THE DODECAHEDRON UNIVERSE

Lost in spaceA multiconnecteddodecahedraluniverse gives theimpression of being 120 times biggerthan it actually is,similar to a hall of mirrors.

Jeff Weeks

Physics World September 2005 3

Feature: Cosmic topologyphysicsweb.org

For centuries the size and shape of space has intriguedthe human race. The Greek philosophers Plato andAristotle claimed that the universe was finite with aclear boundary. Democritus and Epicurus, on the otherhand, thought that we lived in an infinite universe filledwith atoms and vacuum. Today, 2500 years later, cos-mologists and particle physicists can finally addressthese fundamental issues with some certainty.

Surprisingly, the latest astronomical data suggest thatthe correct answer could be a compromise betweenthese two ancient viewpoints: the universe is finite andexpanding but it does not have an edge or boundary. Inparticular, accurate maps of the cosmic microwavebackground – the radiation left over from the Big Bang– suggest that we live in a finite universe that is shapedlike a football or dodecahedron, and which resemblesa video game in certain respects.

In such a scenario, an object that travels away fromthe Earth in a straight line will eventually return fromthe other side of the universe, having been rotated by36° in the process. Space might therefore act like a cos-mic hall of mirrors by creating multiple images of far-away light sources, which raises new questions aboutthe physics of the early universe. However, this is justone possibility and other proposals made by research-ers in the expanding field of cosmic topology includetetrahedral and octahedral spaces, flat doughnuts andan infinite “horn-shaped” universe.

The curvature of spaceThe first testable predictions about the size and shapeof the universe were made by Einstein in 1916 as part ofhis general theory of relativity. In general relativitymassive bodies such as stars change the shape ofspace–time around them, much as a bowling ball wouldchange the shape of a trampoline. Indeed, it is this localdeformation of space–time that is responsible for grav-ity in Einstein’s theory.

The average curvature of space therefore dependson the overall density of matter and energy in the uni-verse. This density is usually expressed in terms of theratio Ω, which is defined as the actual density of the uni-verse divided by the critical density required for spaceto be flat or Euclidean. Space can therefore have threepossible curvatures: zero curvature (Ω = 1), whichmeans that two parallel lines remain a constant dis-

tance apart as they do in the familiar Euclidean space;negative curvature (Ω<1), with parallel lines divergingas they do on the hyperbolic surface of a saddle; or pos-itive curvature (Ω > 1), which means that parallel lineseventually cross one another as they do on the surfaceof a sphere.

In the standard model of cosmology, space has beenflat and infinite ever since the universe underwent ashort period of extremely rapid expansion called infla-tion shortly after the Big Bang. Moreover, we now knowthat the expansion of the universe is actually accelerat-ing due to a mysterious repulsive force caused by “dark”energy (see Physics World May 2004 pp37–42).

In 2003 the Wilkinson Microwave Anisotropy Probe(WMAP) produced a high-resolution map of the cos-mic microwave background that provided clues aboutthe expansion rate of the universe and its curvature.Combined with other astronomical observations, theWMAP data suggest that Ω = 1.02 ± 0.02, whichfavours a spherical universe with positive curvature.The simplest such space is a “hypersphere”, which canbe thought of as the 3D surface of a 4D ball, just as anordinary sphere is the 2D surface of a 3D ball. Hyper-spherical space is therefore finite but it does not have aproblematic boundary (figure 1). However, as we willsee, many other spherical spaces can fit the data betterthan a hypersphere.

The topology of spaceCurvature is clearly central to the large-scale shape ofspace, but it is not the only factor. The global topolog-ical properties of space are also important because theydetermine whether the universe is finite or infinite. Allspherical spaces are finite, for instance, but not allfinite spaces are spherical. Indeed, flat and hyperbolicspaces can have finite or infinite volumes dependingon their topologies.

To illustrate this in two dimensions, think of a squareand identify opposite sides as being the same, as hap-pens in video games where a spaceship disappearing tothe right of the screen reappears on the left. In threedimensions, a spaceship or anything else (such as a par-

Most astronomers think that the universe isinfinite, but recent measurements suggestthat it could be finite and relatively small.Indeed, as Jean-Pierre Luminet describes,we could be living in an exotic universeshaped rather like a football

Jean-Pierre Luminetis at the DépartementUnivers et Théories,Observatoire deParis, Meudon,France, e-mail [email protected]

A cosmic hall of mirrors

There are three possibilities for the curvature of the universe: space can be flat,spherical or hyperbolic

The geometry of the universe depends on its curvature and also on its topology,which governs the way space is connected and so determines if it is finite or infinite

Measurements of the cosmic microwave background constrain the curvature of the universe and provide hints about its topology

Recent data suggest that the universe might be multiply connected, like the left- and right-hand sides of the screen in a computer game

Since the early 1990s the number of cosmo-topologists around the world hasgrown to more than 50

At a Glance: Cosmic topology

Physics World September 20054

Feature: Cosmic topology physicsweb.org

ticle or a photon) that leaves the “fundamental” cubethrough one face re-enters it from the opposite face. Inthis case one can imagine a cubic block of space whoseopposite faces have been “glued” together to producewhat is effectively a 3D torus.

At first glance all the familiar rules of Euclideangeometry hold in both of these examples, and thespaces look infinite to those who live in them. However,unless the spaceship keeps encountering the sameobjects on its travels, there is no way that its crew couldtell if it is moving through an infinite space or throughthe same finite space again and again.

Furthermore, general relativity does not distinguishbetween these possibilities because each of the threeplausible cosmic geometries – flat, hyperbolic andspherical – is consistent with many different topolo-gies. For example, a 3D torus and ordinary flat Eu-clidean space are described by the same equations ingeneral relativity, even though the former is finite andthe latter infinite. Determining the topology of the cos-mos therefore requires some physical understandingbeyond relativity, in particular concerning the way dif-ferent regions of space–time are connected.

Cosmologists usually assume that the universe is sim-ply connected like a plane, which means there is onlyone direct path for light to travel from a source to anobserver. A simply connected Euclidean or hyperbolicuniverse would indeed be infinite, but if the universe ismultiply connected, like a torus, there would be manydifferent possible paths. This means that an observerwould see multiple images of each galaxy and could

easily misinterpret them as distinct galaxies in an end-less space, much as a visitor to a mirrored room has theillusion of seeing a crowd. Could we, in fact, be livingin such a cosmic hall of mirrors?

Topologists have proved that in addition to the ordi-nary, simply connected Euclidean, spherical and hyper-bolic spaces, there are 17 other Euclidean spaces andan infinite number of spherical and hyperbolic spaces –all of which are multiply connected. These spaces differin the shape of their fundamental blocks, which cantake the form of a parallepiped or a hexagonal prismfor a Euclidian space or more complicated polyhedronsfor spherical and hyperbolic spaces. The way the facesof these blocks are glued together also differs betweeneach space. The surprise from the WMAP data is thatthe topology of space seems indeed to be multiply con-nected and described by a special class of shapes thatare called “well proportioned”.

Cosmic harmonicsThe best way to determine the shape of our universe isto go back to its beginning, just after the Big Bang. Theinfant universe is thought to have been crossed byacoustic waves that would have caused tiny density fluc-tuations in the primordial plasma. After about 380 000years, however, the universe had expanded and cooledenough to allow matter and antimatter to decouple.This meant that photons could travel unhinderedthrough space, carrying with them vital informationabout the primordial density fluctuations (which arenow thought to have been the seeds for galaxies andclusters of galaxies to form). Today, 13.7 billion yearsafter the Big Bang, this radiation has cooled to a tem-perature of about 2.7 K, which is in the microwaveregion. And the fluctuations are imprinted as hot andcold spots in this cosmic microwave background.

A good way to understand the connection betweenacoustics and topology is to sprinkle fine sand uni-formly over a drumhead and then make it vibrate. Thegrains of sand will collect in characteristic areas andpatterns that reveal information about the local geom-etry of the drum and about the elasticity of its mem-brane. But the distribution of these spots also dependson the global shape – i.e. the topology – of the drum.For example, the waves will be reflected differentlyaccording to whether the drumhead is infinite or finite,and whether it is shaped like a circle, an ellipse or someother shape.

Just as the vibration of a drumhead may be expressedas a combination of its harmonics, fluctuations in thetemperature of the cosmic background radiation maybe expressed as combinations of the vibrational modesof space itself. When the level of fluctuations is plottedas a function of angle, we therefore find a series ofpeaks that provides a signature of the geometry ofspace 13.7 billion years ago (figure 2). For example, theposition and amplitude of the first peak – i.e. the peakat the largest angle – in this “angular power spectrum”gives the radius of curvature of space.

Different cosmological models predict differentpower spectra, and high-resolution measurements ofthe cosmic microwave background from instrumentssuch as WMAP now allow us to compare different the-ories against real data. However, when WMAP re-

1 How flat is the universe?

horizon sphere

hypersphericaluniverse

The curvature of space and our horizon radius are determined by theaverage density of the universe and its expansion rate. Cosmologistsoften say that space is nearly flat because the observed value of thedensity is close to the critical value for a flat universe. However, if thedensity was 2% more than the critical density, the horizon radius(yellow line) would be about 46 billion light-years and the radius ofcurvature of the corresponding hypersphere would be only 2.6 timesgreater. We would therefore see a modest but non-trivial portion of thehypersphere. If the density is exactly equal to the critical value, spaceis Euclidean, the radius of curvature is infinite and we can only see aninfinitesimal portion of the universe.

Finite well-proportionedspaces open aPandora’sbox for thephysics of theearly universe



leased its first data in 2003, advocates of the standardcosmological model were faced with several surprises.

The position of peaks in the angular spectrum is usu-ally described by their wavenumber or mode = 180°/θ, where θ is the angular distance in the sky.In fact, the lowest mode – the dipole or = 1 mode – isswamped by the far stronger dipole induced by themotion of the solar system relative to the cosmic back-ground, which means that it cannot be measured. Butwhen researchers determined the first observablemode – the = 2 or quadrupole mode – they found thatit was seven times weaker than the predictions for a flat,infinite universe. Furthermore, the octopole or = 3mode was also less than the expected value by a factorof about two-thirds.

For higher modes up to = 900, which correspondto angular scales of just 0.2°, the WMAP data werefairly consistent with the standard model. But a morecareful analysis of the power spectrum also reveals thatthe distribution of temperature fluctuations is not fullyisotropic and that the fluctuations are distributed dif-ferently on different angular scales.

All these anomalies contradict the standard pictureof the universe, which has led some more conservativecosmologists to claim that they are due to errors in thedata analysis. Furthermore, the second round of WMAPdata – originally expected in February 2004 – has beendelayed for more than a year, which may hint at addi-tional trouble to come! Meanwhile, other cosmologistshave taken the problem seriously and proposed newlaws to explain the early universe, some of which haveexotic names such as “vanilla” and “racetrack” inflation.

Cosmo-topologists, on the other hand, have tried tofind a more natural, geometrical explanation for theobserved power spectrum. Put simply, the unusuallylow amplitudes of the quadrupole and octopole modesmeans that long wavelengths (i.e. temperature fluctu-ations over large angular scales) are missing, possiblybecause space is not big enough to sustain them. Thiscan be likened to oscillation of a string fixed at bothends, where the maximum wavelength of an oscillationis twice the string length. The geometrical explanationof the power spectrum thus implies that we live in afinite, multiply connected space that is smaller than theobservable universe.

Dodecahedral spaceSurprisingly, not all small-volume universes suppressthe large-scale fluctuations. In 2003 the present author,Jeff Weeks and co-workers proved that the long-wave-length modes tend to be relatively lowered only in aspecial family of finite, multiconnected spaces that arecalled “well-proportioned spaces” because they havea similar extent in all three dimensions. More specifi-cally, we discovered that the best candidate to fit theobserved power spectrum is a well-proportioned spacecalled the Poincaré dodecahedral space.

This space may be represented by a polyhedron with12 pentagonal faces, with opposite faces being “glued”together after a twist of 36° (figure 3). This is the onlyconsistent way to obtain a spherical (i.e. positivelycurved) space from a dodecahedron: if the twist was108°, for example, we would end up with a radically dif-ferent hyperbolic space. The Poincaré dodecahedral

space is essentially a multiply connected variant of asimply connected hypersphere, although its volume is120 times smaller.

A rocket leaving the dodecahedron through a givenface immediately re-enters through the opposite face,and light propagates such that any observer whoseline-of-sight intercepts one face has the illusion of see-ing a slightly rotated copy of their own dodecahedron.This means that some photons from the cosmicmicrowave background, for example, would appeartwice in the sky.

The power spectrum associated with the Poincarédodecahedral space is different from that of a flat spacebecause the fluctuations in the cosmic microwave back-ground will change as a function of their wavelengths.In other words, due to a cut-off in space corresponding

2 Angular power spectrum

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The evidence that space might have an unusual shape is obvious when we plot the level oftemperature fluctuations in the cosmic microwave background as a function of angle, θ (top).The WMAP data are shown in green (purple depicts earlier data) and the modes of thespectrum are defined as = 180°/θ. The largest angular scales (or lowest modes) are verysensitive to the global shape or topology of space. In this region (bottom), data from WMAP donot agree with theoretical predictions for an infinite Euclidean space (red curve), whereaspredictions based on a Poincaré dodecahedral space (blue curve) do agree. This suggests thatwe live in a multiply connected spherical space.


Feature: Cosmic topology physicsweb.org

to the size of the dodecahedron, one expects fewer fluc-tuations at large angular scales than in an infinite flatspace, but at small angular scales one must recover thesame pattern as in the flat infinite space. In order to cal-culate the power spectrum we varied the mass–energydensity of the dodecahedral universe and computedthe quadrupole and the octopole modes relative to theWMAP data. To our delight, we found a small intervalof values over which both these modes matched theobservations perfectly. Moreover, the best fit occurredin the range 1.01 < Ω < 1.02, which sits comfortablywith the observed value.

The Poincaré dodecahedral space therefore ac-counts for the lack of large-scale fluctuations in themicrowave background and also for the slight positivecurvature of space inferred from WMAP and otherobservations. Moreover, given the observed values of

the mass–energy densities and of the expansion rate ofthe universe, the size of the dodecahedral universe canbe calculated. We found that the smallest dimension ofthe Poincaré dodecahedron space is 43 billion light-years, compared with 53 billion light-years for the“horizon radius” of the observable universe. Moreover,the volume of this universe is about 20% smaller thanthe volume of the observable universe. (There is a com-mon misconception that the horizon radius of a flat uni-verse is 13.7 billion light-years, since that is the age ofthe universe multiplied by the speed of light. However,the horizon radius is actually much larger because pho-tons from the horizon that are reaching us now have

had to cross a much larger distance due to the expan-sion of the universe.)

If physical space is indeed smaller than the observ-able universe, some points on the map of the cosmicmicrowave background will have several copies. As firstshown by Neil Cornish of Montana State Universityand co-workers in 1998, these ghost images wouldappear as pairs of so-called matched circles in the cos-mic microwave background where the temperaturefluctuations should be the same (figure 4). This “lens-ing” effect, which can be precisely calculated, is thuspurely attributable to the topology of the universe.

Due to its 12-sided regular shape, the Poincaré dodec-ahedral model actually predicts six pairs of diametri-cally opposite matched circles with an angular radius of10–50°, depending on the precise values of cosmologi-cal parameters such as the mass–energy density.

Circles in the skyWhen news of our dodecahedral model appeared inNature in October 2003, it was not long before the pressstarted running headlines based on what was beinghailed the new “football-shaped” model of the uni-verse. However, since cosmo-topology is a very com-petitive field, the initial response from other groupswas not always favourable.

For instance, the New York Times ran the headline“Cosmic soccer ball? Theory already takes sharp kicks”,based on an apparently negative search for matched cir-cles in the WMAP data performed by Cornish and co-workers. Using massive computer simulations, theyclaimed to have found no evidence of matching onangular sizes greater than 25° and thus rejected thePoincaré hypothesis the same day it appeared.

In fact, their rejection was rather premature becausethey had only looked for non-rotated matched circlesthat were diametrically opposite one another – a casethat did not test the dodecahedron model at all. After

3 The many shapes of the universe

The Poincaré dodecahedral space (left) can be described as the interior of a “sphere” made from 12 slightly curved pentagons. However, there is one big differencebetween this shape and a football because when one goes out from a pentagonal face, one immediately comes back inside the ball from the opposite face after a 36°rotation. Such a multiply connected space can therefore generate multiple images of the same object, such as a planet or a photon. Other such well-proportioned,spherical spaces that fit the WMAP data are the tetrahedron (middle) and octahedron (right).

The most fundamental issue is to link the topology of space to a quantum origin

Jeff

Wee

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the initial excitement, Cornish and co-workers wentback and reassessed the data. Taking account of the ad-ditional 36° twist took a few additional months of com-puter time, but the matched circles remained elusive.This led them to conclude that there was no reasonabletopology for the universe that had a characteristiclength smaller than the observable horizon.

However, it turned out that the researchers hadtaken a short cut to save computer time. While theycorrectly took into account the possible rotationsbetween matched circles that are implied by mostmulti-connected topologies, they only searched formatched circles that were back-to-back or very nearlyback-to-back. This led them to exclude all likely multi-ply connected spaces. In the mean time, however, wehad proved that in most multiply connected, well-pro-portioned topologies space is not homogeneous. Thismeans that the position of matched circles in the skydepends on the location of the observer, and they arenot, therefore, back-to-back. Only in the simplest oftopologies, such as the hypertorus in flat space and thePoincaré dodecahedron in a spherical space, is spacehomogeneous and the circles back-to-back.

This violates one of the most basic principles of cos-mology, that there is no privileged position in the uni-verse. But this principle could be illusory, like the antin the desert that is convinced the whole world is filledwith sand and dunes. For instance, in a flat-torus uni-verse, any gluing together of the opposite faces com-bined with a screw motion produces pair of circles thatare far from being back-to-back. Unfortunately, theincrease in the number of degrees of freedom thatresults from such a scenario means that a full-circlesearch in the WMAP data is beyond current comput-ing capabilities.

Cosmic hornIn June 2004, however, Boud Roukema and colleaguesat the Torun Centre for Astronomy in Poland inde-pendently searched for circles in the WMAP data. Byonly looking for back-to-back circles within a limitedrange of angular sizes and neglecting all other possiblematches, the computer time was reduced drastically.Remarkably, the Polish team found six pairs ofmatched circles distributed in a dodecahedral patternand twisted by 36°, each with an angular size of about11°. This implied that Ω = 1.010 ± 0.001, which is per-fectly consistent with our dodecahedral model,although the result was much less publicized than theearlier negative results.

In fact, the statistical significance of the match stillneeds to be improved, which means that the validity ofthe Poincaré dodecahedron model is still open todebate. In the last few months, however, there has beenmuch theoretical progress on well-proportioned spacesin general. Early this year, for example, Frank Steinerand co-workers at the University of Ulm in Germanyproposed a multiply connected hyperbolic topologycalled the Picard hyperbolic space. Like the Poincarédodecahedron, this horn-shaped space belongs to thefamily of well-proportioned spaces and it also correctlyfits the low vibrational modes of the WMAP data.However, since the topology requires the density para-meter to have a value of Ω = 0.95, and thus a negatively

curved space, it does not fit the experimental con-straints we already have on the curvature of space.

After studying the horn-shaped topology further,Steiner and co-workers realized that well-proportionedspherical spaces were, in fact, more promising. Theywent on to prove that the fit between the power spec-trum predicted by the Poincaré dodecahedron modeland that observed by WMAP was even better than wehad previously thought. But the German team alsoextended its calculations to well-proportioned tetra-hedral and octahedral spherical spaces in which Ω > 1(see figure 3).

These spaces are somewhat easier to understandthan a dodecahedral space, but they require higher val-ues of the density: Ω > 1.015 for octahedral spaces andΩ > 1.025 for tetrahedral spaces, compared withΩ > 1.009 for dodecahedral spaces. However, thesevalues are still compatible with the WMAP data.Furthermore, Steiner and co-workers found that thesignal for pairs of matched circles could have be missedby current analyses of the cosmic microwave back-ground due to various measurement effects that dam-age or even destroy the temperature matching.

Another active area of cosmic topology is “cosmiccrystallography”, which was initially devised by the pre-sent author and co-workers in 1996 and is now beingpursued by, among others, Germán Gomero of theUniversidade Estadual Paulista in Brazil and MarceloReboucas of the Brazilian Center for Research inPhysics. In cosmic crystallography researchers look forrepeating patterns in the 3D distribution of high-red-shift sources, such as galaxy clusters and quasars, muchlike the repeating patterns of atoms observed in crys-tals. By building so-called pair-separation histograms,cosmologists are in most cases able to detect a multi-connected topology of space in the form of spikes thatclearly stand out above the distribution expected forthe simply connected case.

4 Simulated circle matching

The topology of the universe describes how different regions are connected and could thereforeleave its imprint on the cosmic microwave background. For example, if our physical space issmaller than the observable universe (as recent data suggest it is) then the horizon spherewraps around the universe and intersects itself. As a result, duplicated images of the cosmicmicrowave background (in which the colours represent temperature fluctuations) will intersectalong a circle and we would observe this circle on different sides of the sky.

Mapping spaceLaunched in 2001,the WilkinsonMicrowave AnisotropyProbe has changed our view of the cosmos.

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Feature: Cosmic topology

A Pandora’s box for physicsFinite well-proportioned spaces, especially the Poin-caré dodecahedron, open something of a Pandora’sbox for the physics of the early universe. The standardmodel of cosmology relies in the main on the hypothe-sis that the early universe underwent a phase of expo-nential expansion called inflation, which produceddensity fluctuations on all scales. In the simplest in-flationary models, space is supposed to have becomeimmensely larger than the observable universe. There-fore, a positive curvature (i.e. Ω > 1), even if weak,implies a finite space and sets strong constraints oninflationary models.

It is possible to build “low scale” inflationary uni-verses in which the inflation phase ends more quicklythan it does in general inflationary modes, leading to adetectable space curvature. In other words, even if spaceis not flat, a multiconnected topology does not contra-dict the general idea of inflation. However, no convinc-ing physical scenario for this has yet been proposed.

Perhaps the most fundamental challenge is to linkthe present-day topology of space to a quantum origin,since general relativity does not allow for topologicalchanges during the course of cosmic evolution. A quan-tum theory of gravity could allow us to address thisproblem, but there is currently no indication about howsuch a unified theory might actually describe the emer-gence of multiply connected spaces.

Data from the European Planck Surveyor, which isscheduled for launch in 2007, will be able to determine

Ω with a precision of 1%. A value lower than 1.01 willrule out the Poincaré dodecahedron model, since thesize of the corresponding dodecahedron would becomegreater than the observable universe and would notleave any observable imprint on the microwave back-ground. A value greater than 1.01, on the other hand,would strengthen the models’ cosmological pertinence.

Whether or not some multiply connected model ofspace such as the Poincaré dodecahedron is refuted byfuture astronomical data, cosmic topology will con-tinue to remain at the heart of our understanding aboutthe ultimate structure of our universe.

More about: cosmic topologyW Aurich et al. 2005 CMB anisotropy of the Poincarédodecahedron arXiv.org/abs/astro-ph/0412569N Cornish et al. 2004 Constraining the topology of the universePhys. Rev. Lett. 92 201302J P Luminet et al. 2002 Is space finite? The Once and FutureCosmos, Scientific American (special edition) pp58–65J P Luminet et al. 2003 Dodecahedral space topology as anexplanation for weak wide-angle temperature correlations in thecosmic microwave background Nature 425 593A Riazuelo et al. 2004 Cosmic microwave background inmulticonnected flat spaces Phys. Rev. D 69 103518B Roukema et al. 2004 A hint of Poincaré dodecahedraltopology in the WMAP first year sky map Astron. Astrophy.423 821J Weeks 2001 The Shape of Space (New York, Dekker)WMAP results: map.gsfc.nasa.gov

physicsweb.org

The secondround of WMAPdata may hintat additionaltrouble tocome

THE DODECAHEDRON UNIVERSE

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