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Measuring the Shape of the Neil J. Cornish and Jeffrey R. Weeks

Introduction of positive are necessarily finite, Since the dawn of civilization, humanity has grap- spaces of negative or zero curvature may be either pled with the big questions of existence and cre- finite or infinite. In order to answer questions ation. Modern seeks to answer some of about the global of the universe, we need these questions using a combination of math- to know both its curvature and its . Ein- ematics and measurement. The questions people stein’s equation tells us nothing about the topol- ogy of , since it is a local equation relating hope to answer include How did the universe the spacetime curvature at a point to the matter begin?, How will the universe end?, Is finite density there. To study the topology of the uni- or infinite?. After a century of remarkable progress, verse, we need to measure how space is connected. cosmologists may be on the verge of answering at In doing so we will not only discover whether space least one of these questions: Is space finite? Using is finite but also gain insight into physics beyond some simple geometry and a small NASA satellite . set for launch in the year 2000, the authors and The outline of our paper is as follows: We begin their colleagues hope to measure the size and with an introduction to cosmology, fol- shape of space. This article explains the math- lowed by a review of some basic concepts in geom- ematics behind the measurement and the cos- etry and topology. With these preliminaries out of mology behind the observations. the way, we go on to describe the plan to measure Before setting out, let us first describe the broad the size and shape of the universe using detailed picture we have in mind. Our theoretical framework observations of the afterglow from the big bang. is provided by Einstein’s theory of general relativity and the hot big bang model of cosmology. General Big Bang Cosmology relativity describes the universe in terms of geom- The big bang model provides a spectacularly suc- etry, not just of space, but of space and time. Ein- cessful description of our universe. The edifice is stein’s equation relates the curvature of this space- supported by three main observational pillars: (1) time geometry to the matter contained in the the uniform expansion of the universe, (2) the universe. abundances of the light elements, (3) the highly uni- A common misconception is that the curvature form background of microwave radiation. of space is all one needs to answer the question, The primary pillar was discovered by Edwin Is space finite or infinite?. While it is true that Hubble in the early 1920s. By comparing the spec- tral lines in starlight from nearby and distant galax- Neil J. Cornish is a Research Fellow at the University of ies, Hubble noticed that the vast majority of dis- Cambridge, United Kingdom. His e-mail address is tant galaxies have their spectra shifted to the red, [email protected]. or long wavelength, part of the electromagnetic Jeffrey R. Weeks is a freelance geometer living in Canton, spectrum. Moreover, the was seen to be NY. His e-mail address is [email protected]. larger for more distant galaxies and to occur uni-

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formly in all directions. A simple explanation for see the universe contracting and the temperature this observation is that the space between the increasing. Roughly t 300, 000 years from the ' galaxies is expanding isotropically. By the princi- start, the temperature has reached several thou- ple of mediocrity—i.e., we do not live at a special sand degrees Kelvin. Electrons get stripped from point in space—isotropic expansion about each the atoms, and the universe is filled with a hot point implies homogenous expansion. Such a ho- plasma. Further back in time, at t 1 second, the ' mogeneous and isotropic expansion can be char- temperature gets so high that the atomic nuclei acterized by an overall scale factor a(t) that de- break up into their constituent protons and neu- pends only on time. As the universe expands, the trons. Our knowledge of nuclear physics tells us 10 wavelength λ of freely propagating light is this happens at a temperature of 10 ◦K. At this stretched so that point let us stop going back and let time move for- ward again. The story resumes with the universe a(t0) (1) λ(t0)=λ(t) , filled by a hot, dense soup of neutrons, protons, a(t) and electrons. As the universe expands, the tem- where t0 denotes the present day and t denotes perature drops. Within the first minute the tem- 9 the time when the light was emitted. Astronomers perature drops to 10 ◦K, and the neutrons and define the redshift z as the fractional change in the protons begin to fuse together to produce the nu- wavelength: clei of the light elements deuterium, helium, and λ(t0) λ(t) lithium. In order to produce the abundances seen (2) z = − . λ(t) today, the nucleon density must have been 18 3 Since we expect atoms to behave the same way in roughly 10 cm− . Today we observe a nucleon density of 10 6 cm 3, which tells us the universe the past, we can use atomic spectra measured on ∼ − − Earth to fix λ(t). Using equation (1), we can relate has expanded by a factor of roughly 18 6 1/3 8 the redshift to the size of the universe: (10 /10− ) =10 . Using equation (4), we therefore expect the gas today to be at a a0 (3) a = . temperature of roughly 10 K. made (1 + z) ◦ this back-of-the-envelope prediction in 1946. We have adopted the standard shorthand In 1965 Penzias and Wilson discovered a highly a0 = a(t0) for denoting quantities measured today uniform background of cosmic microwave radia- and a = a(t) for denoting quantities measured at tion at a temperature of 3◦ K. This cosmic a generic time t. By measuring the redshift of an microwave background (CMB)∼ is quite literally the object, we can infer how big the universe was when afterglow of the big bang. More refined nucleo- the light was emitted. The relative size of the uni- synthesis calculations predict a photon tempera- verse provides us with a natural notion of time in ture of 3◦ K, and more refined measurements of cosmology. Astronomers like to use redshift z as the CMB∼ reveal it to have a black body spectrum a measure of time (z =0 today, z = at the big at a temperature of T =2.728 0.010 K. Typical ∞ 0 ◦ bang), since, unlike the time t, the redshift is a mea- cosmic microwave have± wavelengths surable quantity. roughly equal to the size of the letters on this A photon’s energy E varies inversely with its page. wavelength λ. A gas of photons at temperature T The CMB provides strong evidence for the ho- contains photons with energies in a narrow band mogeneity and of space. If we look out in centered at an energy E that is proportional to the any direction of the sky, we see the same mi- temperature. Thus T E λ 1 , and the temper- crowave temperature to 1 part in 104. This implies ∼ ∼ − ature of a photon gas evolves as the curvature of space is also constant to 1 part 4 T E λ0 a0 in 10 on large scales. This observed homogene- (4) = = = =1+z. ity lets cosmologists approximate the large-scale T0 E0 λ a structure of the universe, not by a general space- This equation tells us that the universe should time, but by one having well-defined spatial cross- have been much hotter in the past than it is today. sections of constant curvature. In these Friedman- It should also have been much denser. If no par- Robertson-Walker (FRW) models the spacetime ticles are created or destroyed, the density of or- manifold is topologically the product R , dinary matter is inversely proportional to the oc- where the Mreal line represents time and ×rep- 3 R cupied volume, so it scales as ρm a . If no ∼ − resents a 3-dimensional space of constant curva-Σ photons are created or destroyed, the number of ture.1 The metric on the spacelike slice Σ(t) at 3 photons per unit volume also scales as a− . How- time t is given by the scale factor a(t) times the ever, the energy of each photon is decreasing in standard metric of constant curvatureΣ accordance with equation (4), so that the energy 4 density of the photon gas scales as ργ a . 1 ∼ − Even though elementary particle theory suggests the Starting at the present day, roughly 10 or 15 bil- universe is orientable, both the present article and the re- lion years after the big bang, let us go back through search program of the authors and their colleagues per- the history of the universe. With time reversed we mit nonorientable as well.

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k =+1, 0, 1. The sectional curvature is k/a(t)2, so current theories of the very early universe, in- − when k =1, the scale factor a(t) is the curvature cluding the inflationary paradigm. Vacuum energy | | radius; when k =0, the scale factor remains arbi- with density ρ =3 /8πG mimics the cosmolog- trary. ical constant , which Einstein introduced into The function a(t) describes the evolution of the his field equationsΛ inΛ 1917 to avoid predicting an universe. It is completely determined by Einstein’s expanding or Λcontracting universe and later re- field equation. In general Einstein’s equation is a tracted as “my greatest blunder.” tensor equation in spacetime, but for a homoge- In a universe containing only ordinary matter neous and isotropic spacetime it reduces to the or- ( γ = =0), the mass density scales as 3 dinary differential equation ρ = ρ0(a0/a) . Substituting this into equation (5), Λ 2 oneΩ mayΩ find exact solutions for a(t). These solu- a˙ k 8πG tions predict that if < 1, the universe will expand (5) + 2 = ρ.  a  a 3 forever; if > 1, the expansion will slow to a halt Here G is Newton’s gravitational constant, ρ is the and the universe willΩ recontract; and in the bor- mass-energy density, a˙ = da/dt; and we have cho- derline caseΩ =1, the universe will expand for- sen units that make the c =1. ever, but at a rate a˙ approaching zero. These pre- The first term in equation (5) is the Hubble pa- dictions makeΩ good intuitive sense: under the rameter H = a/a˙ , which tells how fast the uni- definition of as the ratio 8πGρ/3H2, > 1 verse is expanding or contracting. More precisely, means the mass density ρ is large and/or the ex- it tells the fractional rate of change of cosmic dis- pansion rate HΩ is small, so the gravitationalΩ at- tances. Its current value H0, called the Hubble con- traction between galaxies will slow the expansion stant, is about 65 (km/sec)/Mpc.2 Thus, for exam- to a halt and bring on a recollapse; conversely, ple, the distance to a galaxy 100 Mpc away would < 1 means the density is small and/or the ex- be increasing at about 6,500 km/sec, while the pansion rate is large, so the galaxies will speed away distance to a galaxy 200 Mpc away would be in- fromΩ one another faster than their “escape veloc- creasing at about 13,000 km/sec. ity”. Substituting H = a/a˙ into equation (5) shows Cosmologists have suffered from a persistent that when k =0, the mass-energy density ρ must misconception that a negatively curved universe be exactly 3H2/8πG. Similarly, when k =+1(resp. must be the infinite hyperbolic 3-space H3 . This k = 1), the mass-energy density ρ must be greater has led to the unfortunate habit of using the term − than (resp. less than) 3H2/8πG. Thus, if we can “open universe” to mean three different things: measure the current density ρ0 and the Hubble “negatively curved”, “spatially infinite”, and “ex- constant H0 with sufficient precision, we can de- panding forever”. Talks have even been given on duce the sign k of the curvature. Indeed, if k =0, the subject of “closed open models”, meaning fi- 6 we can solve for the curvature radius nite hyperbolic 3-manifolds (assumed to be com- plete, compact, and boundaryless). Fortunately, as (6) finite manifolds are becoming more widely un- 1 k 1 k a = = ,k=0, derstood, the terminology is moving toward the fol- H s8πGρ/3H2 1 H s 1 6 lowing. Universes of positive, zero, or negative − − spatial curvature (i.e., k =+1, 0, 1) are called − where the density parameter is Ωthe dimension- “spherical”, “flat”, or “hyperbolic” respectively. less ratio of the actual density ρ to the critical den- Universes that recollapse, expand forever with 2 sity ρc =3H /8πG. Ω zero limiting velocity, or expand forever with pos- The universe contains different forms of mass- itive limiting velocity are called “closed”, “criti- energy, each of which contributes to the total den- cal”, or “open” respectively. (Warning: This conflicts sity: with topologists’ definitions of “closed” and ρ ργ + ρm + ρ “open”.) (7) = = = γ + m + , ρc ρc Λ Messengers from the Edge of Time Ω Ω Ω ΩΛ where ργ is the energy density in radiation, ρm is In the early 1990s the COBE satellite detected small the energy density in matter, and ρ is a possible intrinsic variations in the cosmic microwave back- vacuum energy. Vacuum energy appears in many 5 Λ ground temperature, of order 1 part in 10 . This small departure from perfect isotropy is thought 2 The abbreviation Mpc denotes a megaparsec, or one to be due mainly to small variations in the mass million parsecs. A parsec is one of those strange units in- distribution of the early universe. Thus the CMB vented by astronomers to baffle the rest of us. One par- photons provide a fossil record of the big bang. The sec defines the distance from Earth of a star whose an- gular position shifts by 1 second of arc over a 6-month field of “cosmic paleontology” is set for a major period of observation. i.e., a parsec is defined by paral- boost in the next decade as NASA plans to launch lax with two Earth-Sun radii as the baseline. A parsec is the Microwave Anisotropy Probe (MAP) and ESA, the about three light-years. Surveyor. These satellites will produce clean

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at a redshift of z 1400, the entire uni- verse was filled 'with an electron-ion- photon plasma similar to the outer lay- ers of a present-day star. In contrast to a gas of neutral atoms, a charged plasma is very efficient at scattering light and is therefore opaque. We can see back to, but not beyond, the surface of last scatter at z 1200. ' Once the plasma condensed to a gas, the universe became transparent, and the photons have been travelling largely unimpeded ever since. They are distrib- uted homogeneously throughout the uni- verse and travel isotropically in all di- rections. But the photons we measure with our instruments are the ones ar- riving here and now. Our position defines Figure 1. (Left) A block of space at the time of last scatter sliced open to a preferred point in space, and our age show the surface of last scatter seen by us today. The dot marks the point defines how long the photons have been where the Earth will eventually form. (Right) A cut-away view showing the travelling to get here. Since they have all spherical shell we refer to as the last scattering surface. been travelling at the same speed for the same amount of time, they have all travelled the same distance. Conse- quently, the CMB photons we measure today originated on a 2- of fixed radius, with us at the center (see Figure 1). An alien living in a galaxy a billion light years away would see a different sphere of last scatter. To be precise, the universe took a fi- nite amount of time to make the transi- tion from opaque to transparent (be- tween a redshift of z 1400 and z 1200), so the sphere of' last scatter is 'more properly a spherical shell with a finite thickness. However, the shell’s thickness is less than 1% of its radius, so cosmologists usually talk about the sphere of last scatter rather than the shell of last scatter. Figure 2. The temperature variations in the CMB measured by the COBE Now that we know where (and when) satellite. The northern and southern hemispheres of the celestial sphere the CMB photons are coming from, we have been projected onto flat circular disks. Here the equator is defined by can ask what they have to tell us. In a per- the galactic plane of the Milky Way. The black pixels correspond to portions fectly homogeneous and isotropic uni- of the sky where the data were badly contaminated by galactic emissions. verse, all the CMB photons would arrive here with exactly the same energy. But all-sky maps of the microwave sky with one fifth this scenario is ruled out by our very ex- of a degree resolution. In contrast, the COBE satel- istence. A perfectly homogeneous and isotropic lite produced a very noisy map at ten degree res- universe expands to produce a perfectly homoge- neous and isotropic universe. There would be no olution [1]. But where are the CMB photons com- galaxies, no stars, no planets, and no cosmolo- ing from, and what can they tell us about the gists. A more plausible scenario is to have small curvature and topology of space? inhomogeneities in the early universe grow via The first thing to realize about any observation gravitational collapse to produce the structures we in cosmology is that one cannot talk about “where” see today. These small perturbations in the early without also talking about “when”. By looking out universe will cause the CMB photons to have into space, we are also looking back in time, as all slightly different energies: photons coming from forms of light travel at the same finite speed. Re- denser regions have to climb out of deeper po- call that about 300,000 years after the big bang, tential wells and lose some energy while doing so.

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There will also be line-of-sight ef- fects as photons coming from dif- ferent directions travel down dif- ferent paths and experience different energy shifts. However, the energy shifts en route are typ- ically much smaller than the in- trinsic energy differences im- printed at last scatter. Thus, by making a map of the microwave sky, we are making a map of the density distribution on a 2-di- mensional slice through the early universe. Figure 2 shows the map produced by the COBE satellite [1]. In the sections that follow we will explain how we can use such maps Figure 3. (a, left) Flat . (b, right) Closed hyperbolic manifold. to measure the curvature and topology of space. But first we need y y +1, z z +1. Similar constructions yield → → to say a little more about geometry and topology. hyperbolic and spherical manifolds. In the spher- ical and hyperbolic cases, the connection between Geometry and Topology the geometry and the topology is even tighter than in two dimensions. For spherical and hyperbolic Topology Determines Geometry 3-manifolds, the topology completely determines For ease of illustration, we begin with a couple of the geometry, in the sense that if two spherical or 2-dimensional examples. A flat torus (Figure 3a) hyperbolic manifolds are topologically equivalent may be constructed as either a square with oppo- (homeomorphic), they must be geometrically iden- site sides identified (the “fundamental domain” pic- tical (isometric) as well.3 In other words, spherical ture) or as the Euclidean plane modulo the group and hyperbolic 3-manifolds are rigid. However, of motions generated by x x +1 and y y +1 this rigidity does not extend to Euclidean 3-man- (the “quotient picture”). Similarly,→ an orientable→ ifolds: a 3-torus made from a cube and a 3-torus surface of genus two (Figure 3b) may be con- made from a parallelepiped are topologically equiv- structed as either a regular hyperbolic octagon alent but geometrically distinct. The final section with opposite sides identified or as the hyperbolic plane modulo a certain discrete group of motions. of this article will explain how the rigidity of a In this fashion, every closed surface may be given closed spherical or hyperbolic universe may be a geometry of constant curvature. used to refine the measured radius of the last scat- Note that in the construction of the torus the tering sphere. square’s four corners come together at a single When we look out into the night sky, we may point in the manifold itself, so it is crucial that the be seeing multiple images of the same finite set of square’s angles be exactly (2π)/4=π/2. In other galaxies, as Figure 3 suggests. For this reason cos- words, it is crucial that we start with a Euclidean mologists studying finite universes make heavy use square; a hyperbolic square (with corner angles less of the “quotient picture” described above, model- than π/2) or a spherical square (with corner an- ling a finite universe as hyperbolic 3-space, Eu- gles greater than π/2) would not do. Similarly, in clidean 3-space, or the 3-sphere, modulo a group the construction of the genus-two surface, the oc- of rigid motions. If we could somehow determine tagon’s eight corners come together at a single the position and orientation of all images of, say, point in the manifold itself, so it is crucial that the our own galaxy, then we would know the group of angles be exactly (2π)/8=π/4. A Euclidean or rigid motions and thus the topology. Unfortu- spherical octagon would not do. In fact, even a nately, we cannot recognize images of our own smaller (resp. larger) hyperbolic octagon would galaxy directly. If we are seeing it at all, we are see- not do, because its angles would be greater (resp. ing it at different times in its history, viewed from less) than π/4. More generally, in any constant cur- different angles; and we do not even know what it vature surface the Gauss-Bonnet theorem looks like from the outside, in any case. Fortu- k dA =2πχ forces the sign of the curvature k nately, we can locate the images of our own galaxy | | indirectly, using the cosmic microwave background. toR match the sign of the Euler number χ. The constructions of Figure 3 generalize to three The section “Observing the Topology of the Uni- dimensions. For example, a 3-torus may be con- verse” will explain how. structed as either a cube with opposite faces iden- 3 tified or as the 3-dimensional E 3In the hyperbolic case this result is a special case of the modulo the group generated by x x +1, Mostow Rigidity Theorem. →

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Geometric Models stant H0 and the density parameter 0. If 0 =1, Elementary linear algebra provides a consistent space is flat. Otherwise, equation (6) shows the cur- way to model the 3-sphere, hyperbolic 3-space, vature radius a0 . The parameters H0Ωand Ω0 may and Euclidean 3-space. be deduced from observations of “standard can- Our model of the 3-sphere S3 is the standard dles” or the CMB. Ω 4 one. Define Euclidean 4-space E to be the Standard Candle Approach 4 vector space R with the usual inner product Astonomers observe objects like Cepheid variables u, v = u0v0 + u1v1 + u2v2 + u3v3. The 3-sphere is and type Ia supernovae whose intrinsic luminos- h i the set of points one unit from the origin, i.e., ity is known. In a static Euclidean space the ap- 3 = v v,v =1 . The standard 4 4 rotation parent brightness of such standard candles would S { |h i } × and reflection matrices studied in linear algebra fall off as the square of their distance from us. In naturally represent rigid motions of S3, both in the- an expanding, curved universe the distance-bright- oretical discussions and in computer calculations. ness relationship is more complicated. Different These matrices generate the orthogonal group values of H0 and 0 predict different relation- O(4). ships between a standard candle’s apparent bright- Our model of the hyperbolic 3-space H3 is ness and its redshiftΩ z. Sufficiently good ob- F formally almost identical to our model of the servations of and z for sufficiently good 1,3 F 3-sphere. Define E to be standard candles will tell us the values of H0 and 4 the vector space R with the inner product 0 and thence the curvature radius a0 . u, v = u0v0 + u1v1 + u2v2 + u3v3 (note the The results of standard candle observations are h i − minus sign!). The set of points whose squared “dis- stillΩ inconclusive. Some recent measurements [2] tance” from the origin is 1 is, to our Euclidean have yielded results inconsistent with the as- − eyes, a hyperboloid of two sheets. Relative to the sumption of a matter-dominated universe and Minkowski space metric, though, each sheet is a point instead to a vacuum energy exceeding the copy of hyperbolic 3-space. Thus, our formal def- density m of ordinary matter! More refined mea- 3 Λ inition is = v v,v = 1,v0 > 0 . The rigid surements over the next few yearsΩ should settle H { |h i − } motions of H3 are represented by the “orthogo- this issue.Ω nal matrices” that preserve both the Minkowski CMB Approach space inner product and the sheets of the hyper- The temperature fluctuations in the CMB (recall Fig- boloid. They comprise an index-2 subgroup of the ure 2) are, to a mathematician, a real-valued func- Lorentz group O(1, 3). tion on a 2-sphere. As such they may be decom- The tight correspondence between our models posed into an infinite series of spherical harmonics, 3 3 for S and H extends only partially to the Eu- just as a real-valued function on a circle may be 3 clidean 3-space E . Borrowing a technique from the decomposed into an infinite series of sines and 3 computer graphics community, we model E as the cosines. And just as the Fourier coefficients of a 4 hyperplane at height 1 in E and represent its sound wave provide much useful information isometries as the subgroup of GL4(R) that takes about the sound (enabling us to recognize it as, say, the hyperplane rigidly to itself. the note A [ played on a flute), the Fourier coeffi- Natural Units cients of the CMB provide much useful informa- has a natural unit of length. tion about the dynamics of the universe. In par- Commonly called a radian, it is defined as the 3- ticular, they reflect the values of H0, 0, , and sphere’s radius in the ambient Euclidean 4-space other cosmological parameters. E4. Similarly, also has a nat- In the year 2002, when data from theΩ MAPΩΛ satel- ural unit of length. It too should be called a radian, lite become available, one of two things will hap- because it is defined as (the absolute value of) the pen: either the spectrum of Fourier coefficients will ’s radius in the ambient Minkowski match some choice of H0, 0, etc., and we will space E1,3. The scale factor a(t) introduced in the know the basic cosmological parameters to un- “Big Bang Cosmology” section connects the math- precedented accuracy, or MotherΩ Nature will sur- ematics to the physics: it tells, at each time t, how prise us with a spectrum inconsistent with our many meters correspond to one radian. In both current understanding of big bang physics. The spherical and hyperbolic geometry the radian is reader may decide which possibility is more ap- more often called the curvature radius, and quan- pealing. tities reported relative to it are said to be in cur- vature units. has no natural Observing the Topology of the Universe length scale, so all measurements must be re- In a finite universe we may be seeing the same set ported relative to some arbitrary unit. of galaxies repeated over and over again. Like a hall of mirrors, a finite universe gives the illusion of Measuring the Curvature of the Universe being infinite. The illusion would be shattered if To determine the curvature of the universe, cos- we could identify repeated images of some easily mologists seek accurate values for the Hubble con- recognizable object. The difficulty is finding ob-

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jects that can be recognized at different times and at different orientations. A more promis- ing approach is to look for correlations in the cosmic microwave background radiation. The CMB photons all originate from the same epoch in the early universe, so there are no aging ef- fects to worry about. Moreover, the shell they originate from is very thin, so the surface of last scatter looks the same from either side. The importance of this second point will soon be- come clear. Consider for a moment two different views of the universe: one from here on Earth and the other from a faraway galaxy. As mentioned earlier, an alien living in that faraway galaxy would see a different surface of last scatter (see Figure 4). The alien’s CMB map would have a different pattern of hot and cold spots from Figure 4. (Left) A block of space at the time of last scatter sliced open ours. However, so long as the alien is not too to show two different surfaces of last scatter. The dot marks our far away, our two maps will agree along the cir- vantage point, and the triangle marks the alien’s vantage point. cle defined by the intersection of our last scat- (Right) An outside view showing how our shell of last scatter tering . Around this circle we would intersects the alien’s. both see exactly the same temperature pat- tern, as the photons came from exactly the same place in the early universe. Unless we get to exchange notes with the alien civilization, this correlation along the matched circles plays no role in cosmology. But what if that faraway galaxy is just an- other image of the Milky Way, and what if we are the aliens? (Recall the flat torus of Figure 3a, whose inhabitants would have the illusion of living in an infinite Euclidean plane con- taining an infinite lattice of images of each ob- ject.) Cornish, Spergel, and Starkman [3] real- ized that in a finite universe the matched circles can transform our view of cosmology, for then the circles become correlations on a single Figure 5. The northern and southern hemispheres of the CMB sky in a copy of the surface of last scatter: i.e., the 3-torus universe. The thirteen matched circle pairs are marked by matched circles must appear at two different black lines. locations on the CMB sky. For example, in a uni- verse with 3-torus topology we would see shown in Figure 5. The model has a cubical 3-torus matched circles in opposite directions on the sky. topology and a scale invariant spectrum of density More generally, the pattern of matched circles perturbations. The nearest images are separated varies according to the topology. The angular di- by a distance equal to the radius of the last scat- ameter of each circle pair is fixed by the distance tering surface. Consequently, there are thirteen between the two images. Images that are displaced matched circle pairs and three matched points from us by more than twice the radius of the last (circles with angular diameter 0). The matched cir- scattering sphere will not produce matched circles. By searching for matched circle pairs in the CMB, cles are indicated by black lines. With good eyes we may find proof that the universe is finite. and a little patience, one can follow the tempera- At present we do not have a good enough CMB ture pattern around each pair of matched circles map to perform the search, but this will soon and convince oneself that the temperatures at cor- change. The CMB map produced by COBE (repro- responding points are correlated. duced here in Figure 2) has a resolution of 10 de- An automated search algorithm has been de- grees, and 30 percent of what one sees is noise, not veloped [3] to search for matched circle pairs. The signal. However, by 2002 the MAP satellite will computer searches over all possible positions, di- have furnished us with a far superior map at bet- ameters, and relative phases. On a modern super- ter than 0.5◦ resolution. In the interim we can test computer the search takes several hours at 1◦ res- our search algorithms on computer-generated sky olution. Our prospects for finding matched circles maps. One example of a synthetic sky map is are greatly enhanced if the universe is highly

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The fundamental domain we construct is a spe- cial type known as a Dirichlet domain. Imagine in- flating a huge spherical balloon whose center is fixed on our galaxy and whose radius steadily in- creases. Eventually the balloon will wrap around the universe and meet itself. When it does, let it keep inflating, pressing against itself just as a real balloon would, forming a planar boundary. When the balloon has filled the entire universe, it will have the form of a polyhedron. The polyhedron’s faces will be identified in pairs to give the original man- ifold. Constructing a Dirichlet domain for the uni- verse, starting from the list of circle pairs, is quite easy. Figure 6 shows that each face of the Dirich- let domain lies exactly halfway between its center Figure 6. To construct a Dirichlet domain, (our galaxy) and some other image of its center. The inflate a balloon until it fills the universe. previous section showed that each circle-in-the-sky also lies exactly halfway between the center of the curved. The rigidity described in the previous sec- SLS (our galaxy) and some other image of that cen- tion means that the distance between images is ter. Thus, roughly speaking, the planes of the cir- fixed by the curvature scale and the discrete group cles and the planes of the Dirichlet domain’s faces of motions. We will be most interested in hyper- coincide! We may construct the Dirichlet domain bolic models, since observations suggest 0 0.3. as the intersection of the corresponding half ≈ 4 In most low-volume hyperbolic models our near- spaces. est images are less than one radian away.Ω The cru- Finding the rigid motions (corresponding to the cial quantity then becomes the radius of the last quotient picture in the “Geometry and Topology” scattering surface expressed in curvature units: section) is also easy. The MAP satellite data will de- termine the geometry of space (spherical, Euclid- Rsls 2 1 0 ean, or hyperbolic) and the radius for the SLS, as (8) η = arcsinh − . a0 ' p 0 well as the list of matched circles. If space is spher- Ω ical or hyperbolic, the radius of the SLS will be given In a universe with 0 =0.3 we find η 2.42, so Ω ≈ in radians (cf. the subsection “Natural Units”); if all images less than 2η 4.84 radians away will ≈ space is Euclidean, the radius will be normalized produce matched circleΩ pairs. However, we may to 1. In each case, the map from a circle to its mate have difficulty reliably detecting matched circles defines a rigid motion of the space, and it is with angular diameters below θ =10◦; we there- straightforward to work out the corresponding fore restrict ourselves to images within a ball of matrix in O(4), O(1, 3), or GL ( ) (recall the sub- 3 4 R radius 4.6 radians. In the universal cover H , a section “Geometric Models”). sphere of radius 4.6 encloses a volume of about Why bother with the matrices? Most impor- 15,000; so if the universe is a hyperbolic manifold tantly, they can verify that the underlying data are of volume less than about 100, there will be an valid. How do we know that the MAP satellite mea- abundance of matched circles. sured the CMB photons accurately? How do we know that our data analysis software does not Reconstructing the Topology of the contain bugs? If the matrices form a discrete group, Universe then we may be confident that all steps in the If at least a few pairs of matching circles are found, process have been carried out correctly, because they will implicitly determine the global topology the probability that bad data would define a dis- of the universe [4]. This section explains how to crete group (with more than one generator) is zero. convert the list of circle pairs to an explicit de- In practical terms, the group is discrete if the prod- scription of the topology, both as a fundamental uct of any two matrices in the set is either another domain and as a quotient (cf. the section “Geom- etry and Topology”). The fundamental domain pic- ture is more convenient for computing the man- 4All but the largest circles determine planes lying wholly ifold’s invariants (such as its volume, homology, outside the Dirichlet domain, which are superfluous in the etc.) and for comparing it to known manifolds, intersection of half spaces. Conversely, if some face of the Dirichlet domain lies wholly outside the SLS, its “corre- while the quotient picture is more convenient for sponding circle” will not exist, and we must infer the face’s verifying and refining the astronomical obser- location indirectly. The proof that the Dirichlet domain cor- vations. Assume for now that space is finite and rectly models the topology of the universe is, of course, that all circles have been observed with perfect simplest in the case that all the Dirichlet domain’s faces accuracy. are obtained directly from observed circles.

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matrix in the set (to within known error bounds) Acknowledgments or an element that is “too far away” to yield a cir- Our method for detecting the topology of the uni- cle. More spectacularly, the matrices correspond- verse was developed in collaboration with David ing to the dozen or so largest circles should pre- Spergel and Glenn Starkman. We are indebted to dict the rest of the data set (modulo a small number Proty Wu for writing the visualization software of errors), giving us complete confidence in its va- used to produce Figure 5 and to the editor for his lidity. generous help in improving the exposition. One of We may take this reasoning a step further and us (Weeks) thanks the U.S. National Science Foun- use the matrices to correct errors. Missing matri- dation for its support under Grant DMS-9626780. ces may be deduced as products of existing ones. References Conversely, false matrices may readily be recog- nized as such, because they will not fit into the [1] C. L. BENNETT et al., Four year COBE-DMR cosmic mi- crowave background observations: Maps and basic structure of the discrete group; that is, multiply- results, Astrophys. J. 464 (1996), L1–L4. ing a false matrix by almost any other matrix in [2] P. M. GARNAVICH et al., Constraints on cosmological the set will yield a product not in the set. This ap- models from the Hubble Space Telescope: Observa- proach is analogous to surveying an apple orchard tions of high-z supernovae, Astrophys. J. 493 (1998), planted as a hexagonal lattice. Even if large por- L53–L57. tions of the orchard are inaccessible (perhaps they [3] N. J. CORNISH, D. N. SPERGEL, and G. D. STARKMAN, Cir- cles in the sky: Finding topology with the microwave are overgrown with vines), the locations of the background radiation, Class. Quant. Grav. 15 (1998), hidden trees may be deduced by extending the 2657–2670. hexagonal pattern of the observable ones. Con- [4] J. R. WEEKS, Reconstructing the global topology of the versely, if a few extra trees have grown between universe from the cosmic microwave background the rows of the lattice, they may be rejected for not radiation, Class. Quant. Grav. 15 (1998), 2599–2604. fitting into the prevailing hexagonal pattern. Note that this approach will tolerate a large number of inaccessible trees, just so the number of extra trees is small. This corresponds to the types of er- rors we expect in the matrices describing the real About the Cover universe: the number of missing matrices will be An artist’s impression of a finite universe large because microwave sources within the Milky showing galaxies inside a fundamental domain Way overwhelm the CMB in a neighborhood of the of the Weeks manifold. The galaxy field is taken galactic equator, but the number of extra matrices from the Hubble Deep Field image (http://www. stsci.edu/ftp/science/hdf/hdf.html will be small (the parameters in the circle match- )—our deep- est and most detailed optical view of the uni- ing algorithm are set so that the expected number verse. It is possible that one of the galaxies seen of false matches is 1). In practice, the Dirichlet do- in this image is our own Milky Way, and the light main will not be computed directly from the cir- we receive from it has made a complete trip cles, as suggested above, but from the matrices, to around the universe. take advantage of the error correction. —Neil J. Cornish Like all astronomical observations, the mea- Jeffrey R. Weeks sured radius Rsls of the sphere of last scatter will have some error. Fortunately, if space is spherical or hyperbolic, we can use the rigidity of the geom- etry to remove most of it! Recall that the hyper- bolic octagon in Figure 3b had to be just the right size for its angles to sum to 2π. The Dirichlet do- main for the universe (determined by the circle pairs; cf. above) must also be just the right size for the solid angles at its vertices to sum to a multi- ple of 4π. More precisely, the face pairings bring the vertices together in groups, and the solid an- gles in each group must sum to exactly 4π. If the measured solid angle sums are consistently less than (resp. greater than) 4π, then we know that the true value of Rsls must be slightly less than (resp. greater than) the measured value, and we revise it accordingly. The refined value of Rsls lets us re- fine 0 as well, because the two variables depend on one another. Ω

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