Closed Loop Universe and the Topology of Totality

Abstract

The author suggests a cosmology based on just two principles: (1) no boundaries to spacetime and (2) no physical infinities. On the hypothesis that these are exactly true, and that charge conjugation, parity and time reversal invariance (CPT) is a perfect symmetry at the big bang, answers are provided to many puzzles, such as what happened to the antimatter, what happened before the big bang, and what will happen in the far future? Consequences are that our universe $U$ is closed (with positive curvature) and so the total energy, angular momentum and charge of the universe must be zero. At the big bang a universe-antiuniverse pair, $U\overline{U}$ is created as a virtual fluctuation in what we may call Outer SpaceTime, along with a vast number of other disconnected universes—the multiverse. The CPT-conjugate $\overline{U}$ has identical physics (and matter content) to $U$; other $U\overline{U}$ pairs in the multiverse have different physics parameters. A consequence is that time at the fundamental particle level à la Feynman must be distinct from thermodynamic time. The $U\overline{U}$ forms a closed loop.

1. Introduction

If charge conjugation, parity and time reversal invariance (CPT) is a perfect symmetry applied to the universe, the question has long been asked, and is still considered a major puzzle: “Where is the antimatter?” or now more usually: “What happened to the antimatter?” as we assume that the origin was perfectly symmetric.

Already in 1956, shortly after the discovery of the antiproton, Goldhaber published [1] a “speculation” to preserve particle—antiparticle symmetry that “… the cosmos and its possible counterpart, the “anticosmos,” are somehow separated from the very beginning of their existence.”

In 1972, Omnes discussed the problem [2] and argued that there should exist as many antigalaxies as galaxies. No credible matter–antimatter separation mechanism was proposed and he listed many problems, for example, the consequences of galaxy-antigalaxy collisions.

Later that year Davies proposed a model [3] in which time reverses between successive cycles of an oscillating universe. In 1973, the author proposed [4] that CPT conservation at the bounces leads to a different interpretation, that our universe $U$ and an antiuniverse $\overline{U}$ came into existence together, and will eventually come back together in a common “big crunch.” A baryon world-line is a closed loop, and the author noted that this is “reminiscent of fluctuations of the vacuum in quantum electrodynamics, in which a particle-antiparticle pair appear in space and then annihilate.” This $U\overline{U}$ pair has total charge, baryon number and lepton number all equal to zero, but the matter is all in $U$ and the antimatter all in $\overline{U}$. There is no need for some “Great Annihilation” of antimatter after the big bang. The total energy also has to be zero, because space is closed, and so electrical and gravitational fields must be confined. (That is not of course the case for a virtual $e^{+}{e}^{-}$ pair, as proven by the Lamb shift, etc.) This suggestion answered the question: “What happened to the antimatter?” It is in the parallel $\overline{U}$, disconnected from $U$ except at the big bang and a future big crunch. (Related work in the 1960’s by Sakharov [5] had been published only in Russian.) This implies an “Outer SpaceTime” in which such fluctuations occur.

Tryon then published a note [6] in which he also suggested that the universe might be a vacuum fluctuation with total energy $E\left(U\right)$ = 0. His model differed from Ref. [4] in that our universe should consist equally of matter and antimatter, as previously discussed by Omnes and others.

There has been enormous progress in cosmology in the intervening 50 years, and we can now state with confidence that all of the universe presently accessible to observation grew out of a big bang some 13.8 $\times {10}^9$ years ago, that it has been expanding ever since, and that the expansion appears to be accelerating.

2. Two hypothetical principles

If one supposes two simple principles to be absolutely true, a particular topology of the universe follows naturally. These principles are:

1. P1: Spacetime has no boundaries. Einstein’s idea that space may finite but has no boundaries is familiar [7], but I extend this to spacetime.

2. P2: No physical infinities. The realm of infinities should be confined to mathematics. While ${10}^{183}$ is a very big number it is not “almost infinity.”

Strictly applying P1 and P2 has many radical consequences. Singularities (with infinite density) are forbidden. While these principles are not obviously falsifiable by experiment, they lead to a cosmology that provides answers to some of the mysteries: the matter dominance of our universe, what happened before the big bang, and what is its ultimate fate in “deep time”? The proposed model can be called Closed Loop Universe (CLU). This will require making a distinction between the “thermodynamic time” of everyday experience and the “world-line time” of elementary particles.

3. P1: Spacetime has no boundaries

The concept that the three dimensions of space can form a finite volume without boundaries, analogous to the two dimensions of the surface of a sphere, was proposed by Einstein [7], the mathematical form having been invented in 1854 by Riemann. Being unable to formulate boundary conditions for “spatial infinity,” Einstein said: “If it were possible to regard the universe as a continuum which is finite (closed) with respect to its spatial dimensions, we should have no need at all of any such boundary conditions.” The total volume of a closed spherical 3-space $S^3$ of radius $R$ is $V=2{\pi }^2{R}^3$.

We extend the notion of closed and finite 3-space to 4-dimensional spacetime, à la Minkowski. Let $x_1=x,x_2=y,x_3=z$, and $x_4=\mathit{ict}$. While the three spatial coordinates have arbitrary orientation but are orthogonal to each other, time is orthogonal to the dimensions of space in the same sense that imaginary and real coordinates are orthogonal in the complex plane. We suppose that not only do $x_1,x_2,x_3$ form a closed spherical space $S^3$, but $x_1,x_2,x_3$, and $x_4$ form a closed (Euclidean) spacetime $S^3{T}^1$. In x,y,z,t coordinates this is a toroid.

Consider a tube of radius r and length L and join the ends to form a torus. On its 2-dimensional outer surface, the circumference (length $2\pi$r) represents one space dimension and the orthogonal coordinate along L represents time, t, a topology $S^1{T}^1$. Replacing the one space dimension with three gives a topology $S^3{T}^1$, in which time, like space, is finite but unbounded. The three space dimensions form the 3-dimensional surface of a 4-dimensional hypersphere. Without changing the topology, make r(t) a function of time by constricting the tube at one time to be very small (at the big bang) but not zero (no infinite curvature of spacetime, no singularities!). The concept of the CLU is shown in Figure 1, with only one space dimension shown; it is not just closed space but closed spacetime.

Figure 1.

Sketch of the toroidal topology of the CLU version of the universe, with world-line time as a closed loop. The big bang marks the origin of our universe and the symmetrical antiuniverse. Only one of the three space dimensions is shown. Thermodynamic time is different from world-line time and increases away from the big bang in both universe and antiuniverse.

The universe is expanding with the radius r(t) apparently increasing faster than linearly at the present time, attributed to a cosmological constant or dark energy; if true this must be temporary as “forever” is not allowed. Consider the world-line of each lepton or quark—in a Feynman diagram of the whole $U\overline{U}$ pair! Fermion world-lines have zigzags corresponding to pair creations and annihilations.¹

In Ref. [4], the author had suggested a future collapse to a big crunch, which was identified with the big bang. The problems of entropy at that event, together with the hypothesis of closed loop time, can be avoided by instead identifying the far future of our universe with the far future (deep time) of the “parallel” antiuniverse.

This CLU is a “closed timelike loop,” but unlike closed timelike loops that have been much discussed and found to raise major issues in causality, this does not. It is not possible to travel back in time. Hawking’s “Chronology Protection Conjecture” [8] applies.

This does not mean that any event is repeated; world-line time is cyclic but there is just one cycle, and that is it. It is easy to see how, in this topology, the questions: “What happened before the big bang?” and “Does time have an end?” have no more meaning than “Where is the edge of the Earth’s surface?” The future of our universe is finite, as is that of the parallel antiuniverse, and in deep time they will become one.

4. P2: No physical infinities

Infinities abound in mathematics, but the author proposes that they do not occur in the physical universe. If this hypothesis P2 is true, there cannot be an infinite number of particles (or universes). The total volume of space in the universe must be finite. To say that the volume is finite, we must define a measure. The only natural candidate is the Planck volume $V_{\mathit{\text{Planck}}}\sim L_{\mathit{\text{Planck}}}^3\sim 4\times {10}^{-105}{\mathrm{m}}^3$, where $L_{\mathit{\text{Planck}}}=\sqrt{\mathrm{\hslash }{G}_N/c^3}$. Today the volume of the observable universe is $\sim {10}^{78}$${\mathrm{m}}^3\sim {10}^{183}\times V_{\mathit{\text{Planck}}}$. The number is increasing as the Universe expands, but cannot become infinite. The accelerated expansion attributed to dark energy must be temporary. With no infinite quantities, we should have no infinite densities or singularities, neither at the big bang nor at the center of black holes.

If fermion world-lines are continuous through the big bang, there are $\gtrsim {10}^{80}$ fermion lines threading the constriction.²If each fermion has to occupy a distinct Planck volume (which may not be true), they could fit in a volume with $R_{\text{neck}}\approx {10}^{-8}$ m.

It has been suggested by Pegg [9] that a very few fermions (or even only one of each type) could thread the big bang some ${10}^{80}$ times. In the extreme limit, every electron could actually be the same one. When Feynman drew world lines as zigzags in spacetime, he noted (probably very tongue-in-cheek) this possibility within our universe, but to be true it needed an equal amount of antimatter and matter, and therefore required an unlikely large-scale separation mechanism. It would have the advantage of explaining why all electrons (and up quarks, etc.) have the same mass. They are the same particle!

In 1967, Sakharov [5] gave three conditions that should be satisfied to account for the matter–antimatter asymmetry in our universe following a big bang: CP-violation, baryon number violation and non-equilibrium dynamics. While non-equilibrium dynamics no doubt prevailed at the big bang, his first two principles are circumvented by considering not just “our universe” but the $U\overline{U}$ pair. The CP-violation observed in weak interactions may have little or nothing to do with baryon number violation.

Models in which the big bang is preceded by a “big crunch” with a bounce have to address the Second Law of Thermodynamics. What happens to the entropy? This difficulty is avoided with this radical proposal of a distinction between (a) “world-line time” in Feynman diagrams and with his idea [10] that positrons can be considered as electrons moving backwards in time, and (b) “thermodynamic time” with entropy increasing, teacups breaking and our memories forming. The author will comment on thermodynamic time in Section 6, but for the rest of this chapter by “time” the author means world-line time unless otherwise stated.

This closed space, really closed spacetime, is different from that of a black hole, which has mass-energy and can have electric charge, and so has external electromagnetic and gravitational fields.

The author will say nothing about any possible small extra dimensions except that they should obviously all be closed.

5. The big bang and inflation

A motivation for inflation [11] was the so-called “flatness problem” that the overall density of the universe $\rho$ seems to be equal to the critical density for closure ${\rho }_{\mathit{\text{crit}}}$. Possibly inflation (but not eternal inflation) can be incorporated into the CLU model. In 1982, Vilenkin [12, 13] also proposed that the universe is created by quantum tunneling from nothing, again noting an analogy with $e^{+}{e}^{-}$ pair production. He said: “The only verifiable (in principle) prediction of the model is that the universe must be closed. However Guth has argued [14] that the inflationary universe almost certainly overshoots, so that $\rho ={\rho }_{\mathit{\text{crit}}}$ with a very high accuracy even at the present time. This means that we shall have to wait for a long time (the author’s italics) until the sign of $\left(\rho -{\rho }_{\mathit{\text{crit}}}\right)$ can be determined experimentally.” In the CLU model, the density parameter $\rho$ exceeds ${\rho }_{\mathit{\text{crit}}}$ but by a tiny amount that will never be measured if inflation is correct. Guth’s long time should be greater than the proton lifetime, for reasons discussed in the next section.

The present data is consistent with $\rho /{\rho }_{\mathit{\text{crit}}}=1$ but with about 1% uncertainty, that is, visible 3-space is consistent with being flat overall. Local regions of curvature are caused by matter, but in inflationary scenarios the visible universe is a tiny fraction of the whole. The closed 2D surface of the Earth has regions that are locally flat and other regions with positive curvature (hills) and negative curvature (saddles). If the 3-space of the whole universe is perfectly flat (ignoring density fluctuations such as galaxies!) or has negative curvature, it must be infinite, which violates P2 and is incompatible with this model.

According to CLU at the big bang there cannot be a “singularity”; the volume is not zero but the total energy is zero, with positive matter energy canceled by negative gravitational potential energy. In inflationary theories the total energy of the universe can be zero [11, 15, 16].³

6. Deep time: The far future

Since in the CLU fermion world-lines are endless and continuous through the common big bang the total number of leptons and quarks in $U$ must be the same as the number of antileptons and antiquarks in $\overline{U}$. The physics in the “parallel” antiuniverse is the same as in our universe. Both $U$ and $\overline{U}$ are closed 3-spaces, with no connection except at the common big bang and the common far future, deep time.

How can $U$ and $\overline{U}$ be connected in the far future? While the physics is identical in $U$ and $\overline{U}$ the evolution will be very different. Galaxies and stars and planets of antimatter (according to us!) may form, and even life, if inevitable, will begin and evolve with as much variety as life on Earth and elsewhere in our universe. If protons are absolutely stable, planets and anti-planets would likely persist to deep time (black holes will presumably not swallow them all due to the expansion) and such a matching would be impossible.

That impossible matching scenario can be avoided if there are no complex objects, such as teacups, molecules or even protons surviving, but only stable elementary particles. Supposing the complete list to be: $e^{+},e^{-},{\nu }_i,{\overline{\nu }}_i$, photons and maybe gravitons. In that case, a matching of $U$ and $\overline{U}$ seems to require proton decay, $p\to e^{+}+$ photons (directly or via light mesons). This violates separate baryon and lepton number conservations, but those “laws” are not fundamental; they were invented to account for the stability of matter. But fermion conservation still applies; fermion world-lines have no ends. But in the absence of any interactions between them, they should not be considered as localized, but described by their probability waves $\psi \left(x,y,z,t\right)$. Quantum mechanics reigns! In this very cold far future, the wavelengths are vast, even electrons and positrons (and neutrinos) will have probability waves extending over parsecs! Locality (and matching of fermions and antifermions) would be less of an issue. Entropy will have become irrelevant.

Hawking [17] once proposed that “time reverses” when the universe has reached its maximum expansion. He said: “What would happen if and when the universe stopped expanding and began to contract? Would the thermodynamic arrow reverse and disorder begin to decrease with time? … broken cups gathering themselves together off the floor…?” Later he wrote [18] that he was convinced that this was his “greatest mistake in physics: the universe would not return to a smooth state in the collapse. This would mean that the arrow of time would not reverse.” By making a distinction between thermodynamic time and world-line time, that statement can be true of thermodynamic time.

Steinhardt and Turok [19, 20, 21] proposed a cyclic model of a universe that is infinite and flat. An “accelerated expansion, caused by dark energy, is necessary to dilute the entropy, black holes and other debris.” They said in deep time “the universe is returned to its original pristine vacuum state before it begins to contract (due to a negative potential energy), bounce and begin a new cycle.” In the CLU model, the entropy is diluted, even becomes meaningless, allowing the $U$ and $\overline{U}$ to merge and complete the closed time loop—not only world-line time but thermodynamic time.

7. Some other related cosmologies

In recent years, several authors have developed in detail the concept of universe-antiuniverse pair creation. These have similarities with the CLU model but generally involve infinite time.

In 1974, Gott [22] also considered a time-symmetric cosmology, treating the three Minkowskian regions of spacetime at an initial singularity (absolute past, absolute future and “elsewhere”) as if they were distinct universes. The absolute past is C-, P- and T-inverted (antimatter). This differs from the CLU in that his time is infinite in both directions (absolute past and future), and his “elsewhere” is a tachyonic universe.

Hawking has also discussed [17] a finite spacetime with no boundaries in the context of a quantum theory of gravity. He said: “One could say: “The boundary condition of the universe is that it has no boundary.” The universe would be completely self-contained and not affected by anything outside itself. He went on to say: “I’d like to emphasize that this idea that time and space should be finite without boundary is just a proposal: it cannot be deduced from some other principle. Like any other scientific theory, it may initially be put forward for aesthetic or metaphysical reasons, but the real test is whether it makes predictions that agree with observations.”

The requirement that the total energy-momentum, angular four-momentum and electric charge of a closed space are zero restricts the classes of allowed universes (solutions to the equations of general relativity) that can arise from vacuum fluctuations. Ferrando, Lapiedra and Morales [23] showed that closed and flat FRW universes can satisfy these conditions irrespective of conformal transformations, including dilatations, rotations and translations on all spacelike 3-surfaces. However, this is not the case for all universes, e.g., open FLRW universes or the Bianchi V family of universes [24]. It would be interesting to analyze in this way other metrics, such as the Lemaître-Tolman-Bondi universe, which may explain the accelerated expansion without the need for dark energy [25], Stephani universes [26], etc.

Boyle, Finn and Turok [27] applied CPT as an exact symmetry at the big bang, such that the thermodynamic arrow of time (defined by entropy increasing) reverses.

In 1994, Rosen [15] and Cooperstock and Israelit [16, 28] considered the total energy-momentum of open and closed universes in general relativity, in the absence of a cosmological constant, and found $E$ = 0 for any homogeneous isotropic universe described by a FLRW metric. Recently, several authors [29, 30, 31] have supported the idea that the net energy of the universe must be equal to zero.

Robles-Pérez has also recently developed [32, 33] ideas about the quantum creation of universe-antiuniverse pairs.

The CLU is incompatible with an ultimate “big rip” [34], a future singularity suggested in an eternally accelerating universe.

8. Black holes

The “no hair” theorem says that black holes may have electric charge, four-momentum and angular momentum but no other external attributes. An outside observer cannot say anything about the baryon number of a black hole, as there is no long range force that is different for baryons and antibaryons; matter and antimatter black holes look identical. A closed universe is not a black hole; there is nothing external. Hawking proposed that black holes have an associated temperature given by $T_{\mathit{BH}}=\mathrm{\hslash }{c}^3/8\pi GM{k}_b$, and will evaporate if in a heat bath at a lower temperature. The thermal radiation emitted is equally matter and antimatter, and if that were the case with complete evaporation a black hole formed of matter would violate baryon number conservation on a large scale. However, this conclusion is perhaps not inescapable; indeed, Hawking himself said [17] “What happens when the mass of the black hole becomes extremely small is not quite clear, but the most reasonable guess (my italics) is that it would disappear completely …” and “total evaporation is just an assumption.” Arguments about the “inevitable singularity” at the center of a black hole rest on untested assumptions about gravitation and other possible interactions at the Planck scale. As fermion conservation would forbid complete evaporation of matter, stable cores would remain, perhaps very small.

9. Entropy considerations

The concept of an oscillating universe with multiple (even an infinite number of) bounces, but without closed loop time, is old [35]. A serious issue [36] has always been the sacrosanct Second Law of Thermodynamics, stating that the entropy of a closed system, if not in equilibrium, must always increase. In models where the universe reverses its expansion in the far future and collapses down to a “big crunch” which is identified with a future big bang (or one in our past), we would be forced to abandon the Second Law. Also, in Ref. [4] there is no symmetry between the early histories of $U$ and $\overline{U}$.

The CLU model avoids those issues. The $U$ and $\overline{U}$ are matched in the far future. This requires making a distinction between “world-line time” and thermodynamic time. In Feynman diagrams of particle interactions, electrons are shown as moving forward in time, and positrons are shown as moving backward [10]. This is purely conventional and no fundamental asymmetry is implied that would violate CPT. That time at the level of elementary particles can be called “world-line time.” A Feynman diagram of the whole CLU would show no distinction between the upper ($U$) and lower ($\overline{U}$) branches in Figure 1. The time in the arena of complex objects, from molecules to breaking teacups to brains (and hence consciousness), can be called thermodynamic time and flows from left to right in Figure 1 in both $U$ and $\overline{U}$. But this must (in the CLU model) become irrelevant in the far future common to both $U$ and $\overline{U}$.

10. Experimental tests

While the principles of no physical infinities and no boundaries to spacetime could only be disproved by observing such infinities or boundaries, assuming them to be true imposes constraints on cosmology that can in principle be tested. Closed space requires the cosmological density parameter $\rho /{\rho }_{\mathit{\text{crit}}}>$ 1.0, consistent with observations; if it is found to be definitely less than 1.0 it is likely that P1 and P2 are wrong. The assumption of baryon and lepton number conservation is being actively tested by many experiments; we note that they may theoretically be violated [37] even in the Standard Model. But the supposition of endless fermion world-lines and that singularities are forbidden predicts that black holes cannot evaporate to nothing; microscopic remnants carrying the full baryon number and lepton number remain. They will have little or no electric charge and would be largely invisible except through their (very strong at small distances) gravitational fields. If they exist such cores might be detectable and would be dark matter candidates.

Even though it may be practically impossible to either disprove or prove the CLU model, its strength is that it can potentially explain many current mysteries. In this respect, it is similar to other ideas about the multiverse and is based on only P1 and P2 with CPT conservation.

11. Outer SpaceTime: Totality

This section is quite different in that I speculate about matters outside our closed spacetime and therefore in principle unknowable; that need not prevent us from thinking about it. If our $U\overline{U}$ is a fluctuation, there should be many other $u\overline{u}$ pairs in some “Outer SpaceTime.” What is the topology of Outer SpaceTime? It can also be finite with no boundaries, as in Figure 2. Perhaps the finite closed spacetime I have presented is everything, and that is it. If so it seems highly improbable that the laws and constants (forces, masses of particles, etc.) of physics are just right for life (the Goldilocks Principle applied not to exoplanets but to the universe). This is the basis for the many ideas about a multiverse.

Figure 2.

Closed Loop model of Outer SpaceTime, with just one Outer Space dimension shown. A vast number of $u\overline{u}$ pairs appear on the surface as virtual fluctuations. The directions of the arrows is arbitrary (or meaningless!).

As Einstein realized for our universe, a matter-dominated universe without a cosmological constant is unstable and must expand or contract. That will not be the case for Outer SpaceTime, because they $u\overline{u}$ have zero total energy-momentum and there may be no interaction between them. Therefore, the “radius” of Outer Space can be constant all around the Time-loop. Remarkably, this picture of Outer Space is something like the old “Steady State” model of Bondi, Gold and Hoyle [38, 39], being (on average) the same at all Times and Places. This is the Perfect Cosmological Principle, that the Universe should look the same at all times and places (apart from local fluctuations) and it is satisfied by the CLU model, but only in Outer SpaceTime. It achieves this without having to invent a mechanism to continuously create protons from an expanding vacuum, without having an infinitely long time duration, and without having to explain such facts as the cosmic microwave background and primordial light element formation.

Outer Space can even be 1-Dimensional, a Line. Join the ends to make a Ring, and propagate through Time to make a Tube, then join the ends to make a Torus; $u\overline{u}$ fluctuations on the Line are on the surface of that Torus. There may be 2, 3, or any number of Outer Space dimensions (but they are harder to draw!).

In the CLU with the universe and antiuniverse coming into existence together at $t_0$ and matching in the far future at $t_{\mathit{max}}$, in a sense $U$ and $\overline{U}$ exist as parallel but disconnected universes at the present (psychological) time $t$ (i.e., now). Returning to our analogy with a virtual $e^{+}{e}^{-}$ world-line loop in our spacetime, arising as a vacuum fluctuation thanks to the uncertainty principle $\mathrm{\Delta }E>h/\mathrm{\Delta }t$. The analogy is that $u\overline{u}$ arise as virtual fluctuations but in some Outer SpaceTime, a different arena. Naturally, there would be a vast number of such fluctuations, that is, other $u\overline{u}$ pairs; a version of the multiverse idea, shown in Figure 2 as a cartoon. Having no connections between them all, physical parameters can be different; ours just happening to be one where they are just right for life.

To quote Vilenkin [12], who gave a mathematical description of the process of universe creation by tunneling of a de Sitter-Hawking-Moss instanton from “nothing” ($\equiv$ Outer Space): “Nothing is the realm of unrestrained quantum gravity; it is a rather bizarre state in which all our basic notions of space, time, energy, entropy etc., lose their meaning.”

The Perfect Cosmological Principle is also satisfied by Linde’s theory of eternal chaotic inflation [40] and by Aguirre and Gratton’s cosmology [41, 42]. One key difference is that these multiverses are “eternal” while here we forbid eternity. The direction of Time around the torus can be reversed and nothing changes (there is no entropy in Outer SpaceTime). It is possible that the $U\overline{U}$ could be connected by a network of “filaments,” perhaps even with a fractal pattern such as proposed by Linde. In the CLU such connections are unnecessary and unwelcome, as they destroy the independence of the universes necessary for the Goldilocks Principle to apply in our universe and allow life.

12. Conclusions

The Closed Loop Universe model presented here is based on the two principles, assumed exact, of no boundaries to spacetime and no physical infinities. The CLU model answers the question “What happened to the antimatter?” It is all there in a distinct closed space, a parallel universe disconnected from ours except at the common big bang and in the future at $t\gtrsim {10}^{32}$ (years).

The CLU relates to ideas of Einstein, Feynman, Hawking, Steinhardt and Hoyle among many others. The topology of our universe is a $S^3{T}^1$ toroid with world-line time (à la Feynman) as a closed loop. Thermodynamic, entropic or experienced time are not the same as world-line time. The universe and antiuniverse came into existence together, analogous to a virtual fermion loop but on a cosmic scale. Among the many radical consequences are statements about what happened before the big bang and what will happen in the far future, the accelerated expansion (attributed to dark energy) must be transitory, the total energy, angular momentum and charge of the universe must be zero, and the cosmological density parameter $\rho /{\rho }_{\mathit{\text{crit}}}=1.0+\epsilon$ with $\epsilon$ positive but unmeasurably small. Baryon number and lepton number are not separately conserved, protons must decay to positrons + photons, but fermion world-lines have no ends. Black holes cannot evaporate completely, and the cosmic photon:baryon ratio can be explained without a “Great Annihilation.”

Finally, and very speculatively, the author described a closed and finite toroidal Outer SpaceTime in which the CLU is a zero-energy fluctuation, along with a huge number of others (the multiverse) with different physical parameters. Outer SpaceTime need not expand or contract; if static it is a revised version of the “steady state model” obeying the Perfect Cosmological Principle of Hoyle.

Acknowledgments

The author acknowledges valuable discussions with A.J. Stebbins III, and the Fermi National Accelerator Laboratory for support.