An important question today in cosmology is how much mass is contained in the universe. If there were no matter filling the universe, the universe would expand forever and the recession velocity of objects at rest with respect to the expansion of the universe would not change as the universe expands.

We know, of course, that the universe is not empty but filled with matter, and ordinary matter through gravity attracts other matter, causing the expansion of the universe to slow down. If the density of the universe exceeds a certain threshold known as the critical density, this gravitational attraction is strong enough to stop and later reverse the expansion of the universe, causing it eventually to recollapse in what is known as the "Big Crunch." On the other hand, if the average density of the Universe falls short of the critical density, the universe expands forever, and after a certain point the expansion proceeds much as if the universe were empty. A critical universe lies precariously balanced between these two possibilities.

For quite some time it has been known that the mean density of
our universe agrees with the critical density to within better than a
factor of ten. Even with such large margin of error this agreement
is remarkable. Establishing initial conditions so that the mean
density remains close to the critical density for more than a
fleeting moment is much like trying to balance a
pencil on its point.
A universe initially with slightly subcritical density
rapidly becomes increasingly subcritical and soon virtually
indistinguishable from an empty universe. Similarly, an
ever so slightly supercritical universe rapidly
collapses into a Big Crunch, never reaching the old
age of our universe---somewhere around twelve billion years.
To obtain a universe like ours seems to require fine
tuning of the initial density to agree with the
critical density to an accuracy around one part in
10^{60}!

For a long time it was regarded simplest and aesthetically most pleasing to postulate that our universe is now of exactly critical density. The versions of inflation developed in the early 1980s provided a mechanism for setting the density of the universe near the critical density with nearly unlimited precision. For many years an exactly critical universe was touted as one of the few firm predictions of inflation.

In Einstein's General Theory of Relativity, formulated in 1915, gravity is understood in terms of geometry rather than as just another ordinary force. Matter tells spacetime how to curve and the resulting spacetime curvature tells bodies how to move. For the special case of an expanding universe, idealized as filled with a uniform density of matter, a good approximation on large scales, General Relativity establishes an intimate connection between the density of the universe in comparison with the critical density and its geometry. A universe of critical density (at constant cosmic time) has the familiar Euclidean geometry so well known to us from every experience and from classical perspective as taught in art class. However, a universe of subcritical or supercritical density has a non-Euclidean geometry---hyperbolic if the density is subcritical, or spherical if the density is supercritical.

On small scales these different geometries are much alike. An ant on the surface of an apple might view its immediate surrounding as quite flat and might experience difficulty in figuring out that the apple is round. Likewise, if the curvature of the universe would become apparent only on scales beyond several billion light years we might be deceived into believing that its geometry is Euclidean. Only on large scales---larger than the so-called curvature scale---do the differences between the geometries become large effects.

The following three plates illustrate the difference in perspective between the three possible geometries: a hyperbolic geometry, a Euclidean geometry, and a spherical geometry. In all three cases, space is divided into identical cells, whose edges are indicated by the rods. The balls within the cells are of identical size, and increasing distance is indicated by reddening.

In the Euclidean geometry space is divided into cubes and one experiences the ordinary, familiar perspective: the apparent angular size of objects is proportional to the inverse of their distance.

Hyperbolic space shown here is tiled with regular dodecahedra. In Euclidean space such a regular tiling is impossible. The size of the cells is of the same order as the curvature scale. Although perspective for nearby objects in hyperbolic space is very nearly identical to Euclidean space, the apparent angular size of distant objects falls off much more rapidly, in fact exponentially, as can be seen in the figure.

The spherical space space shown here is tiled with regular dodecahedra. The geometry of spherical space resembles the surface of the earth except here a three-dimensional rather than two-dimensional sphere is being considered. Perspective in spherical space is peculiar. Increasingly distant objects first become smaller (as in Euclidean space), reach a minimum size, and finally become larger with increasing distance. This behavior is due to the focusing nature of the spherical geometry.

[The three above figures were prepared by Stuart Levy of the University of Illinois, Urbana-Champaign and by Tamara Munzer of Stanford University for Scientific American. Copyrighted and reprinted with permission.]

During the 1980s observations remained sufficiently crude so that a universe of critical density was quite plausible. But more recent observations have made it increasingly difficult to reconcile a critical universe with the observations.

It is known that in addition to the luminous matter seen in the form of stars the universe contains a large amount of "dark" matter, in particular in the halos around galaxies. The presence of this dark matter is inferred from its gravitational pull on the surrounding matter. Since the dark matter is distributed in a less clustered manner than the luminous matter, the apparent average density seems to increase as larger and larger scales are probed. For a long time it was hoped probing sufficiently large scales would uncover a critical density of dark matter.

Today it seems unlikely that this hope will ever be realized. It is now possible to probe the average density of the universe on scales large enough to compromise a fair sample of the universe. We present the so-called "cluster baryon fraction" as one illustrative example of the strong evidence in favor of a universe of subcritical density. Rich clusters of galaxies are the largest gravitationally bound systems in the universe. Although rare, these systems are excellent laboratories for studying the composition of the matter filling the universe.

Using nuclear physics one can determine the baryon density of the universe. With the density of baryonic matter known, the total density can be determined from measuring the baryon fraction. The baryonic mass of a cluster can be determined by adding the masses of the constituent galaxies inferred from their light to the mass of the hot intracluster gas, which can be determined from X-ray observations of emission from the gas. The total mass can be determined by a variety of methods. The motions of the constituent galaxies allow one to determine the depth of the potential well and hence the total mass of the cluster. X-ray observations allow the same to be done with the gas, and gravitational lensing of background objects by the gravitational field of the cluster, resulting in the distortion in appearance of background galaxies, provides a completely independent check of the total mass.

These techniques, and a number of independent techniques as well, suggest a universe with approximately one third of the critical density. Although a universe of critical density cannot yet be ruled out definitively, the possibility of a critical universe now appears like quite a long shot.

If the universe is in fact of subcritical density, does this require abandoning inflation? If a flat universe really is a "prediction" of inflation as once claimed, one would have to give up inflation.

There however exists an escape from this dilemma. Inflation within a single bubble can create a smooth universe with a hyperbolic geometry, just as is required for a universe of subcritical density.

Single bubble open inflation, based on ideas of S. Coleman and F. de Luccia and of J.R. Gott, III, in the early 1980s, was further developed in the mid-1990s by M. Bucher, A.S. Goldhaber, and N. Turok and later by M. Sasaki, T. Tanaka, and K. Yamamoto.

Inflation smooths the universe by postulating an early epoch of extremely rapid expansion during which whatever irregularities may have existed prior to inflation are virtually erased. In ordinary inflation, as developed by Guth, Linde, Albrecht, and Steinhardt, this smoothing flattens the universe as well, yielding a universe of critical density. In ordinary inflation, a critical universe could in principle be avoided by shortening the amount of inflation, but in that case the smoothness on large scales remains a mystery, causing inflation to lose most of its appeal.

The Creation of a Single Bubble Open Universe.The vertical direction indicates time and the horizontal directions are spatial. The value of the inflaton field is constant on the various slices and the colors indicate the cooling down of the universe as one passes into the bubble interior. The bubble is expanding into the surrounding inflating spacetime stuck in the false vacuum. We live inside the bubble interior.

In single bubble open inflation there are two epochs of inflation. In inflation the rate of expansion is controlled by a scalar field, known as the inflaton field. The inflaton field wants to roll down the hill to the bottom and as the field descends the rate of expansion of the universe decreases, eventually ending the epoch of inflationary expansion. In open inflation the inflaton field at first remains stuck in a local minimum of the potential. While the field is stuck there, a first epoch of inflationary expansion takes place during which the universe is smoothed. In fact during this epoch the symmetry of the spacetime is so large that no particular time direction is preferred over any other.

According to classical physics, once stuck in the local minimum the inflaton field never escapes. However, quantum mechanics allows the field to tunnel through the barrier. This tunneling occurs through the nucleation of a bubble that subsequently expands, somewhat as an expanding bubble in a pot of boiling water.

Subsequently, the bubble expands at the speed of light. It cannot have any velocity other than the speed of light, for else a preferred time direction would be required to exist. The surfaces on the bubble interior on which the scalar field is constant have a hyperbolic spatial geometry, and these are the surfaces that we inside the bubble later perceive as surfaces of constant cosmic time. As one passes inside the bubble, the interior continues to inflate, creating a universe with a large curvature radius. Further inside the bubble the energy of the inflaton field is converted into ordinary matter and radiation, and the hyperbolic universe continues to expand and cool down.

Microwave Anisotropy as a Function of Angle.Plotted is the level of anisotropy as a function of angle and various measurements thereof. The curves indicate theoretical predictions for various models. The solid curve indicates a universe of critical density whereas the dot-dash-dot-dash curve indicates a low density universe. Note how the position of the first peak shifts to the right to smaller angular scales in the low density universe.

The best hope for testing open inflation derives from measuring the geometry of the universe, which can be determined through observing the ripples in the cosmic microwave background radiation.

The 3K cosmic microwave background radiation emanates from an epoch approximately three hundred thousand years after the Big Bang, when the universe was approximately one thousandth its present size. At this time the electrons, because of the cooling of the universe, combined with protons and other nuclei to form neutral hydrogen and other elements. Because of this change in composition from a highly ionized plasma to a neutral gas, the formerly opaque universe becomes virtually transparent. The non-uniformities in the microwave background provide a snapshot of the ripples at that time, which later developed into galaxies and the structure that we observe today.

Inflation in general, and open inflation on scales much shorter than the curvature scale, imprints essentially scale free fluctuations on the matter filling the universe. At recombination, however, the physics at that time, believed to be well understood, introduces a preferred scale of known length on which the first acoustic oscillations of the plasma occur. This scale is of known physical size, and from its angle subtended in the sky today, we can determine the geometry of the universe.

The above models for open inflation provide a counter-example to the standard lore on inflation, but they rely upon the presence of a local minimum in the potential energy of the inflaton field. At our present level of understanding, we simply cannot tell whether this is what is predicted by a more fundamental theory such as M-theory or supergravity. But in the model theories for which we can calculate the inflaton potential energy, such local minima do not usually appear.

Hawking-Turok Instanton.A bubble universe emanates from a Hawking-Turok instanton. The vertical direction indicates time and the horizontal directions are spatial. E indicates the Euclidean region, where time becomes spacelike, and I is the bubble interior. The heavy line to the left indicates the mild singularity occurring in these solutions.

Last year, Hawking and Turok realised that open inflation was in fact much more general, and could even occur in a theory where there is no local minimum in the inflaton potential energy. In fact, they showed that for essentially any potential energy function allowing inflation, an open universe similar to that obtained in the expanding bubble described above could be formed.

Hawking and Turok's calculation was performed in the framework of a proposal for the initial conditions made in 1983 by Hawking and James Hartle. They proposed that the initial condition for the universe should be that it possessed no initial boundary. One can picture the spacetime of an expanding universe as the surface of a cone, placed vertically with its sharp tip down. Time runs up the cone: space runs around it. Time and space end at the sharp tip. The tip is `singular' in mathematical terms and if this were a model of the universe we would find all our equations break down there. Instead, Hartle and Hawking proposed that the tip be rounded off.

This rounding off is only possible if the nature of spacetime changes in the vicinity of the tip. In effect, all directions must become `horizontal' near the tip, which is to say that all directions are spacelike. This is just what we need to explain how time began. In effect the distinction between space and time is blurred and space is then rounded off.

The region where time becomes spacelike is technically termed the instanton region. Instantons are solutions to the equations of general relativity and matter (here, the inflaton field) which have four spacelike directions. Hawking and Turok showed that for essentially any theory which allows inflation, there is a family of instanton solutions each one of which describes the formation of an inflating open universe. The Hawking-Turok instantons do actually possess a singularity, but only at a single point. Unlike the singularity in the standard hot big bang, which is so severe that we cannot predict anything that happened in its presence, the singularity in the Hawking-Turok instantons is so mild that, as for the singularity in the electric field at the centre of a hydrogen atom, it does not affect our ability to make predictions.

The beauty of the instanton solutions is that they not only enable one to compute the probability of formation of open universes from first principles, but one can also compute the spectrum of quantum fluctuations present in the open universes, predicted by the no boundary proposal. Turok and DAMTP students Steven Gratton and Thomas Hertog have recently completed these calculations. The calculations have revealed a potential observational signature in the cosmic microwave sky that will, if the universe has less than critical density, enable one to check which form of open inflation (i.e., with or without a local minimum of the potential) was actually involved.

S.W. Hawking, *A Brief History of Time,* New York: Bantam, 1998.

M. Bucher and D. Spergel, "Inflation in a Low Density Universe,"
*Scientific American,* January 1999.

N. Turok,
"Before the Big Bang,"
in *The Daily Telegraph,* Saturday,
March 14, 1998.