Hannu Karttunen et al. (Eds.), Cosmology.In: Hannu Karttunen et al. (Eds.), Fundamental Astronomy, 5th Edition. pp. 393–414 (2007)DOI: 11685739_19 © Springer-Verlag Berlin Heidelberg 2007

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After the demise of the Aristotelian world picture, ittook hundreds of years of astronomical observations

and physical theories to reach a level at which a satis-factory modern scientific picture of the physical universecould be formed. The decisive steps in the developmentwere the clarification of the nature of the galaxies in the1920’s and the general theory of relativity developed by

Einstein in the 1910’s. Research in cosmology tries to an-swer questions such as: How large and how old is theUniverse? How is matter distributed? How were the ele-ments formed? What will be the future of the Universe?The central tenet of modern cosmology is the model ofthe expanding universe. On the basis of this model, it hasbeen possible to approach these questions.

19.1 Cosmological Observations

The Olbers Paradox. The simplest cosmological ob-servation may be that the sky is dark at night. This factwas first noted by Johannes Kepler, who, in 1610, used itas evidence for a finite universe. As the idea of an infinitespace filled with stars like the Sun became widespread inconsequence of the Copernican revolution, the questionof the dark night sky remained a problem. In the 18thand 19th centuries Edmond Halley, Loys de Cheseauxand Heinrich Olbers considered it in their writings. Ithas become known as the Olbers paradox (Fig. 19.1).

The paradox is the following: Let us suppose theUniverse is infinite and that the stars are uniformly dis-tributed in space. No matter in what direction one looks,

Fig. 19.1. The Olbers paradox. If the stars were uniformlydistributed in an unending, unchanging space, the sky shouldbe as bright as the surface of the Sun, since each line of sightwould eventually meet the surface of a star. A two-dimensional

analogy can be found in an optically thick pine forest wherethe line of sight meets a trunk wherever one looks. (Photo M.Poutanen and H. Karttunen)

sooner or later the line of sight will encounter the surfaceof a star. Since the surface brightness does not dependon distance, each point in the sky should appear to beas bright as the surface of the Sun. This clearly is nottrue. The modern explanation of the paradox is that thestars have only existed for a finite time, so that the lightfrom very distant stars has not yet reached us. Ratherthan proving the world to be finite in space, the Olbersparadox has shown it to be of a finite age.

Extragalactic Space. In 1923 Edwin Hubble showedthat the Andromeda Galaxy M31 was far outside theMilky Way, thus settling a long-standing controversyconcerning the relationship between the nebulae andthe Milky Way. The numerous galaxies seen in pho-

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tographs form an extragalactic space vastly larger thanthe dimensions of the Milky Way. It is important forcosmology that the distribution and motions of the ba-sic components of extragalactic space, the galaxies andclusters of galaxies, should everywhere be the same asin our local part of the Universe. Galaxies generally oc-cur in various systems, ranging from small groups toclusters of galaxies and even larger superclusters. Thelargest structures observed are about 100 Mpc in size(see Sect. 18.6). They are thus significantly smaller thanthe volume of space ( a few thousand Mpc in size) inwhich the distribution of galaxies has been investigated.One way of studying the large-scale homogeneity of thegalaxy distribution is to count the number of galaxiesbrighter than some limiting magnitude m. If the galaxiesare uniformly distributed in space, this number shouldbe proportional to 100.6m (see Example 17.1). For exam-ple, the galaxy counts made by Hubble in 1934, whichincluded 44,000 galaxies, were consistent with a galaxydistribution independent of position (homogeneity) andof direction (isotropy). Hubble found no “edge” of theUniverse, nor have later galaxy counts found one.

Similar counts have been made for extragalacticradio sources. (Instead of magnitudes, flux densi-ties are used. If F is the flux density, then becausem = −2.5 lg(F/F0), the number count will be propor-tional to F−3/2.) These counts mainly involve verydistant radio galaxies and quasars (Fig. 19.3). The re-sults seem to indicate that the radio sources were eithermuch brighter or much more common at earlier epochsthan at present (Sect. 18.8). This constitutes evidence infavour of an evolving, expanding universe.

In general the simple geometric relation betweenbrightness and number counts will only hold for objectsthat are uniformly distributed in space. Local inho-mogeneities will cause departures from the expectedrelationship. For more distant sources the geometry ofthe Universe as well as cosmic evolution will changethe basic 100.6m behaviour.

Hubble’s Law (Fig. 19.4). In the late 1920’s, Hub-ble discovered that the spectral lines of galaxies wereshifted towards the red by an amount proportional totheir distances. If the redshift is due to the Doppler ef-fect, this means that the galaxies move away from eachother with velocities proportional to their separations,i. e. that the Universe is expanding as a whole.

In terms of the redshift z = (λ−λ0)/λ0, Hubble’slaw can be written as

z = (H/c)r , (19.1)

where c is the speed of light, H is the Hubble constantand r the distance of the galaxy. For small velocities(V c) the Doppler redshift z = V/c, and hence

V = Hr , (19.2)

which is the most commonly used form of Hubble’s law.For a set of observed “standard candles”, i. e. galaxies

whose absolute magnitudes are close to some mean M0,Hubble’s law corresponds to a linear relationship be-tween the apparent magnitude m and the logarithm ofthe redshift, lg z. This is because a galaxy at distance rhas an apparent magnitude m = M0 +5 lg(r/10 pc), andhence Hubble’s law yields

m = M0 +5 lg

(cz

H ×10 pc

)= 5 lg z +C , (19.3)

where the constant C depends on H and M0. Suitablestandard candles are e.g. the brightest galaxies in clus-ters and Sc galaxies of a known luminosity class. Someother methods of distance determination for galaxieswere discussed in Sect. 18.2. Most recently, type Ia su-pernovae (Sect. 13.3) in distant galaxies have been usedto determine distances out to the redshift z = 1, wheredepartures from Hubble’s law are already detectable.

If the Universe is expanding, the galaxies were oncemuch nearer to each other. If the rate of expansion hadbeen unchanging, the inverse of the Hubble constant,T = H−1, would represent the age of the Universe. Ifthe expansion is gradually slowing down, the inverseHubble constant gives an upper limit on the age of theUniverse (Fig. 19.4). According to present estimates,60 km s−1 Mpc−1<H<80 km s−1 Mpc−1, correspond-ing to 11 Ga< T< 17 Ga. In fact, current indications(discussed later in this chapter) are that the rate of ex-pansion is accelerating at present. In that case the ageof the Universe may also be larger. However, H−1 willstill be an estimate for the age of the Universe.

Fig. 19.2. The quasar 3C295 and its spectrum. The quasarsare among the most distant cosmological objects. (PhotographPalomar Observatory)

19.1 Cosmological Observations

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Fig. 19.3. Hubble’s law for type Ia supernovae. The solid curverepresents the “concordance” model. The other curves showthat models with a vanishing cosmological constant can beexcluded. (R.A. Knop et al. 2003, ApJ 598,102; taken fromhttp://supernova.lbl.gov/)

One reason for the difficulty in determining the valueof the Hubble constant is the uncertainty in extragalacticdistances. A second problem is that the measured val-ues of the velocity V , corrected to take into account themotion of the Sun within the Local Group, contain a sig-nificant component due to the peculiar motions of thegalaxies. These peculiar velocities are caused by localmass concentrations like groups and clusters of galax-ies. It is possible that the Local Group has a significantvelocity towards the centre of the Local Supercluster(the Virgo Cluster). Because the Virgo Cluster is of-ten used to determine the value of H , neglecting thispeculiar velocity leads to a large error in H . The sizeof the peculiar velocity is not yet well known, but it isprobably about 250 km s−1.

The most ambitious recent project for determin-ing H used the Hubble Space Telescope in orderto measure cepheid distances to a set of nearbygalaxies. These distances were then used to calibrateother distance indicators, such as the Tully–Fisherrelation and type Ia supernovae. The final resultwas H = (72±8) km s−1 Mpc−1. The largest remain-ing source of error in this result is the distance to

Fig. 19.4. If the expansion of the Universe is slowing down,the inverse Hubble constant gives an upper limit of the age.The real age depends on the variable rate of expansion

Fig. 19.5. A regular expansion according with Hubble’s lawdoes not mean that the Milky Way (O) is the centre of theUniverse. Observers at any other galaxy (O′) will see thesame Hubble flow (dashed lines)

the Large Magellanic Cloud, used for calibrating thecepheid luminosity.

The form of Hubble’s law might give the impressionthat the Milky Way is the centre of the expansion, inapparent contradiction with the Copernican principle.Figure 19.5 shows that, in fact, the same Hubble’s lawis valid at each point in a regularly expanding universe.There is no particular centre of expansion.

The Thermal Microwave Background Radiation.The most important cosmological discovery since Hub-ble’s law was made in 1965. In that year Arno Penzias

19.1 Cosmological Observations

397

and Robert Wilson discovered that there is a universalmicrowave radiation, with a spectrum corresponding tothat of blackbody radiation (see Sect. 5.6) at a temper-ature of about 3 K (Fig. 19.6). For their discovery, theyreceived the Nobel prize in physics in 1979.

The existence of a thermal cosmic radiation back-ground had been predicted in the late 1940’s by GeorgeGamow, who was one of the first to study the ini-tial phases of expansion of the Universe. Accordingto Gamow, the Universe at that time was filled withextremely hot radiation. As it expanded, the radiationcooled, until at present, its temperature would be a fewkelvins. After its discovery by Penzias and Wilson,the cosmic background radiation has been studied atwavelengths from 50 cm to 0.5 cm. The first detailedmeasurements, made from the COBE (Cosmic Back-ground Explorer) satellite showed that it correspondsclosely to a Planck spectrum at 2.725±0.002 K. Morerecently the CMB has been mapped in evene greaterdetail by the WMAP satellite.

The existence of the thermal cosmic microwavebackground (CMB) gives strong support to the beliefthat the Universe was extremely hot in its early stages.The background is very nearly isotropic, which sup-ports the isotropic and homogeneous models of theUniverse. The COBE and WMAP satellites have alsodetected temperature variations of a relative amplitude6×10−6 in the background. These fluctuations are in-

Fig. 19.6. Observations of the cosmic microwave backgroundradiation made by the COBE satellite in 1990 are in agreementwith a blackbody law at 2.7 K

terpreted as a gravitational redshift of the backgroundproduced by the mass concentrations that would latergive rise to the observed structures in the Universe.They are the direct traces of initial irregularities in thebig bang, and provide important constraints for theoriesof galaxy formation. Perhaps even more importantly,the amplitude of the fluctuations on different angularscales have provided crucial constraints on the cos-mological model. We shall return to this question inSect. 19.7.

The Isotropy of Matter and Radiation. Apart from theCMB, several other phenomena confirm the isotropy ofthe Universe. The distribution of radio sources, the X-ray background, and faint distant galaxies, as well asHubble’s law are all isotropic. The observed isotropyis also evidence that the Universe is homogeneous,since a large-scale inhomogeneity would be seen as ananisotropy.

The Age of the Universe. Estimates of the ages of theEarth, the Sun and of star clusters are important cos-mological observations that do not depend on specificcosmological models. From the decay of radioactiveisotopes, the age of the Earth is estimated to be 4600 mil-lion years. The age of the Sun is thought to be slightlylarger than this. The ages of the oldest star clusters inthe Milky Way are 10–15 Ga.

The values thus obtained give a lower limit to the ageof the Universe. In an expanding universe, the inverseHubble constant gives another estimate of that age. Itis most remarkable that the directly determined agesof cosmic objects are so close to the age given by theHubble constant. This is strong evidence that Hubble’slaw really is due to the expansion of the Universe. It alsoshows that the oldest star clusters formed very early inthe history of the Universe.

The Relative Helium Abundance. A cosmological the-ory should also give an acceptable account of the originand abundances of the elements. Even the abundanceof the elementary particles and the lack of antimat-ter are cosmological problems that have begun to beinvestigated in the context of theories of the earlyUniverse.

Observations show that the oldest objects containabout 25% by mass of helium, the most abundant

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element after hydrogen. The amount of helium pro-duced is sensitive to the temperature of the Universe,which is related to that of the background radiation.The computations made for the standard models ofthe expanding Universe (the Friedmann models) yielda helium abundance of exactly the right size.

19.2 The Cosmological Principle

One hopes that as ever larger volumes of the Universeare observed, its average properties will become simpleand well defined. Figure 19.7 attempts to show this.It shows a distribution of galaxies in the plane. Asthe circle surrounding the observer O becomes larger,the mean density inside the circle becomes practicallyindependent of its size. The same behaviour occurs, re-gardless of the position of the centre of O: at closedistances, the density varies randomly (Fig. 19.8), butin a large enough volume, the average density is con-stant. This is an example of the cosmological principle:apart from local irregularities, the Universe looks thesame from all positions in space.

Fig. 19.7. The cosmologi-cal principle. In the smallcircle (A) about the ob-server (O) the distributionof galaxies does not yetrepresent the large-scaledistribution. In the largercircle (B) the distributionis already uniform on theaverage

The cosmological principle is a far-reaching assump-tion, which has been invoked in order to put constraintson the large variety of possible cosmological theories.If in addition to the cosmological principle one alsoassumes that the Universe is isotropic, then the onlypossible cosmic flow is a global expansion. In that case,the local velocity difference V between two nearbypoints has to be directly proportional to their separation(V = Hr); i. e. Hubble’s law must apply.

The plane universe of Fig. 19.7 is homogeneous andisotropic, apart from local irregularities. Isotropy at eachpoint implies homogeneity, but homogeneity does notrequire isotropy. An example of an anisotropic, homoge-neous universe would be a model containing a constantmagnetic field: because the field has a fixed direction,space cannot be isotropic.

We have already seen that astronomical observationssupport the homogeneity and isotropy of our observ-able neighbourhood, the metagalaxy. On the groundsof the cosmological principle, these properties may beextended to the whole of the Universe.

The cosmological principle is closely related tothe Copernican principle that our position in the

19.3 Homogeneous and Isotropic Universes

399

Fig. 19.8. The galaxies seem to be distributed in a “foamlike” way. Dense strings and shells are surrounded by relatively emptyregions. (Seldner, M. et al. (1977): Astron. J. 82, 249)

Universe is in no way special. From this principle,it is only a short step to assume that on a largeenough scale, the local properties of the metagalaxyare the same as the global properties of the Uni-verse.

Homogeneity and isotropy are important simplify-ing assumptions when trying to construct cosmologicalmodels which can be compared with local observations.They may therefore reasonably be adopted, at least asa preliminary hypothesis.

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19.3 Homogeneousand Isotropic Universes

Under general conditions, space and time coordinatesin a universe may be chosen so that the values of thespace coordinates of observers moving with the matterare constant. It can be shown that in a homogeneous andisotropic universe, the line element (Appendix B) thentakes the form

ds2 = − c2dt2 + R2(t)

×[

dr2

1− kr2+r2(dθ2 + cos2 θdφ2)

],

(19.4)

known as the Robertson–Walker line element. (The ra-dial coordinate r is defined to be dimensionless.) R(t) isa time-dependent quantity representing the scale of theUniverse. If R increases with time, all distances, includ-ing those between galaxies, will grow. The coefficient kmay be +1, 0 or −1, corresponding to the three possiblegeometries of space, the elliptic or closed, the parabolicand the hyperbolic or open model.

The space described by these models need not beEuclidean, but can have positive or negative curvature.Depending on the curvature, the volume of the universemay be finite or infinite. In neither case does it havea visible edge.

The two-dimensional analogy to elliptical (k = +1)geometry is the surface of a sphere (Fig. 19.9): its sur-face area is finite, but has no edge. The scale factor R(t)represents the size of the sphere. When R changes, thedistances between points on the surface change in the

Fig. 19.9. The two-dimensional analogues of the Friedmannmodels: A spherical surface, a plane and a saddle surface

same way. Similarly, a three-dimensional “spherical sur-face”, or the space of elliptical geometry, has a finitevolume, but no edge. Starting off in an arbitrary direc-tion and going on for long enough, one always returnsto the initial point.

When k = 0, space is flat or Euclidean, and the ex-pression for the line element (19.4) is almost the same asin the Minkowski space. The only difference is the scalefactor R(t). All distances in a Euclidean space changewith time. The two-dimensional analogue of this spaceis a plane.

The volume of space in the hyperbolic geometry(k = −1) is also infinite. A two-dimensional idea ofthe geometry in this case is given by a saddle surface.

In a homogeneous and isotropic universe, all phys-ical quantities will depend on time through the scalefactor R(t). For example, from the form of the line ele-ment, it is evident that all distances will be proportionalto R (Fig. 19.10). Thus, if the distance to a galaxy is rat time t, then at time t0 (in cosmology, the subscript 0refers to the present value) it will be

R(t0)

R(t)r . (19.5)

Similarly, all volumes will be proportional to R3. Fromthis it follows that the density of any conserved quantity(e. g. mass) will behave as R−3.

It can be shown that the wavelength of radiation in anexpanding universe is proportional to R, like all otherlengths. If the wavelength at the time of emission, cor-responding to the scale factor R, is λ, then it will be λ0

Fig. 19.10. When space expands, all galaxy separations growwith the scale factor R: r ′ = [R(t′0)/R(t0)]r

19.4 The Friedmann Models

401

when the scale factor has increased to R0:

λ0

λ= R0

R. (19.6)

The redshift is z = (λ0 −λ)/λ, and hence

1+ z = R0

R; (19.7)

i. e. the redshift of a galaxy expresses how much thescale factor has changed since the light was emitted.For example, the light from a quasar with z = 1 wasemitted at a time when all distances were half theirpresent values.

For small values of the redshift, (19.7) approaches theusual form of Hubble’s law. This can be seen as follows.When z is small, the change in R during the propagationof a light signal will also be small and proportional tothe light travel time t. Because t = r/c approximately,where r is the distance of the source, the redshift willbe proportional to r. If the constant of proportionalityis denoted by H/c, one has

z = Hr/c . (19.8)

This is formally identical to Hubble’s law (19.1).However, the redshift is now interpreted in the senseof (19.7).

As the universe expands, the photons in the back-ground radiation will also be redshifted. The energy ofeach photon is inversely proportional to its wavelength,and will therefore behave as R−1. It can be shown thatthe number of photons will be conserved, and thus theirnumber density will behave as R−3. Combining thesetwo results, one finds that the energy density of the back-ground radiation is proportional to R−4. The energydensity of blackbody radiation is proportional to T 4,where T is the temperature. Thus the temperature ofcosmic background radiation will vary as R−1.

19.4 The Friedmann Models

The results of the preceding section are valid in any ho-mogeneous and isotropic universe. In order to determinethe precise time-dependence of the scale factor R(t)a theory of gravity is required.

In 1917 Albert Einstein presented a model of theUniverse based on his general theory of relativity. Itdescribed a geometrically symmetric (spherical) spacewith finite volume but no boundary. In accordance withthe cosmological principle, the model was homoge-neous and isotropic. It was also static: the volume ofspace did not change.

In order to obtain a static model, Einstein hadto introduce a new repulsive force, the cosmologicalterm, in his equations. The size of this cosmologicalterm is given by the cosmological constant Λ. Ein-stein presented his model before the redshifts of thegalaxies were known, and taking the Universe to bestatic was then reasonable. When the expansion of theUniverse was discovered, this argument in favour ofthe cosmological constant vanished. Einstein himselflater called it the biggest blunder of his life. Nev-ertheless, the most recent observations now seem toindicate that a non-zero cosmological constant has tobe present.

The St. Petersburg physicist Alexander Friedmannand later, independently, the Belgian Georges Lemaitrestudied the cosmological solutions of Einstein’s equa-tions. IfΛ= 0, only evolving, expanding or contractingmodels of the Universe are possible. From the Fried-mann models exact formulas for the redshift andHubble’s law may be derived.

The general relativistic derivation of the law of ex-pansion for the Friedmann models will not be givenhere. It is interesting that the existence of three typesof models and their law of expansion can be derivedfrom purely Newtonian considerations, with results incomplete agreement with the relativistic treatment. Thedetailed derivation is given on p. 411, but the essentialcharacter of the motion can be obtained from a simpleenergy argument.

Let us consider a small expanding spherical regionin the Universe. In a spherical distribution of matter,the gravitational force on a given spherical shell de-pends only on the mass inside that shell. We shall hereassume Λ= 0.

We can now consider the motion of a galaxy ofmass m at the edge of our spherical region. Accord-ing to Hubble’s law, its velocity will be V = Hr and thecorresponding kinetic energy,

T = mV 2/2 . (19.9)

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19. Cosmology

The potential energy at the edge of a sphere of mass Mis U = −G Mm/r. Thus the total energy is

E = T +U = mV 2/2− G Mm/r , (19.10)

which has to be constant. If the mean density of the Uni-verse is ρ, the mass is M = (4πr3/3)ρ. The value of ρcorresponding to E = 0 is called the critical density, ρc.We have

E = 1

2m H2r2 − G Mm

r

= 1

2m H2r2 − Gm

4π

3

r3ρc

r

= mr2(

1

2H2 − 4

3πGρc

)= 0 ,

(19.11)

whence

ρc = 3H2

8πG. (19.12)

The expansion of the Universe can be comparedto the motion of a mass launched vertically from thesurface of a celestial body. The form of the orbit de-pends on the initial energy. In order to compute thecomplete orbit, the mass M of the main body and theinitial velocity have to be known. In cosmology, thecorresponding parameters are the mean density and theHubble constant.

The E = 0 model corresponds to the Euclidean Fried-mann model, the Einstein–de Sitter model. If the densityexceeds the critical density, the expansion of any spher-ical region will turn to a contraction and it will collapseto a point. This corresponds to the closed Friedmannmodel. Finally, if ρ < ρc, the ever expanding hyperbolicmodel is obtained. The behaviour of the scale factor inthese three cases is shown in Fig. 19.11.

These three models of the universe are called thestandard models. They are the simplest relativistic cos-mological models for Λ= 0. Models with Λ = 0 aremathematically more complicated, but show the samegeneral behaviour.

The simple Newtonian treatment of the expansionproblem is possible because Newtonian mechanics isapproximately valid in small regions of the Universe.However, although the resulting equations are formallysimilar, the interpretation of the quantities involved (e.g.the parameter k) is not the same as in the relativistic

Fig. 19.11. The time dependence of the scale factor fordifferent values of k. The cosmological constant Λ= 0

context. The global geometry of the Friedmann modelscan only be understood within the general theory ofrelativity.

Consider two points at a separation r. Let theirrelative velocity be V . Then

r = R(t)

R(t0)r0 and V = r = R(t)

R(t0)r0 , (19.13)

and thus the Hubble constant is

H = V

r= R(t)

R(t). (19.14)

The deceleration of the expansion is described by thedeceleration parameter q, defined as

q = −RR/R2 . (19.15)

The deceleration parameter describes the change of therate of the expansion R. The additional factors havebeen included in order to make it dimensionless, i. e.independent of the choice of units of length and time.

The value of the deceleration parameter can be ex-pressed in terms of the mean density. The densityparameter Ω is defined as Ω = ρ/ρc, so that Ω = 1corresponds to the Einstein–de Sitter model. From theconservation of mass, it follows that ρ0 R3

0 = ρR3. Us-ing expression (19.12) for the critical density, one thenobtains

Ω = 8πG

3

ρ0 R30

R3 H2. (19.16)

19.5 Cosmological Tests

403

On the other hand, using (19.26), q can be written

q = 4πG

3

ρ0 R30

R3 H2. (19.17)

Thus there is a simple relation between Ω and q:

Ω = 2q . (19.18)

The value q = 1/2 of the deceleration parameter corre-sponds to the critical densityΩ = 1. Both quantities arein common use in cosmology. It should be noted thatthe density and the deceleration can be observed inde-pendently. The validity of (19.18) is thus a test for thecorrectness of general relativity with Λ= 0.

19.5 Cosmological Tests

A central cosmological problem is the question of whichFriedmann model best represents the real Universe. Dif-ferent models make different observational predictions.Recently there has been considerable progress in the de-termination of the cosmological parameters, and for thefirst time there is now a set of parameters that appearcapable of accounting for all observations. In the fol-lowing, some possible tests will be considered. Thesetests are related to the average properties of the Uni-verse. Further cosmological constraints can be obtainedfrom the observed structures. These will be discussedin Sect. 19.7.

The Critical Density. If the average density ρ is largerthan the critical density ρc, the Universe is closed. Forthe Hubble constant H = 100 km s−1 Mpc−1, the valueof ρc = 1.9×10−26 kg m−3, corresponding to roughlyten hydrogen atoms per cubic metre. Mass determina-tions for individual galaxies lead to smaller values forthe density, favouring an open model. However, the den-sity determined in this way is a lower limit, since theremay be significant amounts of invisible mass outsidethe main parts of the galaxies.

If most of the mass of clusters of galaxies is dark andinvisible, it will increase the mean density nearer thecritical value. Using the virial masses of X-ray clustersof galaxies (Sect. 18.2), one finds Ω0 = 0.3. Consider-ations of the observed velocities of clusters of galaxiesindicate that the relative amount of dark matter does notincrease further on even larger scales.

It may be that the neutrino has a small mass (about10−4 electron mass). A large neutrino backgroundshould have been produced in the big bang. In spite ofthe small suggested mass of the neutrino, it would stillbe large enough to make neutrinos the dominant form ofmass in the Universe, and would probably make the den-sity larger than critical. Laboratory measurement of themass of the neutrino is very difficult, and the claims fora measured nonzero mass have not won general accep-tance. Instead much recent work in cosmology has beenbased on the hypothesis of Cold Dark Matter (CDM),i. e. the idea that a significant part of the mass of theUniverse is in the form of non-relativistic particles ofan unknown kind.

The Magnitude-Redshift Test. Although for smallredshifts, all models predict the Hubble relationshipm = 5 lg z +C for standard candles, for larger redshiftsthere are differences, depending on the decelerationparameter q. This provides a way of measuring q.

The models predict that galaxies at a given redshiftlook brighter in the closed models than in the open ones(see Fig. 19.4). Measurements of type Ia supernovaeout to redshifts z = 1 using the Hubble Space Telescopehave now shown that the observed q is inconsistent withmodels havingΛ= 0. AssumingΩ0 = 0.3 these obser-vations require ΩΛ = 0.7, where ΩΛ is defined below((19.30)). The Hubble diagram for Type Ia supernovaeis shown in Colour Supplement Plate 34.

The Angular Diameter–Redshift Test. Along with themagnitude-redshift test, the relation between angular di-ameter and redshift has been used as a cosmologicaltest. Let us first consider how the angular diameter θof a standard object varies with distance in static mod-els with different geometries. In a Euclidean geometry,the angular diameter is inversely proportional to the dis-tance. In an elliptical geometry, θ decreases more slowlywith distance, and even begins to increase beyond a cer-tain point. The reason for this can be understood bythinking of the surface of a sphere. For an observer atthe pole, the angular diameter is the angle between twomeridians marking the edges of his standard object. Thisangle is smallest when the object is at the equator, andgrows without limit towards the opposite pole. In a hy-perbolic geometry, the angle θ decreases more rapidlywith distance than in the Euclidean case.

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19. Cosmology

In an expanding closed universe the angular diametershould begin to increase at a redshift of about one. Thiseffect has been looked for in the diameters of radiogalaxies and quasars. No turnover has been observed,but this may also be due to evolution of the radio sourcesor to the selection of the observational data. At smallerredshifts, the use of the diameters of clusters of galaxieshas yielded equally inconclusive results.

Basically the same idea can be applied to the an-gular scale of the strongest fluctuation in the cosmicmicrowave background. The linear size of these de-pends only weakly on the cosmological model and cantherefore be treated as a standard measuring rod. Theirredshift is determined by the decoupling of matter andradiation (see Sect. 19.6). Observations of their angularsize have provided strong evidence that Ω0 +ΩΛ = 1,i. e. the Universe is flat.

Primordial Nucleosynthesis. The standard model pre-dicts that 25% of the mass of the Universe turned intohelium in the “big bang”. This amount is not sensi-tive to the density and thus does not provide a strongcosmological test. However, the amount of deuteriumleft over from helium synthesis does depend stronglyon the density. Almost all deuterons formed in the bigbang unite into helium nuclei. For a larger density thecollisions destroying deuterium were more frequent.Thus a small present deuterium abundance indicatesa high cosmological density. Similar arguments applyto the amounts of 3He and 7Li produced in the bigbang. The interpretation of the observed abundancesis difficult, since they have been changed by laternuclear processes. Still, present results for the abun-dances of these nuclei are consistent with each otherand with a density corresponding to Ω0 about 0.04.Note that this number only refers to the mass in theform of baryons, i. e. protons and neutrons. Since thevirial masses of clusters of galaxies indicate that Ω0 isabout 0.3, this has stimulated models such as the CDMmodel, where most of the mass is not in the form ofbaryons.

Ages. The ages of different Friedmann models can becompared with known ages of various systems. The

age t0 of a Friedmann model with given Ω0 and ΩΛ isobtained by integrating equation (19.30). This gives

t0 = H−10

1∫0

da(Ω0a−1 +ΩΛa2 +1−Ω0 −ΩΛ

)−1/2.

(19.19)

This age is required to be larger than the ages of theoldest known astronomical objects.

If the density is critical andΛ= 0, t0 H0 = 2/3. Thusif H0 = 75 km s−1 Mpc−1, the age is t0 = 9 Ga. Largervalues ofΩ0 give smaller ages, whereas positive valuesof Λ lead to larger ages. It has been a source of em-barrassment that the best values of H have tended togive an age for the Universe only marginally consistentwith the ages of the oldest astronomical objects. Withthe introduction of a positive cosmological constant anda slight downward revision of stellar ages this problemhas disappeared. The best current parameter values give13–14 Ga for the age of the Universe.

The “Concordance” Model. In summary there hasbeen a remarkable recent convergence between differentcosmological tests. The resulting model has a positivecosmological constant, and most of the matter is coldand dark. It is thus referred to as the ΛCDM model.The best parameter values are H = 70 km s−1 Mpc−1,ΩΛ = 0.7, Ω0 = 0.3, with cold dark matter making up85% of the total density.

The concordance model is by no means definitive.In particular the reason for the cosmological constantis a major puzzle. In order to allow for the possibilitythat Λ is variable it has become customary to refer to itas dark energy, saving the term cosmological constantfor the case of a constant Λ. Even if some alternativemechanism can produce the same effect as a non-zeroΛ,finding at least one set of acceptable parameters is animportant step forward.

An additional set of constraints on the cosmologi-cal model comes from the large-scale structures of theUniverse. These constraints, which contribute furthersupport for the concordance model, will be consideredin Sect. 19.8.

19.6 History of the Universe

405

19.6 History of the Universe

We have seen how the density of matter and of radiationenergy and temperature can be computed as functionsof the scale factor R. Since the scale factor is knownas a function of time, these quantities can be calculatedbackwards in time.

At the earliest times, densities and temperatures wereso immense that all theories about the physical pro-cesses taking place are highly conjectural. Neverthelessfirst attempts have been made at understanding the mostfundamental properties of the Universe on the basis ofmodern theories of particle physics. For example, noindications of significant amounts of antimatter in theUniverse have been discovered. Thus, for some reason,the number of matter particles must have exceeded thatof antimatter particles by a factor of 1.000000001. Be-cause of this symmetry breaking, when 99.9999999%of the hadrons were annihilated, 10−7% was left laterto form galaxies and everything else. It has been spec-ulated that the broken symmetry originated in particleprocesses about 10−35 s after the initial time.

The breaking of fundamental symmetries in the earlyUniverse may lead to what is known as inflation of theUniverse. In consequence of symmetry breaking, thedominant energy density may be the zero-point energyof a quantum field. This energy density will lead toinflation, a strongly accelerated expansion, which willdilute irregularities and drive the density very close tothe critical value. One may thus understand how thepresent homogeneity, isotropy and flatness of the Uni-verse have come about. In the inflationary picture theUniverse has to be very nearly flat, Ω0 +ΩΛ = 1. Theinflationary models also make specific predictions forthe form of the irregularities in the CMB. These pre-dictions are in general agreement with what has beenobserved.

As the Universe expanded, the density and tempera-ture decreased (Fig. 19.12) and conditions became suchthat known physical principles can be applied. Dur-ing the hot early stages, photons and massive particleswere continually changing into each other: high-energyphotons collided to produce particle-antiparticle pairs,which then were annihilated and produced photons. Asthe Universe cooled, the photon energies became toosmall to maintain this equilibrium. There is a thresholdtemperature below which particles of a given type are

Fig. 19.12. The energy densities of matter and radiation de-crease as the Universe expands. Nucleon–antinucleon pairsannihilate at 10−4 s; electron–positron pairs at 1 s

no longer produced. For example, the threshold tem-perature for hadrons (protons, neutrons and mesons)is T = 1012 K, reached at the time t = 10−4 s. Thus thepresent building blocks of the atomic nuclei, protons andneutrons, are relics from the time 10−8–10−4 s, knownas the hadron era.

The Lepton Era. In the time period 10−4 –1 s, the leptonera, the photon energies were large enough to producelight particles, such as electron–positron pairs. Becauseof matter-antimatter symmetry breaking, some of theelectrons were left over to produce present astronomi-cal bodies. During the lepton era neutrino decouplingtook place. Previously the neutrinos had been kept inequilibrium with other particles by fast particle reac-tions. As the density and temperature decreased, so didthe reaction rates, and finally they could no longer keepthe neutrinos in equilibrium. The neutrinos decoupledfrom other matter and were left to propagate throughspace without any appreciable interactions. It has beencalculated that there are at present 600 such cosmolog-ical neutrinos per cubic centimetre, but their negligibleinteractions make them extremely difficult to observe.

The Radiation Era. After the end of the lepton era,about 1 s after the initial time, the most important formof energy was electromagnetic radiation. This stageis called the radiation era. At its beginning the tem-perature was about 1010 K and at its end, about onemillion years later, when the radiation energy densityhad dropped to that of the particles, it had fallen to

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19. Cosmology

about 40,000 degrees. At the very beginning of the ra-diation era within a few hundred seconds helium wasproduced.

Just before the epoch of helium synthesis, the numberratio of free protons and neutrons was changing becauseof the decay of the free neutrons. After about 100 sthe temperature had dropped to about 109 K, which islow enough for deuterons to be formed. All remainingneutrons were then incorporated in deuterons; these, inturn, were almost entirely consumed to produce heliumnuclei. Thus the amount of helium synthesized was de-termined by the number ratio of protons and neutronsat the time of deuterium production t = 100 s. Calcu-lations show that this ratio was about 14 : 2. Thus, outof 16 nucleons, 2 protons and 2 neutrons were incorpo-rated in a helium nucleus. Consequently 4/16 = 25% ofthe mass turned into helium. This is remarkably closeto the measured primordial helium abundance.

Only the isotopes 2H, 3He, 4He and 7Li were pro-duced in appreciable numbers by nuclear processes inthe big bang. The heavier elements have formed later instellar interiors, in supernova explosions and perhaps inenergetic events in galactic nuclei.

Radiation Decoupling. The Matter Era. As we haveseen, the mass density of radiation (obtained from theformula E = mc2) behaves as R−4, whereas that of or-dinary matter behaves as R−3. Thus the radiation massdensity decreases more rapidly. At the end of the radia-tion era it became smaller than the ordinary mass den-sity. The matter era began, bringing with it the formationof galaxies, stars, planets and human life. At present, themass density of radiation is much smaller than that ofmatter. Therefore the dynamics of the Universe is com-pletely determined by the density of massive particles.

Soon after the end of the radiation era, radiationdecoupled from matter. This happened when the tem-perature had dropped to a few thousand degrees, andthe protons and electrons combined to form hydro-gen atoms. It was the beginning of the “dark ages” atredshifts z = 1000–100, before stars and galaxies hadformed, when the Universe only contained dark matter,blackbody radiation, and slowly cooling neutral gas.

At present, light can propagate freely through space.The world is transparent to radiation: the light from dis-tant galaxies is weakened only by the r−2 law and by theredshift. Since there is no certain detection of absorp-

Fig. 19.13. The motion of the Milky Way in relation to themicrowave background can be seen in the measurements ofWMAP. One side of the sky is darker (colder), and the otherside is lighter (warmer). The horizontal stripe is the densestpart of the Milky Way. (Photo NASA)

tion by neutral gas, there must have been a reionisationof the Universe. It is thought that this occurred aroundz = 5–10.

19.7 The Formation of Structure

As we go backward in time from the present, the dis-tances between galaxies and clusters of galaxies becomesmaller. For example, the typical separation betweengalaxies is 100 times their diameter. At the redshiftz = 101 most galaxies must have been practically incontact. For this reason galaxies in their present formcannot have existed much earlier than about z = 100.Since the stars were presumably formed after the galax-ies, all present astronomical systems must have formedlater than this.

It is thought that all observed structures in theUniverse have arisen by the gravitational collapse ofsmall overdensities. Whereas the presently observablegalaxies have undergone considerable evolution, whichmakes it difficult to deduce their initial state, on largerscales the density variations should still be small andeasier to study. These are the structures considered inthe present section. The later evolution of galaxies isdiscussed in Sect. 18.8, and the formation of the MilkyWay in Sect. 17.5.

The Statistical Description of Large-scale Structure.The departures from strict homogeneity in the Universeare random in character, and must therefore be described

19.7 The Formation of Structure

407

using statistical methods. Perhaps the most straightfor-ward way of doing this is to take regions of a given size,specified in terms of their mass, and give the probabilitydistribution for relative density variations on that scale.

A second method is to consider the spatial separa-tions between individual objects such as galaxies orclusters. The distribution of these separations is used todefine the correlation function, which is a measure ofthe clustering of the objects in question.

A third method to describe large-scale fluctuationsis by means of the power spectrum. Here the densityvariations (in space or in projection on the sky) arerepresented as sum of waves. The power spectrum isthe squared amplitude of these waves as a function ofwavelength.

All three methods are representations of the densityvariations in the Universe, and they are theoreticallyclosely related. However, in practice they are observedin different ways, and therefore, which representation ismost suitable depends on what kind of observations arebeing analysed. The density variations are usually de-scribed by means of a spectral index n and an amplitudeσ8 to be introduced below.

The Growth of Perturbations. In order to describe thegrowth of structures in the Universe, consider a givenregion containing the mass M. If its denity is slightlylarger than the mean density, its expansion will beslightly slower than that of the rest of the Universe, andits relative overdensity will grow. The rate of growthas a function of mass depends on the relative impor-tance of the material components of the Universe, darkmatter, radiation, and ordinary baryonic matter.

It is assumed that there is an initial distribution of per-turbations where the fluctuations with mass M have anamplitude that is proportional to M−(n+3)/6. The spectralindex n is a cosmological parameter to be determinedfrom observations.

The first step in structure formation is when a givenmass comes within the horizon, i. e. when there hasbeen enough time since the big bang for light signalsto cross the given region. During the radiation era thehorizon mass grows proportionally to t3/2, and thus thetime at which the mass M comes within the horizonwill be proportional to M2/3 . Any perturbation will ini-tially be larger than the horizon mass, and while this isthe case it grows in proportion to t. Once a matter per-

turbation comes inside the horizon its amplitude willremain constant. This constant amplitude will behaveas tM−(n+3)/6, which is proportional to M−(n−1)/6. Ifn = 1, the perturbations enter the horizon with an am-plitude that is independent of mass. We shall see thatthe observed value of n is in fact very close to 1.

Perturbations of both dark and baryonic matter den-sity will behave as described above during the radiationera. At the end of the radiation era at time tEQ, when themass densities of radiation and (non-relativistic) mat-ter become equal, the amplitude of the perturbationswill be given by the horizon mass MEQ ≈ 1016 M forM MEQ, and will be proportional to M−(n+3)/6 forM MEQ. After equality the dark matter perturbationswill be free to start growing again as t2/3, independentof mass.

Unlike dark matter, ordinary baryonic matter per-turbations cannot grow as long as the Universe remainsionised. Instead there is a minimum mass of a collapsinggas cloud given by the Jeans mass MJ:

MJ ≈ P3/2

G3/2ρ2, (19.20)

where ρ and P are the density and pressure in the cloud(see Sect. 6.11). The value of MJ before decoupling was

MJ = 1018 M (19.21)

and after decoupling

MJ = 105 M . (19.22)

The reason for the large difference is that before decou-pling, matter feels the large radiation pressure (P = u/3,see *Gas Pressure and Radiation Pressure, p. 238). Afterdecoupling, this pressure no longer affects the gas.

The large Jeans mass before decoupling means thatoverdense regions of normal gas cannot start growingbefore z = 1000. Rather than growing they oscillate likesound waves. After decoupling a large range of massesbecome Jeans unstable. By then density perturbationsof dark matter have had time to grow, and the gas willtherefore rapidly fall into their potential wells. The firststars will start forming in the collapsing regions, andwill reionise the Universe.

Because the expansion of the Universe works againstthe collapse, the density of Jeans unstable regions grows

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19. Cosmology

rather slowly. In order to produce the observed systems,the density perturbations at decoupling cannot be toosmall. In models without dark matter the variations inthe CMB predicted on this basis tended to be too large.In the CDM model the predicted variations are of theexpected amplitude.

In the CDM model the amplitude of fluctuations onscales about MEQ and smaller depends only weakly onmass. This is why the CDM model leads to an hierar-chical description of structure formation. In this picture,systems of all masses above 105 M begin forming af-ter decoupling. Because smaller systems will collapsemore rapidly, they are the first to form, at redshiftsabout 2 0. Once the first sources of light, starburstsor AGNs, had formed, they could reionise the gas.This marked the ending of the dark ages at redshiftsz = 10–5.

The redshift of reionisation is still not well known,and is therefore treated as a parameter to be determinedin tests based on large-scale structures. It is usuallyexpressed by means of τ , the optical depth to electronscattering of the background radiation. A larger valueof τ corresponds to a higher electron density, implyingreionisation at a higher redshift. Since galaxies can beseen at redshifts larger than 6, the corresponding valueof τ = 0.03 represents a minimum. The result of the firstyear of observations with WMAP gave τ = 0.16, whichwould have corresponded to a redshift 17.

One finally has to ask where the initial perturba-tions came from. An attractive feature of the inflationarymodel is that it makes specific predictions for these ini-tial perturbations, deriving them from quantum effectsat very early times. In this way the observed propertiesof the largest astronomical systems contain informationabout the earliest stages of our Universe.

Fluctuations of the Cosmic Microwave Background.One important way of studying the large-scale structureof the early Universe is by means of the irregularitiesof the cosmic microwave background. The overdensi-ties that were later to give rise to observed structuresshould also give rise to temperature variations of theCMB.

The temperature variations in the microwave back-ground have been mapped, first by the COBE satellite,and later by WMAP. A map of the CMB according toWMAP is shown in Colour Supplement Plate 34. The

observed variations are in qualitative agreement withthe scenario for structure formation described above.

A more quantitative view of the observations is pro-vided by the power spectrum of the observations shownin Fig. 19.14. This shows the amplitude of the tempera-ture variations as a function of angular scale on the sky.

The physical processes we have described give riseto the features in the power spectrum. Thus the firstpeak is produced by perturbations that have just hadtime to collapse to a maximum density since equalityof matter and radiation, before bouncing back. The lin-ear size of this peak does not depend strongly on theexact parameters of the model, and its postion (angu-lar scale) can therefore be used as a standard measuringrod in the diameterâ“redshift test. The second and thirdpeaks in the power spectrum, the “acoustic peaks”,correspond to perturbations that have bounced back andcollapsed again, respectively. The positions and ampli-tudes of these features in the power spectrum depend onthe cosmological parameters. Thus the WMAP observa-tions of the CMB can be fitted using a model containingsix free parameters: the Hubble constant, the densitiesof dark matter and baryons, the optical depth τ , and theamplitude σ8 and shape n of the initial perturbations.The curvature of the Universe, 1−Ω0 −ΩΛ can be setequal to zero, which determines the value of ΩΛ. Theresults of three years of observations with WMAP havegiven values for these parameters in full agreement withthe concordance model. In particular, the optical depthτ = 0.1, corresponding to a redshift 7–12.

Very Large Scale Structure. After decoupling struc-tures on different scales are free to grow. The importantdividing line is whether a given overdensity is still ex-panding with the rest of the Universe or whether ithas had time to recollapse. Systems that have recol-lapsed will fairly rapidly virialise, i. e. settle down intoa stationary state. Such systems can be considered realastronomical systems.

The largest astronomical systems are clusters ofgalaxies. For some time there was controversy aboutwhether even larger structures, superclusters, existed.The controversy was settled in the 1970’s when it wasrealized that this largely depended on what was meantby superclusters. Regions of higher density existed onscales larger than clusters of galaxies (a few Mpc), allthe way up to 100 Mpc, but they did not really form in-

19.7 The Formation of Structure

409

Fig. 19.14. Angular power spectrum of the temperature varia-tions of the cosmic microwave background from the three-yearresults the WMAP satellite. This gives the square of the ampli-

tude of the temperature variations on different angular scales.(http://map.gsfc.nasa.gov/m_mm.html)

dividual structures in approximate equilibrium like theclusters of galaxies.

Between these two alternatives are structures that areonly now turning round and beginning to recollapse.One way of specifying the amplitude of the initial fluc-tuations is to give the scale of this transition point. Thisscale corresponds roughly to a present linear size of8 Mpc, and for this reason the fluctuation amplitude iscommonly given by means of σ8, the amplitude of thefluctuations at present scale 8 Mpc. Because of the wayit has been chosen σ8 should be close to 1.

Smaller systems will already have collapsed. Thereare many ways of statistically describing their large-scale distribution. One of the most studied descriptors

of clustering is the correlation function, ξ(r). Considertwo infinitesimal volumes dV1 and dV2 separated bythe distance r. If there were no clustering, the prob-ability of finding galaxies in these volumes wouldbe N2dV1dV2, where N is the number density ofgalaxies. Because of clustering this probability is actu-ally N2(1+ ξ(r))dV1dV2. The correlation function thusmeasures the higher probabilty of finding galaxies neareach other.

Although there are ways of estimating the correlationfunction from the distribution on the sky, a more reli-able estimate can be made by mapping the distributionof galaxies in three dimensions, using redshifts to de-termine their distances. The distribution from one such

410

19. Cosmology

Fig. 19.15. Correlation function for galaxies in the 2dF GalaxyRedshift Survey (see Fig. 18.14). The galaxies have been di-vided into passive and actively star-forming types based ontheir spectra, roughly corresponding to the division into early(E and S0) and late (spiral and irregular) Hubble types. Over alarge range of separations the correlation functions are well de-scribed by power laws with slopes 1.93 and 1.50, respectively.(D.S. Madgwick et al. 2003, MNRAS 344, 847, Fig. 3)

survey, the Two-degree Field Galaxy Redshift Survey(2dFGRS) is shown in Fig. 19.15.

It has been known since the 1970’s that the correla-tion function is approximately proportional to rγ . Theconstant of proportionality is related to the amplitudeσ8 and the exponent γ is related to the index n of theinitial perturbations. The observed value of γ is about1.8, depending to some extent on the sample of objectsstudied.

In order to determine the cosmological parametersthe (three-dimensional) power spectrum of the galaxydistribution is usually used. It is then compared to the-oretical power spectra that depend on the parameters inorder to find the best-fitting model. The optical depthparameter τ does not affect these tests. The resultshave been completely consistent with the concordancemodel.

The Concordance Model Again. We have discussedthe use of the cosmic microwave background andthe large-scale distribution of galaxies in order todetermine the cosmological parameters. Both meth-ods independently give consistent results. Combiningthem produces even more (formally) accurate values.Furthermore, these values are in agreement with theones obtained from the traditional cosmological testsdiscussed in Sect. 19.5.

There are eight basic parameters describing theconcordance model. Currently their values, with anaccuracy of about 10%, are: Hubble constant H0

73 km s−1 Mpc−1, baryon density parameter Ωb 0.04,mass density parameter Ω0 0.24, optical depth τ 0.1,spectral index n 0.95, fluctuation amplitude σ8 0.75.

Other, less comprehensive tests, such as the grav-itational lensing by cosmic structures, the velocitiesinduced by mass concentrations, and the number ofclusters of galaxies, have added support to the con-cordance model. This remarkable agreement betweenmany independent determinations of the cosmologicalparameters have earned the model its name. Once theparameters necessary for a general description of theUniverse are reasonably well known, other components,such as gravitational waves or neutrinos, can be includedin the cosmological model. This new development hasbeen called the era of precision cosmology. However, itshould still not be forgotten that in the model one has toassume the presence of both dark matter and dark en-ergy, neither of which is based on any evidence apartfrom their role in cosmology.

19.8 The Future of the Universe

The standard models allow two alternative prospectsfor the future development of the Universe. Expansionmay either go on forever, or it may reverse to a contrac-tion, where everything is eventually squeezed back toa point. In the final squeeze, the early history of the Uni-verse would be repeated backwards: in turn, galaxies,stars, atoms and nucleons would be broken up. Finallythe state of the Universe could no longer be treated bypresent-day physics.

In the open models the future is quite different. Theevolution of the stars may lead to one of four end results:a white dwarf, a neutron star or a black hole may be

19.8 The Future of the Universe

411

formed, or the star may be completely disrupted. Afterabout 1011 years, all present stars will have used up theirnuclear fuel and reached one of these four final states.

Some of the stars will be ejected from their galaxies;others will form a dense cluster at the centre. In about1027 years the central star clusters will become so densethat a black hole is formed. Similarly the galaxies inlarge clusters will collide and form very massive blackholes.

Not even black holes last forever. By a quantum me-chanical tunnelling process, mass can cross the eventhorizon and escape to infinity – the black hole is said to“evaporate”. The rate of this phenomenon, known as theHawking process, is inversely proportional to the massof the hole. For a galactic-mass black hole, the evapora-tion time is roughly 1098 years. After this time, almostall black holes will have disappeared.

The ever expanding space now contains black dwarfs,neutron stars and planet-size bodies (unless the pre-dictions of a finite proton lifetime of about 1031 yearsare confirmed; in that case all these systems will havebeen destroyed by proton decay). The temperature ofthe cosmic background radiation will have dropped to10−20 K.

Even further in the future, other quantum phenomenacome into play. By a tunnelling process, black dwarfscan change into neutron stars and these, in turn, intoblack holes. In this way, all stars are turned into blackholes, which will then evaporate. The time required hasbeen estimated to be 101026

years! At the end, onlyradiation cooling towards absolute zero will remain.

It is of course highly doubtful whether our cur-rent cosmological theories really are secure enoughto allow such far-reaching predictions. New theoriesand observations may completely change our presentcosmological ideas.

* Newtonian Derivation of a Differential Equationfor the Scale Factor R(t)Let us consider a galaxy at the edge of a massive sphere(see figure). It will be affected by a central force due togravity and the cosmological force

mr = −4πG

3

r3ρm

r2+ 1

3mΛr ,

or

r = −4π

3Gρr + 1

3Λr . (19.23)

In these equations, the radius r and the density ρ arechanging with time. They may be expressed in terms ofthe scale factor R:

r = (R/R0)r0 , (19.24)

where R is defined to be R0 when the radius r = r0.

ρ = (R0/R)3ρ0 , (19.25)

where the density ρ = ρ0 when R = R0. Introducing(19.24) and (19.25) in (19.23), one obtains

R = − a

R2+ 1

3ΛR , (19.26)

where a = 4πG R30ρ0/3. If (19.26) is multiplied on both

sides by R, the left-hand side yields

R R = 1

2

d(R2)

dt,

and thus (19.26) takes the form

d(R2)= − 2a

R2dR + 2

3ΛRdR . (19.27)

Let us define R0 = R(t0). Integrating (19.27) from t0 tot gives

R2 − R20 = 2a

(1

R− 1

R0

)

+ 1

3Λ(R2 − R2

0) .

(19.28)

The constants R0 and a can be eliminated in favour ofthe Hubble constant H0 and the density parameter Ω0.Because ρc = 3H2

0 /8πG,

2a = 8πG R30ρ0/3

= H20 R3

0ρ0/ρc = H20 R3

0Ω0 ,(19.29)

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19. Cosmology

where Ω0 = ρ0/ρc. Using expression (19.29) andR0 = H0 R0 in (19.28), and defining ΩΛ =Λ/(3H2

0 ),one obtains

R2

H20 R2

0

=Ω0R0

R+ΩΛ

(R

R0

)2

+1−Ω0 −ΩΛ

(19.30)

as the basic differential equation governing R(t).For simplicity we now set ΩΛ = 0. Then the time

behaviour of the scale factor R depends on the valueof the density parameter Ω0. Because R2 > 0 always,according to (19.30)

Ω0R0

R−Ω0 +1 ≥ 0 ,

or

R0

R≥ Ω0 −1

Ω0. (19.31)

If Ω0 > 1, this means that

R ≤ R0Ω0

Ω0 −1≡ Rmax .

When the scale factor reaches its maximum value Rmax,then according to (19.30), R = 0, and the expansionturns into contraction. If Ω0 < 1, the right-hand side of(19.30) is always positive and the expansion continuesforever.

The equation for the time dependence of the scalefactor in the general theory of relativity contains theconstant k which determines the geometry of space:

R2 = 8πG R30ρ0

3R− kc2 . (19.32)

Equations (19.32) and (19.28) (or (19.30)) can be madeidentical if one chooses

H20 R2

0(Ω0 −1)= kc2 .

Thus, complete agreement between the Newtonian andthe relativistic equation for R is obtained. The valuesof the geometrical constant k = +1, 0, −1 correspondrespectively to Ω0 > 1, = 1 and < 1. More generally,the condition for a flat model, k = 0, corresponds toΩ0 +ΩΛ = 1.

When k = 0, the time dependence of the expansionis very simple. Setting Ω0 = 1 and using (19.32) and

(19.29), one obtains

R2 = H20 R3

0

R.

The solution of this equation is

R =(

3H0t

2

)2/3

R0 . (19.33)

It is also easy to calculate the time from the beginningof the expansion: R = R0 at the time

t0 = 2

3

1

H0.

This is the age of the Universe in the Einstein–de Sittermodel.

* Three Redshifts

The redshift of a distant galaxy is the result of threedifferent mechanisms acting together. The first one isthe peculiar velocity of the observer with respect to themean expansion: the Earth moves about the Sun, theSun about the centre of the Milky Way, and the MilkyWay and the Local Group of galaxies is falling towardsthe Virgo Cluster. The apparatus measuring the lightfrom a distant galaxy is not at rest; the velocity of theinstrument gives rise to a Doppler shift that has to becorrected for. Usually the velocities are much smallerthan the speed of light. The Doppler shift is then

zD = v/c . (19.34)

For large velocities the relativistic formula has to beused:

zD =√

c+vc−v −1 . (19.35)

The redshift appearing in Hubble’s law is the cosmo-logical redshift zc. It only depends on the values of thescale factor at the times of emission and detection of theradiation (R and R0) according to

zc = R0/R −1 . (19.36)

19.9 Examples

413

The third type of redshift is the gravitational redshift zg.According to general relativity, light will be redshiftedby a gravitational field. For example, the redshift ofradiation from the surface of a star of radius R andmass M will be

zg = 1√1− RS/R

−1 , (19.37)

where RS = 2G M/c2 is the Schwarzschild radius ofthe star. The gravitational redshift of the radiation fromgalaxies is normally insignificant.

The combined effect of the redshifts can be calculatedas follows. If the rest wavelength λ0 is redshifted by theamounts z1 and z2 by two different processes, so that

z = λ2 −λ0

λ0= λ2

λ0−1 = λ2

λ1

λ1

λ0−1 ,

or

(1+ z)= (1+ z1)(1+ z2) .

Similarly, the three redshifts zD, zc and zg will combineto give an observed redshift z, according to

1+ z = (1+ zD)(1+ zc)(1+ zg) . (19.38)

19.9 Examples

Example 19.1 a) In a forest there are n trees perhectare, evenly spaced. The thickness of each trunkis D. What is the distance of the wood not seen forthe trees? (Find the probability that the line of sight willhit a trunk within a distance x.) b) How is this relatedto the Olbers paradox?

a) Imagine a circle with radius x around the observer.A fraction s(x), 0 ≤ s(x) ≤ 1, is covered by trees.Then we’ll move a distance dx outward, and draw an-other circle. There are 2πnx dx trees growing in theannulus limited by these two circles. They hide a dis-tance 2πxnD dx or a fraction nD dx of the perimeterof the circle. Since a fraction s(x) was already hid-den, the contribution is only (1− s(x))nD dx. Weget

s(x +dx)= s(x)+ (1− s(x))nD dx ,

which gives a differential equation for s:

ds(x)

dx= (1− s(x))nD .

This is a separable equation, which can be integrated:

s∫0

ds

1− s=

x∫0

nD dx .

This yields the solution

s(x)= 1− e−nDx .

This is the probability that in a random direction wecan see at most to a distance x. This function s is a cu-mulative probability distribution. The correspondingprobability density is its derivative ds/dx. The meanfree path λ is the expectation of this distribution:

λ=∞∫

0

x

(ds(x)

dx

)dx = 1

nD.

For example, if there are 2000 trees per hectare, andeach trunk is 10 cm thick, we can see to a distance of50 m, on the average.

b) The result can easily be generalized into three di-mensions. Assume there are n stars per unit volume,and each has a diameter D and surface A = πD2

perpendicular to the line of sight. Then we have

s(x)= 1− e−n Ax ,

where

λ= 1/n A .

For example, if there were one sun per cubic parsec,the mean free path would be 1.6×104 parsecs. If theuniverse were infinitely old and infinite in size, theline of sight would eventually meet a stellar surface inany direction, although we could see very far indeed.

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19. Cosmology

Example 19.2 Find the photon density of the 2.7 Kbackground radiation.

The intensity of the radiation is

Bν = 2hν3

c2

1

ehν/(kT )−1

and the energy density

uν = 4π

cBν = 8πhν3

c3

1

ehν/(kT )−1.

The number of photons per unit volume is found bydividing the energy density by the energy of a singlephoton, and integrating over all frequencies:

N =∞∫

0

uνdν

hν= 8π

c3

∞∫0

ν2 dν

ehν/(kT )−1.

We substitute hν/kT = x and dν = (kT/h)dx:

N = 8π

(kT

hc

)3 ∞∫0

x2 dx

ex −1.

The integral cannot be expressed in terms of elementaryfunctions (however, it can be expressed as an infinitesum 2

∑∞n=0(1/n

3)), but it can be evaluated numerically.

Its value is 2.4041. Thus the photon density at 2.7 K is

N = 16π

(1.3805×10−23 J K−1 ×2.7 K

6.6256×10−34 Js×2.9979×108 m s−1

)3

×1.20206

= 3.99×108 m−3 ≈ 400 cm−3 .

19.10 Exercises

Exercise 19.1 The apparent diameter of the galaxyNGC 3159 is 1.3′, apparent magnitude 14.4, and radialvelocity with respect to the Milky Way 6940 km s−1.Find the distance, diameter and absolute magnitude ofthe galaxy. What potential sources of error can you thinkof?

Exercise 19.2 The radial velocity of NGC 772 is2562 km s−1. Compute the distance obtained from thisinformation and compare the result with Exercise 18.1.

Exercise 19.3 If the neutrinos have nonzero mass, theuniverse can be closed. What is the minimum massneeded for this? Assume thatΛ= 0, the density of neu-trinos is 600 cm−3, and the density of other matter isone tenth of the critical density.