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A MAP OF THE UNIVERSE -0004-637X_624_2_463

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A MAP OF THE UNIVERSE J. Richard Gott III, 1 Mario Juric´, 1 David Schlegel, 1 Fiona Hoyle, 2 Michael Vogeley, 2 Max Tegmark, 3 Neta Bahcall, 1 and Jon Brinkmann 4 Receiv ved 2003 November 18; accepted 2005 January 3 ABSTRACT We have produced a newconformal map of the universe illustrating recent discoveries, ranging fromKuiper Belt objects in the solar system to the galaxies and quasars from the Sloan Digital Sky Survey. This map projection, based on th
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  A MAP OF THE UNIVERSE J. Richard Gott III, 1 Mario Juric´, 1 David Schlegel, 1 Fiona Hoyle, 2 Michael Vogeley, 2 Max Tegmark, 3 Neta Bahcall, 1 and Jon Brinkmann 4  Receiv  v  ed 2003 No v  ember 18; accepted 2005 January 3 ABSTRACTWe have producedanewconformal mapofthe universeillustratingrecentdiscoveries,rangingfromKuiper Belt objects in the solar system to the galaxies and quasars from the Sloan Digital Sky Survey. This map projection, basedonthelogarithmmapofthecomplexplane,preservesshapeslocallyandyetisabletodisplaytheentirerangeof astronomical scales from the Earth’s neighborhood to the cosmic microwave background. The conformal natureof the projection, preserving shapes locally, may be of particular use for analyzing large-scale structure. Prominent inthemapisaSloanGreatWallofgalaxies1.37billionlight-yearslong,80%longerthantheGreatWalldiscovered by Geller and Huchra and therefore the largest observed structure in the universe. Subject headin g  g   s:  large-scale structure of universe — methods: data analysis1. INTRODUCTIONCartographers mapping the Earth’s surface were faced withthe challenge of mapping a curved surface onto a plane. Nosuch projection can be perfect, but it can capture important fea-tures. Perhaps the most famous map projection is the Mercator  projection(presentedbyGerhardusMercatorin1569).Thisisaconformal projection which preserves shapes locally. Lines of latitude are shown as straight horizontal lines, while meridiansof longitude are shown as straight vertical lines. If the Mercator  projection is plotted on an (  x ,  y )-plane, the coordinates are plot-ted as  x ¼ k  and  y ¼ ln ½ tan(  /4 þ  /2)  , where    (positive if north, negative if south) is the latitude in radians, while  k  (pos-itive if easterly, negative if westerly) is the longitude in radians(see Snyder [1993] for an excellent discussion of this and other map projections of the Earth). This conformal map projection preserves angles locally, and also compass directions. Localshapes are good, while the scale varies as a function of latitude.Thus, the shapes of both Iceland and South America are shownwell, although Iceland is shown larger than it should be relativeto South America. Other map projections preserve other prop-erties. The stereographic projection, which, like the Mercator  projection, is conformal, is often used to map hemispheres. Thegnomonic map projection (effectively from a ‘‘light’’ at thecenter of the globe onto a tangent plane) maps geodesics intostraight lines on the flat map but does not preserve shapes or areas.Equal-areamapprojectionsliketheLambert,Mollweide,and Hammer projections preserve areas but not shapes.A Lambert azimuthal equal-area projection, centered onthe North Pole, has in polar coordinates ( r  ,   )   ¼ k ,  r  ¼ 2 r  0  sin ½ (  /2   )/2  , where  r  0  is the radius of the sphere. This projection preserves areas. The Northern Hemisphere is thusmapped onto a circular disk of radius  ffiffiffi 2 p   r  0  and area 2  r  02 . Anobliqueversionofthis,centeredatapointontheequator,isalso possible.The Hammer projection shows the Earth as a horizontal el-lipsewith2 : 1axis ratio.Theequator isshownas astraighthori-zontal line marking the long axis of the ellipse. It is produced inthe followingway.Mapthe entire sphere onto its western hemi-sphere by simply compressing each longitude by a factor of 2. Now map this western hemisphere onto a plane by the Lambert equal-area azimuthal projection. This map is a circular disk.This isthen stretched by a factor of 2 (undoingthe previouscom- pression by a factor of 2) in the equatorial direction to make anellipse with a 2 : 1 axis ratio. Thus, the Hammer projection pre-serves areas. The Mollweide projection also shows the sphereas a 2 : 1axis ratio ellipse. The (  x ,  y ) coordinates on the map are  x ¼ (2  ffiffiffi 2 p   /   ) r  0 k  cos   and  y ¼  ffiffiffi 2 p   r  0  sin  , where 2  þ sin 2  ¼   sin  . This projection is equal area as well. Latitude lines onthe Mollweide projection are straight, whereas they are curvedarcs on the Hammer projection.Astronomers mapping the sky have also used such map pro- jections of the sphere. Gnomonic maps of the celestial sphereonto a cube date from 1674. In recent times, Turner & Gott (1976) used the stereographic map projection to chart groups of galaxies (utilizing its property of mapping circles in the skyonto circles on the map). The  COBE   satellite map (Smoot et al.1992) of the cosmic microwave background (CMB) used anequal-area map projection of the celestial sphere onto a cube.The  WMAP   satellite (Bennett et al. 2003) mapped the celestialsphereontoarhombicdodecahedronusingtheHealpixequal-areamapprojection(Go´rskietal.2002).Itsresultsweredisplayedalsoon the Mollweide map projection, showing the celestial sphere asan ellipse, which was chosen for its equal-area property and thefact that lines of constant Galactic latitude are shown as straight lines.De Lapparent et al. (1986) pioneered use of slice maps of theuniverse to make flat maps. They surveyed a slice of sky, 117  long and 6   wide, of constant declination. In three dimensionsthis slice had the geometryof a cone, and they flattened this ontoa plane. (A cone has zero Gaussian curvature and can therefore be constructed from a piece of paper.A cone cut along a line andflattened onto a plane looks like a pizza with a slice missing.) If the cone is at declination    , the map in the plane will be  x ¼ r   cos ( k cos   ),  y ¼ r   sin ( k cos   ), where  k  is the right ascension(in radians) and  r   is the comoving distance (as indicated by theredshift of the object). This will preserve shapes. Many times a360   slice is shown as a circle with the Earth in the center,where 1 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544. 2 DepartmentofPhysics,DrexelUniversity,3141ChestnutStreet,Philadelphia,PA 19104. 3 Department of Physics, University of Pennsylvania, 209 South 33rdStreet, Philadelphia, PA 19104. 4 Apache Point Observatory, 2001 Apache Point Road, P.O. Box 59, Sunspot, NM 88349. 463 The Astrophysical Journal , 624:463–484, 2005 May 10 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.   x ¼ r   cos  k ,  y ¼ r   sin  k . If   r   is measured in comoving distance,this will preserve shapes only if the universe is flat ( k   ¼ 0) andthe slice is in the equatorial plane (   ¼ 0) (if     6¼ 0, structures[such as voids] will appear lengthened in the direction tangentialto the line of sight by a factor of 1/ cos    ). This correction is im- portantfor study of the Alcock-Paczynski effect, which saysthat structures such as voids will not be shown in proper shape if wetake simply  r  ¼  z   (Alcock & Paczynski 1979). In fact, Ryden(1995) and Ryden & Melott (1996) have emphasized that thisshape distortion in redshift space can be used to test the cosmo-logical model in a large sample such as the Sloan Digital SkySurvey(Yorketal.2000;Gunnetal.1998;Fukugitaetal.1996).If voids run into each other, the walls will on average not havesystematic peculiar velocities; therefore, voids should have ap- proximately round shapes (a proposition that can be checked indetail with  N  -body simulations). Therefore, it is important toinvestigate map projections that will preserve shapes locally. If one has the correct cosmological model and uses such a con-formalmapprojection,isotropicfeatures inthelarge-scalestruc-ture will appear isotropic on the map.Astronomers mapping the universe are confronted with thechallenge of showing a wide variety of scales. What should amapoftheuniverseshow?Itshouldshowlocationsofallthefa-mous things in space: the  Hubble Space Telescope , the  Inter-national Space Station , other satellites orbiting the Earth, thevan Allen radiation belts, the Moon, the Sun, planets, asteroids,Kuiper Belt objects, nearby stars such as    Centauri and Sirius,stars with planets such as 51 Peg, stars in our Galaxy, famous black holes and pulsars, the Galactic center, Large and SmallMagellanic Clouds, M31, famous galaxies like M87, the Great Wall, famous quasars like 3C 273 (Schmidt 1963) and the grav-itationallylensedquasar0957(Kundicetal.1997),distantSloanDigitalSkySurveygalaxiesandquasars,themostdistantknownquasar and galaxy, and finally the CMB radiation. This is quite achallenge. Perhaps the first book to address this challenge was Cosmic View: The Uni v  erse in 40 Jumps  (Boeke 1957). This brilliant book started with a picture of a little girl shown at 1/10scale. The next picture showed the same little girl at 1/100 scale,whonowcouldbeseensittinginherschoolcourtyard.Eachsuc-cessive picture was plotted at 10 times smaller scale. The eighth picture, at a scale of 1/10 8 , shows the entire Earth. The 14th picture, at a scale of 1/10 14 , shows the entire solar system. The18th picture, at a scale of 1/10 18 , includes    Centauri. The 22nd picture,atascaleof1/10 22 ,showsalloftheMilkyWay.The26thand finalpictureinthesequenceshows galaxies outtoadistanceof750millionlight-years.Afurthersequenceofpictureslabeled0,  1,  2,  : : :  ,  13,startingwithalife-sizepictureofthegirl’shand, shows a sequence of microscopic views, each 10 timeslargerinsize,endingwithaviewofthenucleusofasodiumatomat a scale of 10 13 /1. A modern version of this book,  Powers of  Ten  (Morrison & Morrison 1982), is probably familiar to most astronomers. This successfully addresses the scale problem but is an atlas of maps, not a single map. How does one show theentire observable universe in a single map?The modern  Powers of Ten  book described above is based ona movie,  Powers of Ten , by Charles and Ray Eames (Eames &Eames 1977), which in turn was inspired by Kees Boeke’s book.Themovieisarguablyanevenmorebrilliantpresentationthan Kees Boeke’s srcinal book. The camera starts with a pic-ture of a couple sitting on a picnic blanket in Chicago, and thenthe camera moves outward, increasing its distance from themexponentially as a function of time. Thus, approximately every10 s, the view is from 10 times farther away and corresponds tothe next picture in the book. The movie gives one long con-tinuous shot, which is breathtaking as it moves out. The movieis called  Powers of Ten  (and recently an IMAX version of thisidea has been made, called ‘‘Cosmic Voyage’’), but it couldequally well be titled Powers of Two, or Powers of   e , becauseits exponential change of scale with time produces a reduction by a factor of 2 in constant time intervals and also a factor of   e in constant time intervals. The time intervals between factors of 10, factors of 2, and factors of   e  in the movie are related by theratios ln10:ln2:1. Still, this is not a single map that can bestudied all at once or that can be hung on a wall.Wewanttoseethelarge-scalestructureofgalaxyclusteringbut are also interested in stars in our own Galaxy and the Moon and planets.Objectsclosetousmaybeinconsequentialintermsofthewholeuniverse,but theyare importanttous.Itreminds one ofthefamous cartoon New Yorker cover ‘‘View of the World from 9thAvenue’’ of 1976 May 29, by Saul Steinberg (Steinberg 1976). It humorouslyshows a New Yorker’s viewofthe world.The traffic,sidewalks, and buildings along 9th Avenue are visible in the fore-ground. Behind is the Hudson River, with New Jersey as a thinstrip on the far bank. Then at even smaller scale is the rest of theUnited States with the Rocky Mountains sticking up like smallhills. In the background, but not much wider than the HudsonRiver, is the entire Pacific Ocean with China and Japan in the dis-tance.Thisis,ofcourse,a parochialview,butitisjustthatkindof viewthatwewantoftheuniverse.Wewouldlikeasinglemapthat would equally well show both interesting objects in the solar sys-tem, nearby stars, galaxies in the Local Group, and large-scalestructure out to the CMB.2. COMOVING COORDINATESOur objective here is to produce a conformal map of theuniverse that will show the wide range of scales encounteredwhile still showing shapes that are locally correct.Consider the general Friedmann metrics ds 2 ¼ dt  2 þ a 2 t  ðÞ  d   2 þ sin 2   d   2 þ sin 2  d   2    ;  k  ¼þ 1 ;  dt  2 þ a 2 t  ðÞ  d   2 þ  2 d   2 þ sin 2  d   2    ;  k  ¼ 0 ;  dt  2 þ a 2 t  ðÞ  d   2 þ sinh 2   d   2 þ sin 2  d   2    ;  k  ¼ 1 ; 8><>:  ð 1 Þ where  t   is the cosmic time since the big bang,  a ( t  ) is the ex- pansion parameter, and individual galaxies participating in thecosmic expansion follow geodesics with constant values of    ,  ,and  .These three are calledcomovingcoordinates.Neglect-ing peculiar velocities, galaxies remain at constant positions incomoving coordinates as the universe expands. Now  a ( t  ) obeysFriedmann’s equations, a ; t  a   2 ¼  k a 2  þ  3  þ 8  m 3  þ 8  r  3  ;  ð 2 Þ 2  a ; tt  a   ¼  2  3   8  m 3   16  r  3  ;  ð 3 Þ where   ¼ const is the cosmological constant,   m  / a  3 is theaverage matter density in the universe, including cold dark mat-ter,and  r   / a  4 istheaverage radiation densityintheuniverse, primarily the CMB radiation. The second equation shows that the cosmological constant produces an acceleration in the ex- pansion while the matter and radiation produce a deceleration.Per unit mass density, radiation produces twice the decelerationof normal matter because positive pressure is gravitationallyattractiveinEinstein’stheoryandradiationhasapressureineachof the three directions (  x ,  y ,  z  ) that is  13  the energy density.GOTT ET AL.464 Vol. 624  We can define a conformal time     by the relation  d    ¼ dt  /  a ,so that    ( t  ) ¼ Z   t  0 dt a  :  ð 4 Þ Light travels on radial geodesics with  d    ¼  d   , so a galaxyat a comoving distance   from us emitted the light we see todayat a conformal time    ( t  ) ¼   ( t  0 )   . Thus, we can calculatethe time  t   and redshift   z  ¼ a ( t  0 )/  a ( t  )  1 at which that light wasemitted. Conversely, if we know the redshift, given a cosmo-logical model (i.e., values of   H  0 ,  ,   m ,   r  , and  k   today), we cancalculate the comoving radial distance of the galaxy from usfrom its redshift, again ignoring peculiar velocities. For a moredetailed discussion of distance measures in cosmology, seeHogg (2001).The  WMAP   satellite has measured the CMB in exquisitedetail (Bennett et al. 2003) and combined these data with other data (Percival et al. 2001; Verde et al. 2002; Croft et al. 2002;Gnedin & Hamilton 2002; Garnavich et al. 1998; Riess et al.2001; Freedman et al. 2001; Perlmutter et al. 1999) to produceaccurate data on the cosmological model (Spergel et al. 2003).We adopt best-fit values at the present epoch,  t   ¼ t  0 , based onthe  WMAP   data of   H  0   a ; t  a  ¼ 71 km s  1 Mpc  1 ;      3  H  20 ¼ 0 : 73 ;  r  ¼ 8 : 35  ;  10  5 ;  m   8  m 3  H  20 ¼ 0 : 27   r  ; k  ¼ 0 : The WMAP  dataimplythat  w  1fordarkenergy(i.e.,  p vac  ¼ w  vac    vac ), suggesting that a cosmological constant is anexcellent model for the dark energy, so we simply adopt that.The current Hubble radius  R  H  0  ¼ cH   10  ¼ 4220 Mpc. The CMBis at a redshift   z  ¼ 1089. Substituting, using geometrized units inwhich c ¼ 1,andintegratingthefirstFriedmannequation,wefindthat the conformal time can be calculated as    t  ðÞ¼ Z   t  0 dt a ¼ Z   a t  ðÞ 0  ka 2 þ 8  3  a 4  m  a ð Þþ  r   a ð Þ½ þ  3  a 4    1 = 2 da ; ð 5 Þ where   m  / a  3 and   r   / a  4 . This formula will accuratelytrack the value of     ( t  ), provided that this is interpreted as thevalue of the conformal time since the end of the inflationary pe-riod at the beginning of the universe. (During the inflationary period at the beginning of the universe, the cosmological con-stant assumed a large value, different from that observed today,and the formula would have to be changed accordingly. So wesimply start the clock at the endof the inflationary periodwherethe energydensityinthe false vacuum [large cosmological con-stant]isdumpedintheformofmatterandradiation.Thus,whenwe trace back to the big bang, we are really tracing back to theendoftheinflationaryperiod.Afterthat,themodeldoesbehave just like a standard hot Friedmann big bang model. This stan-dard model might be properly referred to as an inflationary big bang model, with the inflationary epoch producing the big bangexplosion at the start.) Now,  a ( t  ) is the radius of curvature oftheuniverseforthe k   ¼þ 1and k   ¼ 1cases,butforthe k   ¼ 0case,whichwewillbeinvestigatingfirstandprimarily,thereisnoscaleand so we are free to normalize, setting  a ( t  0 ) ¼  R  H  0  ¼ cH   10  ¼ 4220 Mpc. Then,    measures comoving distances at the present epoch inunits ofthe current Hubbleradius  R  H  0 . Thus, for the  k   ¼ 0 case, using geometrized units, we have    a ð Þ¼    a t  ðÞ½ ¼ Z   a 0 aa 0  m þ  r  þ  aa 0   4   #  1 = 2 daa 0 ;  ð 6 Þ where   m ,    , and   r   are the values at the current epoch.Given the values adopted from  WMAP  , we find    a 0 ð Þ¼ 3 : 38 :  ð 7 Þ That means that when we look out now at   t   ¼ t  0  (when a ¼ a 0 ), we can see out to a distance of   ¼ 3 : 38  ð 8 Þ or a comoving distance of    R  H  0  ¼ 3 : 38  R  H  0  ¼ 14 ; 300 Mpc :  ð 9 Þ This is the effective particle horizon, where we are seeing particles at the moment of the big bang. This is a larger radiusthan 13.7 billion light-years, i.e., the age of the universe (thelook-back time) times the speed of light, because it shows thecomoving distance that the most distant particles we can ob-servenowwillhavefromuswhentheyareasoldaswearenow,i.e.,measuredatthecurrentcosmologicalepoch.Wecancalculatethe value of      as a function of   a , or equivalently as a function of observedredshift   z  ¼ ( a 0 /  a )  1. Recombinationoccursat   z  rec  ¼ 1089, which is the redshift of the CMB seen by  WMAP  :    z  rec ð Þ¼ 0 : 0671 :  ð 10 Þ Thus, the comoving radius of the CMB is   R  H  0  ¼    0    rec ð Þ  R  H  0  ¼ 14 ; 000 Mpc :  ð 11 Þ That is the radius at the current epoch, so at recombination the WMAP   sphere has a physical radius that is 1090 times smaller or about 13 Mpc.According to Sloan Digital Sky Survey (SDSS) luminosityfunction data (M. Blanton 2004, private communication),  L   inthe Press-Schechter luminosity function in the  B  band is 7 : 1  ; 10 9  L   for   H  0  ¼ 71 km s  1 Mpc  1 , and the mean separation between galaxies brighter than  L   is 4.1 Mpc. The Milky Wayhas 9 : 4  ; 10 9  L   in  B . Since the radius of the observable uni-verse (outtothe CMB) is 14Gpc,that means thatthe number of  brightgalaxies(moreluminousthan  L  )formingwithinthecur-rently observable universe is of the order of 170 billion. If our galaxy has of the order of 200 billion stars, the mean blue stel-lar luminosity is of the order of 0.05  L   and the mean number density of stars is at least of the order of 2 : 6  ; 10 9 Mpc  3 . El-liptical and S0 galaxies have a higher number of stars per solar luminosity than the Milky Way, so a conservative estimate for the mean number density of stars might be 5  ; 10 9 stars Mpc  3 .Thus,thecurrentlyobservableuniverseishometooftheorderof 6  ;  10 22 stars.We can compute comoving radii  r  ¼   R  H  0  for different red-shifts, as shown in Table 1. We can also calculate the value of MAP OF THE UNIVERSE  465No. 2, 2005    ( t   ¼1 ) ¼ 4 : 50, which shows how far a photon can travel incomoving coordinates from the inflationary big bang to the infi-nitefuture. Thus, if wewait until the infinitefuture, we will even-tually be able to see out to a comoving distance of  r  t  ¼1  ¼ 4 : 50  R  H  0  ¼ 19 ; 000 Mpc :  ð 12 Þ This is the comoving future visibility limit. No matter how longwe wait, we will not be able to see farther than this. This is sur- prisinglyclose.Thenumberofgalaxieswewilleventuallyeverbeable to see is only larger than the number observable today by afactor of ( r  t  ¼1 /  r  t  0 ) 3 ¼ 2 : 36.This calculation assumes that the false vacuum state (cosmo-logical constant) visible today remains unchanged. (The  WMAP  data are consistent with a value of   w ¼ 1, indicating that thevacuum state[dark energy]today iswell approximated by a pos-itivecosmologicalconstant.Thisfalsevacuumstate[with  p vac  ¼ w  vac  ¼ 1  vac ] may decay by forming bubbles of normal zerodensity vacuum [  ¼ 0] or even decay by forming bubbles of negative energy density vacuum [   <  0]. If the present falsevacuum is only metastable, it will eventually decay by the for-mation of bubbles of normal or negative energy density vacuumand eventually one of these bubbles will engulf the comovinglocation of our galaxy. However, if these bubbles occur late[>10 100 yr],asexpected,theywillmakeanegligiblecorrectiontohow far away in comoving coordinates we will eventually beabletosee.ForafullerdiscussionseeGottetal.[2003]andrefer-ences therein.)Linde (1990) and Garriga & Vilenkin (1998) have pointedout that if the current vacuum state is the lowest stable equi-librium, then quantum fluctuations can form bubbles of high-density vacuum that will start a new inflationary epoch, new baby universes growing like branches off a tree. Still, as in theabove case,weexpecttobe engulfedbysucha newinflatingre-gion only at late times (say, at least 10 100 yr from now), and theobserver will still be surrounded by an event horizon with alimit of future visibility in comoving coordinates in our uni-versethatisvirtuallyidenticalwithwhatwehaveplotted.Thus,although the future history of the universe will be determined by the subsequent evolution of the quantum vacuum state (asalso noted by Krauss & Starkman 2000), in practice we expect the current vacuum tostay as is for considerablylonger than theHubbletime,andinmanyscenariosthisleavesuswithalimitof future visibility that is for all practical purposes just what wehave plotted.If we send out a light signal now, by  t   ¼1  it will reach aradius   ¼   ( t   ¼1 )    ( t  0 ) ¼ 4 : 50  3 : 38 ¼ 1 : 12, or  r  ¼ 4740 Mpc ;  ð 13 Þ which we refer to as the ‘‘outward limit of reachability.’’ Wecannot reach (with light signals or rockets) any galaxies that arefarther away than this (Busha et al. 2003). What redshift doesthiscorrespondto?Galaxiesweobservetodaywitharedshiftof   z  ¼ 1 : 69 are at this comoving distance. Galaxies with redshiftslarger than 1.69 today are unreachable. This is a surprisinglysmall redshift.Wecanseemanygalaxiesatredshiftslargerthan1.69thatwewill never be able tovisit or signal. Inthe acceleratinguniverse,these galaxies are accelerating away from us so fast that we cannever catch them. The total number of stars that our radio sig-nals will ever pass is of the order of 2  ; 10 21 .3. A MAP PROJECTION FOR THE UNIVERSEWe choose a conformal map that will cover the wide rangeof scales from the Earth’s neighborhood to the CMB. First, weconsider the flat case ( k   ¼ 0), which the  WMAP   data tell us isthe appropriate cosmological model. Our map will be two-dimensionalsothatitcanbeshownonawall chart.DeLapparent et al. (1986) showed with their slice of the universe just howsuccessful a slice of the universe can be in illustrating large-scale structure. The SDSS should eventually include spectraand accurate positions for about 1 million galaxies and quasarsin a three-dimensional sample (for SDSS scientific results seeStoughton et al. 2002; Abazajian et al. 2003; Strauss et al. 2002;Richards et al. 2002; Eisenstein et al. 2001; for further technicalreference see Blanton et al. 2003a; Hogg et al. 2001; Smith et al.2002; Pier et al. 2003). However, virtually complete already isan equatorial slice 4   wide (  2   <  <  2  ) centered on the ce-lestial equator covering both the northern and southern Galactichemispheres. This shows many interesting features, includingmany prominent voids and a Great Wall longer than the Great Wall found by Geller & Huchra (1989).Since the observed slice is already in a flat plane ( k   ¼ 0model, along the celestial equator), we can project this slice di-rectly onto a flat sheet of paper using polar coordinates with r  ¼   R  H  0  being the comoving distance and    being the right ascension. (CMB observations from BOOMERANG, DASI,MAXIMA, and  WMAP   indicate that the case  k   ¼ 0 is the ap- propriate one for the universe. For mathematical completenesswealsoconsiderthe k   ¼þ 1and k   ¼ 1casesinAppendixA.)We wish to show large-scale structure and the extent of the ob-servable universe out to the CMB radiation including all theSDSS galaxies and quasars in the equatorial slice. In Figure 1one can see the CMB at the surface of last scattering as a circle.Its comoving radius is 14.0 Gpc. (Since the size of the universeattheepochofrecombinationissmallerthanthatatpresentbyafactor of 1 þ  z  ¼ 1090, the true radius of this circle is about 12.84Mpc.)SlightlybeyondtheCMBincomovingcoordinatesis the big bang at a comoving distance of 14.3 Gpc.(Imagine a point on the CMB circle. Draw a radius aroundthat point that is tangent to the outer circle labeled big bang, asshown in the figure; in other words, a circle that has a radiusequaltothedifferenceinradiusbetweentheCMBcircleandthe big bang circle. That circle has a comoving radius of 283 Mpc.Thatis thecomovinghorizonradiusatrecombination.Ifthe big bang model [without inflation] were correct, we would expect a point on the CMB circle to be causally influenced only bythings inside that horizon radius at recombination. The angular  TABLE  1 Comoving Radii for Different Redshifts  z r  (  z  )(Mpc) Notes 1 .............. 14283 Big bang (end of inflationary period)3233.......... 14165 Equal matter and radiation density epoch1089.......... 14000 Recombination6................ 84225................ 79334................ 73053................ 64612................ 52451................ 33170.5............. 18820.2............. 8090.1............. 413 GOTT ET AL.466 Vol. 624
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