ge-scale - Federation of American Scientists · (b) A galaxy cluster—a structure of intermediate size—in the constellation Leo. (Photograph courtesy of Palomar/California Institute

58

Los Alamos Science Number 22 1994

Cold dark matter, illustrated here in blue and black, isshown forming halos around galaxies and connecting thegiant collection of galaxies known as the Great Wall. Thepicture is an artist’s conception, as cold dark matter hasnever been observed. Its existence is called for by onetheory of how gravity caused the growth of large-scalestructures such as those pictured here. The latest simula-tion “experiments” on massively parallel computers havesufficient speed and accuracy to test the cold-dark-matterscenario.

Acritical limitation in astrophysics is the impossibility oftesting theories with controllable and repeatable experi-

ments. Cosmologists face the worst version of this problem.Not only are the data restricted to observations of uncontrolledevents, but also the experiment was performed only once—thereis only one universe! Moreover, only a fraction of the universethat is in principle observable (given the finite speed of light andthe finite age of the universe) is accessible in practice to observation. Cosmology has suffered from havingtoo little hard data and too much freedom to build theoreticalmodels, so that even the most basic questions—about the size ofthe universe, the age of the universe, the kinds of matter in theuniverse—are still only partially answered.

and the puzzle of large-scale structures Wojciech H. Zurek and Michael S. Warren

59Los Alamos Science Number 22 1994

This situation is changing throughadvances in observation and computa-tion. Instruments such as the HubbleSpace Telescope and the Cosmic Back-ground Explorer satellite are providingnew data, some relating to the veryearly history of the universe, that helpconstrain cosmological theories. Fur-thermore, computer simulations carriedout on the most advanced massivelyparallel machines now contain enoughphysics and are sufficiently accurate topredict detailed consequences of themany proposed models. Here we pre-sent high-resolution computer simula-tions that address one of the outstand-ing puzzles in modern cosmology:How did the observed distribution ofgalaxies—the so-called large-scalestructure of the universe—arise?

Cosmologists once thought that thematter distribution on large scales isfairly uniform, that galaxies like oursare sprinkled evenly throughout thecosmos. Over the last few decades,however, observations out to distancesof billions of light-years* have revealeda “froth-like” structure to the universe.Large numbers of galaxies are groupedinto clusters, and those clusters appearto be interconnected by thin sheets andfilaments of galaxies surrounding largelow-density cells, or “voids,” approxi-mately a hundredth of the radius of theuniverse in size. Figure 1 illustratessome of these large-scale structures,from individual galaxies with brightcomponents on the order of 10 kilopar-secs in radius (a parsec is about 3.3light-years), to clusters of galaxies onscales of 1 to 10 megaparsecs, and fi-nally to sheets or filaments of galaxy

clusters that surround large voids onscales of 30 to 50 megaparsecs. Bycomparison, the current size of the ob-servable universe is over a hundredtimes larger, roughly 10,000 mega-parsecs across.

In confronting the data on large-scale structure, the task of a cosmolo-gist is somewhat similar to the task of aprosecutor trying to prove that the ac-cused (a cosmological model) is indeedguilty of the crime (the creation of thelarge-scale structure), while his obser-vational colleagues are trying either tosupport the case by supplying the ob-servational evidence (which is nearlyalways circumstantial) or to side withthe defense and provide the alibi. Thecrime has, of course, happened onlyonce, and over the years many suspectshave been rounded up. It is conceiv-able, though, that the guilty party hasnot yet come under suspicion. Howev-er, many possible models have alreadybeen exonerated.

Consider the idea that nearly all thematter in the universe is contained invisible, luminous stars and that gravityis the force primarily responsible forthe clumping of matter into structuresof various sizes. After all, that ideadoes indeed describe our solar system.But on galactic and larger scales, thatnatural model has been “found inno-cent.” As we shall see below, it cannotaccount for the internal dynamics ofspiral galaxies. Instead those galaxiesmust contain about ten times moremass than has so far been observed toexplain the rapid motion of the stars in-side them.

More important for our discussion, itis difficult to explain how gravitationalforces alone produced large-scale struc-ture unless we assume the presence ofstill greater amounts of invisible mass—what is usually referred to as dark mat-ter. It seems that there is neitherenough mass in the luminous matter

nor enough time since the beginning ofthe universe for primordial fluctuationsin the density of that matter to havegrown under the influence of gravityinto the structures now observed. Thesize of the primordial fluctuations, asinferred from inhomogeneities in thecosmic background radiation, is just notlarge enough. Thus, observations ongalactic and larger scales present clearevidence that the universe containsmuch more mass than “meets the eye.”

What is this dark matter that domi-nates the mass content of the universe?Astrophysicists have responded to thispuzzle with a variety of ideas. Somepostulate that the dark matter is more orless ordinary (that is, made of the samestuff as our sun and the planets) but, forsome reason, simply invisible. Calcula-tions of the primordial synthesis of thelight elements, however, place an upperlimit on the amount of ordinary matterthat might be present, a limit that isprobably too low for ordinary matteralone to explain large-scale structure.Other astrophysicists suggest that neu-trinos with a small, but non-negligible,mass are the main ingredient of darkmatter. The latter postulate leads to theso-called hot-dark-matter (HDM) mod-els: Massive but not-too-heavy neutri-nos remain relativistic or “hot” (theymove with velocities comparable to thevelocity of light), until fairly late in thehistory of the universe. Their move-ments wipe out density perturbations ongalactic scales and thereby determine acertain predictable course for the devel-opment of structure, which turns out tobe too slow to match observations.

Perhaps the best-defined model, andthe one we have tested through comput-er simulation, is the so-called cold-dark-matter (CDM) model. The mainingredient of dark matter in the CDMmodel is not specified beyond the re-quirements that it be very weakly inter-acting with ordinary matter and suffi-

Experimental Cosmology and the Puzzle of Large-Scale Structure

60 Los Alamos Science Number 22 1994

*These distances and all the other absolute dis-tances in this article are uncertain by approxi-mately a factor of 2 for various reasons, primari-ly the uncertainty in the Hubble parameter.Relative distances can however be expressed in amanner independent of those uncertainties byusing the redshift, which will be introduced inthe following section.


Number 22 1994 Los Alamos Science 61

Figure 1. Inhomogeneity of the UniverseThe distribution of matter in the universe is marked by clumps and voids on all scales. (a) The spatial distribution of galaxies ac-

cording to a survey by the Harvard-Smithsonian Center for Astrophysics. Each point represents a galaxy. The distance from the

vertex (our galaxy) to the outer edge of the region surveyed is roughly 300 megaparsecs. The distances plotted are inferred from an

observed measure of distance called redshift, z, through the use of Hubble’s Law—distance

5 v/H0, where v

< cz for objects with

small redshifts—as will be discussed later in the main text. Here c is the velocity of light and H0 is Hubble’s constant. The elongat-

ed clump of galaxies in the center of the figure includes the rich cluster in the constellation Coma Berenices. It is part of the “Great

Wall,” the large sheet of galaxies running across the image at intermediate distances, which includes more than half the galaxies in

the figure. The Great Wall and the voids beside it are among the largest known structures in the universe. (Figure adapted by per-

mission of Margaret J. Geller, Harvard-Smithsonian Center for Astrophysics.) (b) A galaxy cluster—a structure of intermediate

size—in the constellation Leo. (Photograph courtesy of Palomar/California Institute of Technology.) (c) The nucleus of a single spi-

ral galaxy, catalogued as M100, with a radius on the order of 10 kiloparsecs. Galaxies are the smallest objects we consider in our

study of large-scale structure. (Photograph courtesy of NASA/Space Telescope Science Institute.)

100

200 megaparse

cs

(<600 m

illion lig

ht-years)

(a) Harvard-Smithsonian survey of spatial distribution of galaxies.

(b) A galaxy cluster (c) Nucleus of the spiral galaxy M100

LOCAL SUPERCLUSTER

LOCAL SUPERCLUSTER

T H E G R E A T W A L L

COMA CLUSTER

T H E G R E A T W A L L

COMA CLUSTER

Our galaxy

23 kiloparsecs (<70,000 light-years)

200 kiloparsecs

(<700,000 light-years)

Distance < cz (megaparsecs)

H0

ciently massive to become “cold,” ornonrelativistic (moving at velocitiessmall compared with the velocity oflight), very early in the history of theuniverse, before the universe was abouta thousand years old. The model as-sumes, as an initial condition, a simple,scale-independent spectrum of primor-dial density fluctuations in the colddark matter that are postulated to growunder the influence of gravity to formthe large-scale structures that we nowsee.

The only important free parameter inthe CDM model is the size of the as-sumed primordial density fluctuations.Information about those fluctuations isimprinted in the cosmic background ra-diation, the oldest detectable remnant ofthe early universe. That sea of mi-crowave photons, which fills all space,is thought to have remained essentiallyunscattered since the time when radia-tion decoupled from matter and the uni-verse became transparent to radiation.It would therefore have retained inho-mogeneities present at the time of de-coupling, when, according to standardBig Bang Cosmology, the universe wasabout 10,000 years old.

The size of those inhomogeneitieswas recently determined by data fromthe Cosmic Background Explorer satel-lite. Known as COBE (pronounced torhyme with Toby) the satellite is dedi-cated to measuring various properties ofthe cosmic background radiation. The1992 COBE data show variations in thetemperature of the cosmic backgroundradiation of about one part in 105 ondistance scales of around 1 billion lightyears (hundreds of megaparsecs), de-pending on the spatial direction of themeasurements. The small temperaturefluctuations are a direct measure of theamplitudes of the fluctuations in thematter density present when the cosmicbackground originated. Thus the sizeof the temperature differences measured

by COBE fixes the size of the primor-dial density fluctuations in the CDMmodel.

Our contribution has been to incor-porate the new COBE data into the ini-tial conditions of the CDM model andthen determine, through high-resolution,state-of-the-art simulations, the CDMpredictions for the growth of structurethrough time. Our simulations keeptrack of the long-range gravitationalforces among 17 million point masseswith a precision sufficient to resolvedensity contrasts of six orders of mag-nitude on scales that differ by as muchas four orders of magnitude. Our pro-gram has undergone continual develop-ment since it received the 1992 GordonBell Prize for practical parallel-process-ing research. It is described in “A FastTree Code for Many-Body Problems,”immediately following this article.

The high resolution that we haveachieved has allowed much more de-tailed comparison with observationsthan was previously possible. Our re-sults discussed on page 78 suggest thatthe CDM model, recently abandoned bymany, should still be taken seriously.

Before presenting those comparisons,we will explain the initial conditions ofthe simulations and the basic assump-tions of the CDM model through a re-view of standard Big Bang cosmology.The CDM model and all other recentmodels of structure formation are de-fined within that basic framework, aframework that has been strengthenedby all recent observations.

The Expansion of the Universe

Big Bang cosmology grew out of themost far-reaching and well-acceptedcosmological observation: The uni-verse is expanding. In other words, toany observer in the universe, distant ob-jects such as galaxies appear to be re-

ceding from the observer at velocities vproportional to l, the distance from theobject to the observer. This relation,called Hubble’s law, was discovered byEdwin Hubble in 1929. It is written

v

5 H0l.

The Hubble “constant,” H0, is short-hand for H(t0), which means the valueof the Hubble parameter H(t) measurednow, at t 5 t0. Although Hubble’s lawwas deduced from observations carriedout from our vantage point, the planetEarth, it is generalized to the cosmosby assuming that at any given time theuniverse looks qualitatively the same toall observers regardless of their loca-tions and the directions in which theylook. This assumption, called the Cos-mological Principle, is fundamental tothe study of cosmology.

Figure 2 shows a traditional way ofvisualizing the expansion of the uni-verse. The three-dimensional universeis replaced by an analogous two-dimen-sional surface, the surface of a sphere.As the sphere expands, any observerconfined to the surface sees other pointsreceding with speeds proportional totheir distances from the observer, as ob-served by Hubble.

Hubble discovered this proportionali-ty, and thus the expansion of the uni-verse, by measuring both the apparentvelocities v of nearby galaxies and theirdistances l. Measuring the distances togalaxies is difficult, involving a ladderof distance measurements from nearbystars (tens of parsecs away) to moredistant stars in our own galaxy (up to10 kiloparsecs away) to stars in nearbygalaxies (1 to 10 megaparsecs away)and on to more distant galaxies. Theprocess remains problematic, so thateven today uncertainties in the distancemeasurements yield estimates of H0ranging from 50 to 100 kilometers persecond per megaparsec. A convenient



way to represent this uncertainty is todefine a parameter h such that H0 5h 100 (km/s)/Mpc, where 1/2 # h # 1(Mpc is the abbreviation for a mega-parsec). To ensure that our calculationsdo not assume a universe too young tobe consistent with other predictions ofastrophysics, we have taken the valueof h to be 1/2, or the value of H0 to be50 (km/s)/Mpc, throughout this articleand in our simulations.*

In contrast to the distance, the appar-ent velocity of a receding galaxy is reli-ably and easily determined by measur-ing the Doppler shift—in this case theredshift—of the light that it emits.More specifically, one identifies the ab-sorption and emission lines in the spec-trum of the galaxy and measures theshift of those features toward longerwavelengths, in the direction from bluetoward red. (Here again is an applica-tion of the Cosmological Principle:One is assuming that the matter (atoms)and the wavelengths of the light itemits are the same throughout the uni-verse.) The size of the redshift z is re-lated to the apparent recession velocityv through the Doppler-shift formula:

11z ; 5 1 21/2

,

in which lemitted is the wavelength emit-ted by the galaxy, lobserved is the wave-length of the observed signal, and c isthe speed of light. For galaxies withsmall redshifts, or with small recessionvelocities v ,, c, the redshift z < v/c, orthe recession velocity is given by

v < zc.

Thus the velocities of nearby galaxies

are directly proportional to their ob-served redshifts. The formula applieseven to the most distant galaxies shownin Figure 1. They have redshifts of0.05 and by Hubble’s law are at dis-tances of l < zc/H0, or 300 mega-parsecs for h 5 1/2.

In contrast, the most distant objectobserved to date is a quasar with a red-shift of nearly 5. The full Doppler-shiftformula implies that that object is rush-ing away from us at the enormous ve-locity of 0.95c or about 285,000 kilo-meters per second, and, by Hubble’slaw, is at a distance of approximately5700 megaparsecs.

In general, the higher the redshift ofan object, the greater is its distancefrom us, the faster is its recession ve-locity, and the longer its signal took toreach us. The observed redshift of anobject thus indicates the “look-backtime,” a fact that is used again andagain to interpret cosmological observa-tions and to piece together the historyof the universe.

Cosmologists typically relate theredshift not to the look-back time butrather to the relative size of the uni-verse, because the latter relationship issimpler. The redshift of a light signal(more exactly, 1 1 z) is inversely pro-

11v/c12v/c

lobservedlemitted



*The new corrective optics recently installed inthe Hubble Space Telescope should help put thecontroversy about the value of H0 to rest by skip-ping several of the rungs in the ladder of distancemeasurements.

astrofig2.adb 7/26/94

R(t)

t

t0

R(t0)

l(t0)

l(t)

Figure 2. The Expansion of the UniverseThe Hubble flow and the expansion of the universe are depicted here in a spherical

fragment of the real universe. The size of the sphere, R (t ), changes with time, as do

the distances l(t ) between the objects (such as galaxies) inside. If all of the objects

started expanding from the same point at different velocities v, then their present loca-

tions will depend on how fast they are moving. This dependence yields a proportional-

ity relation, l 5 v/H, which is the content of Hubble’s law. The velocity of a galaxy can

usually be inferred from the redshift, z, of the light emitted by it, and when z ,, 1 the

velocity is given by v 5 zc. Because the light travels at a finite velocity, the redshift is

also a measure of the “look-back time,” which in turn is related to the relative size of

the universe at the time when the photons were emitted: Expansion “stretches” the

wavelengths of the photons and the size of the universe at the same rate, so that the

photons emitted at the instant t and detected now, at t0, will be redshifted—the photon

wavelengths will be longer by the redshift factor 1 1 z 5 R (t0)/R (t ). It is convenient to

define a universal scale factor a(t )—the factor by which all of the distances in the real

universe need to be rescaled to account for the effect of the universal expansion.

Thus R(t0)/R(t )5 l(t0)/l(t ) 5 a(t0)/a(t ). By definition a(t0) 5 1. Thus a(t ) 5 1/[1 1 z(t )].

portional to the relative size, or scale,of the universe at the time that signaloriginated. Here the size of the uni-verse is measured by the distances be-tween nearby objects, l(t), or by the ra-dius, R(t), of a spherical fragment ofthe universe. As depicted in Figure 2,both quantities increase with the univer-sal expansion. The relative size—theuniversal scale factor—is then definedas the ratio

a(t) ; l(t)/l(t0) ; R(t)/R(t0).

In other words, the universal scale fac-tor is defined to be one at present,a(t0) 5 1, and its value decreases tozero as we go back in time.

The inverse proportionality betweenthe redshift and the universal scale fac-tor can be derived by reinterpreting theorigin of the redshift as due to the over-all expansion of the universe. As theuniverse expands, the distances betweenobjects co-moving with the universalexpansion increase proportionally toR(t). In addition, the wavelengths ofphotons traversing the universe at ve-locity c are stretched by the expansionin proportion to R(t). The ratio ofwavelengths in the redshift formula cantherefore be written as

5 ,

where t0 is now, the time at which thesignal is observed, and t is the time atwhich the signal originated. Rewritingthis equation in terms of the universalscale factor shows that the redshift in-creases as the scale factor decreases:

1 1 z(t) 5 5 .

This equation means, for example,

that when we observe a quasar with aredshift of 5 (or 1 1 z 5 6), we arelooking back in time to when distancesbetween astronomical objects were ap-proximately 1/(1 1 z) 5 1/6 of theirpresent values. According to standardBig Bang cosmology, the cosmic back-ground photons, which contain the old-est imprint of large-scale structure,have been redshifted by a factor ofabout 1000 (z < 1000), which meansthat those photons decoupled from mat-ter at the time when the scale factorwas only one-thousandth of its presentsize. Our simulations of the clumpingof matter into large-scale structuresbegin later, at z < 100, or when thescale factor was approximately one-hundredth of its present size.

The Big Bang and theDefinition of V

Extrapolating the expansion of theuniverse all the way back in time sug-gests that initially any point was arbi-trarily close to any other point, orR(t50) < 0 and a(t50) < 0. Such apicture is consistent with solutions tothe equations that arise when generalrelativity is applied to the entire cos-mos; it is also consistent with observa-tions. Therefore cosmologists nowgenerally believe that the universe ex-panded and cooled from an initial stateof extremely high temperature and den-sity. Cosmologists can trace the historyback no farther than the time when thedensity of the universe was the so-called Planck density (roughly 5 31033 grams/centimeter3). At that densi-ty quantum-gravity effects begin todominate. The initial explosive expan-sion is called the Big Bang.

How much time has elapsed sincethe Big Bang? We need to estimate theage of the universe to determine thetime available for the development of

large-scale structure. The simplest esti-mate is made by assuming that theHubble parameter has remained con-stant at its present value of H0 and thusthe speed of expansion has also re-mained constant since the Big Bang.Then t0, the time since the Big Bang, isapproximated by

t0 <

and, by Hubble’s law, that time is

t0 < < 20 billion years

where we have taken the value of theHubble constant to be 50 (km/s)/Mpc.

The true age of the universe must besomewhat less than 20 billion years be-cause the initial expansion at t 5 0must have slowed down over time dueto the mutual gravitational attraction ofthe total energy content, or equivalentmass content, in the universe. The the-ory of relativity relates all forms of en-ergy E, even pure radiation, to anequivalent mass m through the formulaE 5 mc2. Thus both matter and radia-tion are sources of gravitation and con-tributed to the slowing down of the ini-tial expansion. As the universe hasexpanded, the Hubble parameter H(t)has been decreasing continuously to itspresent value H0.*

The time history of the expansion istypically described in terms of the Hub-ble parameter where

1H0

distance expandedspeed of expansion

1a(t)

R(t0)R(t)

R(t0)R(t)

lobservedlemitted



*For H0 5 50 (km/s)/Mpc and V 5 1 (seebelow), the age of the universe is 14 billionyears, approximately the age of the oldest globu-lar clusters inferred from stellar evolution andprimordial abundances of the light elements. Alarger value of H0, or equivalently h, wouldimply a younger universe and could thus lead toa contradiction, or at the very least, indicate alow value for V.



Figure 3. Evolution of the UniverseEach curve shows a possible history of the size of the universe as a function of time,

depending on whether the average equivalent mass density rw of the universe is greater

than, equal to, or less than the critical value, rcritical, or equivalently on whether V is

greater than, equal to, or less than 1. The curves can be obtained by using the New-

tonian approximation to predict the motion of a spherical fragment of the universe of

radius R (t) and containing mass M equal to 4pR3rw /3. The Newtonian approximation

can be derived from Einstein’s equations when GM/Rc2 ,, 1, that is, when R (t ) is large

compared to the Schwarzschild radius of the mass M. Then the radius of the sphere

satisfies the familiar equation

5 2 ,

which can be used in turn to derive the equally

familiar statement of conservation of energy in

a gravitating system:

1 22

5 1 C.

The constant C on the right-hand side appears

in the course of the integration involved in the

transition from the first equation to the second.

Its sign determines the balance between the ki-

netic energy of expansion (proportional to the

left-hand side of this equation) and the potential

energy (on the right-hand side). By dividing

both sides of equation 2 by R2 (to express the

right-hand side in terms of the density rw(t )

rather than mass M) and re-expressing R(t ) as

R0a (t ), one eventually arrives at the equation

H(t )2 ; 1 22

5 2 .

Here k is the constant C from equation 1, rescaled so that it can assume only the val-

ues 61 and 0. It is known as the curvature constant. It should be emphasized that the

above derivation is perfectly correct, and not merely a Newtonian analogue: The con-

dition R .. GM/c2 can always be satisfied by adopting a sufficiently small radius.

Each curve is a solution to the equation obtained by using one of the different values

of k (and by assuming that matter, not radiation, makes the dominant contribution to

the energy density). Regardless of the value of k and the composition of the universe,

solutions to the equation have R 5 0 at t 5 0; that is, they entail a Big Bang at the be-

ginning of the universe. Also, the equation implies that k 5 11 if V ; rw(t )/rcritical(t ) .

1, where rcritical(t ) ; 3(H (t ))2/8pG. Likewise k 5 0 if V 5 1, and k 5 21 if V , 1. The

value of k, or equivalently the value of V, determines whether the universe is open, flat,

or closed, as explained in the main text.

kc2

R2

8pGrw(t )

3dadt

1a2

2GM

RdRdt

GMR2

d2Rdt2

R ( t )Open universe Ω < 1

Flat universe Ω = 1

Closed universe Ω > 1

Time (t )

Rad

ius

of c

urva

ture

of t

he u

nive

rse


Ω =ρ

ρcritical

ρ(t 0)

3H 2 /8πG0

=

0 The

Big Bang

(2)

(1)

(3)

H(t) ; 5 ,

and a(t) is the universal scale factor in-troduced above. The relevant differen-tial equation from general relativity ispresented in the caption for Figure 3.This equation yields three differenttypes of universe, depending on the av-erage density of the universe, or moreexactly, on the value of V, the dimen-sionless density parameter. This para-meter is defined as the ratio of the aver-age equivalent mass density in theuniverse, rw(t), to a critical density:

V ;

where

rcritical(t) ; ,

and G is Newton’s gravitational con-stant. Figure 3 shows the three possi-ble histories of the expansion of theuniverse depending on whether thevalue of V is greater than, equal to, orless than one.

In Newtonian cosmology, which ap-plies when most energy is in the formof matter, V has a simple interpretation.It is just the ratio of the magnitude ofthe gravitational potential (or binding)energy of matter to the kinetic energyof the universal expansion:

V 5 .

Thus, if the potential energy is lessthan the kinetic energy (V , 1), galax-ies have enough energy to escape thepull of gravity and will travel to infini-ty. Alternately, if the potential energyis greater than the kinetic energy (V .

1), galaxies are gravitationally bound,the Hubble expansion will eventuallystop, and the universe will contractback on itself.

The same criterion defines the divid-ing line between two entirely differenttypes of universes allowed by generalrelativity. As shown in Figure 3, ifV , 1, the universe is said to be open;it is spatially infinite and will continueexpanding forever. If V . 1, the uni-verse is said to be closed; it is spatiallyfinite (being curved like the sphere inFigure 2) and will eventually stop ex-panding and begin to contract [H(t) willeventually become negative]. A thirdsolution also exists. Namely, if V isexactly equal to 1, the universe is saidto be flat; it is infinite in space andtime, but differs from an open universein that the recession speeds of galaxieseventually approach zero rather than apositive constant.*

Observational evidence suggests thatV should lie within the generousbounds 0.2 , V , 2.0. This range isobtained from observations that usuallyallow one to estimate V more directlythan by comparing the present averagedensity of the universe, rw(t0), with thecritical density:

rcritical(t0) ;

, 10229 h2 g/cm3.

This number, uncertain by a factor of 4because of the uncertainty in the valueof the Hubble constant, corresponds tothe mass density of a few hydrogenatoms per cubic meter.

Determinations of V tend to yieldlarger values when they employ obser-vations on larger scales. Thus V

inferred from galactic scales tends to beon the order of 0.1 or less, while clus-ters of galaxies (scales of a few mega-parsecs) indicate an V of 0.2 or more.On still larger scales of 30 to 50 mega-parsecs, V as large as 1 is necessary toexplain coherent flows of galaxies suchas the “Great Attractor.”

Cosmologists must assume a valuefor V to investigate the development ofstructure in the expanding universe.Although the present density of lumi-nous matter, rw luminous(t0), appears tobe less than one percent of the presentcritical density, that is Vluminous ,0.01, most theorists—for both estheticand theoretical reasons—take the caseof the flat universe (V 51) very seri-ously. In that case, most of the mass inthe universe must be in the form ofdark matter.

Formation of Structure in aCold-Dark-Matter Universe

A major event in the history sincethe Big Bang was the shift from the ra-diation-dominated era, when the uni-verse was so hot that most of the ener-gy was in the form of radiation ratherthan matter, to the matter-dominatedera, when the universe had cooleddown enough that most of the energywas in the form of matter. The mainconstituents of matter then were thetwo most stable baryons, namely pro-tons and neutrons, enough electrons tobalance the charge on the protons, andperhaps, as proposed by the CDMmodel, an exotic brand of noninteract-ing, cold, dark matter. [Note that dur-ing the first few minutes after the BigBang, nuclear synthesis reactions hadcaused most neutrons to combine withprotons (hydrogen nuclei) to form heli-um nuclei.] The transition to a matter-dominated universe (described in “BigBang Cosmology and the Microwave

3H02

8pG

Epotential

Ekinetic

3(H(t))2

8pG

rw(t)rcritical(t)

dRdt

1R(t)

dadt

1a(t)



*Note that if V is equal to 1 at any time, it re-mains constant at that value. Otherwise its valuechanges with time, but because the total energyof the universe is conserved, V cannot changefrom being less than 1 to being greater than 1 orvice-versa.

Background”), about ten thousand yearsafter the Big Bang, marks the timewhen, at least in principle, the force ofmutual gravitation attraction couldbegin causing matter to collapse intothe large-scale structures we see.

A later major event was recombina-tion, when nuclei of hydrogen and heli-um combined with electrons to formstable neutral atoms. At that time thebackground radiation, consisting ofphotons in thermal equilibrium with thematter, decoupled from the neutralatoms and expanded freely to become

the cosmic background radiation nowobserved. As explained in the sidebar“Big Bang Cosmology and the Micro-wave Background,” any density pertur-bation on a distance scale l present atthe time of recombination, trecomb,should be imprinted on the cosmicbackground as a temperature variationon a distance scale l/a(trecomb). Thusthe cosmic background provides a di-rect measure of the matter distributionat the time of recombination and an ob-servational constraint on models oflarge-scale structure formation.

So far the resolution of the instru-ments on the COBE satellite limitsmeasurements of inhomogeneity to verylarge distance scales, too large to differ-entiate the various models of large-scale structure formation. Nevertheless,density inhomogeneity was seen and itsamplitude was large enough to be con-sistent with the idea that gravity wasdecisively responsible for thee forma-tion of structure.

Initial conditions of the CDMmodel. The standard CDM model pos-



Figure 4. Evidence for the Existence of Dark MatterThe figure shows observations of two galaxies from which their rotations can be inferred; the inferred rotations provide evidence for

the existence of dark matter. On the left are photographs of the galaxies NGC 801 and UGC 2885. The images in the center are pho-

tographs of their spectra made by allowing light from the galaxies to diffract through a horizontal slit. Therefore each photograph is

the spectrum of a horizontal strip through the center of the galaxy; vertical position indicates wavelength. The heavy vertical lines

in the center are the spectra of the galactic nuclei, which are highly luminous at all wavelengths. The step-shaped horizontal lines

show the wavelengths of spectral lines of hydrogen and nitrogen as a function of distance from the center. Their shapes result from

the Doppler blueshifts and redshifts due to the rotations of the galaxies; the wavelengths of these lines are shorter on the side of

the galaxy rotating toward us and longer on the side rotating away from us. Thus the average orbital speeds of stars as a function

of radius can be deduced from the displacements of those lines as a function of distance from the center of the galaxies. The re-

sulting “rotation curves,” combining data from both sides of the galaxies, are on the right. (The vertical axis shows the rotational

velocity in kilometers second21) Note that beyond a few kiloparsecs the velocity is nearly constant. This velocity distribution is typi-

cal of spiral galaxies. An entirely different velocity distribution, decreasing to zero as r 2 1/2, would be expected if the masses of the

galaxies were concentrated at their centers, as are their distributions of luminous matter. The only convincing explanation of the

observed velocity distribution is that the galaxies are embedded in massive halos of unobservable “dark matter.” (From an article

by Vera Rubin, The rotation of spiral galaxies, Science 220: 1339–1344, 24 June 1983. Copyright AAAS. The photographs of the

galaxies were made by B. Carney using the 4-meter telescope at Kitt Peak National Observatory.)

tulates very specific initial conditionsfor the development of large-scalestructure. First it assumes that V 5 1,as suggested by inflationary models ofthe very early universe. Second, in linewith the upper limit on the density ofordinary, or baryonic, matter allowedby primordial nucleosynthesis calcula-tions (see “Big Bang Cosmology andthe Microwave Background”), the stan-dard CDM model assumes that 95 per-cent of the matter is nonbaryonic darkmatter that is essentially noninteractingwith ordinary matter. It also assumesthat the dark matter was cold, or mov-ing at nonrelativistic speeds, by the endof the radiation-dominated era. Finally,it postulates a scale-free spectrum ofprimordial density fluctuations—similarto the spectrum predicted by inflation-ary models. However, nonbaryoniccold dark matter has never been detect-ed. So why should anyone take theseassumptions about dark matter seriously?

Evidence for dark matter. The existence of dark matter (not necessari-ly cold) was first proposed by FritzZwicky in 1933 to explain how high-velocity galaxies observed in the verydense cluster known as the Coma clus-ter could remain gravitationally bound.More mass must exist in that clusterthan was visible as luminous matter. Inthe 1970s a similar type of observationwas made on the scale of single galax-ies. Figure 4 shows the circular com-ponents of the orbital velocities of starsand gas clouds in two spiral galaxiesdeduced from very careful redshiftmeasurements; the data are plotted as afunction of radius r from the centers ofthe galaxies. In each galaxy the circu-lar velocities, vcirc(r), outside a radiusof a few kiloparsecs are all approxi-mately equal to a constant, vconst. Sim-ilar results are found in all spiral galax-ies. On the other hand, quite adifferent prediction follows from the

concentration of luminous matter to-ward the centers of those galaxies. Thecentrifugal force on a star orbiting thegalactic center at a radius r must equalthe gravitational force, or

= ,

where m is the mass of the star andM(<r) is the mass of the galaxy insidethe radius r. If the mass of the galaxyis distributed similarly to the luminousmatter, one would conclude that the ve-locities of stars in roughly circular or-bits far from the center, vcirc(r), areproportional to r21/2, as is the case forplanets in our solar system. But in spi-ral galaxies (including our own—seeFigure 4) this is never the case!

To resolve the contradiction, astro-physicists postulated the existence ofinvisible (dark) matter distributed suchthat M(<r) increases approximatelywith the radius:

M(<r) < ~ r.

This distribution is less concentrated to-ward the center of the galaxy than thedistribution of luminous matter and pro-vides the additional gravitational forceneeded to keep the stars in orbit at theobserved velocities.

The dark matter in a typical brightgalaxy would form an invisible halo, asshown in the opening spread of this ar-ticle, and would have to be at least tentimes more massive than the luminousmatter to explain the observed motionsof stars in the galaxy. The estimate ofV would therefore increase from about0.01 (deduced from luminous matter) toabout 0.1. All that matter might bemade of baryons and still not violatethe constraints on baryon density pro-

vided by primordial nucleosynthesis.Indeed, recent observations of a phe-nomenon called gravitational microlens-ing suggest that dark matter in the formof old stars or dense planet-like objects,presumably made of baryonic matter, ispresent in galactic halos.

On scales of megaparsecs, the trans-lational motions of galaxies, especiallythose in clusters of galaxies, seem toimply the existence of still more darkmatter, enough to bring the value of Vto 0.2 6 0.1. On the largest scales ob-served (30 to 100 megaparsecs), thedensities deduced from comparison ofthe translational motions with the distri-bution of matter suggest that V must beat least as large as 0.3, and probablyclose to 1, which is the value assumedin the CDM model. Perhaps the mostcompelling argument in support of alarge amount of cold dark matter is thatits existence may well explain the ob-served structure of the universe withthe fewest assumptions and the mostnatural physics.

The composition of cold dark matter. What might be the composi-tion of cold dark matter? Particle theo-rists have helped attack this interestingproblem by offering up a whole list ofpossible candidates. Those candidates,such as axions and photinos, have beenpredicted in theoretical models that ex-tend the standard model of particlephysics. They are called weakly inter-acting massive particles (WIMPS) andhave both properties necessary to con-stituents of cold dark matter: First,they would interact very, very weaklywith the rest of matter and radiation; infact they would be practically unde-tectable. Second, they would have suf-ficiently large masses (typically, muchlarger than the mass of the proton) thattheir thermal velocities near the end ofthe radiation-dominated era would havebeen far slower than the speed of

v2constrG

GmM(<r)

r2

mv2circr





Figure 5. Horizon Crossing and the “Scale-Free” Spectrum The log-log plot shows the wavelengths of the Fourier modes of the primordial density fluctuations, l, as they stretch out because

of the expansion of the universe. Here the expansion is measured by the scale factor, a(t ). Also shown is the Hubble radius, RH(t )

(red), as a function of a(t ). As noted in the sidebar “Big Bang Cosmology and the Microwave Background,” a (t ) increases exponen-

tially with time (as exp(Hinflationt ) during inflation, as t1/2 during the radiation-dominated era, and as t 2/3 during the matter-dominated

era. Since the Hubble radius is defined as RH(t ) ; c/H(t ) ; ca (t )/(da/dt ), it is equal to the constant c/Hinflation during inflation; it in-

creases as (a(t ))2 during the radiation-dominated era, and it increases as (a (t ))3/2 during the matter-dominated era. Therefore after

inflation the wavelength of each mode increases more slowly than RH(t ). The wavelength of each mode crosses the Hubble radius

at a particular time (circled on the graph), depending on the initial wavelength of that mode. At that time the fluctuation is said to

“enter the horizon.” The Harrison-Zel’dovich spectrum is defined to be scale-free in the sense that each mode has the same ampli-

tude at the time it enters the horizon, or dr /r l 5 RH(t ) 5 K, where K is the same for all modes.

10-27 1

100

10-5

Hu

bb

le r

ad

ius

RH

an

d w

ave

len

gth

of

fluct

ua

tion

s (m

eg

ap

ars

ecs

)

Scale of the universe a(t ) ~ 1/z

Radiation-dominated RH ∝ (a(t ))2

Matter-dominated RH ∝ (a(t ))3/2

Inflation RH = constant

10-4 10-3

10-50

Recombination Now

Point at which a Fourier mode enters the horizon. At that point,

λ = RH, and the amplitude of the Fourier mode is

δρ ρ λ = R H

= K, where K is the same for all fluctuations.

103 Mpc—fluctuations seen by COBE

10 Mpc—intergalactic fluctuations typically observed

10-1 Mpc—fluctuations that form typical halos

105

The Hubble radius RH(t )

The changing wavelength λ of a Fourier mode of a density fluctuation

δρ ρ


light.* Density fluctuations in colddark matter on galactic and smallerscales would therefore not be wiped outby the free streaming of the cold-matterparticles from more dense to less denseregions. Consequently, those fluctua-tions could begin to grow under the in-fluence of gravity at the start of thematter-dominated era, as postulated bythe CDM model.

Primordial fluctuations in colddark matter. The power spectrum is aconvenient and succinct way of charac-terizing density perturbations in many(but not all) cosmological models.Density fluctuations as a function ofposition r can always be expressed interms of a sum of elementary sinusoidalripples, that is, through the Fourier ex-pansion of dr(r) ; r(r) 2 rw, the devia-tion of the density from its average:

dr(r) ~k

ak cos(k ? r 1 fk).

Above, k are the wavevectors of theripples, ak are the amplitudes of differ-ent modes corresponding to differentwavelengths l = 2p/k, and fk arethe phases. In a large class of cosmo-logical models (including CDM), it isassumed that phases do not matter, thatthey are random and uncorrelated be-tween different modes. When this istrue, one can focus solely on the ampli-tudes ak and characterize the fluctua-tions by their average values, that is, bythe power spectrum:

P(k) 5 kak2l,

where k ; k. In addition, the devia-tions of density from the average, dr(r)at various points r, turn out to have aGaussian distribution.

Cold-dark-matter cosmology startswith the assumption that primordial

density perturbations are Gaussian andtheir power spectrum has the form

P(k) ~ k,

which is known as the Harrison-Zel’dovich spectrum. This spectrumfollows approximately from the scaleinvariance of the process of inflation.(Inflation is a postulated period of ex-ponential expansion at very early times,which solves certain cosmological prob-lems related to causality.) This scale-free spectrum was actually proposedbefore inflationary models on the basisof an elegant feature: In a universewith P(k) ~ k and V 5 1, each densityfluctuation has the same amplitude atthe time it enters the horizon of the ob-servable universe, that is, at the time itswavelength l is equal to c times the ageof the universe, or approximately equalto the Hubble radius, c/H(t).

Figure 5 shows the increase in wave-length of each mode with time due tothe expansion of the universe; note thatthe horizon increases faster than thewavelengths of the modes, so as timegoes on, larger and larger modes fulfillthe criterion that their wavelengths areabout equal to c/H(t), at which timesthey enter the observable universe.Such a universe has the appealing fea-ture of being cosmologically scale-invariant: At any time the only impor-tant scale is defined by the size of thehorizon at that time.

Linear growth of density fluctua-tions. The initial density fluctuationspostulated by the CDM model wouldgrow under the influence of gravity be-cause, in a distribution of matter that isnearly uniform in density except forsmall “ripples,” the gravity of denserregions tends to attract more mass fromnearby regions, so the ripples becomelarger. As they become larger, they be-come even more effective at attracting

matter, and so they continue to grow,possibly becoming progenitors of galax-ies or galaxy clusters. In the CDMmodel and most models other thanHDM, gravitational collapse occurredon small scales first and then on largerscales. Thus globular clusters formedbefore galaxies, and galaxies beforeclusters of galaxies.

Until the time of recombination andfor a long time afterward, the growth ofthe density fluctuations can be modeledanalytically because their average am-plitude, or dr/rw, was small enough forthe growth to be linear. Linear growthis calculated simply by computing theindependent growth of each mode—theevolution of the power spectrum withtime. In this linear regime the fate ofeach mode depends on whether it entersthe horizon in the radiation-dominatedor in the matter-dominated era.

Modes that enter the horizon in theradiation-dominated era are, in effect,ripples in the density of a plasma that isdominated by photons. The pressure ofthe plasma prevents gravitational devel-opment of these ripples and causesthem to oscillate as sound waves. Thisstate of affairs persists until teq, thetime at which the energy density ofmatter equals the energy density of ra-diation. Only after teq, when the radia-tive contribution to the energy densitybecomes negligible, can density pertur-bations begin to grow. Thus, thegrowth of the modes with wavelengthssmaller than the horizon at teq is de-layed, and therefore stunted. As shownin Figure 5, these modes include theones that develop into galaxies.

By contrast, the modes that enter thehorizon in the matter-dominated era,well after teq, begin to grow immediate-ly as a result of gravitational attraction.Thus by the time of recombination,when the fluctuations made a lastingimprint in the cosmic background, thethe shape of the power spectrum P(k)



*Axions are an exception: They are less mas-sive, but move slowly for other reasons.

had changed from the primordial Harri-son-Zel’dovich spectrum: Modes withwavelengths less than c/H(teq) had lesspower relative to modes with wave-lengths greater than c/H(teq) than wasthe case in the primordial spectrum (seeFigure 6).

Details of the CDM power spectrumdepend somewhat on the assumed frac-

tion of the matter that consists ofbaryons because the baryonic densitydistribution is affected by interactionswith radiation during the radiation-dom-inated era, but for reasonable values ofVbaryon those effects are minor. Thecrucial assumption determining theCDM power spectrum is that the darkmatter is composed of cold particles at

the start of the matter-dominated era—when the gravity of dark matter beginsto matter!

Early computer simulations ofnonlinear growth. Given the powerspectrum of density fluctuations, P(k),at the time of recombination, one can,in principle, compute the subsequentevolution of the matter distribution inthe CDM universe for comparison withobservations. The computation requiresvery large-scale computer simulationsfor two reasons. First, on scales of tensof megaparsecs and smaller, the ob-served amplitudes of density fluctua-tions dr/rw are on the order of 1 andlarger, and so their growth must havebeen nonlinear. Second, in order to re-solve inhomogeneities on the relativelysmall scales of galaxies, 10 kiloparsecs,and at the same time show the distribu-tion and motion of matter on the largestobserved scales, many tens of mega-parsecs and greater, the computationmust involve a very large number ofmassive particles.

The early computer studies, initiatedabout ten years ago right after the CDMmodel was proposed, were inconclusivebecause they could not include enoughparticles. Typically, about ten thousandvery massive particles represented allthe matter in a region 10 to 100 mega-parsecs across, and a single particlewith a mass on the order of 1012 solarmasses represented all the stars anddark matter in a galactic halo. Thusthere was no convincing criterion fordeciding which of those particles wereprospective galactic halos that wouldbecome luminous due to star formationand which of those particles representedmatter that would remain dark. Never-theless those early “one-particle-per-galaxy” simulations gave hope that theCDM cosmology was right: The valueof the normalization of the power spec-trum could be chosen so that the



Figure 6. The Density-fluctuation Spectrum of Cold Dark MatterPlotted is the density-fluctuation power spectrum, P (k), of cold dark matter at the time

of recombination. (For comparison with present structures, the horizontal scales give

the wavenumbers k and wavelengths l of the modes now, when the universe is

a (t0)/a (trecomb) < 1000 times bigger.) The spectrum is normalized to agree with the

COBE observations; the parallelogram at left shows the range allowed by the COBE

data. Also plotted is the Harrison-Zel’dovich spectrum, P (k) ~ k (gray), with the same

normalization. As discussed in the text, the CDM spectrum has less power at small

scales (large k) than the Harrison-Zel’dovich spectrum, because fluctuations on those

scales enter the horizon during the radiation-dominated era, when they cannot grow.

The relatively small scale labeled “intergalactic” is roughly the scale used for observa-

tions of the galaxy distribution (also shown in Figure 5). The comparison of the galaxy

distribution to the CDM spectrum will be discussed in the text.

105

104

103

102

10

1

104 103 102 10 1

10-3 10-2 10-1 1 10k

ha(t )(megaparsec-1)

P(k

)h3 (

meg

apar

sec

3)

CDM spectrum at recombination normalized to COBE data

Harrison-Zel'dovich spectrum normalized to COBE data


ha(t ) λ (megaparsecs)

Galactic scale

Intergalactic scale



simulations reproduced the observedrelative motions of galaxies on smallintergalactic scales (on the order of 1megaparsec). They also reproduced theobserved spatial distribution of galaxiesprovided one introduced a bias, an adhoc assumption that the nonuniformityof galaxies exaggerates the nonunifor-mity of dark matter by some factor bgreater than 1. More formally, if wedefine the number of galaxies in agiven volume as N and the total massin that volume as M, then on average

dN/Nw 5 b dM/Mw.

Those early studies required a bias b of2.5, or equivalently only particles invery dense regions (on scales of 1megaparsec) were identified as prospec-tive galaxies. Later, smaller values ofthe bias and correspondingly larger val-ues for the normalization of the powerspectrum had to be adopted to accountfor observations on scales around 50megaparsecs such as the famous “GreatAttractor”—an observed flow of galax-ies, including ours—and the distributionof rich clusters of galaxies called Abellclusters.

The introduction of a bias to matchsimulations with the observed spatialdistribution of galaxies is not unreason-able, because the spatial distribution ofgalaxies is determined not just by thedistribution of total matter in the uni-verse, which is modeled by simulations,but also by the highly uncertain processof star formation. The data that moredirectly reflect the underlying matterdistribution, and that therefore must bematched by simulations, are the ob-served peculiar velocities of galaxies,motions that are due not to the expan-sion of the universe, but to gravitationalforces on galaxies from inhomogenei-ties in the local mass distribution, in-cluding dark matter. Therefore peculiarvelocities have long been considered a

better measure of mass inhomogeneitythan correlations of galaxy positions.

Since galaxies are too far away forthe detection of their motion transverseto our line of sight, only the compo-nents along our line of sight of their ve-locities—the components that affectredshift—are observed. Peculiar veloc-ities show up as deviations from Hub-ble’s law—provided there is some wayto determine the distance to galaxies in-dependent of velocity, or redshift, mea-surements. The naive method is to as-sume that galaxies near each other inthe sky are members of the same clus-ter; then differences in their redshiftswould arise from differences in theirpeculiar velocities rather than from dif-ferences in distance.* The actual pro-cedure for determining peculiar veloci-ties is a sophisticated statisticalapplication of the same principle. Theanalysis yields the component along theline between two galaxies of the in-ferred peculiar-velocity difference be-tween them. That component is calledthe pairwise radial velocity. The distri-bution of peculiar velocities is charac-terized by the standard deviation ofpairwise radial velocities, which is writ-ten as sv. Measurements seemed to in-dicate that for pairs of galaxies on theorder of 1 megaparsec apart, sv is 300to 400 kilometers per second.

Before the 1992 COBE results, sci-entists would vary the normalization ofthe power spectrum to achieve agree-ment between the peculiar velocitiespredicted by the simulation and thoseobserved and then vary the bias tomatch the observed spatial distribution

of luminous matter. Since none of thesimulations were able to achieve highresolution on both large and smallscales simultaneously, the results wereambiguous.

Cold Dark Matter, LargeScales, and COBE

The freedom to vary the normaliza-tion of the power spectrum and “fine-tune” the bias disappeared in early 1992after the announcement that the COBEsatellite had detected microwave-background fluctuations of a few partsin 106 on scales of 1000 megaparsecsand higher. Since the shape of theCDM fluctuation spectrum is deter-mined, the COBE measurements fix thenormalization constant and thus the en-tire spectrum.*

The COBE results also determinethe bias b between the distribution ofmatter and the distribution of luminousmatter. One extrapolates the CDMspectrum to the present and comparesthe resulting spectrum to the presentamplitude of density fluctuations. Theresult for scales around 10/h mega-parsecs (the intergalactic scales shownon Figures 5 and 6) is dr/r < 1 with astandard deviation of about 20 percent.One can compare that predicted ampli-tude with the quantity conventionallyused to report the distribution of lumi-nous matter, namely, the amplitudedr/r on the scale of 8/h megaparsecs,known as s8. That is,

s8 ; ul 5 8/h Mpc.drr

*These differences in radial velocities are respon-sible for the distorted appearance of clusters ingalaxy maps based on redshift, such as Figure 1a.The Coma Berenices cluster in that figure proba-bly has an approximately spherical shape, but thepeculiar velocities of its galaxies give them wide-ly different redshifts, elongating the plotted spa-tial distribution of the cluster into a “finger ofGod” pointing directly at Earth.

*There is still some freedom in the baryonic con-tent of the universe and, perhaps more important,in the exponent of the primordial power spectrumof density fluctuations, which would be exactlyscale-invariant only if it were generated by end-less inflation.

The observed value of s8 for luminousmatter is also about 1; thus the COBEdata exclude the possibility of a signifi-cant bias.

High-resolution state-of-the-artsimulations. Spurred by the COBE ob-servations, we recently carried out largecomputer simulations of structure for-mation in a CDM universe. The simu-lations were large enough to resolve theformation of prospective galaxies onkiloparsec scales and to model thestructure and motions of sheets and fila-ments of matter on scales four orders ofmagnitude larger.

The simulations involve either 9 or17 million point particles moving underthe influence of mutual gravitational at-traction in an expanding spherical frag-ment of the universe. (Since the parti-cles have no internal degrees of free-dom, and the only interaction in thesimulation is gravity, no energy is dissi-pated.) The fragment expands duringthe simulation to a final diameter of100 or 250 megaparsecs at a redshift ofz 5 0, which corresponds to the presenttime, t 5 t0. For numerical reasons thesimulation starts at a redshift z between50 and 100, that is, at a time well afterthe time of recombination (z 5 1000)but before the onset of nonlineargrowth. To determine the density-per-turbation spectrum at the start of thesimulation, we use linear theory to cal-culate the spectrum as a function of zfrom the CDM spectrum at recombina-tion under the assumption that the fluc-tuations grow linearly with the scalefactor. The universe is assumed to beflat, the Hubble parameter is assumedto be 50 (km/s)/Mpc, and the normal-ization of the spectrum in each of theruns is selected to bracket the valuesuggested by the COBE measurements.We cannot do this directly, but ratherfine-tune the normalization to yield thedesired value of s8 at z 5 0.

The spectrum of fluctuations at thestarting value of z is built into the sim-ulation by arranging the particles in aregular array and assigning them differ-ent masses selected at random to matchthe predetermined spectrum. To in-clude the effects of the universal expan-sion, the particles are given initial ve-locities in accordance with Hubble’slaw. The simulation then follows themotion of this collection of particles.At each timestep, the changes in theposition and velocity of each particleare found by first calculating the gravi-tational force on each particle producedby the other particles and then integrat-ing Newton’s second law of motionover the duration of each timestep. Theforces are calculated according to thefamiliar formula for Newtonian gravity.

The use of Newtonian gravity to-gether with initial conditions in accor-dance with Hubble’s law may appear tobe a poor man’s version of the generalmodel for a relativistic universe, but itcan be shown (see the caption of Figure3) that, to a good approximation, a re-gion of the universe really does behaveaccording to that description providedthat the region is matter-dominated andis significantly smaller than the Hubbleradius c/H0. Nevertheless, only aftergeneral relativity was understood didscientists become bold enough to applyNewton’s laws to the cosmos.

In a standard simulation of N parti-cles, the time needed to calculate thegravitational force between each pair ofparticles and to sum up the total forceon each particle in each timestep is pro-portional to N2. When N is in the mil-lions, as in our simulations, that time isprohibitively long even on the mostpowerful computers. We have beenable to reduce the time significantly byreplacing pairwise interactions betweendistant particles with the well-knownmultipole approximation for the gravita-tional force exerted by a group of parti-

cles on a distant particle. The approxi-mation is implemented by using a hier-archical algorithm called a treecode inwhich the problem domain is subdivid-ed into cubes of decreasing size, thefirst terms of the multipole expansionare computed for the particles in eachcube, and the multipole approximationof the gravitational force is appliedwhenever the accompanying errors arenegligible. Treecodes calculate forcesinvolving millions of particles thou-sands of times faster than conventionalalgorithms. Their execution time de-pends on the number of particles as Nlog N rather than N2, a very significantreduction when N is large.

Our treecode and its potential appli-cations in other fields are described in“A Fast Tree Code for Many-BodyProblems.” The simulations presentedhere were run on the Intel TouchstoneDelta, a parallel supercomputer ownedpartly by the Laboratory and installedat Caltech.

Figure 7 illustrates the developmentof structure in the simulation. A typi-cal final distribution of particles appearsin Figure 8. Many of the clumpsshown in those figures are made up ofhundreds of particles and have massesin the range characteristic of a galactichalo. We interpret those clumps asprospective galactic halos; the final dis-tribution of halos is shown in Figure 9,and a detailed distribution of both parti-cles and halos in a small area is shownin Figure 10.

Once the halos are identified, we an-alyze local halo dynamics including in-ternal rotation, collapse, and merger.At the same time, since each simulationgenerates approximately 10,000 suchhalos, we can work with statisticallysignificant numbers in investigating therelative motions and overall spatial dis-tribution of halos on scales of tens ofmegaparsecs. The size of the halos andtheir spatial distribution are not in any





Figure 7. The Evolution of Structure in a SimulationThese four frames show consecutive stages of the particle distribution in one of our simulations of the matter distribution. The sim-

ulation starts a few tens of million years after the Big Bang (z 5 64), shortly before density fluctuations began to grow nonlinearly.

The first frame is at z 5 10; the last is at the present. The color of each pixel indicates the logarithm of the particle density along the

line of sight through the computational volume; blue indicates the lowest density, then cyan, green, red, yellow, and white. As the

simulation progresses, the mass becomes more clumped, eventually forming structures qualitatively similar to those observed. The

region of space shown expands as the universe expands; it is about 200 megaparsecs across in the final frame.



Figure 8. Particles at the End of a SimulationThe figure shows the projected location of particles in a sphere of diameter 250 megaparsecs at the end of a simulation. The dis-

tance across the image is about 180 Mpc (part of the sphere is cut off). Only about one in every thirty particles is plotted. The fig-

ure shows that the distribution is inhomogeneous on all scales. The square shows the region enlarged in Figure 10. The treecode

used for the simulation is described in “A Fast Tree Code for Many-Body Problems,” accompanying this article.



Figure 9. Halos at the End of a SimulationA plot of the location of “galactic halos” in the system of particles shown in Figure 7, as determined by an algorithm that locates

clumps of particles. Halos are defined to be regions whose centers are at least 10,000 times denser than the average density of the

universe and contain at least 10 particles. The sizes of the circles are proportional to the masses of the halos. There are about

6000 such halos in this picture. Note that clusters, voids, and sheets (which appear as narrow concentrations on this two-dimen-

sional projection) are present, as in the observations.



Figure 10. Detailed Particle Distribution at the End of a SimulationA high-resolution plot of every particle (about 250,000) in one fiftieth of the total system (indicated by the small square in Figure 8).

The circles represent galactic halos as identified in Figure 9.

obvious disagreement with observation,as can be shown by computing halo-halo correlation functions as well asother measures of the halo distribution,and comparing them with similar quan-tities computed on the basis of observa-tional data. To evaluate the distributionof mass within each halo and checkwhether it is compatible with the obser-vations, we used the positions of thetens or hundreds of particles in eachhalo to determine for each halo M(<r),the mass within a radius r as a functionof r. Then, using the previously dis-cussed formula for circular velocity,vcirc 5 ÏGwMw(<wr)w/rw, we determined theorbital velocities that stars would havearound the center of the halo. Themass distributions of our halos do in-deed look like those inferred from theobservations of spiral galaxies: M(<r)in our simulated halos is roughly pro-portional to r at large r, so orbital ve-locities of stars would be roughly inde-pendent of distance (see Figure 4). Theaverage orbital velocity, or equivalentlythe average mass, of the halos alsoseems to be in general agreement withobservation. Figure 11 is a plot of thenumber of prospective galactic halosversus the inferred orbital velocity of astar at a distance of 100 kiloparsecsfrom the center of the halo. The twocurves in the figure are derived fromsimulations with different normaliza-tions of the density-fluctuation spec-trum, one specified by the COBE valueof s8 and the other by a somewhatlower value. The peaks in both curves,at a velocity of 150 kilometers per sec-ond, are probably an artifact of our res-olution—we cannot resolve halos lessmassive than about one tenth of themass of our galaxy. A more importantfeature is that relatively few orbital ve-locities are much higher than the orbitalvelocity of stars in the Milky Wayaround the galactic center, about 260kilometers per second. In previous

simulations, low resolution had causedhalos to merge into overly large halos,so the inferred orbital velocities wereconsiderably higher than those ob-served.

Apart from that “overmerging” prob-lem, CDM cosmology never had muchdifficulty accounting for spatial struc-ture on megaparsec scales and smaller.

The model ran into serious trouble onlywhen it had to account simultaneouslyfor both the spatial distribution and rel-ative motions of galactic halos. Figure12 shows results from our simulationsfor the root-mean-square pairwise radialvelocities of halos as a function of thedistance separating the halos. (Theroot-mean-square pairwise radial veloc-




3000

2000

1000

200 400 800600 1000Orbital velocity (kilometers per second) at 100 kiloparsecs

Num

ber

of h

alos

σ8 = 0.7 ("undernormalized")

σ8 = 1.0 (COBE normalization)

Orbital velocity of our galaxy

Figure 11. Distribution of Halo Number as a Function of Inferred OrbitalVelocities of Stars in HalosThe graph shows the number of halos in our simulation versus the inferred orbital ve-

locity, vcirc, of a star around the center of each halo at a radius of 100 kiloparsecs.

The radius of 100 kiloparsecs was chosen because orbital velocities of stars around

galactic centers are usually independent of distance at such large distances (as in Fig-

ure 4). The velocity associated with each halo is inferred from the mass distribution

within the halo. The most abundant halos produced in our simulation are those with

the lowest masses that the simulation can resolve, and thus they usually have low or-

bital velocities. Therefore the peaks of the curves shown are artifacts of the simula-

tion, not characteristics to be compared with observations. We show results from runs

of the simulation using two different normalizations of the density-fluctuation spec-

trum: s8 < 1.0 (black curves), the naive prediction from COBE measurements, and a

lower normalization, s8 < 0.7 (red curves), which, as argued in the text, may be a much

better estimate. The vertical line at 260 kilometers per second indicates the velocity of

stars orbiting the center of our galaxy; the highest velocities observed in other galax-

ies are about 600 kilometers per second. Thus the rotational velocities predicted by

our simulation are on the right order of magnitude, indicating that the local mass distri-

bution within halos is about right. The lower normalization gives better agreement with

observations since it does not lead to orbital velocities in excess of 600 kilometers per

second.

ity is closely related to sv, the standarddeviation of pairwise radial velocities.The difference in our simulation isabout 20 percent.) Again the two setsof results derive from the different nor-malizations used in the simulations.For the lower normalization the sv ofhalos separated by 1 to 2 megaparsecsis on the order of 600 kilometers persecond. Figure 13 shows the distribu-tion of pairwise radial velocities forhalos separated by 1 to 2 megaparsecs.

The value of sv of 600 kilometersper second obtained from our high-res-olution simulation is less than half ofthe value indicated by the old “one par-ticle per galaxy” CDM simulations, butstill appears to be too large comparedwith the usually quoted observationalvalue—340 6 40 kilometers per secondon megaparsec scales. At first wethought that this discrepancy ruled outthe CDM scenario. However, havingre-examined observed and simulatedrelative velocities more carefully, wehave concluded that the quoted valuefor sv is not a very reliable diagnostic

for evaluating the global dynamics ofsimulations. In particular, the values de-duced can vary by nearly an order ofmagnitude from dense clusters to un-derdense “backwaters” like our ownlocal group of galaxies. Similarly,large variations in sv (factors of 2 to 5)are seen among fragments of our simu-lated universes containing roughly 1400galaxies. These fragments are compa-rable in size to the regions of the uni-verse usually surveyed to measure sv.

The usually quoted value of sv wasobtained by M. Davis and P. J. E. Pee-bles from their analysis of the sampleof about 1200 galaxies in the northernsky compiled by the Harvard-Smithson-ian Center for Astrophysics. This sam-ple contains the galaxies in the Virgocluster, which move at high speeds andtend to affect relative velocities to anextent disproportionate to their smallnumbers. In addition, overall infall intothe Virgo cluster results in significantinfall velocities quite far from its core.Davis and Peebles employed a specialprocedure to compensate for some of

those effects. When we reanalyzed theobservations without such special treat-ment, we found that the standard devi-ation of the pairwise radial velocitieswas around 500 kilometers per second,in close agreement with our simula-tions, especially those with the lowernormalizations. Thus, contrary to re-cent popular prejudice, the observa-tions of pairwise radial velocities donot seem to rule out CDM cosmology.

Our results for the relative velocitiesof halos and the orbital velocities with-in halos agree particularly well with ob-servations when we take the value ofs8 to be 0.7, rather than 1.0 to 1.3 ashas been inferred from the COBE mea-surements of temperature variations.We would argue that the agreement isnot pure happenstance, but rather thatthe value close to 0.7 is the relevantvalue for galaxy formation. First, thedensity-fluctuation power spectrum re-sulting from inflation is proportionalnot to k but to k12e, where e 5 0.02 to0.05. This small difference is not im-portant at the long wavelengths (smallk) measured by COBE, but it does af-fect the extrapolation of the COBE re-sults to the shorter-wavelength (larger-k) modes relevant to galaxy formation,as it reduces the predicted amplitude ofmodes on megaparsec scales. Second,COBE infers differences in temperaturefrom differences in radiation intensity,but those differences can be caused bygravity waves as well as by densityvariations. Gravity waves do not devel-op into astronomical structures. There-fore the part of the COBE temperature-fluctuation data due to gravity wavesmust be estimated and subtracted to ob-tain the part due to density fluctuations.

These two effects are interrelatedand for small e have similar magni-tudes. Taken together, they imply thatthe primordial density perturbations atshort wavelengths (large values of k)have less power than that usually in-



astro12.adb 7/26/94

1500

1000

500

0

101 102 103

Distance between halos (kiloparsecs)

Pa

irw

ise

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lativ

e v

elo

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σ8 = 1.1 (COBE normalization with Harrison-Zel'dovich CDM spectrum)

σ8 = 0.7 ("undernormalized," or COBE normalization with inflationary corrections to the CDM spectrum)

Particles

Halos

Figure 12. Pairwise Radial Velocities of HalosThe solid lines indicate the root-mean-square value of the pairwise radial velocity—

the differences of the velocities of two halos along the line connecting them, exclud-

ing the Hubble expansion—as a function of separation of the halos. The dashed

lines show the relative velocities of particles in the simulation rather than those of

halos. As in Figure 11, the simulation was run twice, using the COBE normalization

(black curves) and a lower normalization (red curves). The data from the lower-nor-

malization run are in the same range as the observed pairwise radial velocities of real

galaxies, provided the same statistical procedures are applied to the simulated and

observed data (see discussion in the main text).



ferred from the COBE data; extrapolat-ing that result to the scale of our simu-lations implies that s8 is about 20 to40 percent less than the COBE datawould seem to imply. Thus the agree-ment between our “undernormalized”simulations and the observational datamight not be fortuitous; rather it mightbe telling us that inflation really didhappen and that the resulting primor-dial fluctuation spectrum was only ap-proximately of the Harrison-Zel’dovichform. If so (and admittedly we are get-ting carried away with optimism here),our simulations would provide the firstconcrete example in which detailed ef-fects of inflation on the primordial fluc-tuation spectrum have to be taken intoaccount to reconcile theoretical predic-tions with observations!

Nevertheless, the CDM model is stillnot “out of hot water.” Although thediscrepancy of relative velocities ofhalos was considered the most seriousdifficulty with the model, there areother tests that deal more directly withthe spatial distribution of galaxies. Oneobservational uncertainty that will havea major impact on assessing the validityof cosmological models is the presentvalue of the Hubble parameter, H0. Allof the models that assume that V 5 1(as does the standard CDM model)would be seriously endangered if H0turned out to be significantly largerthan the currently favored value, 50(km/s)/Mpc. The value of H0 (in com-bination with the assumption that theuniverse is “flat,” that is, that V 5 1)fixes the age of the universe. Accord-ing to present ideas of stellar evolution,the oldest stars we observe are approxi-mately 14 billion years old, which isabout equal to the age of the universe ifH0 5 50 (km/s)/Mpc and V 5 1. If H0turns out to be 100 (km/s)/Mpc, thosestars would be twice as old as the uni-verse! Such a discrepancy wouldcause, of course, a major “paradigm


100

-2000 20000Pairwise radial velocity (kilometers/second)

Nu

mb

er

of

ha

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10

1

Figure 13. Distribution of Pairwise Relative Velocities of Halos at 1–2Megaparsec Separations The graph shows the number of pairs of halos in which the two halos are separated by

1–2 megaparsecs versus the pairwise relative velocities of the two halos in the pair.

These results are from a simulation with s8 < 0.7. The center of the distribution is not

at 0 but at 2280 km/s; the negative average velocity indicates a net infall, or movement

of halos toward each other, that is, toward local mass concentrations. This infall, due

to gravitational attraction, is similar to observed movements of real galaxies. The local

increases in density as nearby halos move toward each other are examples of the

gravitational growth of inhomogeneity. Note that the graph appears as a triangle on

this log plot, so the distribution of velocities is exponential. Therefore sv at this sepa-

ration is simply proportional to the slopes of the sides of the triangle.

shift” in cosmology, as it could be ac-commodated only by admitting that weare missing some important aspects ofstellar evolution (which appears unlike-ly) or that the universe is open—hasV < 0.1.

Conclusions

The general framework of Big Bangcosmology has been very muchstrengthened by the COBE results. Thecosmic background radiation appears tohave a spectrum (that of emission froma perfect black body) and a temperatureconsistent with the predictions of theBig Bang theory. Furthermore, theCOBE observations showed that theform of the primordial perturbations ofthe density of the universe was consis-tent with an approximately scale-freespectrum and that their amplitude wassufficiently large that they could act asseeds for the gravitational developmentof the currently observed density fluctu-ations, so that no forces other thangravity need to be invoked. The tradi-tional problems with the CDM scenariofollow from the excess of power onsmall scales and the resulting excess ofthe peculiar velocities between thegalactic halos (and, therefore, presum-ably, between galaxies). We believe,however, that the results of our simula-tions are consistent with the observa-tions when the simulations and observa-tions are analyzed in the same way.Therefore there is no reason yet toabandon the CDM scenario.

Nevertheless, there is an almost em-barrassing richness of insufficiently ex-plored alternatives, one of the morepopular being hybrid models involvingboth hot and cold dark matter. Fortu-nately, all of the alternatives will soonconfront even more precise observation-al data, and their success will be evalu-ated by means of much more accurate

simulations. Indeed, the list of modelsruled out by COBE and other observa-tions (with the help of computer simu-lations) is becoming longer than the listof viable models. “Experimental cos-mology,” using parallel supercomputersand sophisticated software, will un-doubtedly help bring about continuedprogress in this exciting and fundamen-tal field. The computer is becoming asimportant as the telescope in shapingour understanding of the universe.

Further Reading

G. Efstathiou, J. R. Bond, and S. D. M. White.1992. COBE background radiation anisotropiesand large-scale structure in the Universe. Month-ly Notices of the Royal Astronomical Society 258:1P–6P.

Margaret J. Geller and John P. Huchra. 1989.Mapping the universe. Science 246: 897–903.

P. J. E. Peebles. 1993. Principles of PhysicalCosmology. Princeton University Press.

Vera C. Rubin. 1983. The rotation of spiralgalaxies. Science 220: 1339–1344.

Frank H. Shu. 1982. The Physical Universe: AnIntroduction to Astronomy. University ScienceBooks.

G. F. Smoot et al. 1992. Structure in the COBEdifferential microwave radiometer first-year maps.The Astrophysical Journal 396: L1–L5.

M. S. Warren, P. J. Quinn, J. K. Salmon, and W.H. Zurek. 1992. Dark halos formed via dissipa-tionless collapse. Astrophysical Journal 399:405–425.

W. H. Zurek, P. J. Quinn, J. K. Salmon, and M.S. Warren. 1993. Large-scale structure afterCOBE. Los Alamos National Laboratory unclas-sified release LA-UR-93-2104. AstrophysicalJournal, in press.

W. H. Zurek, P. J. Quinn, J. K. Salmon, and M.S. Warren. 1993. The second coming of colddark matter? Los Alamos National Laboratoryunclassified release LA-UR-93-4240. In Pro-ceedings of the Ninth IAP Meeting, edited by F.Bouchet, in press.

Michael S. Warren (left) received his B.S. inphysics, engineering, and applied science fromthe California Institute of Technology and hisM.S. and Ph.D. in physics from the University ofCalifornia, Santa Barbara. He came to the Labo-ratory in 1990 as a graduate research assistant inthe Theoretical Astrophysics Group and is cur-rently performing post-doctoral work in the FluidDynamics Group. In 1992 Warren received theGordon Bell Prize and the Intel Grand ChallengeComputing Award for the work described in “AFast Tree Code for Many-Body Problems.”

Wojciech H. Zurek was born in Poland and re-ceived an M.Sc. from the Technical University ofKraków. He received his Ph.D in Physics fromthe University of Texas at Austin. Zurek joinedthe Laboratory’s Theoretical Astrophysics Groupas an Oppenheimer Fellow in 1984 and has re-mained in that group ever since; he is now itsleader. He is also an external faculty member ofthe Santa Fe Institute and an associate of the Cos-mology Section of the Canadian Institute for Ad-vanced Research. His research interests includecosmology, relativistic astrophysics, the quantumtheory of measurements, the foundations of statis-tical physics, and the physics of information. In1988 Zurek was awarded the Los Alamos FellowsPrize for his research on galaxy formation.



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ge-scale - Federation of American Scientists · (b) A galaxy cluster—a structure of intermediate size—in the constellation Leo. (Photograph courtesy of Palomar/California Institute