Time-Symmetric Cosmology

Transcription

1 Time-Symmetric Cosmology Lionel D. Hewett Physics Dept., Texas A&M University-Kingsville, Kingsville, TX 786 April 27, 20 Abstract This article is based upon the assumption that the universe began with a single creation event and utilizes the symmetry of time surrounding that event to derive both a classical and quantum model of cosmology. The classical model is a special case of the flat Friedmann-Lemaître-Robertson-Walker model of cosmology unmodified by any inflationary hypothesis. This classical time-symmetric model is uniquely determined from only four cosmological observations yet correctly predicts the observed values of some twenty additional cosmological variables, including the dark energy content of the universe. The quantum time-symmetric model is built upon the framework of the classical model and predicts that the creation event was immediately followed by an ensemble of first events that created primordial black holes which evaporated through Hawking radiation so as to produce the expected radiation dominated early universe, the excess of baryonic matter over anti-matter, and the observed dark matter content of the universe. The model not only predicts the fundamental properties of the quantized dark matter particles, but also explains why dark matter is cold and why it normally interacts only through gravitation. But the model also predicts that dark matter particles can absorb one another and emit monochromatic radiation when the resulting particle returns to its ground state. If this dark matter glow is sufficiently intense, it would allow direct observation of the distribution, density, temperature, and radial velocity of dark matter. Key topics: creation event, time-symmetry, orders of infinity, TS cosmology, alternative to inflation, dark energy, cosmological parameters, cosmological eras and epochs, quantum gravity, observable physical events, unobservable spacetime, unobservable GR singularities, first events, primordial black holes, black hole continuum, cosmic gravitational background (CGB) radiation, symmetrical Hawking radiation, black hole decay, baryogenesis, cold dark matter, dark matter particles, and dark matter glow.

2 Part : The Classical Time-Symmetric Model of Cosmology Creation Event Time-Symmetric (TS) Cosmology is only one of many alternatives to cosmological inflation []. It differs from the others by adopting an unconventional paradigm that considers space and time to be unobservable rather than observable properties of the physical universe. Only the physical events imbedded within spacetime are subject to physical observation. On the large scale, when events appear to saturate the spacetime continuum, there is the illusion that space and time are physically observable. But on the small scale, one can see that discrete physical events are separated from one another by a spacetime continuum that cannot be detected. 2 For example, classical (non-quantum) special and general relativity consider spacetime to be continuous, able to be subdivided into increasingly smaller and smaller regions even to the limit of a geometrical point in space and time. Such a conceptualized point in spacetime is a geometrical event having no physical significance except as a point of reference within the spacetime continuum. But physical events do have physical significance. Of necessity, they must involve the interactions of particles of matter and/or energy and cannot be resolved to infinite detail because of the limitations of quantum mechanics. The Heisenberg uncertainty principle requires that the more closely a physical event approaches the size of a geometrical event, the greater must be the uncertainty in the energy associated with that physical event. And the period-uncertainty relationship of an event requires that its energy must be greater than the uncertainty in its energy. 4 Therefore, the only way a physical event could ever become the size of a geometrical event (with zero spatial size and zero temporal size) is for that event to exist as a physical singularity of infinite energy and infinite uncertainty in energy. 5 It is hard to imagine that such an event could actually occur in nature, but that is exactly what the Penrose-Hawking Singularity Theorem [2] predicts about the beginning of our universe. Furthermore, logic tells us that if the universe did actually have a beginning and if physical events were involved in the creation process, then our physical universe must have either begun with a single event or with multiple events. If it began with a single physical event, then the expectation value of the time of that very first event would, by definition, be the point at which physical time began (t = 0). Since physical time could have no meaning prior to that time, there would be no uncertainty in that value of time. Since the uncertainty in time would be zero, then both its energy and its uncertainty in energy would have to be infinite. Furthermore, since causality can travel no faster than the speed of light, the spatial extent of that event must also be zero. Therefore, if the universe began with a See Appendix A for a detailed discussion of this paradigm of cosmology and its implications regarding quantum gravity. 2 As an analogy, consider a chalk drawing on a blackboard. The drawing is made of chalk, not the blackboard. The backboard serves only as a canvas holding the chalk in position. From a distance the drawing looks continuous, but up close it is seen to consist of tiny particles of chalk separated from one another by empty (non-chalk) space. The Heisenberg uncertainty relationship may be expressed mathematically as E t / 2, where E is the uncertainty in energy of a quantum system and t is the corresponding uncertainty in time. For an event of minimum uncertainties this relationship becomes E t / 2. Therefore, as t 0, E. 4 The period-uncertainty relationship may be expressed mathematically through the approximation t in order for a quantum system with a period of oscillation to be confined within an event whose uncertainty in time is t. Appendix B (through step 7) shows how this relationship leads to the conclusion that the energy associated with an event must be greater than its uncertainty in energy: E E. 5 Appendix B (through step 9) shows that as a physical event approaches the size of a geometrical event, it must become a physical singularity. 2

3 single creation event, then it must have begun as a physical singularity as described above (having a zero extent in space, a zero extent in time, an infinite energy, and an infinite uncertainty in energy). 6 However, if one assumes that the other possibility is true, namely, that the universe began with multiple events separated either in space or in time, then there is a major causality problem. If the events are separated far enough in time for causality to connect them, then one or the other of the events must have occurred first and the universe would have begun with a single creation event with the properties mentioned above. But if the multiple events are separated in space 7, then they could not have been causally connected and ) the spatial volume of the universe would not have been zero at the time of creation, 2) there would have been no possible way to synchronize the creation process, and ) there would have been no possible way for the distribution of events to have been approximately homogeneous or isotropic. 8 The first of these conclusions means that the universe would have begun without a singularity thereby violating the Penrose-Hawking Singularity Theorem. This presents no problem for those cosmologists who believe the theorem is invalid whenever quantum gravity dominates the situation []. However, it does present a problem for those who believe the theorem remains valid even when quantum gravity is present. The second of the above conclusions means that a group of causally disconnected events must somehow synchronize their clocks sufficiently to begin a spatially separated creation process. Unfortunately, there seems to be no obvious explanation as to how such a causally disconnected synchronization process could possibly occur. In fact, the topic has rarely, if ever, been discussed among the scientific community. However, the third of the conclusions mentioned above has been discussed among the scientific community a discussion greatly influenced by the inflationary hypothesis. Inflation theory asserts that regardless of how the universe actually began (whether from an initial physical singularity of unpredictable outcome or through some unknown primordial process of undefined symmetry), if one assumes that the universe initially expanded slowly enough for causality to permeate a significant region of spacetime and that somewhere around 0 6 seconds after creation it suddenly inflated by a linear factor something like 0 26 over a time period of about 0 seconds [4], then all of the major mysteries surrounding a Big Bang origin would be explained. More specifically, this enormous and rapid inflation would expand the causally connected region beyond the limit of our observable horizon thereby solving the Horizon Problem [5]. It would smooth out any initial spatial irregularities so as to form a homogeneous and isotropic universe thereby explaining the Cosmological Principle [6]. It would stretch out any initial positive or negative curvature thereby solving the Flatness Problem [7]. And it would enlarge the early miniscule quantum fluctuations sufficiently to provide the seeds that would eventually grow to form the current structure of the universe thereby solving the Primordial Fluctuation Problem [8]. 6 See Appendix C for a discussion as to why the creation event is the only physical singularity that can exist in the observable universe. All other singularities predicted by general relativity are unobservable geometric singularities rather than observable physical singularities. 7 Here we are talking about the regular three-dimensional space of our own universe separating simultaneous creation events, not about some hypothetical hyperspace that might separate various hypothetical parallel or bubble universes from one another. 8 If the number of creation events was small, then by a slim chance they could all have been arranged symmetrically, but they could not have been arranged homogeneously and isotropically. If the number was large, then the probability of their being arranged homogeneously and isotropically would be vanishingly small without some kind of causal connection.

4 Nevertheless, if one wishes to construct a model of cosmology that involves primordial physical events, is consistent with the Penrose-Hawking singularity theorem, and also retains causal connectivity from the very beginning, one has little choice but to make the following primary assumption of TS cosmology: Assumption : The universe began as a single physical singularity event of infinite energy and zero spatial extent at time t = 0. This is the initial event of creation, the primary source of all space, time, matter and energy the beginning of the Big Bang and the origin of the whole physical universe. Since there is no spatial extension of the universe at this creation event, the only possible spacetime direction away from the event must be in a pure time direction. 9 In other words, time must radiate outwardly in every possible time direction away from the creation event. 0 Furthermore, every subsequent event occurring anywhere in the universe ) must be time-separated from this creation event, 2) must be causally connected to this creation event, ) must be connected to the creation event by a unique geodesic timeline, and 4) must be associated with a unique local rest frame defined by that timeline. Because every event in the universe is causally connected to the same single creation event, TS cosmology has no Horizon Problem [5]. And because every event in spacetime is associated with a unique local rest frame, TS cosmology predicts that motion through the universe relative to this rest frame should be detectable. This first prediction of TS cosmology has been observed experimentally. In fact, the motion of our local group of galaxies relative to the Cosmic Microwave Background (CMB) radiation of the universe has been measured to be 627 ± 22 km/s [9]. Time Symmetry Assumption 2: There is no preferred time direction. The universe must look the same when viewed along any timeline and time must progress at the same rate in every time direction away from the creation event. 2 Since a spatial rotation (translation perpendicular to time from one timeline to another) must leave the universe unchanged, it follows that the universe must be spatially homogeneous. Since an angular rotation (about any time line) must leave the universe unchanged, it follows that the universe must be 9 Many people have an intuitive concept of what is meant by different spatial directions, but find it difficult to comprehend what is meant by different time directions (especially if they have a Newtonian view of time). But special relativity asserts that observers moving with respect to one another will perceive time differently. Each one views himself to be at rest in his own reference frame with time advancing forward along his worldline. Since the worldlines of such observers lie along different paths in spacetime, their timelines point in different directions of time. 0 As an analogy, consider a given point in -D Euclidean space. The only directions in space away from that point are in the various pure radial directions. Displacements perpendicular to any radial direction are only possible for points that lie outside of that central point. For the creation event, the radial directions correspond to various time directions and the direction in spacetime perpendicular (orthogonal) to time corresponds to space. Therefore, only for events subsequent to the creation event does it become possible to have any spatial displacement. There can be no spatial displacement directly away from the creation event. Actually, every geodesic timeline passing through a subsequent event connects that event causally to the creation event (because the past light-cone of that event converges back to the creation event itself). But only one of these timelines is unique, namely, the one that has the extremal proper time since creation. 2 Let us continue the earlier -D Euclidean analogy. There is no preferred direction away from any given point. Therefore, a geometrical point is said to exhibit spherical symmetry. Everything looks the same when viewed along any radial line away from that point and the spatial distance from that point increases at the same rate in every radial direction. Therefore, a rotation from one radial line to another leaves the situation unchanged and a rotation about any radial line leaves the situation unchanged. Consequently, all surfaces that are everywhere perpendicular to the radial lines (i.e. all concentric spheres about the given point) must be homogeneous and isotropic. 4

5 spatially isotropic. In other words, the spacetime of the universe must be perfectly symmetrical about the creation event. This second prediction of TS cosmology is an affirmation of the Cosmological Principle [6] and requires the universe to be spatially homogeneous and isotropic on the global scale as confirmed to a high degree of accuracy through recent observation of the CMB radiation [0] and the large-scale structure of the universe on the order of billions of light-years []. Notice that this assertion of perfect symmetry can only be valid on the global scale and not on the local scale. The presence of galaxies and stars (and even our own existence) demonstrates that this perfect symmetry must somehow be broken in order to produce the local texture contained in the universe. As mentioned earlier, inflationary cosmology explains the local texture by assuming that the early quantum fluctuations were enlarged by inflation. Eventually they evolve to produce the subtle anisotropies observed in the CMB radiation as recently measured in the COBE and WMAP experiments [0]. They also evolve to produce the structure and distribution of galaxies and stars that we see today. The fact that inflation models have been able to predict the subtle anisotropies observed in the CMB radiation data is one of the great triumphs of inflation theory. Time-Symmetric cosmology explains the local texture of the universe through a completely different mechanism. The classical TS model concludes that perfect symmetry does exist from the very beginning. But the quantum TS model (discussed later) incorporates quantum mechanics and causality to render a statistical ensemble of secondary creation events that break the local symmetry while leaving the global symmetry intact. This break in local symmetry then propagates through the universe as it evolves into what we observe today. Because TS cosmology is still in its infancy, additional work will have to be done to see whether or not it can predict the observed anisotropies of the CMB radiation as successfully as inflation theory. Spatial Flatness It turns out that the constant curvature of a homogeneous and isotropic universe can be positive, zero, or negative [2]. Positive curvature renders a closed spherical universe of finite spatial volume; zero curvature renders a flat universe of infinite spatial volume; and negative curvature renders a hyperbolic universe whose spatial volume is infinitely larger than that of a flat universe. 4 Since quantum mechanics requires a physical singularity to have an infinite energy, then a physical singularity has too much energy to create a spherical universe of finite energy density and too little to create a hyperbolic universe of energy density greater than zero. Therefore, TS cosmology, which begins with a single physical singularity creation event, can only render a spatially flat universe. 5 This means that the spatial curvature of the universe must be zero, space must extend forever in all three spatial dimensions, and the universe must have the precise critical energy density necessary to provide this zero curvature. 6 Finally, the universe must expand forever. Of course, in the -D Euclidean analogy one could ask, But what if the space is non-euclidean? Then the above conclusions would not follow. That is true in the case of the analogy. But in the case of the creation event, that would be like asking, But what if the future is non-symmetrical? Because of the principle of causality, the future cannot affect the past, so the spacetime of the future cannot affect the symmetry of the creation event itself. It is the symmetry of the creation event that affects the spacetime of the future. And there is nothing about the event, itself, that suggests a preference of one time direction over any other. 4 See Appendix D for a discussion of different orders of infinity. 5 See Appendix E for a more detailed discussion of why a physical singularity creation event can only render a flat universe. 6 Time-symmetric cosmology asserts that it is not the finely tuned energy density of the universe that causes space to be flat. It is the flatness of space that causes the energy density to be finely tuned. 5

6 Here again we see a difference between the predictions of the classical TS model and inflationary cosmology. The classical TS model predicts that on the global scale the universe is perfectly flat (that the spatial curvature of the universe remains a constant equal to zero) while inflation cosmology predicts that the global curvature of the universe may not be zero, but after inflation, the spatial curvature of the observable part of the universe can hardly be distinguished from zero. Current observations are inadequate to resolve these predicted flatness differences because both models predict flatness within the limits of current experimental uncertainty. [] Comparison with Conventional Cosmology From this point on, the classical TS model follows the logic of conventional cosmology. In fact, the classical TS model is precisely the same as the well established flat Friedmann- Lemaître-Robertson-Walker (FLRW) model [4]. And since the extremely successful Lambda- Cold-Dark-Matter (CDM) model of cosmology [5] is an extension of the FLRW model [4], then it would also be an extension of the classical TS model except for the fact that the extended CDM model includes a period of inflation [5]. Since TS cosmology contains no inflationary period, it is applicable throughout all time, from the very first event of creation to the most distant conceivable future. Therefore, in the interest of completeness and in order to identify the specific assumptions, predictions, and limitations of the classical TS model, as well as to establish a firm foundation on which to build the quantum model, this article presents the following brief derivation and discussion. Classical TS Metric Equation Since classical TS cosmology predicts a homogeneous, isotropic, and spatially flat universe, its geometry is described in Planck units 7 by the flat FLRW metric equation [4]: 2 d [ dr r ( d sin d )] 2 2 dt a ( t), () where d is the spacetime interval between two neighboring events, t is the proper time coordinate of an arbitrary event, a(t) is an expansion factor expressing the relative size of the universe at time t, and rare the spatial polar coordinates for that event. 8 General Relativity Assumption : The universe evolves according to Einstein s gravitational field equation of general relativity. This equation can be written in tensor form as [7] G g 8T, (2) where G is the Einstein Tensor, is the cosmological constant, g is the metric tensor, and T is the stress-energy tensor. Under our classical TS assumptions, this equation reduces to the coupled equations [8]: 7 In order to simplify the equations used in this article, from here on we will express all relationships in Planck units, where the speed of light c =, the gravitational constant G =, Boltzmann constant k B =, and the reduced Planck constant =. [6] 8 Notice that the metric equation () is valid for all time (including t = 0) for TS cosmology but valid only after the end of inflation (t 0 s) for the CDM model. 6

7 2 a 8 a 8 cm m a cr r 4 a a(0) 0 m r () where a = a(t) is the expansion factor introduced in (), a a(t ) da is its proper time derivative, dt = (t) is the total mass-energy density of the universe, is the constant energy density of the vacuum of space 9, m = m (t) is the density of all forms of matter in the universe, r = r (t) is the density of all forms of radiation in the universe, is Einstein s cosmological constantc m is a constant related to the law of conservation of mass-energy, c r is a constant related to the conservation of the number of radiation quanta (photons), and a(0) = 0 is the boundary condition that the expansion factor (spatial size of the universe) is zero when the time t = 0. Special Solutions Because the coupled equation () constitutes a non-linear first-order differential equation, its general solution cannot be constructed as a simple sum of special solutions. Nevertheless, special solutions do provide insights regarding the general solution, so let us consider the following three special cases where one or the other of the density terms dominates the situation. More specifically, notice that the density of the universe can be written as = /8 + c m /a + c r /a 4, where the first term represents the vacuum energy density, the second term the matter density, and the third term the radiation energy density. It should be obvious that when a is sufficiently small, the third term completely dominates the situation, when a is sufficiently large, the first term dominates; and somewhere between these two extremes, for appropriate values of, c m, and c r, the middle term dominates. In other words, radiation density dominates the early era of the universe, matter density dominates the middle era, and vacuum density dominates the late era. During each of these three special situations, the other two density terms can be neglected and an analytical solution to () can be obtained. Radiation Dominated Universe (RDU) The solution to the early radiation dominated era, where vacuum and matter are negligible, is straightforward and yields: 9 The identification of as the energy density of the vacuum of space is a consequence of allowing the expansion factor a to increase without bound. The resulting physical situation is a pure vacuum and the only remaining energy density term in the coupled equation () is the factor. This vacuum energy density has long been known to exist as a consequence of the quantum fluctuations of virtual particles in a vacuum. However, quantum theory has not yet been able to predict a reasonable value for this constant. Nor have direct measurements (such as measuring the Casimir effect) been able to determine its experimental value. As we shall see, the classical TS model does predict its value to better than two significant figures. 7

8 8 2 2 / 2 / 4 / 2 2 t t a t c a r r r (4) Therefore, the density of the universe during its early radiation dominated era is completely independent of all three parameters (, c m, c r ) as well as the constant of integration. This means that the radiation density of the early universe is completely determined by our previous classical TS assumptions and not by any additional assumptions or any later adjustment of its three parameters in order to fit current observations. Matter Dominated Universe (MDU) During the middle stage of evolution when the matter density term dominates the solution, one obtains the following equations: 2 2 / 2 / / 6 6 t t a t c a m m m (5) Again, the density of the universe is independent of the parameters (, c m, c r ) and the constant of integration. In fact, it is a simple multiple (6/9) times that of the radiation dominated universe because of the fact that both densities decrease with the inverse square of time. Vacuum Dominated Universe (VDU) During the late stage of evolution when a is large and the vacuum energy density dominates the solution, one obtains the following set of equations 20 : 5 8 ) ( 4 / 2 ) ( 4 4 ) ( ) ( H H e a c T e a c e a c e a a t t H r t t H r r t t H m m t t H (6) The constant a is the value of a at time t when the matter and radiation densities first become negligible. Notice that during the late stage of evolution, the expansion factor a increases exponentially with time and the matter density m, radiation density r, and temperature T 20 Here we have included the CMB temperature T and the Hubble expansion factor H (which have not yet been defined in this article) simply in order to illustrate what happens to all of these variables during the final stage of evolution of the universe.

9 decrease exponentially, rapidly becoming truly insignificant once time increases significantly beyond t. On the other hand, the vacuum energy density and the Hubble expansion factor H remain constant. General Solution Notice that the solution to this first order differential equation () renders a model of cosmology with only three adjustable parameters (, c m, c r ) and one arbitrary constant of integration a 0. The limited number of adjustable parameters is a consequence of considering only the three major contributions to energy density. Therefore, the TS model is less flexible and less detailed than those cosmological models that include additional parameters. 2 Nevertheless, the TS model is capable of predicting the outcomes of a surprisingly large number of observations once its three parameters and its constant of integration have been determined. Normalization of the Expansion Factor Assumption 4: The expansion factor today is equal to the age of the universe today. This is really more a matter of a definition than an assumption. It is made possible because the boundary condition a(0) = 0 does not uniquely determine the scale factor for the arbitrary constant of integration of this first order differential equation (). This can be seen by looking at () and noticing that the spacetime interval d is a physical invariant, independent of the coordinate system. Therefore, the product a(t)dr must be independent of the scale factor chosen for the coordinate r. This requires the scale factors for a(t) to be the reciprocal of the scale factor for dr. And since the scale factor for r is completely arbitrary, it may be chosen so as to guarantee that Assumption 4 is valid. The accepted experimental value of the age of the universe today is t 0 = a 0 = a(t 0 ) = 8.06E+60 =.77 Gy [9]. Hubble Expansion The function a H H( t) (7) a is always greater than zero for positive values of a,, c m, and c r. Therefore, the expansion factor a is continually increasing with time and the universe is perpetually expanding. This expansion is called the Hubble expansion and the accepted experimental value of H today is H(t 0 ) = H 0 =.2E-6 = 69. km/s/mpc [9]. Thermal Radiation Assumption 5: The type of radiation affecting the solution to () is thermal black body radiation with a radiation energy density r given in Planck units by 2 T 4 r, (8) 5 where T is the absolute temperature of the black body radiator [20]. Although there are other forms of radiation besides thermal black body radiation (such as gravitational or non-thermal electromagnetic radiation), thermal black body radiation appears to be the dominant form of radiation permeating the whole universe today. This thermal radiation 2 For example, the extremely successful CDM model has some six or seven parameters [9]. 9

10 has been verified experimentally as the CMB radiation and the accepted experimental value of its temperature today is T 0 =.924E-2 = K [2]. Evaluating the Parameters, c m, and c r Assumption 6: The currently accepted experimental values for the age of the universe, the Hubble expansion factor, and the cosmic microwave background temperature are valid. The solution to the non-linear differential equation () can now be obtained through a numerical integration of the equation t 8 ada, (9) a 4 c a c 8 m r with the constants, c m, and c r adjusted until the accepted values of t 0, H 0, and T 0 are obtained. The resulting values of these constants in Planck units are =.4E-22 c m = 2.62E+59 (0) c r =.8E+6. Remember that a different choice of a 0 would affect the values of c m and c r but not. Therefore, the value of has physical significance while the values of c m and c r do not. 22 In essence, they are coordinate dependent parameters that must be combined with our arbitrary choice of a 0 in order to calculate other quantities that do have physical significance. Extending the TS Model of Cosmology The simple three-parameter model of TS cosmology derived so far is able to predict many of the properties of our evolving universe. But it cannot begin to provide the details predicted by other models that are based upon a larger number of parameters. For example, the sevenparameter CDM model is able to evaluate the experimental values of all 9 derived cosmological parameters listed in Table 7 on page 28 of reference [9]. In order for TS cosmology to do the same, it would have to be extended well beyond our simple three-parameter model by making additional assumptions and incorporating additional experimental observations. This article will make no attempt to extend TS cosmology to that extent. However, we will illustrate how this can be done by adding one additional parameter through the following assumption: Assumption 7: The ratio of dark matter to baryonic matter in the universe is equal to the recently observed value of 5.2 and has remained constant from the time these forms of matter came into existence until today. This assumption allows us to extend our three-parameter model (which makes no distinction between the various forms of matter) to a four-parameter model (which is able to distinguish between dark matter and baryonic matter the two major forms of matter recognized to exist today). This distinction will become particularly important when we consider the quantum model of TS cosmology, but it is also informative in the case of the classical model. 22 Although they are related to the laws of conservation of energy and quanta, they do not specify the actual amount of energy or the number of quanta in the universe. Those values, like the size of the universe, are infinite. 0

11 Current Values of Cosmological Variables Once the values of the four cosmological parameters have been determined, the classical TS model of cosmology is uniquely determined and can be used to make numerous predictions. For example, the current values of various cosmological variables as predicted by the four-parameter TS model are listed in Table along with the recently published experimental values for those variables [9]. Since the first four experimental values (in bold font) listed in this table were used to determine the five constants a 0,, c m, c r, and the ratio d b they are required to agree with experiment. However, all of the other predicted values also agree with experiment to within the limits of experimental uncertainty. Table. Current Values of Various Cosmological Variables Time-Symmetric Model Predictions Conventional Cosmological Variable Symbol Planck Units Units Recently Published Experimental Values Conventional Units Expansion Factor a E Gyr Definition Age of Universe t E Gyr.772±0.059 Gyr Hubble Constant H 0.2E km/s/mpc 69.2±0.80 km/s/mpc CMB Temperature T 0.924E K ± K Ratio of Dark Matter to Baryon Matter d b ± 0.22 Hubble Constant Ratio h ± * Cosmological Constant.4E-22.20x0-56 cm -2 (.20 ± 0.0)x0-56 cm -2 * Density of Vacuum.25E x0-0 g/cm (6.44 ± 0.7)x0-0 g/cm * Density of Matter m0 5.0E x0-0 g/cm (2.58±0.08)x0-0 g/cm * Density of Baryonic Matter b0 8.08E x0 - g/cm (4.8±0.06)x0 - g/cm * Density of Dark Matter d0 4.20E x0-0 g/cm (2.7±0.06)x0-0 g/cm * Density of Radiation r E x0-4 g/cm (4.652±0.004)x0-4 g/cm * Critical Density 0.75E-2 9.0x0-0 g/cm (9.0 ± 0.2)x0-0 g/cm * Dark Energy Equation of State w Total Density Ratio tot ± * Curvature Density Ratio k Vacuum Density Ratio Matter Density Ratio m Radiation Density Ratio r ± * Baryon Density Ratio b ± Dark Mass Density Ratio d Physical Matter Density m h ± Physical Baryon Density b h ±.000 Physical Dark Density d h ± Age of photon decoupling t 4K 2.9E+56.75x0 5 yr ( ) yr * *These experimental values were not specifically listed in [9] but were calculated from the data contained therein.

12 Cosmological Eras and Epochs Table 2 lists some of the classical TS model s predictions of various properties of the universe related to the various eras and epochs commonly associated with cosmology. Table 2. Predicted Values for Cosmological Eras and Epochs RADIATION DOMINATED ERA t (s) a (s) m (kg/m ) r (kg/m ) (kg/m ) T (K) H (km/s/mpc) Creation 0 0 Infinity infinity infinity infinity infinity Planck Epoch First Events t.6e E-5 2.E+68.8E+96.8E+96.2E+2 9.9E Planck time t p 5.4E-44.8E-4.6E+67.5E+95.5E E+ 2.9E Black hole continuum fractures t s.2e-4 2.8E-4.0E E E+94 4.E+.2E Primordial black holes decay t f 8.7E-9 7.E-2 5.5E E E+84.6E+29.8E GUT Epoch No inflation begins T no-begin.0e-6 7.8E- 4.5E E E+80.5E+28.5E Strong force decouples t GUT.0E-5 2.5E-0.4E E E E+27.5E No inflation ends T no_end.0e- 2.5E-09.4E E E E+26.5E Quark Epoch Weak force decouples t weak 2.E-0.2E+0.E+7 8.6E E+27.0E+5 6.8E Quarks fuse t Quark.0E E+05 4.E E+6 4.4E+6.5E+2.5E Lepton Epoch Neutrinoes decopule t neutrinos 2.E+00.2E+08.E E E+07.0E+0 6.8E Premordial nucleosynthesis begins t Lepton.0E E E-0 4.4E E+04.5E+09.5E Nuclear Epoch Premordial nucleosynthesis ends t helium.0e+0 2.5E+09.4E E E E+08.5E Radiation/Matter crossover t mr 7.8E+ 7.8E+ 4.4E-6 4.4E-6 8.8E-6.5E E MATTER DOMINATED ERA t (Gy) a (Gy) m (kg/m ) r (kg/m ) (kg/m ) T (K) H (km/s/mpc) Radiation/Matter crossover t mr 2.5E E-0 4.4E-6 4.4E-6 8.8E-6.5E E Atomic Epoch EM radiation decouples t 4K.7E-04.2E-02.9E-8 8.E-9 4.7E-8.E+0.6E Large-scale structure begins t Atom.0E-0 4.4E-0 7.9E-2 4.4E E-2 8.5E+0 6.5E Galactic Epoch Matter/Vacuum crossover t m 9.8E+00.0E+0 6.4E-27.6E-0.E-26.7E E VACUUM DOMINATED ERA t (Gy) a (Gy) m (kg/m ) r (kg/m ) (kg/m ) T (K) H (km/s/mpc) Matter/Vacuum crossover t m E-27.6E-0.E Stellar Epoch Current Age of Universe t E E- 9.0E Exponential growth begins 5 t E+02.E- 6.8E-7 6.4E Remnant Epoch One hundred times the current age of the universe 00 t E+6 9.9E-4 6.0E-7 6.4E E H t H t H t 2

13 Most of these predicted values are consistent with those commonly published in the literature [22]. However, there are three major differences between this table and most other commonly published tables. The first major difference is the inclusion of three important times during the Planck epoch that are predicted by the quantum TS model discussed later in this article. These three times are the average time t of the first events following creation, the average time t S that the black hole continuum fractures, and the time t f that a primordial black hole of average mass decays. Although the times listed here are consistent with the quantum model, some of the other tabulated values are not. This is because the table is based upon the classical model (which predicts that the universe began radiation dominated) rather than the quantum model (which predicts it actually began matter dominated). The second major difference is the inclusion of two important times associated with inflation theory that are assumed to occur during the GUT epoch. These two times are the times that inflation is predicted to begin and end. Since there is no inflationary period in TS cosmology, there is no beginning and ending of inflation. Therefore, these two times are labeled t no_begin and t no_end, respectively. They are included here simply to illustrate how the inflationary model differs from the time-symmetric model. The third major difference is the identification of the Vacuum Dominated Era. Most other tables omit this era and place the Stellar Epoch along with our current age of the universe in the Matter Dominated Era. However, Table 2 clearly shows that we have already passed well beyond the matter/vacuum crossover point and Table shows that the makeup of our universe is already 7.4% vacuum energy. The universe has already begun to accelerate and will continue to do so until it attains a true exponential rate of expansion with the Hubble factor truly constant and with the density of the universe equal to the constant vacuum energy density of empty space. Before the universe even reaches 00 times its current age, the density of matter and radiation will have reached truly insignificant proportions and the CMB temperature will have dropped to essentially absolute zero. We conclude, therefore, that except for a few minor details (primarily in terminology and simplification) the classical time-symmetric model of cosmology predicts the same global properties of the universe as the standard cosmological models when time is significantly greater than the assumed period of inflation. However, prior to the period of inflation the two models differ dramatically. This can be seen by applying quantum mechanics to the time-symmetric model.

14 Part 2: The Quantum Time-Symmetric Model of Cosmology This model of cosmology considers what happens when quantum fluctuations are allowed to break the perfect symmetry of the classical TS model. It utilizes some of the fundamental principles of quantum mechanics, causality, general relativity, and statistical mechanics to derive the statistics and average properties of the first events in the universe that follow the initial creation event. These first events are actually secondary creation events because they are the first events in the universe to involve finite quantities of matter and energy. They set the stage for everything that follows, and hence, provide the detailed boundary conditions from which the universe continues to evolve consistent with the laws of nature. We will begin our derivation of the quantum TS model of cosmology by making the following fundamental quantum mechanical assumption: Heisenberg Uncertainty Principle Assumption 8: All physical events in spacetime must obey the Heisenberg uncertainty relationship for a minimum wave packet: Et / 2 p x / 2 x p y y / 2 p z / 2 z () In essence, this assumption asserts that physical events can occur only when the wave functions of all of the incoming particles collapse as much as is physically possible in order to create an interaction event. In other words, the participating particles must all be at the same place at the same time within the limits set by quantum mechanics. Ensemble of First Events Assumption 9: The first events following creation constitute a statistical ensemble whose uncertainties in energy and time are equal to their respective expectation values. The rationale for this assumption is as follows: If a statistical ensemble is sufficiently nonsymmetrical about its mean value for its positive uncertainty to differ dramatically from its negative uncertainty 2, then a single value of uncertainty cannot possibly provide an appropriate measure of the actual uncertainty inherent in the distribution. 24 Only by restricting our analysis to those distribution functions whose positive and negative uncertainties are reasonably equivalent to one another can we be justified in using a single value of uncertainty a value that is also reasonably equivalent to the other two values. Since neither the energy of an event nor the time since creation can take on negative values, we will also restrict our analysis to statistical distributions functions of positive quantities. Such functions cannot have negative uncertainties greater than their respective mean values. By restricting our analysis to statistical distribution functions of positive quantities that have positive uncertainties reasonably equivalent to their negative uncertainties we can guarantee that 2 The positive uncertainty of a statistical distribution is the uncertainty in the positive direction from the mean value. Similarly, the negative uncertainty is the uncertainty in the negative direction from the mean value. 24 In order to express the true uncertainty of an extremely non-symmetrical distribution, one would need to provide both the positive uncertainty and the negative uncertainty. See Table for examples of how this is done when expressing as accurately as possible the experimental values of highly non-symmetrical physical measurements. 4

15 their maximum uncertainties will be approximately equal to their respective mean values. Under these conditions, Assumption 9 simply states that the uncertainties of the energy and time of the ensemble of first events are as large as physically possible. 25 This assumption immediately leads to the conclusion that during primordial conditions the uncertainty in energy E for the first events must equal the average energy E of those events and the uncertainty in time t for those events must equal the average time t of those events. And since mass and energy are the same in Planck units, it follows that the uncertainty in mass m for the first events must equal the average mass m of those events. Therefore, E E m m t t (2) Combining () and (2) one finds that: E t m t () 2 Therefore, as the average time of the first events increases, the average mass and energy decreases and vice versa. In other words, high energy events occur early in the primordial universe and low energy events occur later. This seems like a reasonable, or even intuitive, conclusion. Self-Connected Causality Assumption 0: An event must be causally connected to itself. In other words, the uncertainty in the spatial size of an event cannot be greater than the distance causality can travel within the uncertainty in the time of the event. As stated previously, TS cosmology asserts that every event in the unbounded universe is causally connected to the creation event. But it does not follow that every event is also causally connected to every other event. Only those events contained within the past and future light cones of an event are causally connected to that event. Since such light cones are bounded by null geodesics where d, the radial boundaries of the light cones can be determined by setting d, d, and d equal to zero in the metric equation () and solving the resulting equation for the radial coordinate r as a function of time t: t dt r( t) r0 (4) a( t) 0 The constant r 0 is the value of the coordinate r where the primordial null geodesic originates at time t = 0. By changing the value of r 0, one can shift the null geodesic curve so as to pass through any radial event in spacetime. By setting r 0 equal to zero and choosing the plus sign, one can obtain the equation for a primordial null geodesic emanating radially outward from an arbitrary central observer. It then follows that the proper radial distance R(t) from that central observer to his primordial null geodesic at time t is simply the radial coordinate r(t) multiplied by the scale factor a(t): 25 Of course, there are many conceivable mathematical distribution functions that do not meet the conditions specified above, but we will ignore such functions in this article and challenge anyone who is interested to explore what would happen if Assumption 9 is relaxed. 5

16 R(t) = a(t) r(t). (5) Therefore, the proper radius R of causality in a primordial universe dominated by radiation or matter can be obtained by substituting (4) or (5), respectively, into (4) and (5). The results are as follows: RDU MDU R = 2 t R = t (6) These equations show that, because of the rapid expansion of space in the primordial universe, causality travels outward from an arbitrary observer twice as fast in a radiation dominated universe as it could travel in a non-expanding universe, and three times as fast in a matter dominated universe. 4 By using the equation V R to get the corresponding spherical volume of the light cone of the causally self-connected first event at arbitrary time t, the equation = E/V to get the corresponding matter-energy density, and () to relate the average energy and average time of the first events, one finds that the average density of the causally self-connected first events under the RDU and MDU conditions are: RDU E 4 4 MDU 2E 4 (7) 9 Correspondence Principle Assumption : The average or expectation values of the variables characterizing the ensemble of first events are equal to the corresponding values we have previously derived for the classical model. 26 Setting the average density of the ensemble of first events as expressed in (7) equal to the respective classical densities of (4) and (5) at average time t provides the following values for the average time, average radius, average mass, and average energy of the ensemble of first events following creation: RDU t R 2.44 m E MDU t R 2 m E (8) 26 Implicit in Assumption is the requirement that there be an infinite number of first events distributed in a uniformly random pattern throughout the whole spatially unbounded universe so as to produce a homogeneous and isotropic universe on the global scale but a statistically random distribution of events on the quantum scale. 6

17 Notice that the values of all of these quantities are in the vicinity of one Planck unit of the respective quantities. This should not be particularly surprising since we are considering the very first events following the creation of the universe under conditions that clearly involve quantum gravitational phenomena at the Planck scale. However, some of these values actually exceed the Planck limit. Such predictions are in direct contradiction to the commonly held scientific paradigm asserting that no physical significance can be attributed to any values of any quantities that exceed the Planck limit. Nevertheless, the predictions are perfectly consistent with the paradigm of cosmology espoused in this article. Primordial Black Hole Assumption 2: A first event will create a classical black hole if its radius is less than the Schwarzschild radius of a black hole of an equivalent mass. In Planck coordinates, the Schwarzschild radius R of a black hole of mass m is R = 2 m. Therefore, on the average, a first event will create a black hole if R < 2 m, or equivalently if R 2m (9) Using the data in (8) and setting the density m / V of a Schwarzschild black hole of mass m S m and volume VS 4R S / equal to the density of the universe in (4) and (5) one finds: RDU MDU S S S R 2m R S R 2.44 S ts t R 2m R S R S ts 8t 2.09 (20) The time t S is equal to the time at which the overall density of the universe { from (4) and (5)} becomes equal to the density of black holes with masses equal to the average mass of the first events { S from (20)}. In other words, it is the time that the universe is statistically just saturated with black holes. Prior to this time the universe was oversaturated, consisting of a continuum of primordial black holes with their event horizons overlapping one another. Conceptually, the universe could be visualized classically as a single all-encompassing black hole continuum 27 expanding outward in time so fast that its event horizon eventually shatters into 27 We are calling this a black hole continuum because it consists of a continuum of black holes with their event horizons overlapping. However, the dynamics of this continuum are actually those of a white hole (where matter and energy can only emerge outward and nothing falls inward) rather than that of a black hole (where everything falls inward and nothing emerges outward). In fact, the whole creation process described in this article is an example of the white hole phenomena emanating from the white hole singularity that we have been calling the creation event. 7