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Can A Man-Made Universe Be Achieved by Quantum Tunneling Without an Initial Singularly?* ALAN H. GUTH Massachusetts Institute of Technology and Harvard~mithsonian Center for Astrophysics ABSTRACT Essentially all modern particle theories suggest the possible existence of a false vacuum state-a metastable state with an energy density that cannot be lowered except by means of a very slow phase transition. Inflationary cosmology, which is anew reviewed, makes use of such a state to drive the expansion of the big bang, allowing the entire observed universe to evolve from an initial mass of only about 10 kilograms. A sphere of false vacuum in our present universe could inflate Into a "child" universe, and general relativity is used to learn where the new universe would go. It is not yet settled, but it seems liked that the known laws of physics permit in principle the possibility of creating a child universe by man-made processes. 'art COSMIC COOKBOOK Dunng me past decade, a radically new picture of cosmology has emerged. The novelly of this picture is particularly striking when one considers the question of what it takes to produce a universe. According to the standard big bang picture of a decade ago, the visible universe could be assembled, at t ~ 1 see for example, by mung approximately 1089 . Based in part on "Inflame and False Vaamm Bubbles, n bar A. H. Guth, in Wed of the Stom Meets (1988 Meeting of the Division of Particles and Fields of the American Physical Society, Stom, Connecticut, August 15 -18, 19883, eds.: K Hailer, D.B. Caldi, M.M. Islam, Rt Mallett, P.D. Mannheim, and M.S. Swanson florid Scientific, Singapore, 1989), pp. 139- 153. 153
154 AMERICAN AND SOVIET PERSPECTIVES photons, 1089 e+e~ pairs, 1089 zig, pairs, 1079 protons, 1079 neutrons, and 1079 unpaired electrons. The total mass/energy of these ingredients Is about 1065 grams ~ 1032 solar masses ~ 10~°x present mass of visible universe. The mass Is much larger than the present mass of the visible universe, because most of the energy is lost to gravitational potential energy as the universe expands. W~ the advent of grand unified theories (GUTS and inflationary cosmology, however, a much simpler recipe for a universe can now be formulated. pRor a review of inflation, see Linde (1984a and l9g7), Bran- denberger (1985), Turner (1987), Steinhardt (1986), Blau and Guth (1987), or Abbott and Pi (19863.] ~ produce a universe at t ~ 10-35 see, the only necessary ingredient is a region of false vacuum. And the region need not be very large. For a typical GUT energy scale of ~ 10~4 GeV (which I will use for all the numerical examples in this paper), the minimum diameter is about 10-24 cm. The total mass/energy of this ingredient is only about 10 kg ~ 10-29 solar masses ~ 10-5t x present mass of visible universe. This recipe sounds so easy that one cannot resist asking whether it is possible to produce a universe by man-made processes. Unfortunately we do not yet have a definitive answer to this question, but in this paper I wB1 try to summarize our current understanding. In considering this question, one difficulty is immediately obvious: the mass density of the required false vacuum is about 1075 ~3. IS mass density is certainly far beyond anything that is technologically possible, either now or in the foreseeable future. Nonetheless, for the purposes of this discussion ~ will whimsically assume that some civilization in the distant future will be capable of manipulating these kinds of mass densities. There are then some very interesting questions of principle that must be addressed in order to decide if the creation of a new universe is possible. While I will discuss these questions in terms of the possibility of man-made creation, I want to emphasize that the same questions will no doubt also have relevance to venous natural scenarios that one could imagine. The outline of this paper will be as follows. The first section wd1 sum rnarize the properties of the false vacuum, and the second section will review the inflationary universe model-readers familiar with inflationary cosmology should either skim these sections or skip them entirely. The third section wD1 discuss the evolution of a false vacuum bubble. In the last section, I will discuss the key question: Do the laws of physics as we know them permit in principle the creation of a new universe by man-made processes? PROPERTIES OF THE FAISE VACUUM A false vacuum is a peculiar state of matter which arises when a
HIGH-ENERGY ASTROPHYSICS 155 PI Vim, FALSE VACUUM L O _ o 11 \ TRUE | \ VAC UUM / . ~ 1~1 / At FIGURE 1 Potential energy density for a scalar field ¢. Ibe form shown contains a very flat plateau, a shape that is suitable for a new inflationary soenano. I.ne true vacuum is the state of rawest energy for the field, and the [else vacuum is the metastable state in which the Solar field has a value at the top of the plateau. particle theory contains a scalar field ~ with a potential energr density similar to that shown in Figure 1. The state ~ = ¢' is the state of minimum energy density, and therefore is the true vacuum. If, however, there is a region of space in which ~ ~ O. this region is called a false vacuum. The false vacuum is obviously not stable, since the scalar field will sooner or later roll off me "hill" of this potential energy diagram. The false vacuum can, however, be highly metastable, if the plateau in the potential energy diagram is eat enough I will assume that ~ am talking about a false vacuum with a lifetime that is long compared to the other time scales of interest for the early universe. The mass deposit of the false vacuum is fixed by me value of the potential energy density for ~ ~ O. which for typical grand unified theory numbers has a value of about
156 AMERICAN AND SOVIET PERSPECTIVES PI ~ (1014 GeV)4 ~ 1074 g~cm3 . (I will generally use units with h = c = 1, but for clarity I will sometimes write formulas with factors of c included.) The false vacuum has the very unusual properly that this mass density is fixed. If a region of false vacuum is enlarged, the mass density does not decrease as it would for a normal material. Instead it is held at this constant value, provided of course that there is not enough time for the scalar field to roll down the hill. This constancy of the mass density implies a peculiar property for the pressure p. 1b see this, consider a chamber filled with false vacuum, as shown in Figure 2. When the chamber is enlarged by the piston moving outward, the volume of the chamber increases by AV. Since the energr density is constant at pf c2, this implies that the agent that pulled the piston out must have done work /`W = pfc2AV. Since the pressure outside the piston is just the pressure of the true vacuum, which is zero, we conclude that the pressure inside must be negative. Since the work done in the expansion of a gas is given by AW =-pAV, one finds immediately that P =-Of c So the pressure of the false vacuum is huge, and negative. If the early universe went through a false vacuum phase, then the evolution can be determined by putting the energy density and pressure of the false vacuum into Einstein's field equations. For those familiar with cosmology, the effect can be described by saying that the false vacuum acts exactly like a (positive) cosmological constant, except of course that the false vacuum is not permanent it wD1 eventually decay. 1b see the consequences more explicitly, note that the gravitational deceleration of the cosmic expansion is, according to general relativity, proportional to 3p PA 2 . Ordinarily the second term is a small relativistic correction. During the radiation-dominated period of the early universe, however, the second term is equal in magnitude to the first term. For the false vacuum, however, the pressure term is negative, and in fact it overwhelms the positive contribution from the mass density. The net effect is to create a huge gravitational repulsion, causing the universe to go into a period of exponential expansion. TlIE NEW INFLATIONARY UNIVERSE While the original form of the inflationary universe model (Gush 1981) failed to provide a smooth ending to inflation, this problem was overcome
HIGH-ENERGY ASTROPHYSICS Before iFolse Vacuum Energy Density True Vocuum :)Energy Density \ Pressure = 0 -~\V 157 =pf c2 =0 FIGURE 2 A thought Averment to derive the pressure of the [else vacuum. The piston chamber is filled with false vacuum, and is surrounded by true vacuum. When the piston is pulled out the energy density remains constant, so the additional energy must be furnished Or the force needed to pull the piston against the negative pressure of the false vacuum. by the introduction of the new inflationary universe scenario (I:inde 1982; Albrecht and Steinhardt 1982~. The scenario begins with a patch of the universe somehow settling into a false vacuum state. The mechanism by which this happens has no influence on the later evolution, and at least three possibilities have been discussed in the literature: 1) Supercoolingirom high temperatures. This was the earliest suggestion (Gush 1981; Linde 1982; Albrecht and Steinhardt 1982~. If we assume that the universe began very hot, as is traditionally assumed in the standard big bang model, then as the universe cooled it presumably went through a number of phase transitions. For many types of scalar field potentials, supercooling into a false vacuum occurs naturally. This scenario has the di~culbr, however, that there is no known mecha- nism to achieve the desired pre-inflationaty thermal equilibrium state. For fields as wealdy coupled as is needed for inflation (Starobinsly 1982; Guth and Pi 1982; Hawking 1982; Bardeen et al 1983), there is not nearly enough time for thermal equill~num to be achieved by the normal dynamical processes. It has been shown, however, that true thermal equil~num is not really necessary: a variety of random
158 AMERICAN AND SOVIET PERSPECTIVES configurations give results that are very similar to those of thermal equilibrium (Albrecht et aL1985~. 23 lDnne~gfom `'no~u~g" Rayon 1973; Vilenldn 1982 and 1985a; Linde 1983a and 1984b; Hartle and Hawking 1983~. These ideas are of course very speculative, since they involve a theory of quantum gravity that does not actually Ernst. The basic idea, however, seems very plausible. If geometry is to described by quantum theory, then the geometry of space can presumably undergo quantum transitions. One can then imagine an initial state of absolute nothingnes~the absence of matter, energy, space, or time. The state of absolute nothingness can presumably undergo a quantum transition to a small universe, which then forms the initial state for an inflationary scenario. 3) Random fluctuaiu~ns in chaotic cosmology. Linde (1983b,c) has ad- vacated a chaotic cosmology in which the scalar field ~ begins in a random state in which all possible values of ~ occur. Inflation then takes place in those regions that have appropriate values of ¢, and these inflated regions dominate the universe at later times. In these models-it is not necessary for the scalar field potential energy function V(~) to have a plateau, since inflation can occur as the scalar field rolls downward, starting from a very large value. As in other models, however, it can be shown that the potential must be very flat In order minimize the density perturbations that result from quantum fluctua- tions (Starobinsly, 1982; Guth and Pi 1982; Hawing 1982; Bardeen et at1983~. Regardless of which of the above mechanisms is assumed, one expects that the correlation length of the scalar field just before inflation is of the order of the age of the universe at the tune. Assuming again a GUT energy scale of about 10~4 GeV, one finds a correlation length of about 10-24 cm. The patch then expands exponentially due to the gravitational repul- sion of the false vacuum. In order to achieve the goals of inflation, we must assume that this exponential expansion results in an expansion factor > 1025. For typical grand unified theory numbers, this enormous expansion requires only about 10-32 see of inflation. During this inflationary penod, the density of any particles that may have been present before inflation is diluted so much that it becomes completely negligible. At the same time, any nonuniformities in the metric of space are smoothed by the enormous expansion. The explanation for this smoothness is identical to the reason why the surface of the earth appears to be flat, even though the earn is actually round any differentiable curve looks like a straight line if one magnifies it enough and looks at only a small segment. The correlation length for the scalar field is stretched by the expansion factor to become at least about 10 cm. If the duration of inflation is more than the minimal
HIGH-ENERGY ASTROPHYSICS 159 value, which seems quite liketr, then the final correlation length could be many orders of magnitude larger. There appears to be no upper limit to the amount of inflation that may have taken place. The false vacuum is not stable, so it eventually decays. If the decay occurs by the usual Coleman~allan (Coleman 1977; Callan and Coleman 1977) process of bubble nucleation, as was assumed in the original version of inflation, then the randomness of the bubble nucleation process would produce gross inhomogeneities in the mass density (Gush and Weinberg 1983; Hawking et at 1982~. This problem is avoided in the new inflationary scenario (Linde 1982; Albrecht and Stemhardt 1982) by introducing a scalar field potential with a flat plateau, as was shown in Figure 1. This leads to a "slow-rollover" phase transition, in which quantum fluctuations destabilize the false vacuum, starting the scalar field to roll down the hill of the potential energy diagram. These fluctuations are initially correlated only over a microscopic region, but the additional inflation that takes place during the rolling can stretch such a region to be large enough to easily encompass the observed universe. When the phase transition takes place, the energy that has been stored in the false vacuum is released in the form of new particles. These new particles rapidly come to thermal equilibrium, resulting in a temperature with kT ~ 10~4 GeV. At this point the scenario rejoins the standard cosmological model The baryons are produced [see, for example, Kolb and lbrner 1983 or Yoshimura 1981] by baryon nonconse~g processes alar inflation Any bacons that may have been present before inflation are simply diluted away by the enormous expansion factor. Thus, inflationary cosmology requires an underling particle theory, such as a grand unified theory, in which bacon number is not conserved. The inflationary universe model has a number of key successes, the most of important of which are the following: 1) It cures the "magnetic monopole problem." In the context of grand unified theories, cosmologies without inflation generally lead to huge excesses of magnetic monopoles. These monopoles are produced at the grand unified theory phase transition, when the GUT Higgs fields acquire their nonzero values. The rapidity of the phase transition implies that the correlation length of the Higgs fields is very short, and the fields therefore become tangled in a high density of knot these knots have the physical properties of superheavy (~ 10~6 GeV) magnetic monopoles ('t Hooft 1974; Polyakov 1974; for a review, see Goddard and Olive l978~. For typical grand unified theories the expected mass density of these magnetic monopoles would exceed
160 AMERICAN AND SOVIET PERSPECTIVES (Zel'donch and Khlopov 1978; Preskill 1979) the mass density of everything else by a factor of about 1012. 23 It explains why the universe is so homogeneous. The most striking evidence for the homogeneity of the universe is seen in the cosmic background radiation, which is Mown to have a temperature that is uniform in all directions to an accuracy of a few parts in 105. This implies that the temperature of the universe was uniform to this accuracy when the background radiation was released, a few hundred thousand years after the big bang. In standard cosmology, the establishment of thermal equilibrium at such an earlier time over such a huge volume would require the transfer of information at approximately lOO times the speed of light. In the inflationary model, on the other hand, thermal equilibrium could have been established an incredibly small region before the onset of inflation. The process of inflation would then take this very small region and magnify it to become large enough to encompass the entire observed universe. 3) It explains why the mass density of the earlier universe was so close to the critical value. The critical mass density, Pc, is defined as that mass density which is just barely sufficient to eventually halt the expansion of the universe. Idday the crucial ratio Q _ P/Pc (where p is We mass density of the universe) is known to lie in the range 0.1 S Q S 2. Despite the breadth of this range, the value of Q at early limes is higher constrained, since Q diverges from one as the universe evolves. At t = 1 see, for example, Q must have been equal to one (Dicke and Peebles 1979) to an accuracy of one part in 10~. Standard cosmology provides no explanation for this fact-it is simply assumed as part of the initial conditions. In the inflationary model, however, Q is driven during the period of inflation very rapidly toward one, regardless of where it begins. 43 It provides a possible origin for the density fluctuations that seed galaxy formation. In standard cosmology an entire spectrum of primordial density fluctuations must be assumed as part of the initial conditions. In the inflationary model, on the other hand, density fluctuations are produced naturally by quantum fluctuations during the inflationary phase transition (Starobinsky 1982; Guth and Pi 1982; Hawking 1982; Bardeen etaLl983). These density fluctuations, moreover, have a spectrum that is at least roughly what is desired for galaxy formation. This success of inflation, however, occurs only at a pnce: for the magnitude of the density fluctuations to turn out correctly, the scalar field that drives inflation must be coupled incredibly wealOy. For a simple A¢4 meow, for example, me value of ~ must be about 10-~2. This incredibly weak coupling is necessary regardless of whether one is using a chaotic inflationary model, or a standard (new) inflationary
HIGH-ENERGY ASTROPlIYSICS 161 model It should also be mentioned that particle physics provides an alternative possibility for generating primordial density perturbations: it could be that the perturbations produced by inflation were negligi- bl~r small, while the important fluctuations developed later from the random formation of cosmic strings (for a review, see ~lenkin 1985b). 5) It explains the origin of essentially all the matter, energy, and entropy in the universe. While this statement may seem to violate known conservation principles, in fact it does not provided that baryon number is not conserved. Energy conservation is no problem. The gravitational contribution to the energy of the universe is negative, and in any model with Q > 1, the gravitational energy precisely cancels the energy of matter (Ityon, 1973~. Any physical theory should be testable, and fortunately there are at least a few tests of inflation that are in principle possible. First, inflation predicts that Q. even today, should be equal to one. More precisely, the prediction is it+ ~2 =1~0~10-4) , where A is the cosmological constant. The term A/3H2 can be thought of as the contribution to Q from the energy density of the vacuum. The uncertainty of 0~10-4) allows for quantum fluctuations, and its magnitude is estimated not from first principles, but instead from the fluctuations that are required for galaxy formation. Second, inflation predicts the spectral form of the primordial density fluctuations. In particular it predicts a scale- invariant Gaussian spectrum known as the Harr~son-Zel'dovich spectrum (Harrison 197~, Zel'dovich 1972~. The scale-invariance can in principle be tested by precise measurements of anisotropies in the cosmic background radiation, and/or by developing a detailed theory of galaxy formation. These tests are of course extremely difficult, but the problems do not appear to be insurmountable. EVOLUTION OF A FALSE VACUUM BUBBLE By a false vacuum bubble, I mean a region of false vacuum surrounded by anything else. I will discuss in detail the simplest case: a spherical region of false vacuum surrounded by true vacuum. In considering the evolution of such a region one is lead immediateb to a paradox. If the region of false vacuum is large enough, one expects that it would undergo inflation. An observer in the outside true vacuum region, on the other hand, would see the false vacuum region as a region of negative pressure. The pressure gradient would point inward, and the observer would not expect to see
162 AMERICAN AND SOVIET PERSPECTrVES the region increase in size. The resolution of this paradox hinges on the dramatic distortion of spacetime that is caused by the false vacuum bubble. Here I wB1 outline the solution to this problem, but a reader interested in the technical details will have to consult the literature (Sato et aLl981 and 1982; Berezin et at 1983, 1985, and 1987; Ipser and Sik~vie 1984; Aurilia et at 1984 and 1985; Lake 1979, Lake and Wevrick 1986). The description given here follows the work that I did with Blau and Guendelman (Blau et at 19~71. The first step in solving the problem is to dissect it, dividing the spacetime into three regions. The exterior region is spherically symmetric emppr space, for which the unique solution, In general relativity, is the well- ~own Schwarzschild metric. The interior region consists of sphencaLly symmetric false vacuum, and is required to be regular at r = 0. This spacetime also has a unique solution: de Sitter space. At the interface between these two regions is a domain wall-a region in which a scalar field is undergoing a transition between its true and false vacuum values. The solution that I will describe uses a thin-wall approximation, In which the thickness of the wall is assumed to be negligibly small compared to any other distance in the problem. In this approximation it can be shown that the surface energy-density is equal to the surface tension and is independent of time, and we take this surface energy density ~ as an additional parameter of the problem. The wall can then be described mathematically by a set of junction conditions (Israel 1966) which are obtained by applying Einstein's equations to an energy-momentum tensor restricted to a thin sheet. (These equations are just the gravitational analogue of the well-known statement that the normal component of an electric field has a discontinuing of 4,r~ at a sheet of surface charge density a.) The evolution is completely determined by using these junction conditions to join the interior and exterior forms of the metric The evolution of the bubble wall can be described by the function rang, where r is the radius of the bubble wall (defined as 1/2,r tunes the circumference), and ~ is the proper time as would be measured by a clock that follows the bubble wall. The junction conditions described above imps an equation of motion for real that can be cast into a form identical to that of a nonrelativistic particle moving in a potential V(r), as shown in Figure 3. The energy of the fictitious particle is related to the mass of the physical false vacuum bubble. As can be seen in the potential energy diagram, there are three lauds of solutions. First, there are '`bounded" solutions in which the bubble grows from r = 0 to a maximum size and then collapses. Second, there are "bounce" solutions. Here the bubble stans at infinite size in the asymptotic past, contracts to a minimum size, and Hen expands without limit. Finally, there are "monotonic" solutions bubbles that start at zero size and grow monotonically. The monotonic solutions require a
HIGH-ENERGY ASTROPHYSICS V(r) 163 Monotonic Bounded ~ ~--~ - -A_ _ \ Bounce \ 1 1 1 ,, 1 1 1 \ 1 o r FIGURE 3 In the thin-wall approximation the trajectory of the bubble wall is equivalent to the modon of a nonrelativistic particle in the potential energy curve shown above. The energy of the fictitious particle is related to the mass of the false vacuum bubble; the energy increases with the mass, and approaches the top line of the diagram as M ~ A. minimum mass, so that the energy of the fictitious particle is high enough to get over the potential barrier in Figure 3. For small values of the surface energy density (a << /, this critical mass is given simply by Mcr = 41rX_3 where X is the rate of the exponential expansion tie., scale factor oc expects ), which is related to the false vacuum energy density pf by X 3 Pf ~ For typical GUT parameters, MCr ~ 10 kg. Having described the evolution of the bubble wall, ~ must still descn~e how the bubble wall is embedded in spacetime. Here I will describe only the behavior of the monotonic solutions. A spacetime diagram for this situation is shown as Figure 4. The true vacuum region, to the right of the bubble wall, is shown in the standard Kruskal~zekeres coordinates. The false vacuum region, to the left of the bubble wall, is shown in peculiar coordinates designed solely to allow the two halves of the diagram to fit together in the plane. The diagram is constructed so that lightlike lines lie at 45° to the vertical, but the metric is highly distorted. In particular
164 AMERICAN AND SOVIET PERSPECTIVES I it) singular , mu. False \/acuum r=0 / non singular r=0 singular ~. ~ FIGURE 4 A spacetime diagram of a monotonic false vacuum bubble solution. Angular coordinates are suppressed, and the diagram is plotted so that lightlike lines are at 45°. The bubble wall is shown as a heavy line with an arrow on it. I-he true vacuum region (dotted) is to the right of the bubble wall, and the false vacuum region ~onzontal lines3 is to the left. The diagram shows initial (lower) and final (upper) r = 0 singularities, and also a nonsingular r = 0 line Q.e., the center of a spherical coordinate system) that runs along the left edge. the exponential expansion of the false vacuum region, which occurs as one moves upward and to the left in the diagram, is completely hidden by the distortion of the metric. The physical meaning of a spacetime diagram of this type can be seen most clearly by examining a sequence of equal-time slices. Figure 5 shows the positions of four slices, labeled (a), (b), (c), and (d), and Figure 6 shows a representation of each slice. For purposes of illustration, Figure 6 shows only two of the three spatial dimensions. Since the spaces of interest are spherically symmetnc, this results in no loss of information. The two- dimensional sheet is shown embedded in a fictitious third dimension, so that the curvature can be visualized. Figure 6(a) shows a space which is hat at large distances, but which has a singularity at the origin. In Figure 6(b) a small, expanding region of false vacuum has appeared at the center, replacing the singulanbr. The false vacuum region is separated from the rest of space by a domain waR Figure 6(c) shows the false vacuum region beginning to swell Note, however, that the swelling takes place by the
HIGH-ENER~ ~TROP~ICS : r=-O singular r =0 nonsingula r 165 i~ ~ A:: ~ ~ a= ~:~ ~- r=0 - singular FIGURE 5 Horizontal lines indicating spacelike hypersurf~ces to be illustrated in Figure 6. production of new space; the plane of the original space is unaffected. The false vacuum region continues to inflate, and it soon disconnects completely from the original space, as shown in Figure 6(d). It forms an isolated closed universe which Sato et al (1981 and 19823 have dubbed a "child" universe. Note, by the way, that Figure 6 shows clearly how the paradox raised at the beginning of this section is resolved. The net force on the bubble wall points from the true vacuum region to the false vacuum region, as expected. Due to the inversion shown in Figures 6(c3 and 6(d), however, this force causes the bubble wall to expand, rather than contract. ~ summarize, the false vacuum bubble appears from the outside to be a black hole. From me inside, however, it appears lo be an inflating region of false vacuum, with new space being created as the region expands. The region completely disconnects from Me original spacetime, forming a new, isolated closed universe. Although the problem that has been solved is very idealized, it nonethe- less appears to contain the essential physics of more complicated inhomo- geneous spacetimes. The paradox discussed at the beginning of this section will exist wherever an inflating region is surrounded by noninflating regions, and the qualitative behavior of the system seems to be determined by the way in which this paradox is resolved. Thus, one concludes that if inflation
166 r - ~ - v / a ~an_ c AMERICAN AND SOVIET PERSPECTIVES / b / FIGURE 6 The evolution of a monotonic false vacuum bubble solution. Each lettered diagram illustrates a spacelike hype~surface indicated in Figure 5. The diagrams are drawn lay suppressing one dimension of the hypersurface and embedding the resulting two-dimensional surface in a fictitious three~imensional space so that the curvature can be displayed. I.ne false vacuum region is shown as dotted. Note that diagram (d) shows a child universe detaching Mom the original spacetime. Occurred in an inhomogeneous universe, then many isolated child universes would have been ejected. Furthermore, even if inflation somehow began in a completely homo- geneous way, one still expects the universe lo break apart into a host of child universes. The reason stems from the intrinsic nonuniformity, on very large scales, of the decay of the false vacuum (Aryal and Vilenldn, 1987~. This process occurs exponentially, lye most other decay processes, but for inflation to be successful the parameters must be arranged so that the ~ I have studied a simplified but exactly soluble model of a slow-rot/over phase transition with S.-Y. Pi (Gush and Pi 1985).
HIGH-ENERGY ASTROPHYSICS 167 exponential decay constant is slow compared to the exponential expansion rate. This implies that the total volume of false vacuum increases with time. Thus, no matter how long one waits there will still be regions of false vacuum. These regions have no reason to be spherical, but the arguments of the previous paragraph lead one to expect a high likelihood of producing child universes. CAN ONE IN PRINCIPLE CREATE AN INfIATIONARY UNIVERSE IN THE LABORATORY? Figures 4 6 illustrate the creation of a new universe, but there is one undesirable feature. The sequence begins with an initial singularity, shown as the lower r = 0 singularity in Figures 4 and 5, and as part (a) of Figure 6. Although an initial singulantr is often hypothesized to have been present at the big bang, there do not appear to be any initial singularities available today. So we ask whether it is possible to intervene in some way, to modify the early stages of this picture, so that an inflationary universe could be produced without an initial singularity. This question can be addressed at either the classical or quantum levels. At the classical level Farhi and ~ (I arhi and Guth 1987) have shown that the initial singularity cannot be avoided. Any false vacuum bubble which grows to become a universe necessarily begins from an initial singulanty. The argument rests on an application of the Penrose theorem.2 The inflationary solutions are very rapidly expanding, and the Penrose theorem implies that such rapid expansion can result only from an initial singularity. (Lee Penrose theorem is more widely known in a form which is the time- reverse of the present application: if a system is collapsing fast enough, there Is no way to avoid the collapse to a singularity.) The application of the Penrose theorem involves two technical loom holes. First, if the final bubble is not spherically symmetric, then we have not been able to show that the Penrose theorem applies. We believe that this shortcoming, however, is probably the result of our own limitations, and does not provide a way to avoid the theorem. Second, if a material can be found with a pressure that exceeds its energy density, then the Penrose theorem would not apply. ~ quantum field theories it is possible to construct states that have this property, but it is not clear if a large enough region of this type can be attained. At the quantum level, on the other hand, the Penrose theorem does not applier, since it is derived from the classical equations of motion. With E. Fahri and J. Guven (Farhi et al 19903, I have studied the question of 2We thank R. Wald, ~ Israel, J. Bardeen, and W. Unruh for pointing out to us the relevance of this theorem.
168 AMERICAN AND SOVIET PERSPECTTYES whether quantum physics allows the creation of an inflationary universe without an initial singularity. In particular, we have been exploring the following recipe. Suppose a small bubble of false vacuum (with mass less than the critical mass MCr ~ 10 kg) Is created and caused to expand at a moderate rate. Since the bubble is not expanding rapidly, the Penrose theorem does not preclude its production by classical processes, without an initial singularity. We have not explored in detail the mechanisms by which such a region might be created, but presumably it could be created either by supercooling from high temperatures or perhaps by compressing a gas of fermions that couple to the scalar field. If such a bubble were allowed to evolve classically, it would correspond to one of the bounded solutions, as discussed in the context of Figure 3. It would expand to a maximum size and then the pressure gradient would halt the expansion and cause the bubble to collapse. By quantum processes, however, one might imagine that the bubble could tunnel through the potential energy barrier shown in Figure 3, becoming a bounce solution that would continue to grow until eventually the false vacuum decayed. The late-time behavior of this bounce solution would strongly resemble that shown in Figures 6(c) and 6(d). Although no fully satisfactory theory of quantum gravity exists, we have attempted to estimate the tunneling amplitude by using a semiclassical (WKB) approximation. Specifically, we used the same kind of Euclidean field theory technique that was used by Coleman and De Luccia3 (1980) to calculate the decay rate of the false vacuum in curved spacetime. That is, we assume that the amplitude to go from one three-geometry to another is well-appronmated by ei~c~/6 where Ic' is the action of the classical solution to the field equations which interpolates between the two three-geometries. If no real-time solution exists then we seek a Euclidean four-geometry that solves the imaginary tune field equation and whose boundary is the two three-geometries of interest. The tunneling amplitude is then estimated as em/, where IS is the properly subtracted classical action of the Euclidean solution that is, it is the action of the solution, minus the action of a configuration that remains static at the initial state of the tunneling process for the same Euclidean tune as the solution requires for its transit. We have found, however, that no true Euclidean interpolating manifold exists. There is no difficulty or ambiguity in analytically continuing the bubble wall trajectory into the Euclidean regime, but when this trajectory is plotted on a Euclidean spacetime diagram it is found to cross both the initial and final surfaces of the tunneling problem. These intersection points prevent a conventional manifold interpretation. 3See also Section 6 of Guth and Weinberg 19~, which includes a discussion of a spacetime region that was omitted in the original reference.
HIGH-ENERGY ASTROPHYSICS 169 We admit that we are not sure what the absence of a true interpolating manifold implies about the tunneling problem. Perhaps it indicates that the stationary phase method has failed, perhaps it indicates that one cannot extrapolate the thin-wall approximation into the Euclidean regime, or perhaps it Is a suggestion that tunneling Is for some reason forbidden. We find it difficult to believe, however, that the tunneling process is forbidden, since there is no barrier tO constructing a well-defined manifold (with either Lorentzian or Euclidean signature) that interpolates between the initial and final states. Such a manifold is not a solution, but it would constitute a path contn~uting to the functional integral. Furthermore, since any small variation about such a path would also contnbute, the measure of these paths appears naively to be nonzero. The amplitude would then be nonzero unless the venous paths conspire to cancel each other, as they do for an amplitude that violates a conservation principle associated with a symmetry. In the present case, however, there is no apparent symmetry or conservation law at wore We therefore conjecture that the tunneling process is allowed, and that the semiclassical approximation is valid. Although no Euclidean interpolating manifold exists, it is nonetheless possible to generalize the notion of a manifold tO describe a well~efined Euclidean interpolation. In our paper we defined a object that we called a "pseudomanifold," which we defended in two alternative ways. In the simpler description the pseudomanifold closely resembles a true manifold, except that ~ is allowed to vanish and to change sign. We assume that the action of the pseudomanifold can be taken as the usual expression for the Euclidean action, except that ~ is not positive definite. We have used our definition of the action to estimate the tunneling amplitude as a function of the various parameters in the problem, and we have found that it behaves very reasonably: the tunneling action decreases monotonically to zero as the bubble mass M approaches the critical mass MCr at which tunneling would not be necessary, and it diverges monoton- ically as the gravitational constant G ~ O. The action is negative definite by the standard sign conventions, but we argue that, regardless of the sign of the action, the tunneling probability is always exponentially suppressed. In a recent paper, F~schler et al. (1990) have calculated an amplitude for this same process, using a Hamiltonian method somewhat different from the method we used. In their formalism they find no inconsistencies, and their answer is identical to ours. The final result is obtained by a numerical integration, and it is shown graphically in Figure 7. The (subtracted) Euclidean tunneling action IN depends on M/Mcr, and also on the dimensionless parameter 2 ~ = , _ ~ j1 + (pj/6~G~2)
170 2I -GX E AMERICAN AND SOVIET PERSPECTIVES 1.0 ~ 5 - _ O o - ~, - \y= .01 0.2 0.4 0.6 M/ Mcr 0.8 1 FIGURE 7 Graph Of-GX2IE' where IN iS the Euclidean tunneling action. It is shown as a function of M/MCr, for venous values of the parameter A. For typical GUT parameters, ~ ~ 10-4. As a rough estimate, the action In is of order 1/(GX2), as long as Gcr2 Is smaller than or comparable to pf' and M is not too near Mu. For typical grand unified theory parameters, this would give an outrageously small tunneling probability, such as 10-~° . Even with this small probability, however, there might still be a large probability of an event of this sort occurring somewhere in a universe that has undergone a large amount of inflation. Thus, the possibility of a chain reaction by which one universe produces more than one universe is not obviously ruled out by this estimate. On the other hand, if we are talking about creating a universe in a hypothetical laboratory, then a probability this small must be considered equivalent to zero. Thus the production of a universe at the GUT scale seems proh~itive~ unlikely, but it might be possible at energy scales approaching the Planck scale. In any case, I find it fascinating that the creation of a new umveme can even be discussed in scientific terms. If our semiclassical result is correct, then it seems to raise an important issue in quantum gravity: how does a pseudomanifold arise in a quantum gravity path integral? It might mean that such objects occur in the physical
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