chapter 17 energy and matter in the...

Chapter 17

Energy and Matter in the Universe

The history and fate of the Universe ultimately turn onhow much matter, energy, and pressure it contains:

1. These components of the stress–energy tensor allcouple to gravity.

2. This coupling determines how self-gravitation of theUniverse influences the Hubble expansion.

In this chapter we begin to address quantitatively the issueof the matter and energy contained in the Universe andhow that determines its history.

463

464 CHAPTER 17. ENERGY AND MATTER IN THE UNIVERSE

The Universe is mostly empty space, which might suggest thata New-tonian description of gravity (which is valid in the weak gravity limit)is adequate for describing the large-scale structure of theUniverse.

• But whether general relativity effects are important relative to aNewtonian description may be estimated in terms of the ratioofan actual radius for a massive object compared with its radius ofgravitational curvature.

• If we apply such a criterion to the entire Universe, reasonable es-timates for the mass–energy contained in the Universe indicatethat the actual radius of the known Universe and the correspond-ing gravitational curvature radius could be comparable.

• Thus, a description of the large-scale structure of the Universe(cosmology) must be built on a covariant gravitational theory,rather than on Newtonian gravity.

• Even so, we can understand a substantial amount concerningtheexpanding Universe simply by using Newtonian concepts.

17.1. EXPANSION AND NEWTONIAN GRAVITY 465

Earth r

Density = ρ

Homogeneous mass

distribution

Distant

galaxy

Figure 17.1:Newtonian model of the expanding Universe.

17.1 Expansion and Newtonian Gravity

Consider the test galaxy illustrated in Fig. 17.1. The gravitationalpotential acting on the galaxy is

U =−GMm

r,

wherem is the mass of the galaxy and

Total mass within sphere= M = 43πr3ρ ,

which is constant sinceρ decreases with time andr increases but theproductρr3 is constant. Thus

U = −43πGr2ρm.


Earth r

Density = ρ

Homogeneous mass

distribution

Distant

galaxy

If the motion of the galaxy is caused entirely by the Hubble expan-sion, its radial velocity relative to the Earth isv = H0r . This impliesa kinetic energy

T = 12mv2 = 1

2mH20r2,

wherem is the inertial mass of the galaxy, assumed to be equivalentto its gravitational mass. The total energy of the galaxy is then

E = T +U = 12mH2

0r2−

43πGr2ρm

= 12mr2

(

H20 −

83πGρ

)

.

17.2. THE CRITICAL DENSITY 467

17.2 The Critical Density

If the expansion is to halt, we must haveE = 0 and thus

E = 12mr2

(

H20 −

83πGρ

)

−→ H20 = 8

3πGρ .

Solving forρ , thecritical densitythat will just halt the expansion is

ρc =3H2

0

8πG≃ 1.88×10−29h2 g cm−3.

The correspondingcritical energy densityis

εc = ρcc2 = 1.05×10−2h2 MeV cm−3

= 1.69×10−8h2 erg cm−3.

The critical density corresponds to an average concentra-tion of only six hydrogen atoms per cubic meter of spaceor about 140M⊙ per cubic kiloparsec.

We may distinguish three qualitative regimes for the actualdensity(in this simple Newtonian picture)ρ :

1. If ρ > ρc the Universe is said to beclosedand the expansion willstop in a finite amount of time.

2. If ρ < ρc the Universe is said to beopenand the expansion willnever halt.

3. if ρ = ρc the Universe is said to beflat (or euclidean) and theexpansion will halt, but only asymptotically ast → ∞.


Thus, in this simple Newtonian picture the ultimate fate ofthe Universe is determined by its present matter density.

We shall see that this conclusion ismodified—profoundly—by the apparentpresence of dark energy in the actualUniverse.

17.3. BARYONIC AND NON-BARYONIC MATTER 469

17.3 Baryonic and Non-Baryonic Matter

Baryonic matter is “ordinary” matter consisting of protonsand neutrons. Non-baryonic matter consists of particlesthat do not undergo the strong interactions.

For example, neutrinos are one example ofnon-baryonic matter.

Constraints can be placed on thebaryonic matter densityρB by com-paring observed and predicted abundances of light isotopessuch as3He and7Li that are formed in the early Universe.

1. One findsρB ≃ 2×10−31 g cm−3.

2. This is far too small to close the Universe sinceρc ≃ 1.88×10−29h2 g cm−3.

3. However, this does not settle the issue because

• There is substantial evidence that most matter in the Uni-verse isdark matterand isnon-baryonic.

• There is a substantialdark (vacuum) energycontribution tothe current evolution of the Universe.

4. We shall discuss candidates for this non-baryonic matterand therole of vacuum energy in subsequent chapters.


Extending the trend started by Copernicus: we are not thecenter of the Universe, and we aren’t even made up of thedominant matter of the Universe.

Not only are we not the center, we aren’t evenmade of the right stuff!


Radius

Radia

l V

elo

city

Predicted assuming mass

follows luminosity

Observed

Figure 17.2:Schematic velocity curves for spiral galaxies.

17.3.1 Evidence for Dark Matter: Galaxy Rotation Curves

In spiral galaxies, if we balance the centrifugal and gravitational forcesat a radiusR, the tangential velocityv should obey the relation

v =

√

GMR

implied by Kepler’s laws, withR the radius andM the enclosed mass.

• Well outside the main matter distribution, we expectv≃ R−1/2.

• The velocities can be measured using the Doppler effect, bothfor visible light from the luminous matter,and from the21 cmhydrogen line for non-luminous hydrogen.

• For many spirals we find notv≃ R−1/2 but almost constant ve-locity well outside the bulk of the luminous matter.

This is illustrated schematically in Fig. 17.2.


300

200

100

00 20 40 60 80 100 120 140 160

Distance (arcminutes)

vr

(km

s-1

)

Figure 17.3:Rotation curve for the Andromeda Galaxy. White points indicatemeasured velocities (open circles at large distance are RF observations).

Observational data on the rotation curve for the An-dromeda Galaxy (M31) are displayed in Fig. 17.3. Con-verting the angular size to kpc using the distance of 778kpc to Andromeda we see that

• The obvious visible matter lies within about60′ ∼14 kpcof the center.

• RF observations suggest that the rotation curve isconstant out to at least about150′ ≃ 36 kpc.

• Direct measurements suggest constant velocities out to atleast30 kpc in many spirals, and

• Indirect means suggest that constant velocities may extend outto 100 kpc or more in some spirals.

This indicates the presence of substantial gravitating matter distributedin a halo beyond the visible matter.


1"

Aug 1991 Aug 1994

Figure 17.4:Gravitational lensing: the Einstein Cross.

17.3.2 Evidence for Dark Matter: Gravitational Lensing

The path of light is curved in a gravitational field.

• This can causegravitational lensing, where intervening massesact as “lenses” to distort the image of distant objects.

• A spectacular example of gravitational lensing is the EinsteinCross, shown in Fig. 17.4.

• In this image, a single object appears as four objects.

• A very distant quasar is thought to be positioned behind a mas-sive galaxy.

• The gravitational effect of the galaxy has created multiple im-ages through gravitational lensing on the light from the quasar.

The individual stars in the foreground galaxy may also be actingas gravitational lenses, causing the images to change theirrelativebrightness in these two images taken three years apart, as stars changeposition in the lensing galaxy.


Quasarimages

Spiral arms

Bar

Nucleus

Einstein Cross

Faint lensing

galaxy

Figure 17.5:The Einstein Cross and the lensing galaxy. The intensity hasbeendisplayed on a logarithmic scale so that the very bright quasar images and the ex-tremely faint bar and arms of the lensing galaxy can be seen atthe same time.Image courtesy W. Keel, University of Alabama.

This interpretation of the Einstein Cross is bolstered byFig. 17.5, which shows in faint outline the foregroundlensing galaxy surrounding the bright central nucleus ofthe spiral and the four quasar images.

• The lensing galaxy is a relatively nearby barred spi-ral.

• Both the spiral arms and the central bar of the fore-ground galaxy can be seen if one looks carefully (seethe annotated version of the figure in the right panel).


The strength of a gravitational lens depends on thetotalmass contained within it,whether that mass is visible ornot.

• Gravitational lenses can serve as excellent indicatorsof how much unseen matter is present in the regionof the lens.

• Extensive analysis of gravitational lensing by largemasses leads to conclusions similar to those sug-gested above by the rotation curves for spiral galax-ies:

More than 90% of the mass contributing tothe strength of large gravitational lenses isdark.


17.4 Dark Energy

Dark matter may appear exotic by normal standards, since we don’tknow what it is and therefore do not know why it fails to couplestrongly through any force other than gravity.

• However, we shall see in Chapter 18 that there is growing evi-dence that the evolution of the present Universe is being domi-nated by something even more exotic:dark energy.

• Dark energy (also known asvacuum energy) behaves fundamen-tally differently from either normal matter and energy, or darkmatter.

• It appears to cause the force of gravitation to becomerepulsive.

• To understand and to deal adequately with this remarkable no-tion will require a covariant formulation of gravitation.

Therefore, we defer substantial discussion of the evidencefor and roleplayed by dark energy until the following two chapters.

17.5. COSMIC SCALE FACTOR 477

17.5 Cosmic Scale Factor

As we have seen, the Hubble expansion makes it convenient to intro-duce acosmic scale factora(t) that sets the global distance scale forthe Universe.

• If peculiar motion is ignored, the expansion is governed entirelyby a(t) andall distances simply scale with this factor.

• Example: if present time ist0 and present scale factor isa0, awavelength of lightλ emitted at timet < t0 is scaled toλ0 att = t0 by the universal expansion:

λ0

a0=

λa(t)

.

• Likewise, if r0 andρ0 are the present values ofr andρ ,

r(t)r0

=a(t)a0

ρ(t)ρ0

=

(a0

a(t)

)3

.

• This permits us to express all dynamical equations in terms of thescale factor.Example:gravitational force acting on the galaxy

FG = −∂U∂r

= −GMmr2 = −

43πGρrm,

and the corresponding gravitational acceleration is

r =FG

m= −

GMr2 = −

43πGρr.

Then fromr(t)/r0 = a(t)/a0,

r =r0

a0a = −

43

πGρ0a3

0

a3

r0

a0a → a = −

43πGρ0a3

0

(1a2

)

.

(acceleration of the scale factor).


17.6 Density Parameters

It is convenient to introduce thetotal density parameterevaluated atthe present time

Ω ≡ρρc

=8πGρ3H2

0

.

whereρ is thecurrenttotal density coupled to gravity.

• Thus, the closure condition implies thatΩ = 1 (critical density).

• The subscript “0” is often used onΩ andρ to indicate explic-itly that they are evaluated at the present time; we suppressthatsubscript to avoid notational clutter in later equations.

The acceleration of the scale factor may be expressed in terms of thedensity parameterΩ,

a = −43πGρa3

0

(1a2

)

Ω ≡ρρc

=8πGρ3H2

0

→ a =−

12H2

0a30Ω

a2

(where it is understood thatρ ≡ ρ0 andΩ ≡ Ω0 correspond to theircurrent values.)

Anticipating the later treatment of the expansion using general rel-ativity, we may expect that the density parameter gets contributionsfrom three major sources in the current Universe:

1. Matter, including dark matter(with density denoted byρm)

2. Radiation(with density denoted byρr)

3. Vacuum or dark energy(with density denoted byρv or ρΛ).

17.6. DENSITY PARAMETERS 479

These densities may be used to define corresponding density param-etersΩi through

ρr(a) = ρcΩr ρm(a) = ρcΩm ρv(a) = ρcΩv,

where we shall show (Chapter 18) that the total density changes witha(t) according to

ρ(a) = ρc

(Ωr

a4 +Ωm

a3 +Ωv

)

(a(t0) ≡ 1),

• we have assumed the standard convention of normalizing thecurrent value of the scale parametera(t0) to unity.

• We shall make no explicit distinction between mass densityρand the corresponding energy densityε = ρc2, since they arenumerically the same inc = 1 units.

• Note that the different densities scale differently witha(t), andthus differently with time.

For baryonic matter alone, we obtain from the observedρB ≃ 2×10−31 g cm−3 that

Ω = Ωm =ρB

ρc

≃ 0.024. (baryonic matter).

This is well below the critical density (Ω = 1) but, as wehave previously noted, baryonic matter is not the dominantmatter in the Universe and we must include

• the effect of non-baryonic dark matter and

• the effect of dark energy

to determine the true value ofΩ.


Table 17.1: Density parametersSource Value (Ωi = ρi/ρc)

Total matter Ωm = 0.3

Baryonic matter ΩB = 0.024

Total radiation Ωr <∼ 8×10−5

Total vacuum Ωv = 0.7

Curvature Ωc ≤ 0.01

Some estimates of the current density parameters for theradiation, matter, baryonic portion of the matter, and thevacuum energy are given in Table 17.1 (the curvature den-sity entry will be explained in Chapter 18).

17.7. TIME DEPENDENCE OF THE SCALE FACTOR 481

17.7 Time Dependence of the Scale Factor

Identity: a =12

dda

a2

Earlier: a =−

12H2

0a30Ω

a2

−→12

dda

a2 =−

12H2

0a30Ω

a2 .

Solving this forda2 and integrating from the present timet0 back toan earlier timet,

∫ t

t0da2 = −H2

0a30Ω

∫ a

a0

daa2 → a2 = a2

0+H20a3

0Ω(

1a−

1a0

)

,

and sincea0 = a0H0 (Exercise),

a2 = a20H2

0 f (Ω,t),

where we definef (Ω,t) = 1+Ω

a0

a(t)−Ω,

which must obey the condition

f (Ω,t) ≥ 0,

sincea2 can never be negative.

We may use this condition to enumerate different possi-bilities for the history of the Universe.NOTE:Ω ≡ Ω0 inthese equations.


17.8 Expansion Histories for the Universe

Let us consider as an example, dust-filled universes; that is, universescontaining only

• pressureless, non-relativistic matter and

• negligible amounts of radiation or vacuum energy.

Three qualitatively different scenarios for such a Universe, dependingon the value ofΩ ≡ Ω0 = Ωm.

1. Ω < 1 (undercritical): In this case, asa(t) → ∞,

f (Ω,t) = 1+Ωa0

a(t)−Ω −→ 1−Ω > 0.

Thusa never goes to zero(a2 ∝ f (Ω,t) and we live in an open,ever-expanding universe ifΩ < 1.

2. Ω = 1 (critical): For this case, asa(t) → ∞,

f (Ω,t) −→ 0,

but it only reaches 0 att = ∞. Hence, ifΩ = 1, the universeis ever-expanding(constraint: expanding now)but the rate ofexpansion approaches zero asymptotically ast → ∞.

3. Ω > 1 (overcritical): Now ast increases

f (Ω,t) −→ 0,

but in afinite time tmax. Beyond this time we still must satisfy thecondition f (Ω,t) ≥ 0. Thus, ifΩ > 0 the expansion turns into acontractionat timetmax and the universe begins to shrink.

17.8. EXPANSION HISTORIES FOR THE UNIVERSE 483

Time t

Scale

facto

r a

(t)

Ω < 1

Ω = 1

Ω > 1

Open

Flat

Closed

Now

Figure 17.6:Behavior of the scale factora(t) as a function of time for a dust-filleduniverse.

The evolution of the corresponding scale factor is sketchedin Fig. 17.6.


17.9 The Deceleration Parameter

The density of the Universe is clearly related to the rate atwhich the Hubble expansion is changing with time.

If we expand the cosmic scale factor to second order in time,

a(t) ≃ a0+ a0(t − t0)+ 12a0(t − t0)

2

(wherea0 ≡ (da/dt)t=t0, and so on), introduce thedeceleration pa-rameter at the present timeq0 ≡ q(t0) through

q0 ≡−a0

a0H20

= −a0a0

a20

,

and utilizea0

a0= H0,

we obtain

a(t) = a0

1+H0(t − t0)︸︷︷︸

Hubble

−12H2

0q0(t − t0)2 + . . .

.

17.9. THE DECELERATION PARAMETER 485

-24 -20 -16 -12 -8 -4 0 4 8 12 160

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Time (109 years)

0

1

3

510

Redshift

Now

Ω m =1, Ω r =0.5

Ω v =0 q 0 =1.0

Ω m =0, Ω r =0

Ω v =1 q 0 =-1 .0

Ω m =0, Ω r =0

Ω v =0.5 q 0 =-0 .5

Ω m =0, Ω r =0

Ω v =0 q 0 =0

Ω m =1, Ω r =0

Ω v =0 q 0 =0.5

1

0.50

-0.5-1

H0 = 72 km s-1 Mpc-1

Scale

facto

r re

lativ

e to today

Figure 17.7: Quadratic deviations from the Hubble expansion. The differentcurves correspond to different assumed values of the density parameters and thecorresponding deceleration parameterq0. Each curve has the same linear term buta different quadratic (acceleration) term. Positive values of the deceleration param-eter correspond to a slowing of the expansion and negative values to an increase inthe rate of expansion with time.

Quadratic deviations from the Hubble law are illustratedin Fig. 17.7.


17.9.1 Deceleration and Density Parameters

Generally, the deceleration parameterq0 is related to the density pa-rametersΩi through (Exercise, Ch. 18)

q0 =Ωm

2+Ωr −Ωv.

The parameters of Table 17.1 suggest that the deceleration parameterfor the present Universe is negative,

q0 ≃Ωm

2+Ωv ≃−0.55,

and that the expansion is currentlyaccelerating.

17.9. THE DECELERATION PARAMETER 487

-24 -20 -16 -12 -8 -4 0 4 8 12 160

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Time (109 years)

0

1

3

510

Redshift

Now

Ω m =0, Ω r =0

Ω v =0 q 0 =0

Ω m =0, Ω r =1

Ω v =1 q 0 =0

Ω m =1, Ω r =0

Ω v =0.5 q 0 =0

H0 = 72 km s-1 Mpc-1

q0 = 0S

cale

facto

r re

lativ

e to today

Figure 17.8:Different choices of matter, radiation, and vacuum energy densitiesthat give the same deceleration parameter. The curves all agree near the presenttime to second order, but have very different long-time behaviors.

17.9.2 Deceleration and Cosmology

Figure 17.8 illustrates thatH0 andq0 determine the behavior of theUniverse only near the present time.

• The three curves have the sameH0 andq0 = 0, but very differentmixtures of matter, radiation, and vacuum energy densities.

• Within the gray box the curves are essentially indistinguishablebut at redshifts of 1 or larger they are very different.

• For example, these three curves predict ages of the Universe (in-tercepts with the lower axis) that differ by almost a factor of 2.


Until very recently, the primary quest in cosmology wasto determine with precision the Hubble constantH0 andthe deceleration parameterq0. Acquisition of precisioncosmology data through

• The study of high-redshift Type Ia supernovae

• The detailed analysis of the cosmic microwave back-ground

mean that the cosmological data now are beginning to con-strain a broader range of parameters than just these two.We shall discuss this in more detail in Chapter 18.

17.10. LOOKBACK TIMES 489

17.10 Lookback Times

Telescopes are time machines:

• Lookback time:tL how far back in time we are look-ing when we view an object having a redshiftz,

tL = t(0)− t(z),

wheret(z= 0) is the present age of the Universe andt(z) is the age when light observed today with red-shift zwas emitted.

• Example:in a flat universe (Exercise)

t(z)τH

=23(1+z)−3/2 t(0)

τH

=23

and the lookback time is

tL

τH

=23−

23(1+z)−3/2

=23

(

1− 1(1+z)3/2

)

,

whereτH = 1/H0 is the Hubble time.

• Thus light from an object that we observe with a red-shift z∼ 5 was emitted when

1. The Universe was only∼7% of its present age

2. The cosmic scale factora(t) was six timessmaller than it is today.


8

ΩtL

(z=5)

H0

= 72 km/s/Mpc

Figure 17.9:Geometrical interpretation of the lookback timetL for a dust Universewith three different values of the density parameterΩ = Ωm.

The lookback time as a function of redshift is interpretedgeometrically in Fig. 17.9.

17.10. LOOKBACK TIMES 491

Redshift z

Lookback tim

e t

L (1

09

years

)

Ω = 0.1

Ω = 0.5

Ω = 1.0

00

8

10

6

4

2

12

54321

Figure 17.10:Lookback time as a function of redshift for three different assumedvalues of the density parameter in a dust model withH0 = 72 km s−1 Mpc−1.

The lookback time is plotted for various assumed valuesof the density parameterΩ in Fig. 17.10 for a dust model.

• For small redshiftstL ≃ zτH, as we would expect fromthe Hubble law.

• For larger redshifts the curves in Fig. 17.10 differsubstantially from this approximation.


17.11 Problems with Newtonian Cosmology

As promised, we have been able to make considerable headway in un-derstanding the expanding Universe simply by using Newtonian grav-itational concepts. However, the purely Newtonian approach leads tosome problems and inconsistencies. For example,

1. At large distances the expansion leads to recessional velocitiesthat can exceed the speed of light. How are we to interpret this?

2. Newtonian gravitation is assumed to act instantaneously, but be-cause light speed is the limit for signal propagation, thereshouldbe a delay in the action of gravitation.

3. In the Newtonian picture we had a uniform isotropic sphereex-panding into nothing, which causes conceptual problems in in-terpreting the expansion. Alternatively, if the sphere is assumedto be of infinite extent, there are formal difficulties with evendefining a potential.

These and other difficulties suggest that we need a bettertheory of gravitation to adequately describe cosmologiesbuilt on expanding universes. In Chapter 18 we shall de-velop an understanding of the expanding Universe basedof general relativity that will deal with these problems.

chapter 17 energy and matter in the...

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