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Low Density Universes and InflationLow Density Inflationary Universes How Much Matter Does the Universe Contain?An important question today in cosmology is how much massis contained in the universe. If there were no matter fillingthe 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 filledwith matter, and ordinary matter through gravity attracts othermatter, 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 attractionis strong enough to stop and later reverse the expansionof the universe, causing it eventually to recollapse in what is known asthe "Big Crunch." On the other hand, if the average densityof 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 precariouslybalanced between these two possibilities. Why a Universe of Critical Density?For quite some time it has been known that the mean density ofour 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 virtuallyindistinguishable from an empty universe. Similarly, an ever so slightly supercritical universe rapidly collapses into a Big Crunch, never reaching the oldage of our universe---somewhere around twelve billion years.To obtain a universe like ours seems to require finetuning of the initial density to agree with the critical density to an accuracy around one part in 1060!For a long time it was regarded simplest and aesthetically most pleasing to postulate that ouruniverse is now of exactly critical density. The versionsof inflation developed in the early 1980s provided amechanism for setting the density of the universe near the critical density with nearly unlimited precision. For many years an exactly critical universe was toutedas one of the few firm predictions of inflation. Geometry and the Density of the UniverseIn Einstein's General Theory of Relativity, formulatedin 1915, gravity is understood in terms of geometry rather than as just another ordinary force. Mattertells 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 ofmatter, a good approximation on large scales,General Relativity establishes an intimateconnection 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 familiarEuclidean geometry so well known to us from every experience and from classical perspectiveas taught in art class. However, a universe of subcritical orsupercritical density has a non-Euclideangeometry---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 itsimmediate 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 billionlight years we might be deceived into believing thatits geometry is Euclidean. Only on large scales---largerthan the so-called curvature scale---do the differences between the geometries become large effects.The following three plates illustrate the difference inperspective between the three possible geometries:a hyperbolic geometry, a Euclidean geometry, and a spherical geometry. In all three cases, space is divided into identicalcells, whose edges are indicated by the rods. The ballswithin the cells are of identical size, and increasingdistance is indicated by reddening.  In the Euclidean geometry space is divided into cubesand one experiences the ordinary, familiar perspective: the apparentangular size of objects is proportional to the inverse oftheir 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 toEuclidean space, the apparent angularsize of distant objects falls off much morerapidly, 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 spaceresembles 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 ofthe University of Illinois, Urbana-Champaign and by Tamara Munzer of Stanford University for Scientific American. Copyrighted and reprinted with permission.]What is the Geometry of Our Universe?During the 1980s observations remained sufficiently crude so that a universe of critical density was quite plausible.But more recent observations have made it increasinglydifficult 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 amountof "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 mannerthan the luminousmatter, the apparent average density seems to increase as larger and larger scales are probed. For a long time it was hopedprobing sufficiently large scales would uncover a critical densityof dark matter.Today it seems unlikely that this hope will ever be realized.It is now possible to probe the average density of the universeon 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 universeof subcritical density. Rich clusters of galaxies are the largestgravitationally bound systems in the universe. Althoughrare, these systems are excellent laboratories for studying the composition of the matter filling the universe.Using nuclear physics one can determine the baryon densityof the universe. With the density of baryonic matter known,the total density can be determined from measuring thebaryon fraction. The baryonic mass of a cluster can be determined by adding the masses of the constituentgalaxies inferred from their light to the mass of the hot intracluster gas, which can be determined from X-rayobservations of emission from the gas. The total mass can be determined by a variety of methods. Themotions of the constituent galaxies allow one to determine the depth of the potential well and hencethe total mass of the cluster. X-ray observations allowthe same to be done with the gas, and gravitationallensing of background objects by the gravitational fieldof the cluster, resulting in the distortion in appearanceof background galaxies, provides a completely independent checkof the total mass. These techniques, and a number of independent techniques aswell, suggest a universe with approximately one third of thecritical density. Although a universe of critical densitycannot yet be ruled out definitively, the possibility of a critical universe now appears like quite a long shot. Reconciling a Low Density Universe with InflationIf the universe is in fact of subcritical density, does thisrequire 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 createa smooth universe with a hyperbolic geometry,just as is required for a universe of subcritical density.Single bubble open inflation, based on ideasof 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. Turokand later by M. Sasaki,T. Tanaka, and K. Yamamoto. Inflation smooths the universe by postulating anearly epoch of extremely rapid expansion duringwhich whatever irregularities may have existedprior to inflation are virtually erased. In ordinaryinflation, as developed by Guth, Linde, Albrecht,and Steinhardt, this smoothing flattens the universeas well, yielding a universe of critical density. In ordinary inflation, a critical universe couldin principle be avoided by shortening the amountof inflation, but in that case the smoothness on largescales remains a mystery, causing inflation to lose mostof its appeal. The Creation of a Single Bubble Open Universe.The vertical direction indicates time and the horizontal directionsare 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 isexpanding into the surrounding inflating spacetime stuck in the falsevacuum. 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 rolldown the hill to the bottom and as the fielddescends the rate of expansion of the universedecreases, eventually ending the epoch of inflationaryexpansion. In open inflation the inflaton fieldat first remains stuck in a local minimum of the potential. While the field is stuck there, a firstepoch of inflationary expansion takes place during which the universe is smoothed. In factduring this epoch the symmetry of the spacetimeis so large that no particular time directionis preferred over any other.According to classical physics, once stuck in the local minimum the inflaton field never escapes. However, quantum mechanics allowsthe 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. How Can We Test Open Inflation? Microwave Anisotropy as a Function of Angle.Plotted is the level of anisotropy as a function of angleand various measurements thereof. The curves indicate theoreticalpredictions for various models. The solid curve indicates auniverse of critical density whereas the dot-dash-dot-dash curve indicatesa low density universe. Note how the position of the first peak shifts to the right to smaller angular scales in the lowdensity universe.The best hope for testing open inflation derives frommeasuring the geometry of the universe, which can be determined through observing the ripples in thecosmic microwave background radiation. The 3K cosmic microwave background radiation emanatesfrom 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 ofthe universe, combined with protons and other nucleito form neutral hydrogen and other elements. Becauseof this change in composition from a highlyionized plasma to a neutral gas, the formerly opaqueuniverse 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 structurethat we observe today. Inflation in general, and open inflation on scales muchshorter than the curvature scale, imprints essentiallyscale free fluctuations on the matter filling the universe. At recombination, however, the physics atthat time, believed to be well understood, introducesa preferred scale of known length on which the firstacoustic oscillations of the plasma occur. This scaleis of known physical size, and from its anglesubtended in the sky today, we can determine the geometryof the universe. More General Open Inflation The above models for open inflation provide a counter-exampleto the standard lore on inflation, but they rely upon the presence of a local minimum in the potentialenergy of the inflaton field. At our present level of understanding,we simply cannot tell whether this is what is predicted bya more fundamental theory such as M-theory or supergravity. But in the model theories for which we can calculate the inflatonpotential 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 leftindicates 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 expandingbubble 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 andJames Hartle. They proposed that the initialcondition for the universe should be that it possessed no initialboundary. One can picture the spacetime of an expanding universeas the surface of a cone, placed vertically with its sharp tip down.Time runs up the cone: space runs around it. Time andspace end at the sharp tip. The tip is `singular' in mathematicalterms and if this were a model of theuniverse 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 changesin the vicinity of the tip. In effect, all directions must become`horizontal' near the tip, which is to say that all directions arespacelike. This is just what we need to explain how time began.In effectthe distinction between space and time is blurred andspace is then rounded off.The region where time becomes spacelike is technically termedthe instanton region. Instantons are solutions to the equationsof general relativity and matter (here, the inflaton field) which have four spacelike directions. Hawking and Turok showedthat for essentially any theory which allows inflation, there is a family of instanton solutions each one of whichdescribes 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 thestandard hot big bang, which is so severe that we cannotpredict anything that happened in its presence, the singularity in the Hawking-Turok instantons is so mildthat, as for the singularity in the electric fieldat the centre of a hydrogen atom, it does not affect our ability to make predictions. The beauty of the instanton solutions is that they notonly 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 haverecently completed these calculations.The calculations have revealed a potentialobservational signature in the cosmic microwavesky 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 actuallyinvolved. Further reading: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. |
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