
The First T + 100 Planck Seconds Phillip I. Good, Ph.D., drgood@statcourse.com Information Research, 205 W. Utica Ave., Huntington Beach CA 92648, USA.


Contents

The night sky as seen from a telescope in space may be likened to a vast ocean on which a few archipelagos of light can be observed. These archipelagos or superclusters can be further resolved into clusters of galaxies and then into the galaxies themselves. This analogy with an ocean breaks down when we observe that the individual archipelagos are drifting further and further apart as was to be expected of a still expanding scalar field, while the islands (galaxies) within each archipelago (supercluster) are moving closer and closer together under the influence of gravity. The distribution of superclusters like that of islands in Earth’s own oceans is far from homogeneous. Yet the cosmic microwave background generated in the early universe is almost uniform. To explain why, Guth^{1} posits the following: · Our universe began as the expansion of a scalar field of constant energy density. Presumably, this field expanded at the speed of light, though it would be more accurate to say that the speed of light was determined by the rate of expansion of this field. · At random points in spacetime, quantum tunneling within the field resulted in the creation of a Planck volume of ordinary space. The immediate results are as follows::
· As the scalar field expands at the speed of light, these regions are never able to catch up. According to Guth^{1}, firstorder phase transitions occurred when the temperature dropped below the grandunification and WeinbergSalam energies and a Planck volume of a new phase in which the symmetry between the gravitational and other forces was broken formed surrounded by the scalar field. Let T represent the time after the origin of the universe from a point source that the first transition occurred. As Planck second succeeded Planck second thereafter, the Planck volume of the brokensymmetry phase would have extended itself asymmetrically, a Planck volume at a time, via a selfavoiding random walk along the points of a lattice. The scalar field would have continued to expand at the speed of light to occupy a sphere (T+n)^{3} units in size after n further Plank seconds of time. As it expanded, it would generate or coexist with an expanding gravitational field. As shown in the next section, the brokensymmetry phase, also limited by light speed, would only have been able to fill a fractal volume of n^{2.6 }units. The result would have been a filamentlike expansion of the brokensymmetry phase rather than nucleation (as in the formation of bubbles of steam in superheated water) as described by Coleman^{3} and others. Subsequently, when a further Planck volume of the brokensymmetry phase formed after a random time period in a random region of the scalar field, it would have zero probability of interconnecting with the first region or with subsequent regions of brokensymmetry that formed within the scalar field. The result of such independent development is consistent with the observed honeycomblike distribution of galaxies in the observable universe with large regions empty of observable matter interrupted by smears of lightyielding regions.^{4} We assume the existence of an exponentially expanding Higgs Field of constant energy density in the early universe^{1,5}. This field has both a relative minimum (or false vacuum) and an absolute minimum. Quantum tunneling from the relative to the absolute minimum results in the production of a Planck volume of ordinary space, the nonsymmetric phase in which the gravitational is separated from the other forces. Figure 1 illustrates one possible realization of the initial expansion of a nonsymmetric phase within the scalar field, albeit in only two dimensions. The assumptions on which this figure is based are three in number: 1. Expansion proceeds in discrete units.^{6} 2. Each unit expands separately. 3. Expansion proceeds in a random direction.
Figure 1: Hypothetical expansion in discrete units of EMRproducing space during the first four Planck seconds.
By using fluorescence microscopy Haber et al.^{7} observed that DNA molecules extend themselves in just such a fashion. See Figures 2 and 3.
Figure 2. Nonspherical shape of a randomwalk polymer, as observed by Haber et al^{7}
Figure 3. Growth of a randomwalk polymer, as observed by Haber et al^{7} As conjectured by Kuhn^{8} and Flory^{9} and tabled by Hughes^{10}, selfavoiding random walks in 3dimensions have an expected length after N steps of N^{0.6} rather than the N^{0.5 }expected of the simple random walk or diffusion. Instead of a bubble N^{0.5} in radius one would expect to see an oblong form of shape N^{0.6} N^{0.51}N^{0.4}. The radiation emanating from such an irregular region would also be irregular, elliptical rather than spherical. Figure 4 depicts the hypothetical probability distribution of the aspect ratios, that is, maximum diameter to minimum diameter, expected of such irregular figures or smears.
Figure 4. Hypothetical distribution of the aspect ratios of an expanding volume of discrete space resulting from a selfavoiding random walk. Reproduced from Haber et al^{7}; based on their observations of the growth of a randomwalk polymer.
“[The growth of the nonsymmetric phase]…closely parallel the boiling of a superheated fluid…bubbles of the vapor phase materialize in the fluid phase...the bubble expands until it converts the available fluid to vapor.”^{3} The use of the term “bubble” in past descriptions of the initial development of the nonsymmetric phase is unfortunate as it implies a shape symmetric in three dimensions of minimum surface area and maximum volume. Instead, expansion of the highenergy phase would take place a Planck volume at a time, expanding in an arbitrary direction each time, behaving as a selfavoiding random walk rather than as a diffusion process. Eventually, the multibranched chain of Planck volumes would assume a shape with maximum surface area and minimum volume resulting in the maximum dissipation of energy. The radiation emanating from such an irregular region would also be irregular, elliptical rather than spherical. A bubble of steam continues to be fed by highenergy molecules from its surroundings until, in Coleman’s words, “it converts the ordinary fluid to vapor.” A “bubble” of ordinary space, launched full borne, is repelled by the gravitational field of the surrounding false vacuum. It would not, as Coleman proposes, spread “through the universe converting false vacuum to true.”^{11 } ^{ }Indeed, did such conversion take place, there would be a constant influx of new energy at the boundaries of each supercluster, leading within a few hours to the production of new matter something which has never been observed.^{ } Instead, the early universe would witness expansion in the form of filaments (much like a malignant growth in a normal tissue) rather than nucleation (as in the formation of bubbles of steam in superheated water). Thermodynamic effects might ultimately replace the quantum, but by then the asymmetric shape of the “bubble” or “smear” would be established. An attempt at proof of the “bubble’s” spherical nature^{12} upon which Coleman^{2} relies, rests on two unproven assumptions: first, that the form of the original “bubble,” a point or Plank volume, can be represented by a wave function analogous to that of a moving particle, and, second, that this function is continuously differentiable in all its arguments. The assumption of a wave function fails for the same reason Gertrude Stein despaired of Oakland, “there is no ‘there’ there” over which a wave function may extend. At the instant the “bubble” is created by quantum tunneling, the only space it can occupy is that of a single Planck volume. And because an isolated Planck volume or countable set of such volumes lack a topology, the corresponding wave function cannot be continuously differentiable. Other proofs of the “bubble’s” spherical nature^{13, 14} rely on analytic continuation of wave functions from the symmetric to the nonsymmetric phase. But if the universe is to be viewed as discrete, built of Planck volumes or loops or strings, ^{15,16,17} then wave functions are only approximations, highly accurate on presentday scales, but hopelessly inadequate for describing the first Planck time units of the observable universe. The tunneling geometry from a Lorentzian space time to a Euclidian one ^{13,14, 3, 18} as with the boundaryless view^{19} requires us to accept the improbable: a continuous universe fullyformed at birth. On the other hand, as the transition from the false vacuum occurs at random in both time and space and with a probability that is the same at all coordinates, our universe is globally (if not locally) homogeneous as described by the “bubble” model. The exception lying in the distribution of background radiation which is largely determined by the first few regions of ordinary space that formed. The universe does have a center, located at the starting point of the expanding scalar field, but we will never be able to know whether this center is located within the universe observable from Earth.
According to Linde^{20} and Hawking et al.^{21}, although no proof is offered, “the energy released by bubble formation would be transformed into the kinetic energy of the bubble walls and would cause the walls to expand outwards with uniform acceleration.” But the decrease in temperature is due only in part to the creation of kinetic energy; the balance will come from dissipation via EMR through the boundary to the surroundings. An oblong shape with maximum surface area for minimum volume would optimize such transfer. Reference is also made by Hawking and his colleagues (and earlier ^{4, 22}), again without proof, to the inevitability of collisions between bubbles. Given that the scalar field continues its expansion at the speed of light and that the proportion of the total universe occupied by the oblong energy/mattercontaining forms is near zero, the volume occupied by the selfavoiding random walk having a fractal dimension between 3 and 2, such collisions are extremely unlikely. The results are few or no monopoles created and baryon asymmetry.^{20} Because the decay of the false vacuum and the formation of fractal cracks is ongoing, one should expect to detect EMR from regions of observable space where no EMR had been observed before, albeit such appearances might be on a scale of millions or tens of millions of years. (The nearest supercluster to the our resident supercluster is some 10 million lightyears distant.) Hoyle^{23} may prove to be right, albeit for a quite different reason. The resultant picture of the universe (Figure 5) is similar to that expected from a percolation model of domain formation with a low bias probability.^{ 24}
Figure 5. Depiction of a hypothetical universe resulting from a selfavoiding random walk consisting of false vacuum (clear regions) interrupted by irregular smears of a nonsymmetric mattercontaining phase (solid regions). This figure is taken from Coulson et al.^{21} and was generated by a computer simulation of a percolation model of domain formation in threedimensions with bias probability p =0.1.^{24}
Though symmetric and nonsymmetric regions interact, the effects of one upon the other are quite different in nature. The gravitational field of the symmetric region repels masspossessing particles from the nonsymmetric region.^{25} It affects their movement within the nonsymmetric region and acts as an absolute barrier to their passage. Photons on the contrary can pass freely through the false vacuum. Because the energy field is uniform throughout the false vacuum, the geodesics are straight lines in Lorentz space. Each nonsymmetric region provides an absolute barrier to the further expansion of the false vacuum in the direction of that region. While it is still true that zc=H_{o}L to a first approximation, the Hubble time is L_{H}/cr where r is the proportion of a ray of length L_{H} outward from a fixed observer that intersects nonsymmetric regions.
Figures 1 through 4 depict the expansion of a random chain. The expansion of the brokensymmetry phase at the expense of the surrounding false vacuum is more accurately depicted as the expansion of a random surface, with each surface element capable of extension at each instant in time. The behavior of the radius of gyration of such random surfaces has been shown to be similar to that of the selfavoiding random walk.^{26–28} Suppose we view one such expanding surface and the surrounding false vacuum as if they were superimposed on a threedimensional lattice each element of which comprises exactly one Planck volume. Let p denote the probability that a Plank volume of the false vacuum will spontaneously decay via quantum tunneling in the next instant into the brokensymmetry phase. Let P denote the probability that a Plank volume of the brokensymmetry phase will expand into a specific point of the immediatelyadjacent false vacuum in the next instant. We assume that P is many orders of magnitude larger than p. The probability that a point of the false vacuum will be adjacent to k Plank volumes of the brokensymmetry phase, k= 0, 1, . . ., 26, is given by the formula: 1–(1p)(1P)^{k}. Thus the process of expansion is selfsmoothing for as k increases, this probability approaches 1. The volume occupied by the expanding brokensymmetry phase, though remaining irregular in shape and decidedly nonspherical, will fill in with the only rough edges on its surface. After a sufficient period of time has passed, say 10^{6} Plank seconds, this volume can be closely approximated by a continuous manifold.
Images from the Hubble telescope and other contemporary sources have given us a portrait of a universe consisting of oblongs or smears of masscontaining lightyielding particles in an otherwise empty void. The model proposed here, a variant of the original inflationary model,^{4} would account for just such a structure. Despite these observations, the immediate and continuing reaction to Guth’s proposal was and is one of disbelief, most authorities refusing to accept, “a universe comprised of isolated empty bubbles of spontaneous symmetrybreaking (SSD) vacuum separated by exponentially expanding regions of symmetric metastable vacuum.”^{29}^{ } Ignoring the words of John von Neumann, "With four parameters I can fit an elephant and with five I can make him wiggle his trunk," cosmologists proceed to offer alternative models of greater and greater complexity introducing not one but two cosmic constants, both of which lack a causal basis^{30}. This unnecessary complexity has arisen from three sources:
Yet the stillexpanding scalar field accounts for two sets of phenomena, one connected with the early universe that was to determine all that was to follow and one whose impact was to be observed only after several minutes and then for all time. At the origin of our universe^{1}, this field provides an explanation for the homogenous expansion of the universe and the nearuniformity of the cosmic microwave background that we observe today. The lack of spherically symmetry of this background is explained by the lack of spherical symmetry in the first regions of ordinary space to form. Later, the gravitational field induced by the scalar field would be responsible for all the phenomena associated, erroneously, with dark matter such as galactic rotation. The cold darkmatter model cannot simultaneously fit largescale and smallscale galaxy distributions.^{33} The present model accounts for the differences between the two. The largescale distribution of galaxies results from the creation of independent regions of ordinary SSD space. The smallscale distribution results from gravitational attraction within superclusters and gravitational repulsion from the false vacuum without. As the archipelagos built of discrete Planck volumes appear at random in the false vacuum, the fundamental homogeneity and flatness of the universe achieved by inflation is not affected. Gravitational attraction between and within the regions of masscontaining space will slow the apparent expansion of our current observable portion of the universe, because the regions are drawn closer together. The masses of the superclusters will vary in accordance with a Poisson distribution. For while the probability that two or more Planck volumes would be produced in adjoining regions of timespace is small, it is not infinitesimal. Some regions of ordinary space will have more energy available to them than others; consequently the masses of superclusters will be expected to vary. As recent observations confirm, the universe as a whole will continue to expand indefinitely ^{34}. Limited by the speed of light, the majority of this expansion will take place outside the portion of the universe we are capable of observing. The negative gravity of the surrounding voids affects the movement of material within the SSDs. The extent of the scalar field is such that it will have a greater influence on galactic rotation within the smear than the masses in the smear itself. This influence would be proportional to (1–r)^{3} where r is the proportion defined in the preceding section. The present model obviates the need to assume the existence of cold dark matter and accounts for the discrepancy between estimates of the age of the universe based on the Hubble constant and the ages of Type II stars. The cracks or oriented smears of SSD in the false vacuum appear randomly in time and in space thus accounting for the observed nonuniformities in the cosmic microwave background,^{35 }and providing further support for the BigBang theory.^{3}^{6} References

AbstractMost models of a universe that originated from a scalar field of constant energy density have at least two deficiencies: They make use of parameters (cosmic constants) which lack a causal basis; they ignore the discrete nature of ordinary space, a distinction that is crucial when ordinary space consists of a few hundred Planck volumes at most. The model proposed here corrects both these deficiencies. Assuming that the scalar field and regions of ordinary space coexist accounts for the distinction between largeand smallscale galactic distributions, reconciles disagreements in calculating the age of the universe, and obviates the need to assume the existence of cold dark matter.
Keywords: dark matter, early universe, false vacuum, scalar field, inflationary universe, random walk. PACS: 98.80 Bp
