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OCR for page 153
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
OCR for page 154
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
OCR for page 155
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
OCR for page 156
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
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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
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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
OCR for page 159
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
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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
OCR for page 161
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
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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
OCR for page 163
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
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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
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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
OCR for page 166
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).
OCR for page 167
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.
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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.
OCR for page 169
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)
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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
OCR for page 171
HIGH-ENERGY ASTROPHYSICS
171
definition of the path integral, or it might mean that they arise as saddle
points which are obtained by the distortion of integration contours in the
complex plane.
~ summarize: in this paper I have presented several conclusions, some
of which are firmer than others. I believe that the following conclusions
are well established:
· A false vacuum bubble can inflate without limit, detaching from the
original universe to become an isolated, closed "child" universe. From
the "parent" universe, the false vacuum bubble looks like a black hole.
· By the laws of classical physics, a child universe cannot be created
without an initial singularize (provided that all < icy.
In addition, the following conclusions are strongly indicated by present
research, but ambiguities remain to be resolved:
· In any model of inflation, whether of the new inflation or chaotic
type, isolated child umverses are likely to be produced, presumably in
infinite numbers.
o A new universe can in principle be created in a hypothetical laboratory,
without an initial singularity, by a process of quantum tunneling.
Work in these areas is continuing, and we hope to get a better idea of what
exactly is needed in order to create an inflationary universe.
ACKNOWLEDGMENTS
This work is supported in part by funds provided by the U. S. Depart-
ment of Energy (I).O.E.) under contract #DE-AC02-76ER03069.
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Representative terms from entire chapter:
inflationary universe
