a map of the universe - institute of physics

A MAP OF THE UNIVERSE

J. Richard Gott III,1Mario Juric,

1David Schlegel,

1Fiona Hoyle,

2Michael Vogeley,

2

Max Tegmark,3Neta Bahcall,

1and Jon Brinkmann

4

Receivved 2003 November 18; accepted 2005 January 3

ABSTRACT

We have produced a new conformal map of the universe illustrating recent discoveries, ranging fromKuiper Beltobjects in the solar system to the galaxies and quasars from the Sloan Digital Sky Survey. This map projection,based on the logarithmmap of the complex plane, preserves shapes locally and yet is able to display the entire rangeof 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. Prominentin the map is a Sloan GreatWall of galaxies 1.37 billion light-years long, 80% longer than the GreatWall discoveredby Geller and Huchra and therefore the largest observed structure in the universe.

Subject headinggs: large-scale structure of universe — methods: data analysis

1. INTRODUCTION

Cartographers 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 Mercatorprojection ( presented by GerhardusMercator in 1569). This is aconformal projection which preserves shapes locally. Lines oflatitude are shown as straight horizontal lines, while meridiansof longitude are shown as straight vertical lines. If the Mercatorprojection is plotted on an (x, y)-plane, the coordinates are plot-ted as x ¼ k and y ¼ ln ½tan (�/4þ �/2)�, where � (positive ifnorth, 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 othermap projections of the Earth). This conformal map projectionpreserves 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 Mercatorprojection, 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 orareas. Equal-area map projections like the Lambert, 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 ¼2r0 sin ½(�/2� �)/2�, where r0 is the radius of the sphere. Thisprojection preserves areas. The Northern Hemisphere is thusmapped onto a circular disk of radius

ffiffiffi2

pr0 and area 2�r0

2. Anoblique version of this, centered at a point on the equator, is alsopossible.

The Hammer projection shows the Earth as a horizontal el-lipse with 2:1 axis ratio. The equator is shown as a straight hori-zontal line marking the long axis of the ellipse. It is produced inthe following way. Map the 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 Lambertequal-area azimuthal projection. This map is a circular disk.This is then stretched by a factor of 2 (undoing the previous com-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 2:1 axis ratio ellipse. The (x, y) coordinates on the map arex ¼ (2

ffiffiffi2

p/�)r0k cos � and y ¼

ffiffiffi2

pr0 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 ofgalaxies (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 celestialsphere onto a rhombic dodecahedron using theHealpix equal-areamap projection (Gorski et al. 2002). Its results were displayed alsoon 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 straightlines.

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 dimensions

this slice had the geometry of a cone, and they flattened this ontoa plane. (A cone has zero Gaussian curvature and can thereforebe constructed from a piece of paper. A cone cut along a line andflattened onto a plane looks like a pizza with a slice missing.) Ifthe cone is at declination �, the map in the plane will be x ¼r 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 Department of Physics,DrexelUniversity, 3141Chestnut Street, 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-portant for study of the Alcock-Paczynski effect, which says thatstructures 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 (York et al. 2000; Gunn et al. 1998; Fukugita et al. 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. Ifone has the correct cosmological model and uses such a con-formal map projection, isotropic features in the large-scale struc-ture will appear isotropic on the map.

Astronomers mapping the universe are confronted with thechallenge of showing a wide variety of scales. What should amap of the universe show? It should show locations of all the fa-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, famousblack holes and pulsars, the Galactic center, Large and SmallMagellanic Clouds, M31, famous galaxies like M87, the GreatWall, famous quasars like 3C 273 (Schmidt 1963) and the grav-itationally lensed quasar 0957 (Kundic et al. 1997), distant SloanDigital Sky Survey galaxies and quasars, the most distant knownquasar and galaxy, and finally the CMB radiation. This is quite achallenge. Perhaps the first book to address this challenge wasCosmic View: The Universe in 40 Jumps (Boeke 1957). Thisbrilliant 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,who now could be seen sitting in her school courtyard. Each suc-cessive picture was plotted at 10 times smaller scale. The eighthpicture, at a scale of 1/108, shows the entire Earth. The 14thpicture, at a scale of 1/1014, shows the entire solar system. The18th picture, at a scale of 1/1018, includes � Centauri. The 22ndpicture, at a scale of 1/1022, shows all of theMilkyWay. The 26thand final picture in the sequence shows galaxies out to a distanceof 750 million light-years. A further sequence of pictures labeled0,�1,�2, : : : ,�13, starting with a life-size picture of the girl’shand, shows a sequence of microscopic views, each 10 timeslarger in size, ending with a view of the nucleus of a sodium atomat a scale of 1013/1. A modern version of this book, Powers ofTen (Morrison & Morrison 1982), is probably familiar to mostastronomers. This successfully addresses the scale problem butis 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’sbook. The movie is arguably an evenmore brilliant presentationthan Kees Boeke’s original 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 reductionby a factor of 2 in constant time intervals and also a factor of ein constant time intervals. The time intervals between factors of10, factors of 2, and factors of e in the movie are related by theratios ln 10:ln 2:1. Still, this is not a single map that can bestudied all at once or that can be hung on a wall.Wewant to see the large-scale structure of galaxy clustering but

are also interested in stars in our own Galaxy and the Moon andplanets. Objects close to usmay be inconsequential in terms of thewhole universe, but they are important to us. It reminds one of thefamous cartoon New Yorker cover ‘‘View of the World from 9thAvenue’’ of 1976 May 29, by Saul Steinberg (Steinberg 1976). Ithumorously shows a New Yorker’s view of the 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. This is, of course, a parochial view, but it is just that kind ofview thatwewant of the universe.Wewould like a singlemap thatwould 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 COORDINATES

Our 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

ds2¼�dt2þa2 tð Þ d�2þ sin2� d�2þ sin2� d�2

� �� ; k ¼þ1;

�dt2þa2 tð Þ d�2þ�2 d�2þ sin2� d�2� ��

; k ¼ 0;

�dt2þa2 tð Þ d�2þ sinh2� d�2þ sin2� d�2� ��

; k ¼�1;

8><>:

ð1Þ

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 called comoving coordinates. 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

a2þ �

3þ 8��m

3þ 8��r

3; ð2Þ

2a; tt

a

� �¼ 2�

3� 8��m

3� 16��r

3; ð3Þ

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 is the average radiation density in the universe,primarily the CMB radiation. The second equation shows thatthe 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 gravitationallyattractive in Einstein’s theory and radiation has a pressure in eachof the three directions (x, y, z) that is 1

3the energy density.

GOTT ET AL.464 Vol. 624

We can define a conformal time � by the relation d� ¼ dt/a,so that

�(t) ¼Z t

0

dt

a: ð4Þ

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 �(t0)� �. Thus, we can calculatethe time t and redshift z ¼ a(t0)/a(t)� 1 at which that light wasemitted. Conversely, if we know the redshift, given a cosmo-logical model (i.e., values ofH0, �, �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 otherdata (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 t0, based onthe WMAP data of

H0 �a; t

a¼ 71 km s�1 Mpc�1;

��

3H 20

¼ 0:73;

�r ¼ 8:35 ; 10�5;

�m � 8��m3H 2

0

¼ 0:27� �r;

k ¼ 0:

TheWMAP data imply thatw � �1 for dark energy (i.e., pvac ¼w�vac � ��vac), suggesting that a cosmological constant is anexcellent model for the dark energy, so we simply adopt that.The current Hubble radius RH0

¼ cH�10 ¼ 4220 Mpc. The CMB

is at a redshift z ¼ 1089. Substituting, using geometrized units inwhich c ¼ 1, and integrating the first Friedmann equation, we findthat the conformal time can be calculated as

� tð Þ ¼Z t

0

dt

a

¼Z a tð Þ

0

�ka2 þ 8�

3a4 �m að Þ þ �r að Þ½ � þ �

3a4

� �1=2

da;

ð5Þ

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 inflationaryperiod 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 end of the inflationary period wherethe energy density in the false vacuum [large cosmological con-stant] is dumped in the form of matter and radiation. Thus, whenwe trace back to the big bang, we are really tracing back to theend of the inflationary period. After that, the model does behavejust 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 of theuniverse for the k ¼ þ1 and k ¼ �1 cases, but for the k ¼ 0 case,which wewill be investigating first and primarily, there is no scaleand so we are free to normalize, setting a(t0) ¼ RH0

¼ cH�10 ¼

4220 Mpc. Then, � measures comoving distances at the presentepoch in units of the current Hubble radius RH0

. Thus, for the k ¼0 case, using geometrized units, we have

� að Þ ¼ � a tð Þ½ � ¼Z a

0

a

a0�m þ �r þ

a

a0

�4

��

" #�1=2da

a0; ð6Þ

where �m, ��, and �r are the values at the current epoch.Given the values adopted from WMAP, we find

� a0ð Þ ¼ 3:38: ð7Þ

That means that when we look out now at t ¼ t0 (whena ¼ a0), we can see out to a distance of

� ¼ 3:38 ð8Þ

or a comoving distance of

�RH0¼ 3:38RH0

¼ 14;300 Mpc: ð9Þ

This is the effective particle horizon, where we are seeingparticles 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-serve nowwill have from us when they are as old as we are now,i.e., measured at the current cosmological epoch.We can calculatethe value of � as a function of a, or equivalently as a function ofobserved redshift z ¼ (a0/a)� 1. Recombination occurs at zrec ¼1089, which is the redshift of the CMB seen by WMAP:

� zrecð Þ ¼ 0:0671: ð10Þ

Thus, the comoving radius of the CMB is

�RH0¼ �0 � �recð ÞRH0

¼ 14;000 Mpc: ð11Þ

That is the radius at the current epoch, so at recombination theWMAP sphere has a physical radius that is 1090 times smalleror 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 ;109 L� for H0 ¼ 71 km s�1 Mpc�1, and the mean separationbetween galaxies brighter than L� is 4.1 Mpc. The Milky Wayhas 9:4 ;109 L� in B. Since the radius of the observable uni-verse (out to the CMB) is 14 Gpc, that means that the number ofbright galaxies (more luminous than L�) forming within the cur-rently observable universe is of the order of 170 billion. If ourgalaxy 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 numberdensity of stars is at least of the order of 2:6 ; 109 Mpc�3. El-liptical and S0 galaxies have a higher number of stars per solarluminosity than the Milky Way, so a conservative estimate forthe mean number density of stars might be 5 ; 109 stars Mpc�3.Thus, the currently observable universe is home to of the order of6 ; 1022 stars.

We can compute comoving radii r ¼ �RH0for different red-

shifts, as shown in Table 1. We can also calculate the value of

MAP OF THE UNIVERSE 465No. 2, 2005

�(t ¼ 1) ¼ 4:50, which shows how far a photon can travel incomoving coordinates from the inflationary big bang to the infi-nite future. Thus, if we wait until the infinite future, we will even-tually be able to see out to a comoving distance of

rt¼1 ¼ 4:50RH0¼ 19;000 Mpc: ð12Þ

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-prisingly close. The number of galaxieswewill eventually ever beable to see is only larger than the number observable today by afactor of (rt¼1/rt0 )

3 ¼ 2:36.This calculation assumes that the false vacuum state (cosmo-

logical constant) visible today remains unchanged. (TheWMAPdata are consistent with a value of w ¼ �1, indicating that thevacuum state [dark energy] today is well approximated by a pos-itive cosmological constant. This false vacuum state [with pvac ¼w�vac ¼ �1�vac] may decay by forming bubbles of normal zerodensity vacuum [� ¼ 0] or even decay by forming bubbles ofnegative 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[>10100 yr], as expected, theywill make a negligible correction tohow far away in comoving coordinates we will eventually beable to see. For a fuller discussion see Gott et al. [2003] and refer-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, newbaby universes growing like branches off a tree. Still, as in theabove case, we expect to be engulfed by such a new inflating re-gion only at late times (say, at least 10100 yr from now), and theobserver will still be surrounded by an event horizon with alimit of future visibility in comoving coordinates in our uni-verse that is virtually identical with what we have plotted. Thus,although the future history of the universe will be determinedby the subsequent evolution of the quantum vacuum state (asalso noted by Krauss & Starkman 2000), in practice we expectthe current vacuum to stay as is for considerably longer than theHubble time, and in many scenarios this leaves us with a limit offuture visibility that is for all practical purposes just what wehave plotted.

If we send out a light signal now, by t ¼ 1 it will reach aradius � ¼ �(t ¼ 1)� �(t0) ¼ 4:50� 3:38 ¼ 1:12, or

r ¼ 4740 Mpc; ð13Þ

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 doesthis correspond to? Galaxies we observe today with a redshift ofz ¼ 1:69 are at this comoving distance. Galaxies with redshiftslarger than 1.69 today are unreachable. This is a surprisinglysmall redshift.We can see many galaxies at redshifts larger than 1.69 that we

will never be able to visit or signal. In the accelerating universe,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 ; 1021.

3. A MAP PROJECTION FOR THE UNIVERSE

We 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-dimensional so that it can be shown on a wall chart. De Lapparentet 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 GreatWall found by Geller & Huchra (1989).Since the observed slice is already in a flat plane (k ¼ 0

model, along the celestial equator), we can project this slice di-rectly onto a flat sheet of paper using polar coordinates withr ¼ �RH0

being the comoving distance and � being the rightascension. (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 completenesswe also consider the k ¼ þ1 and k ¼ �1 cases in Appendix A.)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 universeat the epoch of recombination is smaller than that at present by afactor of 1þ z ¼ 1090, the true radius of this circle is about12.84Mpc.) Slightly beyond the CMB in comoving coordinatesis the big bang at a comoving distance of 14.3 Gpc.( Imagine a point on the CMB circle. Draw a radius around

that point that is tangent to the outer circle labeled big bang, asshown in the figure; in other words, a circle that has a radiusequal to the difference in radius between the CMB circle and thebig bang circle. That circle has a comoving radius of 283 Mpc.That is the comoving horizon radius at recombination. If the bigbang model [without inflation] were correct, we would expecta 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 epoch

1089.......... 14000 Recombination

6................ 8422

5................ 7933

4................ 7305

3................ 6461

2................ 5245

1................ 3317

0.5............. 1882

0.2............. 809

0.1............. 413


radius of this small circle as seen from the Earth is [283 Mpc/14,000 Mpc] radians or 1N16. If the big bang model without in-flation were correct, we would expect the CMB to be correlatedon scales of at most 1N16. Inflation, by having a short periodof accelerated expansion during the first 10�34 s of the universe,puts distant regions in causal contact because of the slight ad-ditional time allowed when the universe was very small. Hence,with inflation, we can understand why the CMB is uniform toone part in 100,000 all over the sky. Furthermore, random quan-tum fluctuations predicted by inflation add a series of adiabaticfluctuations that are expected to have a peak in the power spec-trum at an angular scale about the size of the horizon radius atrecombination calculated above, 0N86.)

Beyond the big bang circle is the circle showing the futurecomoving visibility limit. If we wait until the infinite future, wewill be able to see out to this circle. ( In other words, in the in-finite future, we will be able to see particles at the future co-moving visibility limit as they appeared at the big bang.)

The SDSS quasars extend out about halfway out to the CMBradiation. The distribution of quasars shows several features.The radial distribution shows several shelves due to selectioneffects as different spectral features used to identify quasarscome into view in the visible. Several radial spokes appear ow-ing to incompleteness in some narrow right ascension intervals.

Two large fan-shaped regions are empty and not surveyed be-cause they cover the zone of avoidance close to theGalactic plane,which is not included in the SDSS. These excluded regionsrun from approximately 3:7 hrP�P 8:7 hr and approximately16:7 hrP�P 20:7 hr. The quasars do not show noticeable clus-tering or large-scale structure. This is because the quasars are sowidely spaced that the mean distance between quasars is largerthan the correlation length at that epoch.

The circle of reachability is also shown. Quasars beyond thiscircle are unreachable. Radio signals emitted by us now willonly reach out as far as this circle, even in the infinite future.

The SDSS galaxies appear as a black blob in the center. Thereis much interesting large-scale structure here, but the field is toocrowded and small to show it. This illustrates the problem ofscale in depicting the universe. If we want a map of the entireobservable universe on one page, at a nice scale, the galaxies arecrammed into a blob in the center. Let us enlarge the centralcircle of radius 0.06 times the distance to the big bang circle by afactor of 16.6 and plot it again in Figure 2. This now shows acircle of comoving radius 858 Mpc. Almost all of these pointsare galaxies from the galaxy and bright red galaxy samples ofthe SDSS. Now we can see a lot of interesting structure. Themost prominent feature is a Sloan Great Wall at a median dis-tance of about 310Mpc stretching from 8C7 to 14h in R.A. There

Fig. 1.—Galaxies and quasars in the equatorial slice (�2� < � < 2

�) of the SDSS displayed in comoving coordinates out to the horizon. The comoving distances to

galaxies are calculated from measured redshift, assuming Hubble flow and WMAP cosmological parameters. This is a conformal map: it preserves shapes. While thismap can conformally show the complete SDSS, the majority of interesting large-scale structure is crammed into a blob in the center. The dashed circle marks the outerlimit of Fig. 2. The circle labeled ‘‘Unreachable’’ marks the distance beyond which we cannot reach (i.e., we cannot reach with light signals any object that is fartheraway). This radius corresponds to a redshift of z ¼ 1:69. As ‘‘Future comoving visibility limit’’ we label the comoving distance to which a photon would travel from theinflationary big bang to the infinite future. This is the maximum radius out to which observations will ever be possible. At 4:50RH0

, it is surprisingly close.


are numerous voids. A particularly interesting one is close in ata comoving distance of 125 Mpc at R:A: ¼ 1C5. At the far endof this void are a couple of prominent clusters of galaxies, whichare recognizable as ‘‘fingers of God’’ pointing at the Earth. Red-shift in this map is taken as the comoving distance indicator as-suming participation in the Hubble flow, but galaxies also havepeculiar velocities, and in a dense cluster with a high velocitydispersion this causes the distance errors due to these peculiarvelocities to spread the galaxy positions out in the radial direc-tion, producing the ‘‘finger of God’’ pointing at the Earth. Numer-ous other clusters can be similarly identified. This is a conformalmap that preserves shapes, excluding the small effects of pecu-liar velocities. The original Harvard-Smithsonian Center for As-trophysics (CfA) survey in which Geller and Huchra discoveredthe Great Wall had a comoving radius of only 211Mpc, which isless than one-quarter of the radius shown in Figure 2. Figure 2 isa quite impressive picture, but it does not capture all of theSDSS. If we displayed Figure 1 at a scale enlarged by a factor of16.6, the central portion of the map would be as you see dis-played at the scale shown in Figure 2, which is adequate, but thebig bang circle would have a diameter of 6.75 feet (2.06 m). Youcould put this on your wall, but if wewere to print it in the journalfor you to assemble, it would require the next 256 pages. Thispoints out the problem of scale for even showing the SDSS all onone page. Small scales are also not represented well. The dis-tance to the Virgo Cluster in Figure 2 is only about 2 mm, and thedistance from theMilkyWay toM31 is only 1/13 of a millimeter

and therefore invisible on this map. Figure 2, dramatic as it is,fails to capture a picture of all the external galaxies and quasars.The nearby galaxies are too close to see, and the quasars are be-yond the limits of the page.We might try plotting the universe in look-back time rather

than comoving coordinates. The result is in Figure 3. The outercircle is the CMB. It is indistinguishable from the big bangas the two are separated by only 380,000 yr out of 13.7 billionyears. The SDSS quasars now extend back nearly to the CMBradiation (since it is true that we are seeing back to within abillion years of the big bang). Look-back time is easier to ex-plain to a lay audience than comoving coordinates and it makesthe SDSS data look more impressive, but it is a misleading por-trayal as far as shapes and the geometry of space are con-cerned. It misleads us as to how far out we are seeing in space.For that, comoving coordinates are appropriate. Figure 3 doesnot preserve shapes: it compresses the large area between theSDSS quasars and the CMB into a thin rim. This is not a con-formal map. The SDSS galaxies now occupy a larger space inthe center, but they are still so crowded together that one cannotsee the large-scale structure clearly. Figure 4 shows the central0.2 radius circle (shown by a dotted circle in Fig. 3) enlarged bya factor of 5. Thus, if we were to make a wall map of the ob-servable universe using look-back time at the scale of Figure 4,it would only need to be 2 feet (0.61 m) across and would onlyrequire the next 25 pages in the journal to plot. This is an ad-vantage of the look-back time map. It makes the interesting

Fig. 2.—Zoom-in of the region marked by the dashed circle in Fig. 1, out to 0.06rhorizon (=858 Mpc). The points shown are galaxies from the main and bright redgalaxy samples of the SDSS. Compared to Fig. 1, we can now see a lot of interesting structure. The Sloan Great Wall can be seen stretching from 8C7 to 14h in R.A.at a median distance of about 310 Mpc. Although the large-scale structure is easier to see, a ‘‘zoom-in’’ like this fails to capture and display, in one map, the sizes ofmodern redshift surveys.


large-scale structure that we see locally (Fig. 4) a factor ofslightly over 3 larger in size relative to the CMB circle than ifwe had used comoving coordinates. Figure 4 looks quite similarto Figure 2. At comoving radii less than 858Mpc, the look-backtime and comoving radius are rather similar. Still, Figure 4 isnot perfectly conformal. Near the outer edges there is a slightradial compression that is beginning to occur in the look-backtime map as one goes toward the big bang. The effects of radialcompression are illustrated in Figure 5, where we have plotted asquare grid in comoving coordinates in terms of look-back timeas would be depicted in Figure 3. Each grid square would containan equal number of galaxies in a flat slice of constant verticalthickness. This shows the distortion of space that is produced byusing the look-back time. The squares become more and moredistorted in shape as one approaches the edge.

Thus, it would be useful to have a conformal map projectionthat would show the whole SDSS, including galaxies, quasars,and the CMB, as well as smaller scales, covering the local super-cluster, the Local Group, the Milky Way, nearby stars, the Sunand planets, the Moon, and artificial Earth satellites. Such a mapis possible.

Consider the complex plane (u; v) ¼ uþ iv, where i ¼ffiffiffiffiffiffiffi�1

p

and u and v are real numbers. Every complex number W ¼uþ iv will be represented as a point in the (u, v)-plane, where uand v are the usual Cartesian coordinates. The function Z ¼i lnW maps the plane (u, v) onto the plane (x, y), where Z ¼xþ iy. The (u, v)-plane represents a slice of the universe in

isotropic coordinates (in this case comoving coordinates sincek ¼ 0), and the (x, y)-plane represents our map of the universe.The inverse function W ¼ exp (Z/i ) is the inverse map thattakes a point in our map plane (x, y) back to the point it repre-sents in the universe (u, v). In the universe it is useful to es-tablish polar coordinates (r, �) where

u ¼ r cos �; ð14Þv ¼ r sin �; ð15Þ

r ¼ (u2 þ v2)1=2 is the (comoving) distance from the center ofthe Earth, and � ¼ arctan (v/u) is the right ascension measuredin radians. Since

cos � ¼ ei� þ e� i�

2; sin � ¼ e i� � e�i�

2i; ð16Þ

it is clear that

W ¼ uþ iv ¼ r cos �þ i sin �ð Þ ¼ rei�; ð17ÞZ ¼ i ln Wð Þ ¼ i ln r þ i�ð Þ ¼ ��þ i ln r ¼ xþ iy; ð18Þ

so

x ¼ ��; ð19Þy ¼ ln r: ð20Þ

Fig. 3.—Galaxies and quasars in the equatorial slice of the SDSS, displayed in look-back time coordinates. The radial distance in the figure corresponds to look-back time. While the galaxies at the center occupy a larger area, this map is a misleading portrayal as far as shapes and the geometry of space are concerned. It is notconformal: it compresses the area close to the horizon (this compression is more explicitly shown in Fig. 5). Also, the galaxies are still too crowded in the center ofthe map to show all of the intricate details of their clustering. Fig. 4 shows a zoom-in of the region inside the dashed circle.


Thus, the entire (u, v)-plane, except the origin (0, 0), is mappedinto an infinite vertical strip of horizontal width 2�, i.e.,

�2� < x 0; �1 < y < 1: ð21Þ

(Fig. 6 shows the complex plane uþ iv mapped onto thexþ iy plane by this map. One can take this map and make it intoa slide rule for multiplying complex numbers. Photocopy themap on this page and cut it out. Tape the left-hand edge to theright-hand edge to make a paper cylinder. The � coordinate nowmeasures longitude angle on that cylinder. Now photocopy themap onto a transparency, cut it out, and again tape the left-handedge to the right-hand edge to make a transparent cylinder. Incutting out the left- and right-hand sides of the map, cheat alittle: cut along the outside edges of the map borderlines so thatthe circumference of the transparency cylinder is just a tiny bitlarger than the paper cylinder and so that it will fit snugly over it.With the paper cylinder snugly inside the transparent cylinderyou are ready to multiply. If you want to multiply two numbersA ¼ aþ bi and C ¼ cþ di, all you do is rotate and slide thetransparent cylinder until the transparency number 1 [i.e. 1þ0i] is directly over the number aþ bi on the paper cylinder, thenlook up the number cþ di on the transparent cylinder, andbelow it on the paper cylinder will be the product AB. The log-arithm of AB is equal to the sum of the logarithms of A and B.Of course, on the real axis, � ¼ 0 [x ¼ 0], the map looks like thescale on a slide rule. Alternatively make a flat slide rule formultiplying complex numbers: make two photocopies of themap on white paper and tape them together to make two cycles

in � from right to left. Then make one photocopy of the maponto a transparency. Lay the number 1 [on the transparency] ontop of the number A in the right-hand cycle of the paper map andlook on the transparency for the number B, and below it on thepaper map will be the product AB.)For convenience on our map of the universe, let r ¼ (�RH0

)/rE(comoving cosmological distance/radius of the Earth) be mea-sured in units of Earth’s equatorial radius rE ¼ 6378 km. Thus,circles of constant radius (r ¼ const) from the center of the Earthare shown as horizontal lines ( y ¼ const) in the map, and rays ofconstant right ascension (� ¼ const) are shown as vertical lines(x ¼ const) in themap. The surface of the Earth (at its equator) is acircle of unit radius in the (u, v)-plane and is the line y ¼ 0 in themap. The region y < 0 in the map represents the interior of theEarth, so one can show the Earth’s mantle and liquid and solidcore. The solid inner core has a radius of about 0.19rE; thus, thelower edge of the map must extend to y ¼ ln (0:19) ¼ �1:66 toshow it. The region y > 0 shows the universe beyond the Earth.The comoving future observability limit at a radius of 19 Gpc is at9:2 ; 1019rE, and so the upper edge of the mapmust extend to y ¼ln (9:2 ; 1019) ¼ 45:97 to show it. Thus, the dimensions of themap are�x ¼ 2� and�y ¼ 47:63. The aspect ratio for the mapis height/width ¼ 7:58. See Figure 7 for a small-scale version ofthis that will fit on one page. [A square map would have dimen-sions 2� ; 2� and would cover a scale ratio from bottom to top ofexp (2�) ¼ 535:49. A map with an aspect ratio height/width ¼7:58 covers a scale ratio from bottom to top of 535.497.58.]At a scale of about 1 radian per inch for the angular scale,

this would make a map about 6.28 inches (15.95 cm) wide by

Fig. 4.—Zoom-in of the region marked by the dotted circle in Fig. 3, showing SDSS galaxies out to 0.2thorizon. The details of galaxy clustering are now displayedmuch better. However, like Fig. 2, it still fails to capture the whole survey in one reasonably sized map.


47.6 inches (120.9 cm) tall, which could be easily displayed as awall chart. We present the map at approximately this scale laterin this article.

This is not the first time logarithmic coordinates have beenused for a map of the universe. The Amoco Map of SpaceMysteries (De Peyng 1958) plotted the curved surface of theEarth and above it altitude (from the surface, not distance fromthe center) marked off in equal intervals labeled 1 mile, 10 miles,100 miles, 1000 miles, 10,000 miles, 100,000 miles, 1 millionmiles, 10 million miles, and 100 million miles. In this rangesolar system objects from the Moon to Venus, Mars, and theSun are plotted properly. But although � Centauri, the MilkyWay, and M31 are shown beyond, they are not shown at correctscale (even logarithmically). The Earth’s surface is plotted wherean altitude of 0.1 miles should have been. In any case, becauseof the curvature of the Earth’s surface in the map, even in therange between an altitude of 10 miles and 100 million miles, themap is not conformal. In 1999 October, National Geographicpresented a map of the universe (which one of us [J. R. G.]participated in producing) which was a three-dimensional viewwith a spherical Earth at the center with equal width shells sur-rounding it like an onion with radii of 400,000 miles, 40 millionmiles, 4 billion miles, 4 trillion miles, 10 light-years, 1000 light-years, 100,000 light-years, 10 million light-years, 1 billion light-years, and 11–15 billion light-years (Sloan et al. 1999). This mapdisplays objects from the Moon to the CMB but is also notconformal.

The map projection we are proposing is conformal becausethe derivatives of the complex function Z ¼ i lnW have nopoles or zeros in the mapped region. If we want to see how alittle area of the universe slice is mapped onto our slice, weshould do a Taylor expansion: the point W þ�W is mappedonto the point Z þ�Z ¼ Z þ (dZ/dW )�W in the limit where�W ! 0 providing that dZ/dW 6¼ 1 so the map does not blowup there and dZ/dW 6¼ 0 so that the second- and higher orderterms in the Taylor expansion can be ignored (providing thatnone of the higher derivatives dnZ/dWn become infinite at thepointW ). In this case, in the limit as�W ! 0, the Taylor seriesis valid using just the first derivative term:

�Z ¼ dZ

dW�W ; ð22Þ

dZ

dW¼ d(i lnW )

dW¼ i

W: ð23Þ

Thus, for W 6¼ 0 and finite (i.e., excluding the center of theEarth and the point at infinity, which are not mapped anyway)dZ/dW is neither zero nor infinity. The higher derivatives (n �2); dnZ/dWn ¼ i(�1)nn!W�nþ1, are also finite when W is finiteand nonzero. Thus, the pointW þ�W is mapped onto the pointZ þ�Z ¼ Z þ dZ/dW�W in the limit where �W ! 0. Char-acterize the point W as

W ¼ rw cos �w þ i sin �wð Þ: ð24Þ

Fig. 5.—Square comoving grid shown in look-back time coordinates. Grid spacing is 0:1RH0¼ 422:24 Mpc. Each grid square would contain an equal number of

galaxies in a flat slice of constant vertical thickness. The distortion of space that is produced by using the look-back time is obvious as the squares become more andmore distorted in shape as one approaches the horizon.


Now the product of two complex numbers

A ¼ ra cos �a þ i sin �að Þ;B ¼ rb cos �b þ i sin �bð Þ

is

AB ¼ rarb cos �a þ �bð Þ þ i sin �a þ �bð Þ½ �; ð25Þ

so

1

W¼ 1

rwcos ��wð Þ þ i sin ��wð Þ½ �; ð26Þ

and since i¼ cos (�/2)þ i sin (�/2),

i1

W¼ 1

rwcos

�

2� �w

�þ i sin

�

2� �w

�� ð27Þ

and

�Z ¼ r�Z cos ��Z þ i sin ��Zð Þ

¼ 1

rwcos

�

2� �w

�þ i sin

�

2� �w

�� ; r�W cos ��W þ i sin ��Wð Þ

¼ r�W

rwcos ��W þ �

2� �w

�þ i sin ��W þ �

2� �w

�� :

ð28Þ

Thus, the vector �W is rotated by an angle �/2� �w andmultiplied by a scale factor of 1/rw. Since any two vectors�W1

and �W2 at the point W will be rotated by the same amountwhen they are put on the map, the angle between them is pre-served in the map, and hence the map projection is conformal.

Fig. 6.—Logarithmic map of the complex plane. The vertical axis is thenatural logarithm of the absolute value of a complex number, while its phase isplotted on the horizontal axis. We plot 4 ; 10 ; 10 numbers per decade, from thefirst three decades. This map is conformal but covers a wide range of scales.

Fig. 7.—Pocket map of the universe. This is a smaller version of the moredetailed map shown in the foldout Fig. 8.


The only place the first derivative (and the higher derivatives)blows up (or goes to zero) is atW ¼ 0 at the center of the Earthor at the point at infinityW ¼ 1. But the Earth’s center does notappear on the map at all (it is at y ¼ �1). This is a set of mea-sure zero. Likewise, the point at infinity W ¼ 1 is not plottedeither (it is at y ¼ þ1). Thus, the map projection is conformalat all points in the map. Shapes are preserved locally.

The conformal map projection for showing the universepresented here was developed by J. R. G. in 1972, and he hasproduced various small versions of it over the years. These havebeen shown at various times, notably to the visiting committeeof the Hayden Planetarium in 1996 and to the staff of the Na-tional Geographic Society in 1999. Recent discoveries within awide range of scales from the solar system objects to the SDSSgalaxies and quasars have prompted us to produce and publishthe map in a large-scale format.

Our large-scale map is shown in Figure 8 (the foldout). Aradial vector �W (pointing away from Earth’s center) at thepoint W points in the direction ��W ¼ �w. This vector in themap is rotated by an angle �/2� �w so that ��Z ¼ ��W þ �/2��w ¼ �/2, so that it points in the vertical direction. Small re-gions in the universe are rotated in the map so that the radialdirection, away from the center of the Earth, is in the verticaldirection in the map. Radial lines from Earth’s center are plottedas vertical lines. Circles of constant radius from the center of theEarth are horizontal lines. The length of the vector �W is mul-tiplied by a scale factor 1/rw. Thus, the scale factor at a givenpoint on the map can be read off as proportional to 1 over thedistance of the point from the Earth, rw. (Objects that are twiceas far away are shown at 1/2 scale, and objects that are 10 timesfarther away are shown at 1/10 the scale, and so forth.)

Radial lines separated by an angle � (in radians), going out-ward from the Earth, will be plotted as parallel vertical linesseparated by a horizontal distance proportional to �. Thus, ob-jects of the same angular size in the sky (as seen from the centerof the Earth) will be plotted as the same size on the map.

The Sun and Moon, which have the same angular size in thesky (0N5), will be plotted as circles of the same size on the map(since their cross sections are circles and shapes are preservedlocally in a conformal map projection).

The map gives us that Earthling’s view of the universe thatwe want. Objects are shown at a size in the map proportional tothe size they subtend in the sky. The Sun and Moon are equallylarge in the sky and so appear of the same size in the map. Ob-jects that are close to us are more important to us, as depicted inthat New Yorker cover. Buildings on 9th Avenue may subtendas large an angle to our eye as the distant state of California. Ourloved ones, important to us, are often only a few feet away andsubtend a large angular scale to our eyes. A murder occurring inour neighborhood draws more of our attention than a murder ofsomeone halfway around the globe. Plotting objects at a sizeequal to their angular scale makes psychological sense. Objectsare shown taking up an area on the map that is proportionalto the area they subtend in the sky (if they are approximatelyspherical, as many astronomical bodies are). The importance ofthe object in the map (the fraction of the map it takes up) isproportional to the chance we will see the object if we look outalong a random line of sight. Indeed, if we look at the map froma constant distance, the angular size of the objects in the mapwill be proportional to the angular size they subtend in the sky.The visual prominence of objects in the map will be propor-tional to their visual prominence in the sky.

Of course, this means that the Moon and Sun and other ob-jects will be shown at their true scale relative to their surround-

ings (i.e., the Moon is shown in correct scale relative to thecircumference of its orbit), which is small because they aresmall in the sky. Suppose we made aMollweide equal-area mapprojection of the sky at a scale to fit on a journal page: an ellipsewith horizontal width 6.28 inches (15.95 cm) and vertical heightof 3.14 inches (7.98 cm). Along the equatorial plane the scale islinear at 1 inch per radian. The diameter of theMoon or Sun is 0N5,or 1/720 of the 360

�length around the equator. Thus, on this sky

map the Moon would have a diameter of (6:28 inches)/720 ¼0:0087 inches (0.022 cm). On our map the Moon would havea similar diameter, for the scale of our map is approximately1 radian ¼ 1 inch (2.54 cm). The Moon and Sun are rather smallin the sky. With a printer resolution of 300 dots per inch the Sunand Moon would then be approximately 3 dots in diameter.

For easier visibility we have plotted the Sun andMoon as cir-cles enlarged by a factor of 18. Similar enlargements of indi-vidual objects like the Sun and the Moon might appear on skymaps appearing on one page. Just as symbols for cities on worldmaps may be larger than the cities themselves. Still, it is in-teresting to note what the true sizes on the map should be since itshows how much empty space in the universe there is. M31, forexample, subtends an angle of about 2

�on the sky and so would

be about 0.035 inches (0.089 cm) on a map with 1 inch per ra-dian scale. If versions of the map were produced at larger scale,as we discuss below, images of the Sun, Moon, and nearbygalaxies could be displayed at proper angular scale and simplyplaced on the map.

The completed map is shown in the foldout (Fig. 8).The map shows a complete sample of objects in the classes

we are illustrating in the equatorial slice (�2� < � < 2�), whichis shown conformally correct. These objects are shown at thecorrect distances and right ascensions. This we supplement withadditional famous objects out of the plane, which are shown attheir correct distances and right ascensions. So this is basicallyan equatorial slice with supplements. For source material, seeAppendix B.

At the bottom, the map starts with an equatorial interior crosssection of the Earth. First, we see the solid inner core of theEarth with a radius of1200 km. Above this is the liquid outercore (1200–3480 km), and above that are the lower (3480–5701 km) and upper mantle (5701–6341 km). The Earth’ssurface has an equatorial radius of 6378 km. There is a linedesignated Earth’s surface (and crust), which is a little thickerthan an ordinary line to properly indicate the thickness of thecrust. The Earth’s surface (and crust) line is shown as perfectlystraight because on this scale the altitude variation in the Earth’ssurface is too small to be visible. The scale at the Earth surfaceline is approximately 1/250,000,000. Themaximum thickness ofthe crust (37 km) is just barely visible on the map at a resolutionof 300 dots per inch, so we have shown themaximum crust depthaccordingly, by the width of the Earth surface (crust) line.

( If we had wished, we could have extended the map down-ward to cover the entire inner solid core of the Earth down to thecentral neutron [or proton] in an iron atom located at the centerof the Earth. Since a neutron has a radius of approximately1:2 ; 10�13 cm or 1:9 ; 10�22rE, the circumference of this cen-tral neutron would be plotted as a straight line at y ¼ �50:0.The outer circumference of the central iron atom [atomic radiusof 1:40 ; 10�8 cm; Slater 1964] would be plotted by a straightline at y ¼ �38:36. Thus, including the entire inner solid coreof the Earth down to the central neutron in an iron atom at thecenter of the Earth would require [at 1 inch per radian scale]about an additional 48 inches (122 cm) of map, approximatelydoubling its length. The central atom and its nucleus would then


occupy the bottom 11.6 inches (29.5 cm) of the map, giving anice illustration of both the nucleus and all the electron orbitals.But since we are primarily interested in astronomy and the keyregions of the Earth’s interior are covered in the map already,we have stopped the map just deep enough to show the extent ofthe inner solid core.)

We have shown the Earth’s atmosphere above the Earth’ssurface. The ionosphere is shown, which occupies an altituderange of 70–600 km. Below the ionosphere is the stratosphere,which occupies an altitude range of 12–50 km. Although therewas not enough space to include a label, the stratosphere on themap simply occupies the tiny space between the lower error barindicating the bottom of the ionosphere and the ‘‘surface of theEarth’’ line. The troposphere (0–12 km) is of such small altitudethat it is subsumed into the Earth’s surface line thickness. Abovethe ionosphere we have formally the exosphere, where the meanfree path is sufficiently long that individual atoms with escape ve-locity can actually escape. Thus, the top of the ionosphere effec-tively defines the outer extent of the Earth’s atmosphere and themap shows properly just how narrow the Earth’s atmosphere isrelative to the circumference of the Earth.

Next we have shown all 8420 artificial Earth satellites in orbitas of 2003 August 12 (at the time of full Moon 2003 August 1204:48 UT). In fact, all objects in the map are shown as of thattime. This is the last full Moon before the closest approach ofMars to the Earth in 2003. The time was chosen for its placementof the Sun, Moon, and Mars. We show all Earth satellites (notjust those 624 in the equatorial slice). These are actual namedsatellites, not just space junk. Some famous satellites are des-ignated by name. ISS is the International Space Station. HST isthe Hubble Space Telescope. These are both in low Earth orbit.

Vanguard 1 is shown, the earliest launched satellite still in orbit.The Chandra X-Ray Observatory is also shown. There are twomain altitude layers of lowEarth orbiting satellites and a scatteringof them above that. There is a quite visible line of geostationarysatellites at an altitude of 22,000 miles above the Earth’s surface.These geosynchronous satellites are nearly all in our equatorialslice. A surprise was the line of GPS (Global Positioning System)satellites at a somewhat lower altitude. These are all in nearly cir-cular orbits at identical altitudes and so also show up as a line onthe map. We had not realized that there were so many of thesesatellites or that they would show up on the map so prominently.The inner and outer Van Allen radiation belts are also shown.

Beyond the artificial Earth satellites and the Van Allen radi-ation belts lies the Moon, marking the extent of direct humanoccupation of the universe. TheMoon is full on 2003 August 12.

Behind the full Moon at approximately 4 times the distancefrom the Earth is the WMAP satellite, which has recently mea-sured the CMB. It is in a looping orbit about the L2 unstableLagrange point on the opposite side of the Sun from the Earth.Therefore, it is approximately behind the full Moon. At theL1 Langrange point, 180� away, is the Solar and HeliosphericObservatory (SOHO) satellite. From L1, SOHO has an unob-structed and uninterrupted view of the Sun throughout the year.

To illustrate the distances from Earth to the nearest asteroids,we plot the 12 closest to Earth as of 2003 August 12 ( labeledNear Earth Objects [NEOs]). Asteroid 2003 GY was closest toEarth at that time. Another two of them are particularly inter-esting. Asteroid 2003 YN107 is currently the only known quasi-satellite of the Earth (Connors et al. 2004) and shall remain suchuntil the year 2006. Asteroid 2002 AA29 is on an interestinghorseshoe orbit (Connors et al. 2002; Belbruno & Gott 2005)that circulates at 1 AU and has close approaches to Earth every95 yr. Such horseshoe orbits are typical of orbits that have escaped

from the Trojan L4, L5 points (Belbruno &Gott 2005). Since thisasteroid may have originated at 1 AU like Earth and the greatimpactor that formed the Moon, it might be an interesting objectfor a sample return mission, as Gott and Belbruno have noted.Mars is shown at approximately 9000rE, or 0.4 AU (as seen

on the scale on the right). Mars is near its point of closest ap-proach (which it achieved on 2003 August 27 when it was ata center-to-center distance of 55,758,006 km). Farther up areMercury, the Sun, and Venus. Venus is near conjunction withthe Sun on the opposite side of its orbit. The Sun is 180� awayfrom the Moon or halfway across the map horizontally, sincethe Moon is full. The Sun is 1 AU away from the Earth. At dis-tances from Earth of between0.7 and5 AU are the main beltasteroids. Here, out of the total of 218,484 asteroids in theASTORB database, we have shown only those 14,183 that arein the equatorial plane slice (�2� < � < 2�). If we had shownthem all, it would have been totally black. By just showing theasteroids in the equatorial slice, we are able to see individualdots. In addition, some famous main belt asteroids, like Ceres,Eros, Gaspra, Vesta, Juno, and Pallas, are shown (even if off theequatorial slice) and indicated by name. The width of the mainbelt of asteroids is shown in proper scale relative to its circum-ference in the map. The belt is closer to the Earth in the antisolardirection since it is an annulus centered on the Sun and the Earthis off-center. Because themain belt asteroids lie approximately inthe ecliptic plane, which is tilted at an angle of 23N5 relative to theEarth’s equatorial plane, there are two dense clusters where theecliptic plane cuts the Earth’s equatorial plane and the density ofasteroids is highest. One intersection is at about 12h and the otheris at 24h.Jupiter is shown in conjunction with the Sun approximately

6 AU from the Earth. On either side of Jupiter we can see thetwo swarms of Trojan asteroids trapped in the L4 and L5 stableLagrange equilibrium points �60� away from Jupiter along itsorbit. From the vantage point of Earth, 1 AU off-center oppositeJupiter in its orbit, the Trojans are a bit closer to the Earth thanJupiter and a bit closer to Jupiter in the sky on each side than60

�. The Ulysses spacecraft is visible near Jupiter. It is in an or-

bit far out of Earth’s equatorial plane, but we have included itanyway. Beyond Jupiter are Saturn, Uranus, and Neptune.Halley’s comet is shown between the orbits of Uranus and

Neptune, as of 2003 August 12.Next we show Pluto and the Kuiper Belt objects. We are

showing all 772 of the known Kuiper Belt objects (rather thanjust those in the equatorial plane). It is surprising how manyof them there are. Recently discovered 2003 VB12 (‘‘Sedna’’),Quaoar, and Varuna, the largest Kuiper Belt objects currentlyknown, are also shown and labeled. Sedna is currently the sec-ond largest known transneptunian object, after Pluto (Brownet al. 2004). The band of Kuiper Belt objects is relatively nar-row because of the selection effect that objects of a given sizebecome dimmer approximately as the fourth power of their dis-tance from the Earth and Sun. The band of Kuiper Belt objectshas vertical density stripes, again owing to angular selectioneffects depending on where various surveys were conducted. Asprinkling of Kuiper Belt objects extends all the way into thespace between the orbit of Uranus and Saturn (the ‘‘Centaurs,’’of which we are only plotting the ones in the equatorial plane).We show Pioneer 10, Voyager 1, and Voyager 2 spacecraft,

headed away from the solar system. These are on their way tothe heliopause, where the solar wind meets the interstellar me-dium. They have not reached it yet.Almost 100 times farther away than the heliopause is the

beginning of the Oort Cloud of comets, which extends from


about 8000 to 100,000 AU. A comet entering the inner solarsystem for the first time has a typical aphelion in this range.

Beyond the Oort Cloud are the stars. The 10 brightest starsvisible in the sky are shownwith large star symbols. The neareststar, Proxima Centauri, is shownwith a small star symbol. Prox-ima Centauri, an M5 star, is a member of the � Centauri triple-star system. � Centauri A, at a distance of a little over 1 pc (seescale on the left) and a solar type star, is one of the 10 brighteststars in the sky. The 10 nearest star systems are also shown: �Centauri, Barnard’s star, Wolf 359, Halande 21185, Sirius, UVand BL Ceti, Ross 154, Ross 248, Eridani, and Lacaille 9352.Of these, Eridani has a confirmed planet circling it.

Stars with known confirmed planets circling them (withM sin i < 10MJ) are shown as dots with circles around them. Ofthese, 95 are solar type stars whose planets were discovered byradial velocity perturbations. Some of the more famous oneslike 51 Peg, 70 Vir, and Eri are labeled. The star HD 209458has a Jupiter-mass planet that was discovered by radial velocityperturbations but was later also observed in transit (Mazeh et al.2000; Henry et al. 1999). The first planet discovered by transitwas OGLE-TR-56, which lies at a distance of over 1 kpc fromthe Earth. Three other OGLE stars were also found to havetransiting planets: TR-111, TR-113, and TR-132. These wereall in the same field (R:A: 10h50m) at approximately the samedistance (1.5 kpc) and so would be plotted at positions on themap that would be indistinguishable. The planet TrES-1 aroundGSC 02652-01324 was recently discovered by transit and con-firmed by radial velocity measurements (Alonso et al. 2004). Aplanet circling the star OGLE 2003-BLG-235 was discoveredby microlensing (Bond et al. 2004). PSR 1257+12 is a pul-sar (neutron star) with three terrestrial planets circling it, whichwere discovered by radial velocity perturbations on the pulsarrevealed by accurate pulse timing (Wolszczan & Frail 1992;Wolszczan 1994). This was the first star discovered to haveplanets. In addition, SO 0253+1652 is shown and labeled on themap. It is the closest known brown dwarf (Teegarden et al.2003), at 3.82 pc.

The first radio transmission of any significant power to escapebeyond the ionosphere was the TV broadcast of the openingceremony of the 1936 Berlin Olympics on 1936 August 1, a factnoted by Carl Sagan in his book Contact. The wave front cor-responding to this transmission is a circle having a radius of1011rE on 2004 August 12 and is indicated by the straight linelabeled ‘‘Radio signals from Earth have reached this far.’’ Radiosignals from Earth have passed stars below this line.

The Hipparcos satellite has measured accurate parallax dis-tances to 118,218 stars (ESA 1997). We show only the 3386Hipparcos stars in the equatorial plane (�2

� < � < 2�).

Other interesting representative objects in the galaxy areillustrated: the Pleiades, the globular cluster M13, the CrabNebula (M1), the black hole Cygnus X-1, the Orion Nebula(M42), the Dumbbell Nebula and the Ring Nebula, the EagleNebula, the Vela pulsar, and the Hulse-Taylor binary pulsar. Ata distance of 8 kpc is the Galactic center, which harbors a2:6 ;106 M� black hole. The outermost extent of the MilkyWay optical disk is shown by a dotted line.

Beyond the Milky Way are plotted 52 currently knownmembers of the Local Group of galaxies. We have included allof these, not just the ones in the equatorial plane. These areindicated by dots or triangles. M31’s companions M32 andNGC 205 are shown as dots but not labeled, since they are soclose to M31.

M81 is the first galaxy shown beyond the Local Group and isa member of the M81–M82 group. Its distance was determined

by Cepheid variables using the HST. Other famous galaxies la-beled includeM101, theWhirlpool galaxy (M51), the Sombrerogalaxy, andM87, in the center of the Virgo Cluster. If we showedall the M objects, many would crowd together in a jumble at thelocation of the Virgo Cluster. M87 harbors in its center thelargest black hole yet discovered with a mass of 3 ; 109 M�.

The dots appearing beyond M81 in the map are the 126,594SDSS galaxies and quasars (with z < 5) in the equatorial planeslice (�2

� < � < 2�). In addition, all 31 currently known SDSS

quasars with z > 5 are plotted, not just those in the equatorialslice. Since these large redshift quasars are shown from all overthe sky, a number of them appear at right ascensions that occurin the zone of avoidance for the equatorial slice.

The upper part of our map can be compared directly withFigures 1 and 2. The map shows clearly and with recognizableshape all the structures shown in the close-up in Figure 2, whilestill showing all the SDSS quasars shown in the full view inFigure 1. The logarithmic scale captures both scales beautifully.On the left (at about 1C5 and 120 Mpc), we can see clearly thelarge circular void visible in Figure 2. To the right, at R.A. of9h–14h and at a distance of 215–370 Mpc, we can see a SloanGreat Wall in the SDSS data, longer than the Great Wall ofGeller and Huchra (the CfA2 Great Wall). The blank regionsare where the Earth’s equator cuts the Galactic plane and inter-sects the zone of avoidance near the Galactic plane (where theinterstellar dust obscures distant galaxies and which the Sloansurvey does not cover). These are empty fan-shaped regionsas shown in Figures 1 and 2, bounded by radial lines pointingaway from the Earth, so on our map these are bounded byvertical straight lines.

The Great Attractor (which is far off the equatorial plane andtoward which the Virgo supercluster has a measurable peculiarvelocity) is shown.

The most prominent feature of the SDSS large-scale structureseen in Figure 8 is the ‘‘Sloan Great Wall.’’ This feature wasnoticed early-on in the Sloan data acquisition process and hasbeen mentioned in passing a couple of times in Sloan reports,accompanied by phrases such as ‘‘large’’ (Blanton et al. 2003b),‘‘striking,’’ ‘‘wall-like,’’ and ‘‘may be the largest coherent struc-ture yet observed’’ (Tegmark et al. 2004). Tomake a quantitativecomparison, we have also shown the Great Wall of Geller andHuchra. This extends over several slices of the CfA2 survey(from 42� to�8N5 declination). Rather than plotting points for ithere, which would be confused with SDSS galaxies, we haveplotted density contours averaging over all the CfA2 slices from42� to �8N5 declination. This volume extends far above theequatorial plane, and since we are plotting it in right ascensioncorrectly, it is not presented conformally but is being lengthenedin the tangential direction by a factor of 1/ cos (21�) ¼ 1:07.Note that since the CfA2 Great Wall is a factor of approximately2.5 closer to us than the Sloan Great Wall, it is depicted at a scalethat is 2.5 times larger. Hence, although the CfA2 Great Wallstretches from 9h to 16C7 (or 7C7 of right ascension), as comparedwith the SDSS Great Wall, which stretches from 8C7 to 13C7(or 5h of right ascension), the latter’s length in comoving coordi-nates relative to the CfA2 Great Wall is, by this simple analy-sis, 2:5 ; (5/7:7) ; 1:07 � 1:74 times as long. This is apparentin the comparison figure supplied in Figure 9, where both areshown at the same scale in comoving coordinates. To make a faircomparison, since the Great Wall is almost a factor of 3 closerthan the Sloan Great Wall, we have plotted a 12

�wide slice from

the CfA2 survey to compare with our 4� wide slice in the Sloan,so that both slices have approximately the same width at eachwall.


Of course, the walls are not perfectly aligned with the x-axisof our map, so one has to measure their length along the curve.The Sloan Great Wall is at a median distance of 310 Mpc. Itstotal length in comoving coordinates is 450 Mpc as comparedwith the total length of the Great Wall of Geller and Huchra,which is 240 Mpc long in comoving coordinates. This indicatesthe sizes the two walls would have at the current epoch. How-ever, the Great Wall is at a median redshift of z ¼ 0:029, so itstrue size at the epoch at which we are observing is smaller bya factor of 1þ z, giving it an observed length of 232.64 Mpc(or 758 million light-years). The Sloan Great Wall is at a red-shift of z ¼ 0:073, so its true observed length is 419 Mpc (or1365 million light-years). For comparison, the CMB sphere hasan observed diameter of (2 ; 14;000)/1090 ¼ 25:7 Mpc. Theobserved length of the Sloan Great Wall is thus 80% greaterthan the Great Wall of Geller and Huchra.

Since we have numerous studies that show that the three-dimensional topology of large-scale structure is spongelike(Gott et al. 1986; Vogeley et al. 1994; Hikage et al. 2002), itshould not be surprising that as we look at larger samples weshould find examples of larger connected structures. Indeed, wewould have had to have been especially lucky to have discov-ered the largest structure in the observable universe in the initialCfA survey, which has a much smaller volume than the Sloansurvey. Simulated slices of the Sloan using flat-lambda mod-els (as suggested by WMAP) show great walls and great wallcomplexes that are quite impressive (Colley et al. 2000). Coleet al. (1998), for example, had a great wall in their �m ¼ 0:4,�� ¼ 0:6 Sloan simulation that is 8% longer than the GreatWall of Geller and Huchra, and so it could be said that theexistence of a GreatWall in the Sloan longer than the GreatWallof Geller and Huchra was predicted in advance. Visual inspec-tion of the 275 PTHalos simulations reveals similar structures tothe Sloan Great Wall in more than 10% of the cases (Tegmarket al. 2004). Thus, it seems reasonable that the Sloan GreatWall can be produced from random phase Gaussian fluctuationsin a standard flat-lambda model, a model that also predicts aspongelike topology of high-density regions in three dimen-sions. Notably, our quantitative topology algorithm applied tothe two-dimensional Sloan slice identifies the Sloan Great Wallas one connected structure (Hoyle et al. 2002). Figure 2 inHoyle et al. (2002) clearly shows this as one connected structureat the median density contour when smoothed at 5 h�1 Mpc in avolume-limited sample where the varying thickness and vary-ing completeness of the survey in different directions are ac-counted for. It is perhaps no accident that both the Sloan GreatWall and the Great Wall of Geller and Huchra are seen roughlytangential to the line of sight. Great Walls tangential to the lineof sight are simply easier to see in slice surveys, as pointed outby Praton et al. (1997). A GreatWall perpendicular to the line ofsight would be more difficult to see because the near end wouldbe lost owing to thinness of the slice and the far end would belost owing to the lack of galaxies bright enough to be visible atgreat distances. Redshift space distortions on large scales, i.e.,infall of galaxies from voids onto denser regions, enhance con-trast for real features tangential to the line of sight. ‘‘Fingersof God’’ also make tangential structures more noticeable bythickening them. When Park (1990) first simulated a volumelarge enough to simulate the CfA survey, a Great Wall was im-mediately seen in the three-dimensional data. When a slice tosimulate the CfA slice seen from Earth, which gets wider as itgets farther from the Earth, was made, the Great Wall in thesimulation was pretty much a dead ringer for the Great Wall ofGeller and Huchra, equal in length, shape, and density. This

was an impressive success for N-body simulations. The SloanGreat Wall and the CfA Great Wall have been found in quitesimilar circumstances, each in a slice of comparable thick-ness, and as illustrated in Figure 9, both are qualitativelyquite similar except that the Sloan Great Wall is simply larger,and as we have noted, our two-dimensional topology algo-rithm (Hoyle et al. 2002) identifies the Sloan Great Wall asone connected structure in a volume-limited survey wherethe varying thickness and completeness of the slice surveyare properly accounted for. The CfA Great Wall is as large astructure as could have fit in the CfA sample, but the SloanGreat Wall is smaller than the size of the Sloan survey, show-ing the expected approach to homogeneity on the very largestscales.The Two Degree Field (2dF) survey (Colless et al. 2001), of

similar depth to the Sloan, completed two slices, an equatorialslice (9h50m < � < 14h50m, �7N5 < � < 2N5) and a southernslice (21h40m < � < 03h40m, �37N5 < � < �22N5). This sur-vey was thus not appropriate for our logarithmic map of theuniverse. The southern slice was not along a great circle in thesky and therefore would be stretched if plotted in our map inright ascension. The equatorial slice was of less angular extentthan the corresponding Sloan Slice and so the Sloan with itsgreater coverage in a flat equatorial slice was used to plot large-scale structure in our map. Indeed, the 2dF survey, because ofits smaller coverage in right ascension, missed the western endof the Sloan Great Wall and so the wall did not show up asprominently in the 2dF survey as in the Sloan. Power spectrumanalyses of the 2dF and the Sloan come up with quite similarestimates. These two great surveys in many ways complementeach other. Perhaps most importantly, the 2dF power spectrumanalysis that was available before Sloan and in time forWMAPallowed estimates of �m that allowed WMAP to refine the cos-mological parameters used in the construction of this map. TheSloan GreatWall contains a number of Abell clusters (including,for example, A1238, A1650, A1692, and A1750, for which red-shifts are known). The spongelike nature of three-dimensional to-pology means that clusters are connected by filaments or walls,but if extended far enough, walls should show holes allowing thevoids on each side to communicate.Indeed, in our map we can see some remnants of the CfA2

Great Wall (a couple of clumps or ‘‘legs’’) extending into theequatorial plane of the Sloan sample. As shown in Vogeley et al.(1994), if extended to the south, the Great Wall develops holesthat allow the foreground and background voids to communi-cate, leading to a spongelike topology of the median densitycontour surface in three dimensions. In three dimensions, theSloan Great Wall may be connected to the supercluster of Abellclusters found by Bahcall & Soneira (1984), whose two mem-bers lie just above it in declination.In the center of the Great Wall is the Coma Cluster, one of the

largest clusters of galaxies known.The quasar 3C 273 is shown by a cross.The gravitational lens quasar 0957 is shown, as well as the

lensing galaxy producing the multiple image. The lensing gal-axy is along the same line of sight but at about one-third thecomoving distance.The gamma-ray burster GRB 990123 is shown; for a brief

period this was the most luminous object in the observeduniverse.The redshift z ¼ 0:76 is shown by a line, which marks the

epoch that divides the universe’s decelerating and acceleratingphase. Objects closer than this line are observed at an epochwhen the universe’s expansion is accelerating, while objects

GOTT ET AL.476

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farther away than this line are observed at an epoch when theuniverse’s expansion is decelerating.

The unreachable limit is shown at z ¼ 1:69. Because of theacceleration of the expansion of the universe, photons sent fromEarth now will never reach objects beyond this line. Galaxiesbeyond this line will never hear our current TV signals. Space-ships from Earth, traveling slower than light, will also find theterritory beyond this line unreachable. This redshift is surpris-ingly low. It is interesting that we can see many objects todaythat are so far away that we can never get to them.

SDSS quasars in the equatorial plane (�2� < � < 2

�) are

shown as points out to a redshift of z ¼ 5:0 using redshiftsdetermined from the SDSS spectra. For quasars with z > 5:0we have shown all 31 quasars from the SDSSwith z > 5 regard-less of declination. There are now a surprisingly large numberof quasars with z > 5 known. Because a number of these are athigh declination, they occur at right ascensions that are in thezone of avoidance for the equatorial plane. Each is shown at itsproper right ascension and distance, including the largest red-shift one with z ¼ 6:42, which is labeled (M. Strauss 2003, pri-vate communication). This is currently the largest redshift quasarknown.

The galaxy SDF J132418.3+271455, with the largest accu-rately measured redshift (z ¼ 6:578), is also shown. This wasdiscovered in the Subaru Deep Field (Kodaira et al. 2003).

The comoving distance at which the first stars are expected toform is shown by a dashed line. The WMAP satellite has foundthat the first stars appear about 200 million years after the bigbang, and the map therefore indicates the distance out to whichstars could be seen in principle.

Finally, there is the CMB radiation discovered by Penzias &Wilson (1965). CMB photons from this surface arrive directlyfrom an epoch only 380,000 yr after the big bang.

A line showing the comoving radius back to the big bangis also shown. This represents seeing back to the epoch just af-ter inflation. The comoving distance between the CMB and thebig bang is shown in correct proportion to the circumference.In comparing with Figure 1, we note that in that map the cir-cumference is a factor of � larger. So in our map, which showsthe 360

�circumference of the CMB as approximately the same

length as the diameter of the circle in Figure 1, the scale at thatpoint is about a factor of � smaller than in Figure 1 and the CMBand the big bang are closer to each other (by a factor of �) thanin Figure 1, as expected.

Last is shown the comoving future visibility limit. If we waituntil the infinite future, we will eventually be able to see the bigbang at the comoving future visibility limit. Stars and galaxiesthat lie beyond this comoving future visibility limit are foreverhidden from our view. Because of the de Sitter expansion pro-duced by the cosmological constant, the universe has an eventhorizon that we cannot see over no matter how long we wait.

It is remarkable how many of the features shown in this maphave been discovered in the current astronomical generation.When one of us (J. R. G.) began studying astronomy at age 8 (in1955), on astronomical maps there were no artificial satellites,no Kuiper Belt objects, no other stars with planets, no browndwarfs, no pulsars, no black holes, no nonsolar X-ray sources,no gamma-ray bursters, no great walls, no great attractors, noquasars, no gravitational lenses, and no observation of theCMB.

Fig. 9.—Sloan Great Wall compared to CfA2 Great Wall at the same scale in comoving coordinates. Equivalent redshift distances cz are indicated. The Sloan sliceis 4� wide and the CfA2 slice is 12� wide, to make both slices approximately the same physical width at the two walls. The Sloan Great Wall extends from 14h to 9h.It consists of one strand at the left, which divides to form two strands between 11C3 and 9C8, which come back together to form one strand again (like a road thatbecomes a divided highway for a while). The CfA2 Great Wall (which includes the Coma Cluster in the center) has been plotted on a cone and then flattened onto aplane. Total numbers of galaxies shown in each slice are also indicated.

MAP OF THE UNIVERSE 479

4. APPLICATIONS

This map shows large-scale structure well. ‘‘Fingers of God’’are vertical, which makes removing them for large-scale struc-ture purposes particularly easy. A test for roundness of voids asproposed by Ryden (1995), the Alcock-Paczynski isotropy test,could be done on this map, as well as on the comoving map. Theability of this test to differentiate between cosmological modelscan be deduced by measuring void isotropy as a function ofcosmological model in the plane of parameters (�m, ��). Forthe correct cosmological model the void pictures will be iso-tropic (since the Earth is not in a special position in the uni-verse). To do this test, we need conformal maps for variouscosmological models, and so we need conformal maps for thek ¼ þ1 and k ¼ �1 cases, as well as the k ¼ 0 case, so that sta-tistical comparisons can be made. The formulae for these pro-jections are given in Appendix A.

A Fourier analysis of large-scale structure modes in the map(kx , ky) can provide information on the parameter ¼ �0:6

m /b,where b is the bias parameter. For fluctuations in the linearregime in redshift space d�/� / (1þ �2), where � ¼ cos �,where � is the angle between the normal to the wave and the lineof sight in three dimensions (Kaiser 1987). Waves tangential tothe line of sight have a larger amplitude in redshift space thanwaves parallel to the line of sight because the peculiar velocitiesinduced by the wave enhance the amplitude of the wave whenpeculiar radial velocities are added to Hubble positions, as oc-curs when galaxies are plotted using redshift at the Hubble flowpositions assuming that peculiar velocities are zero. Imagine aseries of waves isotropic in three dimensions. Then one canshow that the average value of �2 in three dimensions of thosewaves with an orientation �0 ¼ cos � in the equatorial slice ish�2i ¼ 2

3�02. Thus, for waves observed in our equatorial slice,

we expect approximately ��/� / (1þ 23�02). Waves in our

map with constant (kx , ky) represent global logarithmic spiralmodes with constant inclination relative to the line of sight.Of course, this is an oversimplified treatment, since we mustconsider the power spectrum of fluctuations when relating themodes seen at a particular wavelength in the plane and in threedimensions, and the ‘‘fingers of God’’ must be eliminated bysome friends-of-friends algorithm. But in general, we expect thatmodes that are radial (kx ¼ 0) will have higher amplitudes thanmodes that are tangential (ky ¼ 0), and this effect can be usedempirically, in conjunction with N-body simulations, to providean independent check on the value of . One simply adopts acosmological model, checks with N-body simulations the rela-tive amplitude of map modes as a function of �0 in the loga-rithmic map, and compares with the observations assuming thesame cosmological model when plotting the logarithmic map; ifthe cosmological model (�m,��) is correct, the results should besimilar.

Since logarithmic spirals appear as straight lines in our mapprojection, it may prove useful for mapping spiral galaxies.Take a photograph of a face-on spiral galaxy, place the originof the coordinate system in the center of the galaxy, and thenconstruct our logarithmic map of this two-dimensional planarphotograph. The spiral arms (which approximate logarithmicspirals) should then appear as straight lines on the conformalmap. We have tried this on a face-on spiral galaxy photographgiven to us by James Rhoads, and the results were very satis-fying. The spiral arms indeed were beautiful straight lines, andfrom their slope one could easily measure their inclination an-gle. Star images in the picture were still circular in the mapbecause the map is conformal.

5. CONCLUSIONS

Maps can change the way we look at the world. Mercator’smap presented in 1569 was influential not only because it was aprojection that showed the shapes of continents well but be-cause for the first time we had pretty accurate contours for NorthAmerica and South America to show. The Cosmic View and thePowers of Ten alerted people to the scales in the universe we hadbegun to understand. De Lapparent et al. (1986) showed how aslice of the universe could give us an enlightening view of theuniverse in depth. Now that astronomers have arrived at a newunderstanding of the universe from the solar system to the CMB,we hope our map will provide in some small way a new visualperspective on these exciting discoveries.The map presented here is appropriate for use as a wall map.

Its scale is approximately 1 inch per radian. Aversion at twice thescale, 12.56 inches (31.90 cm) wide by 95.2 inches (241.8 cm)tall, would also be appropriate for a wall chart and would runnearly fromfloor to ceiling in a normal roomwith an 8 foot (2.4m)ceiling. If one wanted to show individual objects at 60 times scale,the Moon and Sun would be 1 inch (2.54 cm) across, M31 wouldbe 4 inches (10.16 cm) across, and Mars would be 0.0145 inches(0.037 cm) in diameter and Jupiter 0.0272 inches (0.069 cm).Alternately, the Sun and Moon and Messier objects could beshown at 60 times scale with the planets at 600 times scale toillustrate their usual appearance in small telescopes.Consider some possible (and some fanciful) ways this map of

the universe might be presented for educational use.We have presented the map on the internet in color on astro-

ph. In principle, it would be easy to have such a map on theinternet automatically continuously updated to track the currentpositions of the satellites, Moon, Sun, asteroids, planets, andKuiper Belt objects as a function of time; in fact, we have plot-ted them as of a particular date and time using such programs.New objects could be added to the map as they were discovered.Click on an object, and a 60 times enlarged view of it wouldappear. Two clicks, and a 3600 times enlarged view would ap-pear, and so forth, until the highest resolution picture availablewas presented. Individual images of all the SDSS galaxies andquasars shown on the map could be accessed in this way, as wellas M objects. When an individual object was selected, helpfulinternet links to sites telling more about it would appear.Since the left- and right-hand edges of the map are identical,

the map could be profitably shown as a cylinder. In cylindricalform, the map at the approximate scale and detail of Figure 7 isin perfect proportion to be used on a pencil with the Earth’s sur-face at the eraser end and the big bang at the point end. Perhapsthe best cylindrical form for the map would be a cylinder on theinterior of an elevator shaft for a glass elevator. Every floor youwent up you would be looking at objects that were 10 timesfarther away than the preceding floor. A trip up such an elevatorshaft could be simulated in a planetarium show, with the cy-lindrical map being projected onto the dome showing the viewfrom the elevator as it rose.We have put our map up on Princeton’s flat video wall. This

wall has a horizontal resolution of 4096 pixels. From top to bot-tom of the video wall is about 1600 pixels, so the scale changein the map from top to bottom is about a factor of 10. This showsa small portion of the entire map. We then scan this in real timemoving steadily upward from the Earth to the CMB and the bigbang. This produces a virtual map 17.6 feet (5.4 m) wide and134 feet (40.8 m) tall. With laser beams it would be easy to painta large version of the map on the side of a building. Avery largetemporary version of our map of the universe could also be set


up in a park or at a star party by simply planting markers for thesalient objects.

The map could be produced on a carpet 6 feet (1.8 m) wide by45.5 feet (13.9 m) long for a hallway in an astronomy depart-ment or planetarium. Then every step you took down the hall-way would take you about a factor of 10 farther from theEarth—a nice way to have a walk through the universe.

Wewould like to thankMichael Strauss for supplying us withthe list of SDSS quasars with redshift greater than 5.

This work was supported by JRG’s NSF grant AST 04-06713.Funding for the creation and distribution of the SDSS Archive

has been provided by the Alfred P. Sloan Foundation, the Par-

ticipating Institutions, the National Aeronautics and Space Ad-ministration, the National Science Foundation, the US Depart-ment of Energy, the Japanese Monbukagakusho, and the MaxPlanck Society. The SDSS Web site is http://www.sdss.org.

The SDSS is managed by the Astrophysical Research Con-sortium (ARC) for the Participating Institutions. The Partici-pating Institutions are the University of Chicago, Fermilab, theInstitute for Advanced Study, the Japan Participation Group,Johns Hopkins University, the Korean Scientist Group, LosAlamos National Laboratory, the Max-Planck-Institute for As-tronomy (MPIA), the Max-Planck-Institute for Astrophysics(MPA), New Mexico State University, University of Pitts-burgh, Princeton University, the US Naval Observatory, and theUniversity of Washington.

APPENDIX A

THE k ¼ þ1, k ¼ 0, AND k ¼ �1 CASES

Although the observations suggest that the k ¼ 0 case is appropriate for the universe, for mathematical completeness we considerthe general Friedmann metrics:

ds2 ¼�dt2 þ a2 tð Þ d�2 þ sin2� d�2 þ sin2� d�2

� �� ; k ¼ þ1;

�dt2 þ a2 tð Þ d�2 þ �2 d�2 þ sin2� d�2� ��

; k ¼ 0;

�dt2 þ a2 tð Þ d�2 þ sinh2� d�2 þ sin2� d�2� ��

; k ¼ �1:

8><>: ðA1Þ

Define the conformal time by

� tð Þ ¼Z t

0

dt

a¼

Z a tð Þ

0

�ka2 þ 8�

3a4 �m að Þ þ �r að Þ½ � þ �

3a4

� �1=2

da: ðA2Þ

If we are currently at the epoch t0, then the current conformal time is �(t0). When we look back to a redshift z, we are seeing anepoch when a0/a(t) ¼ 1þ z and out to a comoving distance � given by

� zð Þ ¼ � t0ð Þ � � tð Þ ¼Z a t0ð Þ

a t0ð Þ= 1þzð Þ�ka2 þ 8�

3a4 �m að Þ þ �r að Þ½ � þ �

3a4

� �1=2

da: ðA3Þ

As we have noted, in the flat case k ¼ 0, we are free to adopt a scale for a, so we set a(t0) ¼ RH0.

For the k ¼ þ1 case, a two-dimensional slice through the universe is a sphere. So we need tomake a conformal projection of the sphereonto a plane, and then we can apply our conformal logarithmic projection as before to produce our map. The stereographic map projectionis such a conformal projection of a sphere onto a plane. Adopt coordinates on the sphere of�, �, where the metric on the sphere is given by

ds2 ¼ a2 t0ð Þ d�2 þ sin2� d�2� �

; ðA4Þ

where the angle � is now a longitude. (Thus, � is equivalent to � in the metric given by the second line of eq. [A1], where we areconsidering the equatorial slice with � ¼ const ¼ �/2. So to get the metric above from the metric given by the second line of eq. [A1],we set � ¼ const ¼ �/2 and replace �with �.) Now make a stereographic conformal projection of this sphere (�, �) onto a plane withpolar coordinates (r, �):

r ¼ 2a t0ð Þ tan �

2

�; ðA5Þ

� ¼ �: ðA6Þ

This is a projection from the north pole onto a plane tangent to the south pole. The Earth would be at the south pole of the sphere(where � ¼ 0). A line from the north pole to the plane through the point (�, �) on the sphere will be at an angle � ¼ �/2 relative to anormal to the plane, and also at an angle � ¼ �/2 relative to a normal to the surface of the sphere at the point (�, �); thus, theforeshortening that occurs along the ray from the north pole as it crosses the surface of the sphere is exactly the same as theforeshortening that occurs when it crosses the surface of the plane. Thus, shapes in the surface are locally mapped without distortionlocally onto the plane. The map is conformal. Now we have a conformal map of the sphere and we apply our logarithmic conformalmap to the planar map to get our map of the universe:

x ¼ ��; ðA7Þ

y ¼ lnr

rE

�¼ ln

2a t0ð Þ tan �=2ð ÞrE

� : ðA8Þ


This provides a conformal mapping from (�, �) to (x, y). It is in fact a Mercator projection of the spherical section of the universewith the Earth located at the south pole of the sphere! We are used to seeing the Mercator projection for the surface of the Earth cutoff at the Antarctic Circle, so we do not see that if it were extended much nearer to the south pole (in the limit as � ! 0), it wouldapproximate our logarithmic map of a plane in the region of the south pole. (This projection would be useful for cosmologicalmodels that were slightly closed.)

For the k ¼ �1 case, a two-dimensional slice of the universe is a negatively curved pseudosphere with metric

ds2 ¼ a2 t0ð Þ d�2 þ sinh2� d�2� ��

; k ¼ �1; ðA9Þ

where � is the comoving radius, and as before we have set � ¼ const ¼ �/2 in the metric given by the third line of equation (A1)above to look at an equatorial slice, andwe have replaced � in themetric given by the third line of equation (A1) with �, so that � is nowa longitude. We can conformally project the pseudosphere onto a plane with a map projection that is an analog of the stereographicprojection for the sphere. The metric above for a pseudosphere of radius a(t0) is the metric on the hyperboloid surface:

x2 þ y2 � t2 ¼ �a2 t0ð Þ; ðA10Þ

where t > 0 in a three-dimensional Minkowski space with metric

ds2 ¼ �dt2 þ dx2 þ dy2: ðA11Þ

Define coordinates on this surface (�, �) by

t ¼ a t0ð Þ cosh �; ðA12Þy ¼ a t0ð Þ sinh � sin �; ðA13Þx ¼ a t0ð Þ sinh � cos �: ðA14Þ

With the definitions above it is easy to see that x2 þ y2 � t2 ¼ �a2(t0), since a2(t0) sinh2� cos2�þ a2(t0) sinh

2� sin2��a2(t0) cosh

2� ¼ �a2(t0). Connect each point (�, �) on the surface with the origin (x; y; t) ¼ (0; 0; 0) via a straight worldline. Thisworldline has a velocity relative to the t-axis of v ¼ sinh �/cosh �¼ tanh �. Hence, this worldline has a boost of � relative to the t-axis.This worldline is normal to the surface at the point (�, �) because that boost takes the t-axis to the worldline in question and leaves thehyperbolic surface invariant. Let the intersection of that worldline with the tangent plane t ¼ a(t0) be the gnomonic map projection of thepoint (�, �) onto the plane with polar coordinates (r, �). Then

r ¼ a t0ð Þ tanh �; ðA15Þ� ¼ �: ðA16Þ

This gnomonic projection maps the pseudosphere 0 � 1 into a disk of radius r ¼ a(t0). Why? Because in a time t ¼ a(t0) theworldline with a velocity of v ¼ tanh � travels a distance of r ¼ vt ¼ a(t0) tanh �. This gnomonic projection is not conformal becausethe worldline is normal to the hyperboloid surface but not normal to the plane t ¼ a(t0). This gnomonic projection maps geodesics onthe negatively curved hyperboloid onto straight lines on the plane t ¼ a(t0) because any plane in the space (x, y, t) passing through theorigin intersects the surface in a geodesic and this plane will intersect the tangent plane t ¼ a(t0) in a straight line.

A conformal projection, like the stereographic projection of the sphere, is provided for the pseudosphere by connecting with aworldline each point on the surface (�, �), to the point (x; y; t) ¼ ½0; 0; �a(t0)�, and letting this worldline intersect the plane t ¼ a(t0)at a point (r, �). Now that worldline has a velocity v ¼ sinh �/(cosh �þ 1) ¼ tanh(�/2) (using a half-angle trigonometric identity).Thus, this worldline has a boost relative to the t-axis of �/2 and therefore a boost of �/2 relative to the normal to the plane t ¼ a(t0).When it intersects the hyperbolic surface at the point (�, �), the normal to the hyperbolic surface at that point has a boost of � relativeto the t-axis in the same plane. Thus, the worldline has a boost of �/2 relative to the normal of the tangent plane and also a boost of �/2relative to the hyperbolic surface, so it has an equal boost (and observes equal Lorentz contractions) relative to both (giving identicalforeshortenings as in the stereographic projection of the sphere) and therefore the map projection is conformal. This worldlineconnecting the point (x; y; t) ¼ ½0; 0; �a(t0)� to the point (�, �) on the surface thus intersects the tangent plane t ¼ a(t0) at the point (r,�) given by

r ¼ 2a t0ð Þ tanh �

2

� �; ðA17Þ

� ¼ �: ðA18Þ

This maps the pseudosphere 0 � < 1 into a disk of radius r ¼ 2a(t0). Now we have a conformal map projection of the negativelycurved k ¼ �1 universe onto a plane. This conformal map is the one illustrated in Escher’s famous angels and devils print, which isoften used to illustrate this cosmology (see Gott 2001, p. 175). Next we take this conformal planar map (r, �) and apply our conformallogarithmic mapping function to it. Thus, we produce a conformal map of the universe with coordinates

x ¼ ��; ðA19Þ

y ¼ ln2a t0ð Þ tanh �=2ð Þ

rE: ðA20Þ

Note the similarity with the formula for a spherical k ¼ þ1 universe; in the k ¼ �1 case ‘‘tanh’’ simply replaces ‘‘tan.’’


We can make maps for all three cases. These are useful for comparison purposes. For example, suppose we develop an isotropymeasure for voids as suggested by Barbara Ryden. We could apply this to the voids on our conformal universe map since it preservesshapes locally and the voids are small. The shapes of the voids depend on the assumed cosmological model. This is the Alcock-Paczynski test. If we have the right cosmological model, the voids will be approximately round as suggested by Ryden, and this can betested. Suppose, asWMAP suggests, that the flat k ¼ 0 case is correct with�m þ �� ¼ 1. Then we can plot the same redshift data butconformally assuming slightly closed �m þ �� > 1, k ¼ þ1 models or slightly open �m þ �� < 1, k ¼ �1 models and checkisotropy of the voids in each case. We can thus check how sensitive the isotropy test is in limiting the location of the model in the (�m ,��) parameter plane.

APPENDIX B

SOURCES

In applying the logarithmic map to the problem of showing the universe, we have used a multitude of online sources, privatecommunications, and data published in journals. To illustrate the wealth and variety of data depicted in the map, we have chosen to listthe sources in this section:

Moon phase: United States Naval Observatory Astronomical Almanac, http://aa.usno.navy.mil/data/docs/MoonPhase.html.Earth geological data: Allen’s Astrophysical Quantities (Cox 2000).Artificial satellites: MikeMcCants’s Satellite TrackingWeb sites (alldat.tle file). The file includes orbital elements supplied by OIG

(NASA/GSFC Orbital Information Group) and data on other satellites obtained primarily by amateur visual observations (see http://oig1.gsfc.nasa.gov and http://users2.ev1.net/~mmccants/tles/index.html).

Hubble Space Telescope location: A. Patterson and D. Workman (Science and Mission Scheduling Branch, Operations and DataManagement Division, Space Telescope Science Institute) 2003, private communication.

Chandra X-Ray Observatory location: R. Cameron (CXO Science Operations Team) 2003, private communication.Asteroid catalog: Lowell Observatory Asteroid Database (ASTORB), 2003 April 19 snapshot, with ephemeris calculated for 2003

August 12 (ftp://ftp.lowell.edu/pub/elgb/astorb.html).Minor bodies’ ephemeris: Ephemeris computed using an adapted version of the OrbFit software package. OrbFit is written by the

OrbFit consortium: Department of Mathematics, University of Pisa (Andrea Milani, Steven R. Chesley), Astronomical Observatoryof Brera, Milan (Mario Carpino), Astronomical Observatory, Belgrade (Zoran Knezevic), CNR Institute for Space Astrophysics,Rome (Giovanni B. Valsecchi) (see http://newton.dm.unipi.it /~neodys/astinfo/orbfit).

Quaoar and Sedna information: http://www.gps.caltech.edu/~chad.Comet, Sun, Moon, and planet ephemeris: Generated using JPL HORIZONS Online Solar System Data and Ephemeris Com-

putation Service (see http://ssd.jpl.nasa.gov/horizons.html).Space probes: Voyager 1, Voyager 2, Pioneer 10, andUlysses positions obtained from NASA HelioWebWeb site at National Space

Science Data Center (NSSDC) (http://nssdc.gsfc.nasa.gov/space/helios/heli.html).Heliopause: Distance taken to be approximately 110 AU in the direction of approximately 18h R.A.Oort Cloud: Taken to extend from 8000 to 100,000 AU.Ten brightest stars: Source for positions and parallaxes—SIMBAD Reference database, Centre de Donnees astronomiques de

Strasbourg (http://simbad.u-strasbg.fr/sim-fid.pl).Ten nearest stars: Source for positions and parallaxes—Research Consortium on Nearby Stars, list as of 2004 July 1 (http://

www.chara.gsu.edu/RECONS/TOP100.htm).Extrasolar planets: IAU ‘‘Working Group on Extrasolar Planets’’ list of planets, the ‘‘Extrasolar Planets Catalog’’ of the ‘‘Extra-

solar Planets Encyclopedia’’ maintained by Jean Schneider at CNRS-Paris Observatory (see http://www.ciw.edu/IAU/div3/wgesp/planets.shtml, http://www.obspm.fr/encycl/encycl.html, and http://cfa-www.harvard.edu/planets/OGLE-TR-56.html).

Hipparcos stars: ESA (1997), The Hipparcos and Tycho Catalogues, ESA SP-1200 (http://tdc-www.harvard.edu/software/catalogs/hipparcos.html).

Selected Messier objects: Distances and common names taken from the Students for the Exploration and Development of Space(SEDS) Messier Catalog pages. Positions taken from the SIMBAD reference database (http://www.seds.org/messier).

Milky Way: Disk radius taken to be 15 kpc. Distance to Galactic center taken to be 8 kpc.Local Group: Data taken from a list of Local Group member galaxies maintained by SEDS (http://www.seds.org/~spider/spider/

LG/lg.html).Great Attractor: Location data from SIMBAD, redshift distance cz ¼ 4350 km s�1.Great Wall: Contours based on CfA2 redshift catalog, subset CfA2. The contours are based on galaxies satisfying �8N5 < � < 42N5,

120� < � < 255

�, and 0:01 < z < 0:05 comprising the CfA2’s first six slices (Geller & Huchra 1989; see http://cfa-www.harvard.edu/

~huchra).SDSS data: Plotted from raw SDSS spectroscopy data obtained using David Schlegel’s SPECTRO pipeline (spALL.dat file) dated

2003/01/15 with a z < 5 inclusion cut applied. SDSS quasar data with z > 5 provided by M. Strauss (2004, private communication)(http://spectro.princeton.edu).

Plotting: Plotted using SM software by R. H. Lupton (http://astro.princeton.edu/~rhl/sm).Individual quasar data: SIMBAD database.Cosmological data: WMAP Collaboration publications (Bennett et al. 2003).WMAP location in space at the time of the map (D. Fink et al. 2003, private communication).


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