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Measuring the Shape of the Universe Neil J cornish.qxp 10/13/98 3:07 PM Page 1463 Measuring the Shape of the Universe Neil J. Cornish and Jeffrey R. Weeks Introduction spaces of positive curvature 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 cosmology seeks to answer some of about the global geometry of the universe, we need these questions using a combination of math- to know both its curvature and its topology. Ein- ematics and measurement. The questions people stein’s equation tells us nothing about the topol- ogy of spacetime, 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 space 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 general relativity. 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 big bang 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 redshift was seen to be NY. His e-mail address is [email protected]. larger for more distant galaxies and to occur uni- DECEMBER 1998 NOTICES OF THE AMS 1463 cornish.qxp 10/13/98 3:07 PM Page 1464 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 photon gas today to be at a a0 (3) a = . temperature of roughly 10 K. George Gamow 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 photons 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 isotropy 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 universes as well. 1464 NOTICES OF THE AMS VOLUME 45, NUMBER 11 cornish.qxp 10/13/98 3:07 PM Page 1465 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.
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