  The work was extended by Alpher & Herman and although the big-bang scenario is still burdened with some difficulties it is still considered the most promising approach, evolving into the Standard Cosmological Model (SCM).

The steady-state model proposed by Bondi, Gold & Hoyle introduced spontaneous matter generation and is now largely discredited, but occasionally resurrected.  Both models accounted for the perceived expansion of the universe.

 A rather offbeat model was proposed by Lemaître which could be denoted the "Oscillating Universe Model".  Assuming that the mass density of the universe was sufficient for closure, the collapse from the maximum volume would follow bounce back and expansion from the minimum volume: endless cycles, but only if no information whatsoever could transit the bounce back.  Presumably the universal kinetic energy would be at a maximum and the potential energy at a minimum at the bounce with the situation reversed at maximum expansion.  Tolman took the bounce model seriously enough to investigate its implications.  If entropy were preserved through a bounce each cycle would be longer than the previous cycle, essentially destroying the oscillation concept.  The duration of each cycle would be shorter than the subsequent cycle, requiring an initial cycle, presumably without duration; and likewise the duration of each cycle would be longer than the previous cycle, essentially resulting in a final cycle of endless duration.

The steady state theory proposed a flat universe (K=0) whose matter density remains constant because matter is spontaneously created between galaxies at a constant rate proportional to the constant rate of expansion. This approach thereby avoids the need for a genesis: the universe had forever existed and forever will. This view presumably failed on two counts: it could not account for the relatively anisotropic distribution of galaxies in the observable universe nor the measured microwave background radiation or neutrino background flux.

Although other sources of the background radiation have been proposed by Hoyle, the blackbody distribution of this radiation most strongly suggests a sufficiently-distant past period in which this radiation was in thermal equilibrium with matter: a hot big-bang. However, there may be a more substantial fault with the steady-state universe. According to steady-state theory to maintain the density of the expanding universe constant the aggregate mass of the universe must be increasing, indicating an evolutionary process. In the past the mass of the universe was less than at present, and accordingly at some substantially distant past the mass must have approached zero, destroying any steady-state scenario.


3 STANDARD COSMOLOGICAL MODEL



The SCM in its most elementary form proposes that universal expansion is the precise and proper interpretation of observed red shifts of stellar and galactic spectrum and that physical laws have remained unchanged at least since radiation-mass equilibrium and presumably from Planck time. The universe is larger than it was and perhaps smaller than it will eventually be, depending on the appropriate model of spacetime curvature. Consequently, on extrapolating back the universe would be substantially smaller and far enough backward the universe would have been an immensely dense collection of hot matter and radiation: perhaps the primordial Lemaître quantum singularity from which sprung the present universe.

This evolutionary model has been comprehensively studied and sufficient agreement has been achieved among cosmologist concerning certain basic predictions - particularly the observed abundances of He and D - that this model has been canonize the Standard Cosmological Model (SCM). Frankly it is a little short of astonishing just how well the universe follows the SCM scenario: at least from nucleosynthesis to the furthest extent of the visible universe. However there are certain basic difficulties that have only recently been addressed: principally the closure problem, the horizon problem, the symmetry problem, and the smoothness problem. Attempts have been made to graft the steady-state scenario onto the SCM to somewhat alleviate these difficulties, such as the proposal of King that continuous mass creation within an expansing space-time framework would go far in explaining the apparent distribution of galaxies, a major stumbling block in the SCM.



SCM Closure Problem
    Closure would really not be a problem if cosmologists had not made it one. Using Kepler's 3rd Law, mass-luminosity relationships related to selected galaxies indicates that the present critical density ratio is on the order of Wo=0.01, several magnitudes less than that required for closure. However much of the baryonic mass of the universe may be non-luminous, such as white dwarfs, neutron stars and even planets such as Jupiter. Nevertheless, such bodies might increase the density no more than several fold, which offers little help in attaining criticality: Wo=1.

What must be addressed first however is just why it is so important to cosmologists for Wo-->1. On a visceral level there is just something indescribably unsatisfactory about a hyperbolic K=-1 universe with a definite cosmogony but an uncertain eschatology: a forever-expanding spacetime of diminishing curvature populated by the cold ashes of bygone stars. Perhaps it is simply too assymetrical to be taken seriously: lacking the symmetry of definite beginning and definite end. The flat K=0 universe fares no better, simply attracting more adherents. Thus the incessant search for additional matter might be no more than exhibitions of the predilections of some cosmologist's (Kolb).

Hence, the quest for Wo>1. A very strong probability in this direction is a possible halo of cold matter surrounding galaxies. Wherein the radial velocity of galaxies at radial distance v(r) should diminish with r1/2, instead roughly constant velocities are found out to luminous cut-off, indicating an increase in mass with radial distance r. This mass is undetected, hence denoted cold dark matter. If this matter is baryonic, then Wo might at best increase by a magnitude to Wo=0.1, or perhaps Wo=0.2: still far short of criticality. This is certainly the upper limit of the baryonic density which would have been coupled with radiation in accordance with the SCM, otherwise the superbly predicted abundance of light elements: the great triumph of SCM, would be adversely affected. For W<1 deuterium would be underproduced and helium and lithium overproduced during nucleosynthesis.

What is left is non-baryonic dark matter, which would not have coupled with radiation. Such matter might be relics of the primordial universe: the stable Weakly-Interacting Massive Particles (WIMP). These might range from magnetic monopoles, pyrgons, maximons and newtorites with masses between 1016 and 1019 GeV to quark nuggets with masses up to 1030 GeV: a circus of massive relics proposed without even a shred of emperical support. Although such WIMP can exist within a consistent theoretical framework, the only problem with this scenario is that the physical processes necessary to form them, their lifetimes, and the processes responsible for their annihilation, are unknown. Although for closure the mass of all such existing WIMP would have to be roughly a magnitude greater than all of the detectable mass of the universe, in fact the survival of just the WIMP with masses on the order of the Planck mass would drive Wo to several magnitudes greater than unity, a nonsensical outcome.

As has been noted, there is no empirical evidence whatsoever for the present existence of any such non-baryonic matter. Nevertheless, there are those who's predispositions favor a closed universe and they still infer that such non-baryonic matter exists, much as did the adherents of the luminiferous ether of bygone days (Strauss & Davis). A disinterested observer must assume however from all of the evidence at hand that Wo<<1 and hence the SCM universe is decidedly hyperbolic and will go on expanding forever.



SCM Horizon Problem
    According to the Cosmological Principle there exists no special viewing point in the universe, our position notwithstanding, and in general over a sufficiently large volume of the universe any observed volume is representative of the whole: the universe is essentially isotropic. This supposition however would require that all of the universe was causally related at some time so that any conditions giving rise tooc some possible particularity in some portion of the universe would eventually influence all of the universe, giving rise to the essential isotropy. However, this requirement is not met in the SCM universe: any characteristic distance L was always, and still is, longer than the causal distance Rc inasmuch as Rc µ t while L µ t1/2, where t is cosmic time.

Accordingly regions causally connected are always smaller than the furthest extent of the universe: the event horizon is always smaller than the SCM universe. For example, the essentially uniform temperature of the cosmic background radiation (CBR) in any observed quadrant would naturally lead to the assumption that the conditions which originally lead to the emission and quantization of the radiation extended over the entire universe. However the horizon is too small in the SCM universe at quantization to accommodate such an assumption. Although R --> L in the present epoch, with the horizon at most roughly a kiloparsec, at earlier times the horizon would been significantly smaller than the greatest extent of the universe.

To circumvent this problem a modification of the SCM of the early universe denoted the Inflationary Standard Model (ISM) was originally proposed by Gliner and further developed by Guth. Essentially the breakdown in the electromagnetic and electroweak symmetry in the primordial universe according to the Grand Unification Theories (GUT) is accompanied by a phase transition in the thermodynamic sense. The primordial universe for a short period is isothermal rather than cooling with SCM expansion due to the release of the latent heat of transition, followed by a limited superheating of the universe subsequent to which the SCM scenario is followed to the present epoch. The result of superheating is a sudden and uncharacteristic inflation. To accommodate this inflation, the primordial ISM universe must have been significantly smaller than the classical SCM universe before inflation and hence smaller than the horizon. However, the ISM requires that the critical density Wo=1±106, a condition that is not presently supported by any observational evidence whatsoever, all of which evidence still supports Wo<<1, as has been discussed. Setting aside this detail the ISM has attracted much serious thought.



SCM Symmetry Problem
    This problem involves certain conservation relationships, the sum of whose terms one would expect to vanish, for example charge balance: the number of positive charges and negative charges are supposedly equal in the universe. Without exception all cosmological models arbitrarily provide for charge balance, yielding a charge number C=0. Accordingly, one would then expect a priori that the number of baryons such as protons p+ and antiprotons p- to be exactly equal, along with the number of neutrons and antineutrons, yielding a baryon number B=0. Likewise the number of leptons such as electrons e- and positrons e+ would be equal: yielding a lepton number L=0. Accordingly the baryon and lepton numbers, like the charge numbers of the universe, would be zero, with the universe symmetrical in these regards.

The reason this logical outcome is unacceptable is not due to any convincing observations but rather arises intrinsically from the SCM, according to which matter is in thermal equilibrium with radiation at nucleosynthesis. However the SCM provides no means that could account for the segregation of matter from antimatter under these severe conditions: a necessary requirement for separate matter and antimatter nucleosynthesis without mutual annihilation. Accordingly, a condition of B>0 must be provided prior to nucleosynthesis so that matter density exceeds antimatter density. Thus after annihilation some matter particles will survive to nucleosynthesis: accounting for all of the mass of the observable universe.

Models based on GUT do provide an out: for example non-conservation mechanisms by which anti-baryons can be converted to baryons, thereby allowing an original B=0 condition to subsequently become B>0 before nucleosynthesis. Of course all such processes require a causal connection throughout the early universe, a condition provided by the ISM scenario. What mitigates against this approach, and other mechanisms proposed by which the antimatter density can be diminished is that all such processes involve reactants and products whose chemical potentials are identical, such as equivalent baryons and antibaryons. Any equilibrium process that would favor baryosynthesis would have the same probability of favoring antibaryosynthesis.

Even under nonequilibrium GUT conditions all such models must not only provide for the formation of baryons at the expense of antibaryons, but simultaneously must provide for means of suppressing equivalent antibaryon formation. As a further difficulty, loss of electromagnetic and electroweak symmetry can lead to the formation of WIMP at the assymmetry transition, the number of which would dominate the mass of the universe, but which have not been observed. Needless to say, much difficulty have been encountered with the GUT approach to systematically provide baryon assymmetry, resulting in a plethora of GUT to circumvent these difficulties.



SCM Smoothness Problem
    Essential to the SCM is a homogeneous universe with matter and radiation in thermal equilibrium before decoupling, the strongest evidence for which is the uniformity of the Cosmic Background Radiation (CBR), yet this uniform soup gave rise to the large scale structure of the observed universe. That is, the primordial SCM universe was exceedingly homogeneous yet the present universe, although denoted isotropic, exhibits much large-scale structure. Galaxies are found in clusters and clusters are found in superclusters, the latter separated by vast voids, with the clusters and superclusters themselves are often arranged in chains.

Hence the isotropy of the universe is a matter of scale. Nevertheless there is sufficient anistropy to be of concern. In fact, the CBR now appears to be too uniform. Although perturbations are observed, some are smaller than the resolution of the measuring equipment, being mentally magnified by ingenious investigators using esoteric methodologies to account for the required anisotropy.


4 CONTINUUM COSMOLOGICAL MODEL



It is evident that all of these described problems are interconnected, either requiring special pleading, arbitrary initial conditions, or avoidance of empirical realities; and all of them are intrinsic to the SCM. However the SCM has been so overwhelmingly successful despite these difficulties that any alternative cosmology must at best encompass the Standard Model or at least be consistent with it. What will now be described is such alternative model.

Basic Assumption
    According to the Continuum Cosmological Model (CCM) the big bang singularity did not occur some 15 billion years in the past but rather resides some 15 billion lightyears distance from us.  Essentially, the temporal processes involved before decoupling are assumed to be an on-going concern.  Surprisingly the SCM formalism can be employed to describe the CCM universe, or more accurately the co-moving portion of our observable universe which is assumed to have been expelled some 15 billion years or so ago.

    In terms of lookback time astronomers can only see back a few billion years, which effectively limits the observable extent of our co-moving portion of the universe.  Although claims have been made that objects some 10 billion or so lightyears distance have been observed in the infrared by the HST, these claims has not yet been confirmed.  According to the CCM these latter objects would be relatively young, perhaps five billion years old.  Hence even this period and distance is still much too short to observe the temporal singularity.

World Form
    What will be assumed is that the universe takes the world form shown in Figure 1, a rather inadequate two-dimensional rendition of four-dimensional spacetime. According to the CCM the singularity has a temporal existence. From this singularity spacetime geodesics diverge and eventually converge. Hence the singularity exists not only at the origin of spacetime but for all time: it is the origin from which the discernible universe expanded and the destination to which it will contract. Hence the observable universe is a small portion of a continuum universe of fixed volume with timeless duration.

Although the volume of the CCM universe is fixed, its density is not necessarily uniform. Because a significant fraction of the mass of the CCM universe must be concentrated in close proximity to the singularity, the greatest curvature of spacetime occurs at the past singularity and at the coincidental future singularity.

Before considering an analytical description of the CCM universe, it would be advantageous to further describe the world form because of the difficulty in rendering 4-dimensional spacetime in two dimensions. The most difficult concept to describe is the continuum singularity in asmuch as it has a temporal existance.

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Figure 4.2a. CCM Singularity Concept
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Essentially the past universe in contraction is continuously passing through the singularity to the future universe in expansion. The photonic energy density and hence the temperature increases during contraction to infinite at the singularity and then decreases during expansion.
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Figure 4.2b. CCM Singularity Analogy
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As an analogy consider a light ray passing through an ideal lens system. The ray moves from the past through the focal point (at which point the radiation energy density is infinite) to the future. Consider now an analytical description of the CCM.


Friedmann-Lemaître Spacetime
    The coordinates of the CCM universe described in Friedmann-Lemaître spacetime (FLS) can be so chosen that the line element ds is described by

(4.1)                     ds2  =  dt2  -  R(t)2ds2

where t is local time, R(t) is a measure of the local time-dependent spacetime curvature and ds2 is the line element in Riemannian 3-space. Recognizing this distinction the line element can be described in general terms for both the SCM and the CCM. According to the SCM (c)=sin(c) for the case of K=+1, resulting in a closed system with greatest expansion at c=p/2 where 0<c<p for a single cycle, with convergence at c=p.

(4.2)                   ds2 = dc2 + …2(c/2) (dq2 + sin2qdj2) .

In contrast, for the CCM (c)=sin(c/2) as shown in equation (3), resulting in a closed system with greatest radial extent at the cusp c=p(tc,sc) where 0<c<2p. For continuum spacetime the geodesic folds back on itself: c=0 simultaneously at c=2p. Thus the same singularity exists at c=0 and c=2p.


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Figure 4.3. CCM Geodesic
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Hence a CCM elapsed time measures strictly the local time to from the origin (0,0) of our comoving shell to the present epoch at coordinate (t,s) as shown in Figure 4.3, not the time t since expansion initiated as in the SCM, and can be expressed simply as

(4.3a)                   to = (4pGDo/3) [c/2-sin(c/2)]

where Do is the local density within our local comoving spherical shell at our Present Local Epoch (PLE) as shown in Figure 4.4. The local radius of curvature Ro shown in Figure 4.4 can be expressed as

(4.3b)                    Ro = (4pGDo/3) [1-cos(c/2)]   .

Hence Ro=0 at both c=0 and c=2p. Consequently there exists a singularity from which spacetime diverges (c=0) and to which it converges (c=2p).  Thus, in passing through the singularity at convergence the original point of divergence is reattained.  The characteristic length for each epoch is shown shorter at the earlier local epoch than at the present local epoch and shorter again at the future local epoch as to--> tW.

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.Figure 4.4. Local Co-moving Shell
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The CCM universe is a continuum, with the singularity having a temporal existence.  Hence, as shown in Figure 4.4, only a local time to and distance so can be measured in spacetime coordinates (to,so) from the past singularity (0,0) to the present epoch (to,so) and thence to the future coincident singularity (tW,sW).  The time and distance at which the spacetime geodesics transition from divergence to convergence can be defined as the cusp or maximum radial extent (tc,sc).

Accordingly, any concept of an absolute cosmological time t is meaningless.  All that can be measured is to for our local epoch.  The local epoch (to,s) expands from the past singularity (0,0) to the cusp at (tc,sc) and thence contracts to the future coincident singularity (tW,sW).  The CCM universe is closed because the density D in the vicinity of the singularity at (0,0) and (tW,sW) is sufficiently great that Ro-->0 as to-->tW, with (0,0) and (tW,sW) rotationally equivalent.



Spectral Shift
Depending on the position of our local comoving spherical shell (to,so) in relation to the cusp (tc,sc) our co-moving shell can either be expanding [(to,so)<(tc,sc)] or contracting [(to,so)>(tc,sc)].  Accordingly we must now modify Ehlers working hypothesis that at any time and at any point in the SCM universe there exists a mean motion such that all cosmological properties are isotropic with respect to the reference frame following that mean motion by substituting for SCM universe our CCM comoving spherical shell.  It is our shell that is either expanding or contracting, not the CCM universe.  This done we can now consider the observed stellar and galactic spectrum shift z:

(4.4)                            z  = lo/le  -  1    .

Obviously the SCM interpretation of z does not fit the CCM scenario.  According to the SCM the universe did not only initiate at a single point but at a single time. Unfortunately, the continuous cosmogony of the CCM thoroughly confuses the issue.  Within our co-moving spherical shell all celestial bodies originated at the same time, regardless of their red shift relative to the observer.  Disregarding non-Hubble drift, the observed spectral shift z reveals essentially circumferential motion of bodies orthogonal to the radial motion of our comoving spherical shell. Observed very high spectral shifts could indicate either bodies at great distance from us - but still within our co-moving shell -  where the component of radial motion becomes important, or bodies outside our co-moving shell.  The latter case introduces even more confusion inasmuch as the age of the bodies become important: some would be younger and others older.  That extensive starbirth observed (Barger) in high-redshift galaxies is consistant with the CCM inasmuch as these galaxies may be members of comoving spheres subsequent to our comoving sphere, and consequently of an earlier age.



Singularity
The singularity is the state of the CCM universe at local to=0 and Ro=0, and consequently it is the demarkation between past and future. Equally so, the singularity is the beginning of the past and end of the future.

According to the CCM the singularity is a volume Vs=0 radiator through which no information whatsoever can transit.  Accordingly only photons without wavelength (l=0) can possibly pass through such a singularity without carrying information.  This latter condition will be the basis of the following discussion, although employing the SCM formalism where applicable.

According to the CCM the radiation emitted by the singularity at t=0 has an infinite frequency n=oo with an infinite density Ds=oo and therefore infinite temperature T=oo.  Beyond Vs=0 of course T immediately falls as the radiation expands into the surrounding transparent volume VP presumably with T oct-1/2 and T ocV-1/3.   Hence the CCM singularity is decidedly not the single quantum state as conceived by Lemaître.   The CCM singularity condition immediately precludes fermions transiting the singularity  inasmuch as matter possesses information in the form of mass, charge and spin.   Ostensibly bosons would also be precluded because mesons have mass.



Pseudo-Stellar Fireball
    The CCM singularity is encased within an opaque sphere spewing out matter and energy at a surface temperature of roughly T=3600 K.  Using the SCM formalism this fireball has a diameter of roughly three parsecs and our comoving portion of the universe will have emerged roughly 1010 years past.    At the Hubble distance (roughly 300 megaparsecs at Hoª2.7x10-18 s) the fireball can not be resolved by even the largest telescope as it's disc would subtend an angle of only some 350 microarcseconds.  Hence, if the fireball could be discerned it would appear as a stellar point source, most likely as a K5 star whose thermalized radiation peaks in the near infrared, consistent with CBR observations.
 
 


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Figure 4.5. CCM Temporal Singularity
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In accordance with the SCM after an elapsed time tPl=10-43 s denoted the Planck time the spherical Planck interface will intercept the flux radiating from the singularity.  Thus the volume VP is bound by the Planck interface with a radius rPl=10-34 m, as shown in Figure 4.5.   According to the SCM the radiation temperature at the Planck Interface will have dropped to the Planck Temperature TPl=1020 GeV.    However, according to the CCM this interface has a temporal existence:  it is a permanent feature of the universe.



Planck Interface
This is a crucial juncture because at temperatures T>1029 GeV it is generally thought that the relativistic concepts on which the SCM is based break down without quantum corrections, but no relevant corrections extant are applicable to this region.   Quantizing the gravitational degrees of freedom is a prodigious task.  The most obvious approach is to derive a gravitational analogy to the Schrödinger equation.  The solution would be a wave function of the universe which governs spacetime geometry at T>1019 GeV.   However there are considerable disagreements as to the appropriate boundary conditions, if in fact any boundary conditions are appropriate for any such wave function governing the universe.   Essentially quantum relativity attempts to reconcile two essentially irreconcilable concepts: quantum mechanics and general relativity, which on some level are simply incompatible (Wigner).

Another approach is of course a Grand Unification of the gravitational interactions with the electrostrong, electroweak and electromagnetic interactions. Although the electroweak and electromagnetic interactions have finally been unified, the real stumbling block is the formulation of the electrostrong and electromagnetic interactions in a geometric framework, as are gravitational interactions.  Such a framework would involve some kind of non-spatial but space-like dimensions.  These would have to be quite small (<<10-18 m) because nothing of the sort have been observed.

An alternative approach would be the formulation of a non-geometric theory of gravitational interactions wherein the interactions would be mediated by tensor particles with even-integral spin.  Such particles must have zero-mass and they are still as imaginary as non-spatial dimensions.  A further alternative is that perhaps no such integration of non-gravitational and gravitational fields is possible: we are indeed dealing with apples and oranges.

Hence at T>1019 GeV there is no satisfactory model to describe the processes operating, a temperature above which mass particles in any event could not be distinguished from one another.   A plausible way to resolve this dilemma is to assume that matter does not exist above the Planck Temperature. This is a reasonable assumption considering that the electron radius is presumed to be on the order of 10-17 m, enormous compared to the Planck radius rPl=10-34 m.

According to the CCM a monochromatic flux radiates outwardly from the singularity, striking the Planck interface at Tª1020 GeV, and constitutes the driving force for the ejection process.  The radiation pressure imposed on the Planck imterface is roughly 1077 J/m2, which is equivalent to some 1033 solar masses per unit area.
 

This monochromatic flux striking the Planck interface is either absorbed or reflected back.  Simulanteously energy radiates inwardly from the Planck interface at Tª1019 GeV.  Hence, as shown in Figure 6, the singularity is centered in a steady-state cavity of radiation:
    1) emitted from the singularity g1,
    2) emitted from the Planck interface g2, :
    3) reflected from the Planck interface g3, and
    4) absorbed at the Planck interface g4.

Hence within the cavity ng3+ng4=ng1+ng2+ng3.

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Figure 4.6. Energy Balance at Planck Interface of Pseudo-Stellar Fireball
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Because a temperature gradient exists between the singularity and the Planck Interface black-body conditions are not achieved.



Spacetime Curvature
The spacetime geodesics are essentially time invarient with K=+1 in the CCM rather than expanding as in the SCM with most likely K<0. Because the local mass-density of the universe is greatest at the singularity the greatest curvature occurs about the singularity.  This does not imply however that most of the mass of the CCM universe resides in the vicinity of the singularity. Obviously matter ejected from the singularity at (0,0) must have an initial velocity sufficient to overcome the gravitational attraction of the singularity, as must antimatter, in order to reach the cusp at (tc,sc), beyond which the attractive force of the contraction sigularity at (tW,sW) predominates.  The most significant concern is what proportion of the ejected matter and anti-matter will reach (tc,sc)  rather than falling back towards (0,0). Obviously this is a crucial concern in terms of the validity of the CCM relative to the observed universe.

Let us now consider particle synthesis under CCM conditions as an on-going process rather than a past happening and consider the means by which the present universe survived despite B=0 and L=0.


5 SYNTHESIS



Fermiosynthesis
    According to CCM fermions appears at T<1019  GeV as ongoing processes in pseudo-equilibrium with radiation.  In agreement with the SCM formalism matter appears in the form of three generations of leptons comprising the electron, muon and tau particle with their associated neutrinos together with three generations of quark singlets, all of which are created in matter and antimatter pairs.   The quarks persist to roughly T<1 GeV with the quark-hadron transition to essentially stable quark triplets: essentially neutrons and protons, with simultaneously an equivalent anti-hadron transition.



Leptosynthesis
    Our principle concern however is the lepton production of electron particles e and e by reaction (5.1a) inasmuch as such leptons scatter radiation, where underline indicates an anti-particle.

(5.1a)             2g  -->  e  +  e

Hence the monochromatic flux radiating from the singularity is scattered by the intercepted lepton production shell when T<1019 GeV, as shown in Figure 3b.  Accordingly, the universe becomes opaque with radiation scattering within a shell of inner radius r=10-34 m, with matter produced by reaction (5.1a) and annihilation by reverse reaction (5.1b), both reactions operating simultaneously.

 (5.1b)             e  +  e --> 2g      .

Consequently the radiation expanding beyond r=10-34 m gradually assumes a blackbody energy distribution in thermal equilibrium with the essentially equilibrated leptons permanently existing within this CCM region whose inner boundary is the Planck Interface.  That such a thermalizing process could occur has been affirmed by Hawking and Ellis.

 Although such interactions cannot be considered as truly equilibrium processes because there is a co-moving flow of particles outwards from r=10-34 m under the influence of radiation pressure, nevertheless this region of spacetime divergence can be considered one of steady-state or pseudo-equilibrium processes.  As such the radiation will gradually approach thermalization with increasing distance from the interface, with the result that the particles react in a soup of near-blackbody radiation beyond the Planck interface.  As such leptons and radiation are effectively coupled in essentially thermal equilibrium.

From classical particle collision theory the formation rate of lepton pairs ee per unit volume V according to equation (5a) is kf2g and the annihilation rate according to equation (5b) is kree then the overall lepton pair production rate R is

(5.2a)                R   =   kf2g  - kree

where g indicates number of photons.  The reaction rate constants kf and kr can be equated at equilibrium where Req=0 and therefore kf2geq=Rreeqeeq where 2geq and eeqeeq are the equilibrium number of photon and lepton pairs per unit volume V.  Hence

 (5.2b)              R   =   kf [2g  -  2geq(ee/eeqeeq)]   .

and accordingly when e+e<eeq+eeq reaction (5a) predominates (R>0) and when  e+e>eeq+eeq reaction (5b) predominates (R<0).

The number of particles in equilibrium at unit volume V can be represented by (eeq+eeq)/2geq which is simply kf/kr from particle collision theory. According to Steigman the unit volume V is proportional to

(5.3)                              V  µ  mc2/kT   .

As long at V<1 thermal equilibrium is maintained (ee=eeqeeq) and with decreasing temperature (eeqeeq)/g decreases with decreasing rate of reaction (5a).  At some critical temperature at which V>1 equilibrium is broken and the number of lepton pairs, specifically e and e becomes essentially fixed, with ee<<g.   Because CCM lepton number L=0 inasmuch as no information can pass through the singularity and therefore e = e, primordial leptons are so effectively annihilated by reaction (5.1b) that only an insignificant relic could possibly survive beyond this freeze-out temperature.  Likewise for quarks: after freeze-out q = q and pair annihilation would be essentially complete.  Hence the observed universe could not have evolved, ostensibly a CCM problem shared with the SCM.



Baryosynthesis
    At temperatures on the order of T=1 GeV the quark-hadron transition occurs with the formation of neutrons n and protons p in a specific ratio.  However these particles can interact, changing the ratio of neutrons to protons with decreasing temperature (T<1 GeV) by reversible processes involving neutrino ne interactions:

(5.4a)                                  n  +  ne   <-->   p  +  e

or

(5.4b)                             n  +   e   <-->    p  +  ne      .

These reactions are assumed to proceed by an ordinary reversible first-order mechanism:

(5.5)                                R   =   kf cn  - kr cp

where  R is the neutron reaction rate, cn is the neutron fraction and cp the proton fraction present: cn+cp=1.    From equation (5.5) at equilibrium R=0 and consequently  kr/kf=cn /cp  .

According to Weinberg the ratio of the rate constants takes the form

(5.6)                                       kr/kf    =   e-Q/kT

where Q is the energy difference between electrons and neutrinos.  Combining equations (9) and (10) yields

(5.7a)                               cn=  1/[1+e-Q/kT]

From equation (5.7a) at the quark-hadron transition (T=1 GeV) cn=0.38.  However the temperature is decreasing with expansion and at roughly T=20 MeV thermal equilibrium is broken, with cn=0.16.   The number fraction of protons cp and neutrons cn are subsequently fixed except for the crucial b-decay of the neutrons according to cn oc t/tn where tn is the neutron half-life.

Obviously the relative numbers of neutrons and protons present at nucleosynthesis is crucial to the relative quantities of the nucleides formed, and consequently to the measured value of tn, which is known only to ±3%.  However, as far as anyone can surmise, the chemical potentials of particles are identical to that of their equivalent antiparticles, and consequently an equivalent process to (5.4a) and (5.4b) is simultaneously occurring between antiprotons and antineutrons, resulting in

(5.7b)                                cn   =  1/[1+e-Q/kT]

for

(5.8a)                                    n  +  ne   <-->   p  +  e   .

 Because equivalent particle-antiparticle chemical potentials are identical obviously Q=Q.  Moreover, because proton-neutron interacts also involve antiparticles [equation (8b)] obviously antiproton-antineutron interacts can involve equivalent particles.

(5.8b)                                     p  +  ne  <-->   n  +   e      .

 The only problem with this scenario is that it presupposes an original L=0 and B=0 and moreover that both dL=0 and dB=0.   These conditions are generally considered contradictory.  That is, the SCM harbors a contradiction, ostensibly shared by the CCM.   Under the SCM B=0 and therefore from 1 GeV to 20 MeV equal numbers of baryons and antibaryons are coupled to radiation and are equally reactive because chemical potentials are identical.  Consequently the baryons should share the same fate as the leptons: mutual annihilation.

(5.9)                                B   +   B  -->     2g

 With B=0 the result would be a cold matter-free universe after decoupling.    The general observation that a matter-containing universe supposedly exists ostensibly argues against this eventuality.



Baryon Symmetry Violation
    Seeking means of circumventing this SCM eventuality that a matter-containing universe cannot exist constitutes a full-time effort on the part of a slew of cosmologists: to demonstrate that baryon symmetry-violation is either an initial condition or arises from suppression of the antiquark-->antihadron transition, and thereby in either case eliminating any relevancy to processes (5.8a) and (5.8b).

 Accordingly, in regard to this question of baryons symmetry violation, let us consider:
      1. the requirement for baryon symmetry violation,
      2. the empirical evidence for baryon symmetry violations, and
      3. the mechanisms proposed for baryon symmetry violations.

1. Need for Baryon Symmetry Violation:
         a) Annihilation processes after freeze out at T=20 MeV are so efficient
that reaction (13) would deplete the universe of necessary baryons
were in fact nB/ng=nB/ng.

         b) Statistical fluctuations leading to clustering that might circumvent reaction (5.9) are too small to achieve any kind of significant B and B isolation before freeze out.   Our galaxy (and presumably the universe) contains roughly nB/ng =10-10.    Statistical fluctuations  might achieve locally nB/ng =10-40 but could not possibly attain  parity: nB/ng = nB/n = 10-10.

2. Evidence for Baryon Symmetry Violations:
        a) Asteroids comprising anti-matter have not been observed.  Their interaction with the solar-wind would have permitted their detection  as they would constitute a very luminous source of g-radiation.
        b) Cosmic rays are generally of extra-solar origin, particularly above 100 MeV, with the most abundant antimatter comprising antiprotons  and antihelium nuclei; however their relative abundance in cosmic rays is only p/p = 10-4 and  He/He = 10-5.
       c) Even the number of antimatter particles found in cosmic rays could have originated by collisions, such as p+p=3p+p, rather than from an extra-solar source.
        d)   Gases between matter and antimatter galactic clusters should produce g-radiation.  Such radiation has not been detected.

3. Mechanisms for Baryon Symmetry Violations:
        a) Initially B>0 in the primordial universe.
        b) Initially B=0 wherein processes are conjectured that produce greater numbers of baryons compared to antibaryons nB/ng <nB/ng ; the mutual annihilation of all but the excess resulting in a residuum of baryons and virtual disappearance of antibaryons.

 Let us now consider the implications of the two mechanisms proposed, neither of which arise from the SCM formalism.

Mechanism (a) is purely arbitrary, and although it solves the baryon assymmetry problem, it places an a priori condition on the singularity without further describing or defining the nature of the singularity.  This mechanism is probably not amenable to empirical verification.

 Mechanism (b) introduces baryon assymmetry as a integral part of baryosynthesis.  This is the realm of the Grand Unification Theories (GUT), of which there are a sufficient number to satisfy any condition or eventuality; variations of which are often contradictory.

 A horizon problem is confronted with the GUT, but can be circumvented by modifying the SCM by introducing the inflationary period already discussed which is accompanied by a drastic temperature decrease below the SCM prediction, followed by an abrupt rebound.  During this period the horizon expands to encompass the entire universe, a condition which persists to the present epoch, contrary to the classical SCM.

Accordingly the early universe is causally connected.  The essentially unlimited horizon permits baryosynthesis to occur in all portions of the universe involving supermassive bosons under conditions of sufficiently high temperature that the electroweak and electromagnetic fields become unified.    The collision and decay of these bosons violate B symmetry, shifting B towards a matter-dominated universe.  Because the universe is causally connected such matter domination is global.

For example Kuzmin propose that anomalous baryon and lepton symmetry violating processes can be effective at T=200 GeV: the electroweak phase transition, and as a result |B+L| >0 and |B-L| =0.  However, these processes require that initially |B-L|<0, thus illustrating the flaw in grand unification schemes.  Charge and Parity (CP) violation must be somehow a priori prescribed, and hence the assymmetry that results from such models is an intrinsic outcome of the assymmetry that enters the models.    Worse, no means are provided to create the initial |B-L|*0, although the original assymmetry required would amount to only  B/B~109/(109+1) to result in a present universe of essentially pure matter.   That is, regardless of how miniscule the original B/B assymmetry it still must be accounted for.  This is imperative because without such an a priori assumption B=0 and therefore nB/ng = nB/ng~10-18.   In contrast observations ostensibly indicate instead nB/ng~10-10 with nB/ng =0, a significant difference.

 Although it is argued that both boson and antiboson decay violate CP, they must do so with a subtle difference sufficient to break B symmetry to obtain the observed nB/ng, while suppressing any processes leading towards an equivalent nB/ng .   Not surprisingly, at presently not a single GUT model has been experimentally and unambiguously confirmed to the degree that a definitive mechanism of spontaneous symmetry breaking can even be conditionally accepted, let alone precluding all others candidates.  In fact as simpler GUT models fail, for example by the requirement for proton instability, more complex GUT models must be proposed.  Nevertheless, there is a continuing and almost frantic effort to break baryon symmetry  and the driving force was best summed up by Sakharov more than a quarter of a century ago:

  "The theory of the expanding Universe, which presupposes a superdense initial state of matter, apparently excludes the possibility of macroscopic separation of matter from antimatter; it must therefore be assumed that there are no antimatter bodies in nature, i.e., the Universe is asymmetrical with respect to the number of particles and antiparticles."

 Of course the culprit in all these machinations is the observed existence of antimatter, the nonexistance of which would have greatly simplified cosmology but greatly complicated particle physics.  At present all experimentally observable processes assumed to be involve in the synthesis of leptons and baryons conserve B and L respectively.  Likewise CP  is conserved in all observable processes that might contribute to symmetry breaking, although CP violation is a condition for baryon symmetry-breaking in all of the GUT mechanisms proposed.  Consequently all early-universe symmetry-breaking scenarios must provide a means to restore the presently observed experimental symmetry, and this is conveniently provided by a break-down in grand unification at T = 1014 GeV.  Moreover, if in fact the universe is asymmetrical with respect to B and L, then the relationship between the non-zero values of B and L must be considered.  Obviously charge conservation puts a restriction on any flights of fancy.  Consequently the proviso that |B-L| = 0 is generally accepted, often arbitrarily, as a requirement of certain grand unification theories to obtain the observed nB/ng.

 It is apparent that there are as many problems as solutions involved in the effort to break baryon symmetry to conform with Sakharov's conclusion that "the Universe is asymmetrical with respect to the number of particles and antiparticles."  However, this conclusion arises from an unwarranted assumption that the lack of a suitable alternative mechanism "apparently excludes the possibility of macroscopic separation of matter from antimatter".

 Let us now examine the possibility of a suitable alternative mechanism that improves the possibility of macroscopic separation of matter from antimatter.     Such an assumption first requires that the universe be highly anisotropic on a scale much greater than previously imagined.  Fortunately quantitative observational analyses indicate that galaxies are not even remotely distributed uniformly: that there are in general regions of supercluster chains separated from other clusters by astronomical voids, and these observations are not restricted to any particular quadrant.



Critique of Evidence for Baryon Symmetry Violation
    Because any definitive experimental confirmation of symmetry-breaking models is absent, the only evidence for an asymmetrical universe are those observations discussed in section Evidence for Baryon Symmetry violations.   For the purpose of critiquing this evidence it will be assumed for the sake of argument that the universal baryon number, lepton number and charge number are symmetrical and accordingly the vast voids observed between galactic super-cluster structures separate matter portions of the universe from antimatter portions.  The evidence for L and B symmetry violations will be reconsidered in terms of this conservation assumption.

    a)  That asteroids comprising anti-matter have not been observed would be expected.  Even if asteroids were of extra-galactic origin, they would still in all probability have originated within our local cluster.  The  minuscule possibility that anti-matter asteroids, having reached our particular galaxy within our local cluster from a galaxy within an antimatter cluster, would survive our local cluster matter-field to approach our local star and be observable during some pass-by period is too small to be reasonably considered.

    b)  Conceding that cosmic rays are generally of extra-solar origin,  particularly above 100 MeV, and even conceding that they might  originate from very distance galactic clusters, the probability that any  antimatter particles would survive reaching our galaxy after traversing millions of parsecs through our local cluster matter-field is too small to be reasonably considered.

    c)  The embarrassingly high concentration of antiparticles in cosmic rays  probably have a local origin such as p + p --> 3p + p, rather than  from an extra-solar source.   Conversely, cosmic rays originating from  antimatter galaxies would similarly have a significant matter particle content:  p + p --> 3p + p.

    d)  Although the dust and gases between matter and antimatter galactic  super-clusters would produce g-radiation, the dust and gas particles in these vast regions would have been largely annihilated prior to and after decoupling.  No doubt such regions still are producing g-radiation from dust and gases swept in since then, but at an intensity too small to be detected at such distances.

 It is evident that the observations used to defend baryon assymmetry are unconvincing, and in fact could be used equally well to defend baryon symmetry, particularly Argument (c).  Nevertheless we must still confront Sakharov's concern: thermal equilibrium between baryons and antibaryons before decoupling would result in baryon annihilation, and rather than an observed baryon content of  nB/ng›= 10-10  we would have an inconceivable nB/ng <10-18. Accordingly, means must be found that promote matter-antimatter segregation before decoupling.

Returning now to the CCM, it is evident that baryon number, lepton number and charge number all carry information, and inasmuch as information cannot transit the zero-volume singularity, it follows that B=0, L=0 and C=0 are necessary conditions for the CCM at local time  t=0.  Hence during the later period of thermal equilibrium both nB=nB and nL=nL, a condition which persists to the present epoch according to the CCM, wherein nB/ng = nB/ng ª 10-10. How to resolve this delemma is both a SCM and CCM problem.

 We know that in themselves statistical fluctuations that might circumvent reaction (5.9) are too small to achieve any kind of significant B and B clustering before freeze out.   In terms of nucleation theory, the clusters cannot grow to a critical size beyond which the volume free energy of a cluster exceeds its surface free energy, promoting further growth.  Rather the clusters tend to collapse.   To remedy this situation it will be necessary to reexamine the gravity interactions that exist between matter and antimatter.


6     INTERACTIONS



Gravitational Potentials
    This section is the most speculative portion of this discourse and consequently the most controversial. Although the assumed potentials are not central to the CCM, these potentials go a long way in explaining the large scale structure of our co-moving portion of the universe, and their verification is discussed.  According to the weak equivalence principle for both matter and antimatter inertial mass mI  is equal to gravitational mass mG:

 (6.1)             mI  =  mG  and  mI  =  mG        .

Consequently the gravitational potential V^^(r) between two matter particles m1 and m2 and the potential V^^(r) between two otherwise identical antimatter particles m1 and m2 at displacement distance r are identical:

(6.2)                V^^(r) = V^^(r)       .

 Consistent with, but outside general relativity, the force between the particles m1 and m2 is assumed to be mediated by a tensor particle with even-integral spin, for which the gravitational potentials V^^(r) and V^^(r) can each only be attractive.  Using the Yukawa form of the potential between particles m1 and m2 the tensor particle is a spin-two boson.

Using this formalism there is ostensibly only three possible outcomes, and these have been considered by investigators concerning the gravitational potential V^^(r) between a matter particle m and an antimatter particle m.  The most obvious is of course equivalence

(6.3a)              V^^(r)   =  V^^(r)  = V^^(r)      .

Less obvious is a potential greater than equivalent matter and antimatter potentials arising from quantum gravitational considerations

(6.3b)                 V^^(r)   >  V^^(r)  = V^^(r)

but a distinct possibility being seriously contemplated.

A repulsion possibility that has been considered and generally found wanting, if for no other reason than antigravity presumably violates energy conservation according to Morrison, although a not too convincing argument.

There is however another possibility hitherto not considered, namely that matter and antimatter cause opposing perturbations in spacetime curvature.


Potential Form
    Although the proposed potential form is speculative, ironically it is the concept presented most amenable to verification.  It is assumed that
mass can only be quantitatively described as a complex number: a matter part and an anti-matter part.  However there is no preferential viewpoint.  In our local co-moving portion of the universe naturally we would designate matter rather than anti-matter as the real part of the complex description.  In terms of the conventional Newtonian potential between two particles therefore

                                V(r) = G[(m1+im1)(m2+im2)]/r2

the conventional Newtonian potential relating matter-matter interactions is a special case of a general potential relating matter-matter, antimatter-antimatter and matter-antimatter interactions.  Essentially consistent with the usual Newtonian formalism the universal gravitational potential V(r) will be expanded to comprise the sum of three terms where V^^(r) is our usual matter-matter potential

(6.4)                      V(r)  = V^^(r) + V^^(r) + V^^(r)      ,

 V^^(r) is the usual antimatter-antimatter potential, and V^^(r) is a Newtonian mixing term equal to 2G(mm)/r2.

    For matter-matter interactions m=0 and consequently the second and third terms of potential (6.4) vanish.  Accordingly potential (6.4) takes the expected form

(6.5a)                     V(r)  =  G(mm)/r2

    For antimatter-antimatter interactions m=0 and consequently the first and third terms of potential (6.4) vanish.  Accordingly potential (6.4) takes the expected form

(6.5b)                     V(r)  =  G(mm)/r2

    However for matter-antimatter interactions all the terms of potential (6.4) are operative and consequently on combining terms potential (6.4) takes the form

(6.5c)                      V(r)  =  G(m+m)2/r2

    The implications of equation (6.5c) is discussed in the Verification Section. Nieto & Bonner discuss the validity of alternative potentials.

    According to equation (6.5c) there will exist between a matter (m>0) particle and an equivalent antimatter particle (m<0) a potential null point [V(r)=0] as shown in Figure›6.1a.
 

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Figure 6.1a.  Geometric Representation of Potential Null
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 This view is consistant with the gravitational force being mediated by a spin-two tensor particle with the force always attractive. A possible interpretation of this approach is that there exists an attractive anti-gravity potential identical in all respects to the gravity potential except that they are mediated by opposite spin tensor particles respectively and consequently equal gravity and antigravity potentials cancel for m=|m|.
 
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Figure 6.1b.  Positive Potential Between Particle and Antiparticle
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Hence there exists an attractive potential between matter and anti-matter particles V^^(r)>0 as shown in Figure 6.1b whether or not m>|m| or m<|m| but V^^(r)=0 when m=|m|.



Baryonic Clustering
    Consider the SCM universe after the quark-hadron transition (T=1 GeV) but before decoupling (T=1 eV) in accordance with equation (6.5).  Immediatly after transition the universe is homogeneous and its mass density is largely baryonic.  To effect any kind of clustering that would lead to the segregation of particles from antiparticles requires fluctuations in the isotropic universe.   Essentially, only by clustering can particles be separated from antiparticles, or otherwise particle-antiparticle annihilation will keep nB/ng =nB/ng <10-18, significantly smaller than the ostensibly observed nB/ng =10-10.    However, according to the SCM no mechanism exists to allow clustering before decoupling, and if such clustering does not occur, then essentially all is annihilated and there is nothing to decouple, and the observable universe does not develop.   It is this great failure of the SCM that has given rise to the need for baryon unsymmetry to eliminate the unwanted antiparticles, and the slew of GUT and inflationary models to explain this unsymmetry.

 The conclusion that discrete B and B clustering cannot occur before decoupling is an outcome of the Jeans construction.  Only clusters with a mass M greater than a critical Jeans mass MJ will nucleate and thereafter accrete additional matter and grow.   Accordingly

(6.6)          MJ    =   p dJ [ns(p/GdJ)1/2]3

where dJ is the critical cluster density and ns is the speed of sound within the uniform medium from which clustering occurs.    Because the speed of sound approaches the speed of light before decoupling (T=1 eV) the Jean mass MJ  encompasses a significant portion of the mass of the primordial universe.  This mass is so great that M cannot possibly attain MJ  before decoupling.  Only after decoupling however, when the speed of sound decreases some 105 fold would clustering be possible, which according to the SCM gives rise to the galaxies.  By then however it is too late to save the universe or the SCM.

 This SCM scenario is predicated on the assumption however that the universe is still homogeneous after the quark-hadron transition, and consequently fluctuations required to nucleate clusters have wavelengths far too small compared to that required for M>MJ.  Hence the  SCM universe will not survive the quark-hadron transition and by necessity the assumption of B>0 is made, an excrescence fastened to the SCM to save the model.Although statistical fluctuations that might circumvent the Jeans limitations are too small to achieve any kind of discrete B and B clustering before decoupling, this is not the case under conditions of equation (6.5).

Although the effect of V^^(r)=0 will be inconsequential at high temperatures, presumably before the quark-hadron transition, nevertheless, such a system cannot be homogeneous: discrete particle and antiparticle clustering must occur with matter continuously accreting and dispersing from such clusters.  Although such clusters would be unstable at a sufficiently high temperature, nevertheless the CCM universe is inherently inhomogeneous, conditions under which the Jeans fluctuation criteria is unnecessary.  Clustering occurs by virtue of V^^(r)=V^^(r)>0 as would be expected but also V^^(r)=0, not fluctuations in a homogeneous medium.

 Accordingly a critical temperature Tc must be defined wherein the rates of matter dissipation and matter accretion equilibrate at a cluster mass Mc<MJ.  Therefore at T>Tc clusters are unstable, with matter dispersion exceeding accretion so that cluster masses cannot increase to their critical value Mc.  Only when T<Tc are clusters stable with M>Mc.  Thereafter matter accretion exceeds dispersion and clusters grow, as shown in Figure 6.2.

From baryonic symmetry considerations the practical range of values of Tc have a definite limit.  Baryonic clustering can only occur after the quark-hadron transition (Tc<1 GeV) and must have occurred by virtue of V^^(r)=0 before decoupling (Tc>1 eV), or otherwise there would be essentially nothing to decouple with baryonic symmetry.  However the Tc temperature range must be further limited.  Inhomogenization must initiate before nucleosynthesis (T>20 MeV) otherwise all of the components required for nucleosynthesis would not have collected in discrete B and B clusters.  Hence Tc must lie in the range 1000 MeV>Tc>20 MeV.



Freeze-Out
    Unlike the SCM scenario wherein decoupling is a global event at Tª1 eV, according to the CCM scenario at freeze-out the matter and antimatter particles that were dispersed between the clusters effectively annihilate, thereby eliminating photon coupling between clusters. Hence at freeze-out the region between clusters  becomes  transparent.  However within clusters coupling persists to T~1 eV as shown in Figure 6.2. Hence intercluster transparency occurs between 20 MeV and 1000 MeV before intracluster transparency at T~1 eV.


 Figure 6.2.  Cluster accretion at Critical Temperature
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Cluster accretion will accelerate with increasing cluster mass because of the increasing probably that an unclustered matter particle will be attracted to an increasingly more massive matter cluster (V^^(r)>0) rather than randomly striking an antimatter cluster (V^^(r)=0); with an equivalent situation occurring in regard to antimatter particles and matter clusters.

 Interestingly, a small fraction of quarks could conceivably survive quark-hadron transition as such quarks could only be annihiliated by an equivalent antiquark, and separations would be too great for collisions.  Such primordial quarks might form a stable bond with hadrons at T<200 MeV (Satz).  Alternatively heavy hadrons might form from up, down and residual strange quarks rather than from up and down quarks alone as in ordinary baryons (DeRujula).

 Obvious any extremely massive particles and antiparticles surviving the quark-hadron transition would strongly promote clustering, inasmuch as under such circumstances V^^(r)=V^^(r)>>0 while invarient V^^(r)=0.     Accordingly, such primordial particles would act as powerful nucleating agents, promoting M-->Mc.



Large-Scale Structure
    The matter and antimatter clusters formed after freeze-out but before nucleosynthesis (T>20 MeV) are subject to non-Hubble drift.  Matter-matter clusters drifting into contact will agglomerate, as will antimatter-antimatter clusters.  In contrast matter and antimatter clusters drifting together will rebound under radiation pressure arisng from peripheral particle annihilation at points of contact.  (This scenario is similar in some degree to that of Alfven as a means of segregating matter and antimatter particles.)  Hence large volumes of space will be cleared of clusters by agglomeration of contacting like-clusters and rebounding of contacting unlike-clusters. Consequently separation of matter and antimatter clusters into separate groups of clusters will proceed simultaneously with agglomeration under gravitational attraction to superclusters as shown in Figure 6.2.  The result will be a structured region inhabited by these superclusters before nucleosynthesis.

With rotation these superclusters will evolve into disc-like proto-galaxies after decoupling (Tª1 eV).  Hence structure develops early in the CCM universe despite B=0, with matter and antimatter forming essentially interwoven but separate universes.



Nucleosynthesis
    As the temperature cools through T=20 MeV nucleosynthesis initiates and the cluster accretion rate increases markedly inasmuch as the enhanced gravitational potential of the nuclear clusters will sweep any surrounding matter or antimatter towards their respective clusters, with the concentration of matter and antimatter relative to radiation progressively increasing:  nB/ng =nB/ng -->10-10.

As in the case of the SCM with decreasing temperature nucleosynthesis rapidly progresses with the formation of 1H+, 2D+, 3He++, 4He++ and 7Li+++ within matter clusters and equivalent 1H-, 2D-, 3He=, 4He= and 7Li within antimatter clusters.    With nB/ng =10-10 the nuclear ratios  2D+/1H+=3He++/1H+=10-5 and 4He++/1H+=7Li+++/1H+=10-10 would be expected.   Because the CCM scenario and the SCM are identical as far as physical processes are concerned nB/ng =10-10 will lead to an equivalent ratio of antinuclei: 2D-/1H-=3He=/1H+=10-5 and 4He=/1H-=7Li*/1H-=10-10.



Decoupling
    As T-->1 eV decoupling will progress but only within clusters inasmuch as intercluster particles will have been largely annihilated at freeze-out.  Hence intracluster space will become progressively transparent as the clusters become increasingly isolated.



Dicke Coincidence
    Disregarding the cosmological constant, the expansion rate can be represented as

(6.7)                                    H = [CD + (aR)-2]1/2

where H is the Hubble's constant, D is the mass density, a is the expansion parameter, R is the radius of curvature and C is a constant.  At an earlier epoch the mass density term (1st term) predominated in determining the expansion rate but at present the curvature term (2nd term) predominates.  What is remarkable is that there is such a fine balance between the expansion kinetic energy (KE) dependent on a and the potential energy (PE) dependent on D. If the KE of expansion predominated then the universe would freely expand without matter gathering into galaxies.  In contrast if the PE of expansion predominated matter would gather rapidly, with the universe finally collapsing before galaxies could form.  Rather the KE and the PE are so finely balance that neither eventuality occurred, a fortunate coincidence first discerned by Dicke and accordingly termed the Dicke Coincidence.  This coincidence is so central to the SCM that it guides much of the cosmological interpretation of astronomical obsrvations.

    Often however the requirement for such coincidences result in doubts in the model that require such a natural coincidence and the seeking of alternative models.  Consider the five following conditions in relation to expansion energy:  PE<<KE, PE<KE, PE~KE, PE>KE and PE>>KE. Only the middle condition would allow the SCM universe to evolve into the observed universe.  However any of these energy imbalances  could characterize the successive comoving spherical shells of the CCM universe.  The matter within some shells could rapidly collapse preventing galaxy formation while the matter of other shells would fully disappate, likewise preventing galaxy formation.  For others collapse would be a slow steady process, as would dissipation for others, depending on KE-PE imbalance.



Black Holes, White Holes, and Other Anomalies
    Galaxy formation from rotating proto-galactic clusters would be highly dependent on cluster mass Mcl.  Consequently, within a range of cluster masses Mclmin to Mclmax galaxies would evolve from such clusters while outside this range the clusters would evolve into other entities.

If Mcl<Mclmin centrifugal acceleration of the clustered gases and dust formed after nuleosynthesis would dissipate into interstellar gas and dust clouds, although extra-galactic star formation need not be precluded.

If Mcl>Mclmax gravitational contraction will overcome centrifugal dissipation, resulting in unconstrained contraction. The result would be a rapidly-rotating galactic black hole significantly more massive than  a stellar black hole.  Because the CCM universe is not evolving in the strictest sense, although our local co-moving portion of that universe is certainly doing so, these galactic black holes would be analogous to the contraction singularity at tW,sW, essentially a Secondary Singularity.  Presuming thusly, the galactic black hole would have associated with it a galactic white hole analogous to the expansion sigularity at 0,0: another secondary singularity.  For mass and energy balance these singularities must be contemporaneous, and could be considered a single entity.
 
 


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Figure 6.3,    Secondary Singularity
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Of course there has been much speculation about black holes leading to other universes but such speculation has always been made within the context of the SCM universe wherein the spacetime geodesics are temporal.  In terms of the CCM universe however the spacetime geodesics are largely invarient and fold in on themselves, providing ample opportunities for tunneling from one portion of the universe to another portion. Matter spewing forth from galactic white holes might constitute the core of later galaxies, a most interesting speculation.


Cosmic Background  Radiation
    Consistent with the SCM model the Cosmic Background Radiation (CBR) of the CCM model would have a local temperature of roughly 2.7 K characteristic of our co-moving spherical region.  However, depending on the coordinates of our local region t,s in relation to the maximum radial extent of the universe at tm,sm: if t,s<tm,smthen for our co-moving shell the TCBR would be higher in the direction towards the singularity at (0,0) and lower in the opposite direction. In contrast, if t,s>tm,smthen for our co-moving shell the TCBR would be higher in the direction towards the singularity at tW,sW and lower in the opposite direction.  Considering that the radiation cools with increasing volume of our co-moving shell and warms with decreasing volume, and the volume is proportional to our spacial distance D from the singularity, then presumably the radiation temperature TCBR would vary as

6.7     DTCBR/TCBRµ(DD/D)3

Consequently any gradient DTCBRwould hardly be perceivable, if at all.

    Although the CBR is characteristic of thermalized radiation it is far smoother than would have been expected, considering the required inhomogenity for structure formation in the early universe would have disturbed the CBR spectral distribution  (Peebles). However according to the CCM fireball radiation is an ongoing concern and hence disturbances in the spectral distribution of energy radiating from the fireball will integrate over time, smoothing the spectral distribution.



Local Expansion
    According to the SCM the observed red shift in the spectrum of distant galaxies is indicative of the cosmological expansion of the entire universe rather than a Doppler effect in a static universe.  In contrast,  according to the CCM the observed red shift is indicative of the cosmological expansion of only our co-moving portion of the universe, but only if t,s<tm,sm.  However if t,s>tm,smthen our co-moving portion will be contracting.  In that event the co-moving portior closer to the singularity at tW,sW will be accelerating away from us and meanwhile we will be accelerating away from the co-moving portions furthet from the singularity.   Hence a red shift will still be observed.  Apparently, expansion passes through different phases.  Although this CCM scenario is not what Dicke had in mind, the same analysis would probably be applicable.


7 CCM CONCERNS



Closure
    With all of the matter of the CCM universe distributed essentially as it had been and as it will always be, but with sufficient mass density in the vicinity of the sigularity to force closure.   Obviously matter ejected from the singularity at (0,0) must have an initial velocity sufficient to overcome the gravitational attraction of the singularity, as must antimatter, in order to reach the cusp at (tc,sc), beyond which the attractive force of the contraction sigularity at (tW,sW) predominates.  The most significant concern is what proportion of the ejected matter and anti-matter will reach (tc,sc)  rather than falling back towards (0,0). Obviously this is a crucial concern in terms of the validity of the CCM relative to the observed universe.



Horizon
     Because the CCM spherical shell bound by the planck temperature (T=1019 GeV) and the decoupling temperature (T=1 eV) has a temporal existence, all processes occurring within this opaque volume must be in causal contact as all of the time in the world is available for thermalization, leading to the thermal equilibrium between matter and radiation.  Within the central transparent volume (T>1019 GeV) matter has no existence, and no horizon problem can exist.



Symmetry
     By providing for nucleosynthesis within distinct matter and antimatter clusters baryon, lepton and charge symmetry can be maintained without mutual annihilation of matter.  This CCM view is of course far closer to the SCM than are the inflationary models.



Smoothness
     Because clustering occurred before decoupling,  galactic clusters and other large scale structures are a natural outcome of the CCM.  Genesis is an ongoing concern and hence disturbances in the spectral distribution of energy radiating from the fireball will integrate over time, smoothing the spectral distribution.


Homogeneity
    On the scale of the Hubble length the universe is essentially homogeneous and isotropic.  According to the CCM this observation would be expected.  Although each comoving sphereical shell might in itself be highly heterogeneous and anisotropic, on viewing the universe through successive shells the universe would appear both homogeneous and isotropic. as is observed.  Viewing through a limited number of shells however should reveal the intrinsic heterogeniety of our local portion of the universe, as also is observed.  Evidently, the inhomogeneity of each shell must affect the matter ditribution in successive shells or large scale structure could not form.


8 VERIFICATION



Certain claims are made herein concerning the CCM, namely that:
    1) a temporal singularity exists with zero-volume and infinite temperature;
    2) no information whatsoever can pass through the temporal singularity;
    3) matter cannot exist at temperatures higher than the Planck temperature;
    4) there is sufficient mass in the vicinity of the singularity to effect closure.
    5) no gravitational potential exists between equivalent matter and antimatter particles;
    6) large scale structure forms before nucleosynthesis; and
    7) the baryon number, lepton number, and charge number of the universe are each absolutely zero.

The spacetime geodesics are essentially time invarient with K=+1 in the CCM rather than expanding as in the SCM with most likely K<0. Because the local mass-density of the universe is greatest at the singularity the greatest curvature occurs about the singularity.  This does not imply however that most of the mass of the CCM universe resides in the vicinity of the singularity.

The failure of any of the tabulated claims will of course compromise the CCM model, but evidence of that failure must be empirical.  The only condition amenable to an empirical test at present is condition›(4).  According to Nieto & Goldman such an experiment is not only possible, but practical, albeit with much difficulty.  Such experiments are being prepared at CERN (Nieto).



Lepton Experiment
Time-of-flight determinations were conducted by Fairbanks to compare the gravitational deceleration of electrons and positrons in a vertical drift tube.  The results were inconclusive because stray electric and magnetics fields interact with the positron charge as predicted by Schiff & Barnhill, although attempts were made to reduce stray fields to <10-11 V/m.



Baryon Experiment
    Again using a vertical time-of-flight procedure, described by Holtzscheiter. Antiprotons will be cooled to a few degrees Kelvin and collected.  The deceleration of antiprotons escaping up the drift tube will be measured and compared with the deceleration of protons.  Stray fields unfortunately had the same debilitating effect as on the lepton experiment.

Neutrons have been found by McReynolds to be affected by a gravitational field in the expected manner without being hampered by stray fields.  Antineutrons would solve the stray field problem but they are difficult to collect and thermalize (Brando, Fainberg, Kalogeropoulos, Michael & Tzanakos).  Antihydrogen would be an ideal particle to examine but is difficult to manufacture although experiments are planned at CERN to commence in the next two years. Horizonal trajectory tests may also be considered and would perhaps be superior to vertical drift experiments as a wider range of initial particle accelerations can be tolerated.

In 1957 Fairbanks made the observation that "Nothing is known about the gravitational acceleration of antimatter".  His assessment still stands.



9  DISCUSSION


Cosmogony and Eschatology
     According to the CCM the universe is in a permanent state of unending genesis, with a cosmogony being meaningful only for a specified comoving portion of the universe.  Our portion the universe had evolved from a local to=0 through to=tc and thence on to oblivion at tc=tW: an unending eschatology.

 Following spacetime divergence, the galaxies move towards convergence.  The processes that gave birth to the structure of the universe are then reversed, with the eventual annihilation of matter and antimatter; the resultant energy merging at the singularity as T=oo radiation .   Evidently this process requires that baryon number, lepton number and charge number be zero because no information whatsoever can pass through the singularity at T=oo.



10  SUMMARY

 A Continuum Cosmological Model is presented that in large measure is consistant with the mechanics of the Standard Cosmological Model.  However, according to the Continuum Cosmological Model
 
1) a temporal singularity exists with zero-volume and infinite temperature;
2) no information whatsoever can pass through the temporal singularity;
3) matter cannot exist at temperatures greater than the Planck temperature;
4) there is sufficient mass in the vicinity of the singularity to effect closure.
5) no gravitational potential exists between equivalent matter and antimatter particles;
6) clustering leading to large scale structure forms prior to nucleosynthesis; and
7) the baryon number, lepton number, and charge number of the universe are each absolutely zero.

   According to the SCM the universe is no older than our clocks indicate. No matter how far out we gaze, the age of the universe is no older than our present age.  According to CCM however, our clocks merely measure the age of our co-moving portion of the universe.  Other portions are either older or younger, depending on their timeframe in the CCM Universe.
***
    From the musing of the ancient Assyrians through the discovery of our galaxy to the Standard Cosmological Model mankind has been progressively displaced from his anthropocentric spatial position in the Universe. Adding insult to injury the Continuum Cosmological Model described herein displaces mankind from his anthropocentric temporal position in the universe. Present time is merely the local elapsed time of our co-moving portion of the universe since its inception and Hubble's Constant is solely a measure of the local expansion of our co-moving portion of the universe within the framework of the Continuum Cosmological Model.



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