Qualitatively, Olbers said that, for an infinite uni-

gen and helium can easily be explained in evolv-

verse, every line of sight should eventually end up

ing theories, but not in steady-state theories. In

on the surface of a star. Therefore, the whole

addition, the existence of many quasars some ten

night sky should look like the surface of a star. In

billion years ago, but very few now, argues for the

fact, the whole day sky should appear the same,

conditions in the universe changing over that

since the Sun would just be one of a large num-

past ten billion years.

ber of stars.

You might think that the steady-state theory

It may be easier to understand in terms of the

would have died with the observation of the

quantitative argument, illustrated in Fig. 20.1.

expansion of the universe (Hubble™s law). If the

Suppose we divide the universe into concentric

universe is expanding, its density must be

decreasing. If the density is not constant with

time, we cannot have a steady-state universe.

However, proponents of the steady-state theory

pointed out that it might be possible to create

matter out of nothing. We do not mean creation

of mass from energy, but from nothing. This

theory, called continuous creation, calls for the

violation of conservation of energy. (This viola-

tion would be permanent, not for short times

for the temporary particle“antiparticle creation dr

discussed in Chapter 8.) However, it requires a

violation at a level well below that to which

conservation of energy has been verified exper-

imentally. We know of no mechanism to create

r

this matter, but proponents of this idea say that

there is no experimental evidence to rule it out.

It would require the creation of the mass of one

proton per year in a box of side 1 km (Problem

20.4). Fig 20.1. Olbers™s paradox.We consider the contribution

Evolving theories, called big-bang theories to the night sky from a shell of radius r and thickness dr.

finally took the upper hand with the discovery, in (The regular distribution of the stars is a simpli¬cation to

help view how the number of stars per shell goes up with r.

1965, of the cosmic background radiation (CBR). This

In reality the stars have an irregular distribution, as we have

radiation is a relic of a time when the universe

seen in earlier chapters.)

was much hotter and denser than it is now. We

20 COSMOLOGY 379

spherical shells, centered on the Earth, each of

thickness dr. The volume of each shell is then

4 r2 dr

dV

If there are n stars per unit volume, the num-

ber of stars in each shell is

4 r2 n dr

N (20.1)

The number of stars per shell goes up as r2.

However, the brightness we see for each star falls

as 1/r2. Therefore, the r2 and 1/r2 will cancel, and

the brightness for each shell is the same. If there

are an infinite number of shells, the sky will

appear infinitely bright after we add up the con-

tributions from each shell. Actually, this isn™t

quite the case. The stars have some extent, and

eventually the nearer stars will block the more

distant stars. However, this will not happen until

(a)

the whole sky looks as if it is covered with stars,

with no gaps in between.

There might appear to be some obvious ways

out of this. You might say that our galaxy doesn™t

go on forever. Most lines of sight will leave the

galaxy before they strike a star. Unfortunately,

the argument can be recast in terms of galaxies

instead of stars, and the same problem applies.

Another possible solution is to invoke the

absorption of distant starlight by interstellar

dust. However, if the universe has been around

forever (or even a very long time), the dust will

have absorbed enough energy to increase its

temperature to the same as that of the surface of

a star. If the dust became any hotter than that,

the dust would cool by giving off radiation

which the stars would absorb. Even if the sky was

not bright from the light of stars, it would be

(b)

bright from the light of dust. (Scattering by dust

wouldn™t help because it would make the sky Fig 20.2. (a) How a ¬nite size for the universe helps with

look like a giant reflection nebula, again like the Olbers™s paradox. (b) How a ¬nite age for the universe helps

surface of a star.) with Olbers™s paradox.

The redshift due to the expansion of the uni-

verse is of some help. The energy of each photon have to take the integral of dr/r (see Problem 20.3).

is reduced in proportion to the distance it travels This gives us ln(r), so if r can be arbitrarily large

before we detect it. This adds an additional factor the brightness can be also.

of 1/r to the apparent brightness of each shell, It is possible to get out of the problem if the

meaning that the brightness of each shell falls off universe has a finite size. This is illustrated in Fig.

as 1/r, instead of being constant. Note that this 20.2(a). If there is a finite size, then we cut off our

doesn™t completely solve the problem. If we add integral at whatever that size is. There is another

up the contributions from all of the shells, we way to achieve the same effect. This is if the

380 PART V THE UNIVERSE AT LARGE

universe has a finite age t0, as illustrated in Fig. depend on where you are in the universe.) This

20.2(b). We can only see stars that are close means that

enough for their light to have reached us in this

v¿1r a2 v1r a2 (20.5)

time. That is, we can cut off r at ct0. There may

even be a cutoff before this because it took a cer- Using this, equation (20.4) becomes

v1r 2

tain amount of time for stars and galaxies to

v1r a2 v1a2 (20.6)

form. So, there is a finite cutoff to the number of

shells that can contribute to the sky brightness, This means that v(r) must be a linear function of

and the problem is solved. It is amazing that this r. The only velocity law that satisfies this rela-

simple observation “ that the night sky is dark “ tionship is

leads to the conclusion that the universe has a

v1r 2 H1t 2 r (20.7)

finite size or age (or both).

Note that we haven™t required the expansion (H

20.2.2 Keeping track of expansion could be zero). However, if there is an expansion,

We can show that Hubble™s law follows from the it must follow this law, if the cosmological prin-

assumption of homogeneity. In Fig. 20.3, suppose ciple is correct.

that P observes two positions, O and O , with dis- When we want to keep track of the expansion

tance vectors from P being r and r , respectively. of the universe, it is not convenient to think

The vector from O to O is a, so that about the size of the universe. Instead, we intro-

duce the scale factor which will keep track of the

a r r¿ (20.2)

ratios of distances. We let t0 be the age of the uni-

We let v be a function to give the rate of verse at some reference epoch. (It doesn™t matter

change of length vectors ending at O and v the how this reference is chosen.) We let r(t) be the

corresponding function for length vectors ending distance between two points as a function of

at O . We then have time. (The points must be far enough apart so

v¿1r¿ 2 v1r 2 that their separation is cosmologically signifi-

v1a2 (20.3)

cant.) We define

Using equation (20.2) to eliminate r , this

r1t0 2

r0 (20.8)

becomes

v1r 2 The scale factor R(t) is a scalar, defined from

v¿1r a2 v1a2 (20.4)

R1t 2 r1t 2 r1t0 2

The homogeneity of the universe means that

r1t 2 r0

the functional form of v and v must be the same. (20.9)

(The functional form of the velocity cannot

Note from this definition that R(t0) 1. If the uni-

verse is always expanding, R 1 for t t0, and

a

R 1 for t t0.

O O'

We can rewrite Hubble™s law in terms of the

scale factor. We start by writing Hubble™s law as

H(t) r

dr/dt (20.10)

r Using r(t) R(t) r0 (equation 20.9) makes this

r'

r0 1dR dt 2 H1t 2R1t 2 r0

Dividing by r0 gives

H1t 2 R1t 2

dR dt (20.11)

P

Note that we now only have to deal with a

Fig 20.3. Vectors for locating objects in the universe.

scalar equation, instead of a vector equation. We

20 COSMOLOGY 381

(t0 ), also, since R0

where we have used 1,

can solve equation (20.11) to give the Hubble 0

we have

parameter in terms of the scale factor:

H1t 2 3 1 R1t 2 4 3dR dt 4 (20.12)

1t 2

0

R 1t 2

(20.16)

3

20.3 Cosmology and Newtonian Substituting this into equation (20.15), we have

gravitation

4 G0

$

R (20.17)

3 R2

We can learn a lot about the evolution of an expand- $

ing universe by applying Newtonian gravitation. In Note that if 0 is not zero then R cannot be

the next section we will see how the Newtonian zero. A universe with matter cannot be static. It

results are modified by general relativity. must be expanding or contracting. This is like

The assumption of isotropy is equivalent to saying that if you throw a ball up, and the

saying that the universe appears to be spherically Earth™s mass is not zero, then the ball must be

symmetric from any point. This means that any moving up or down; it cannot be forever

spherical volume evolves only under its own stationary.

influence. The gravitational forces exerted on the To integrate the equation of motion (20.17),

#

volume by material outside the volume sum (vec- we first multiply both sides by R, to give

torally) to zero. If the volume in question has a

4 G0 #

$#

radius r, and mass M(r), the equation of motion RR R

0

3 R2

for a particle of mass m on the surface of the

sphere at position r is Noting that

GM1r2 m ˆ

r

d1R2 2

#

$

mr (20.13) #$

2

r 2RR

dt

r

In this case, we let ˆ be a unit vector in the

this becomes

r-direction (radially outward). The assumption of

homogeneity means that the density is the

1 d1R 2

#2

4 G 0 dR

same everywhere (though it can change with 0

3 R2 dt

2 dt

time). The mass M(r) is given by

Multiplying through by two, and using the fact

43

that 11 R2 2 1dR dt 2

M1r2 r d11 R2 dt, we have

3

Substituting into equation (20.13) gives

cR d

8 G0

d #2

0 (20.18)

3R

dt

4

$

r Gr (20.14)

3 Since its time derivative is zero, the quantity

in the square brackets must be a constant. We

We use equation (20.9) to eliminate r by using the

set it equal to some arbitrary constant k, so

scale factor, to get

that

4

$

R GR (20.15) 8 G0

#

3 R2 k

(20.19)

3R

As the universe expands, any given amount of

mass occupies a larger volume. The density goes Farther integration of equation (20.19)

as 1/volume. The volume is proportional to R3, so depends on whether k is zero, positive or nega-

we have tive. We consider each case separately. The

schematic behavior of R(t) for each case is shown

R3

1t 2

0

R3 1t 2

in Fig. 20.4(a).

0

382 PART V THE UNIVERSE AT LARGE

Equation (20.20) can be rewritten so that we can

integrate it and find an explicit function for R(t).

We take the square root of both sides of the equa-

tion, giving

Scale Factor

8 G o 1>2

a b dt

R1 2 dR

3

We integrate the left side from zero to R and the

right side from zero to t to give

1>2

a bt

8G

2 3>2 0

R (20.21)

3 3

Time

This means that R is proportional to t2/3. The uni-

Now

Open verse always expands, but the rate of expansion

Borderline becomes smaller and smaller.

Closed We now look at the case k 0. Since the first

(a) term on the right-hand side of equation (20.19)

becomes smaller when R increases, a point will be

#

reached eventually, at some finite R, where R 0.

Hubble Time

The expansion stops at some maximum scale fac-

tor Rmax. We #can find Rmax as the value of R,

which makes R 0 in equation (20.19). That is,

Scale Factor

8 G0

k

0

3 Rmax

Solving for Rmax:

8 G0

Rmax (20.22)

3k

Age of Universe

After Rmax is reached, the universe starts to col-

lapse. We say that the universe is closed.

TIME

We next look at the case k 0. If k is negative

then k is positive, and the right-hand side of equa-

Now

No Gravity

tion (20.19) is always positive. As R gets very large,

Gravity

the first term on the right-hand side approaches

(b) #

zero, and R2 approaches k. (Remember, k is a

#

Fig 20.4. (a) The scale factor R as a function of time for positive number.) This means that R approaches

borderline, closed and open universes. (b) How the presence

( k)1/2. The expansion continues forever, and we

of gravitation means that the Hubble time is greater than the

say that the universe is open.

real age of the universe.

We can think of an analogous situation of

throwing a ball up in the air. If the total energy is

We first look at the case k 0. Equation negative, the ball will return to Earth. If the total

(20.19) then becomes energy is positive, the ball will escape, and its

speed will remain positive. If the total energy is

8 G0

#

R2 (20.20) zero, the ball will reach infinity, but its speed will

3R

approach zero. We can think of k as being related

#

to the energy of the spherical region of the uni-

It should be noted that R is always positive, but

verse that we are following.

approaches zero as R approaches infinity.

20 COSMOLOGY 383

In Fig. 20.4(b) we can see how the presence of k 0 We combine equations (20.11) and (20.20)

gravity in an expanding universe affects the to give

relationship between the Hubble time (1/H0)

8 G0

and the true age of the universe. The presence R2H2

3R

of gravity means that the expansion has been

slowing, so the universe was expanding faster in or

the past. That means it took less time to reach

8 G0

its current size than we would estimate from H2

3 R3

the current expansion rate. From Fig. 20.4(b) we

see that H0 is proportional to the current slope 8

G

of R(t). 3 (20.25)

Having established that the universe is

where we have used equation (20.16) in the last

expanding, we would now like to ask whether

step. Substituting equation (20.25) into the equa-

that expansion will continue forever. In other

tion for q (equation 20.24), we have

words, is the universe open or closed? We would

like to have some quantity that we can measure q 12

to tell us. If we look back to our analogy of the

k 0 In this case, q can be arbitrarily large. It

ball thrown up in the air, if we know the position

even approaches infinity as R approaches Rmax

and velocity of the ball at some time, we also #

(since R 0 at Rmax ). This means that any value

need to know its acceleration to know if the ball

of q in the range

has sufficient energy to escape. Since the ball is

slowing down, we want to know the deceleration q712

of the ball.

will produce a closed universe.

For the universe, we define a deceleration

parameter, whose value will tell us whether the k 0 We have already seen that q must be

universe is open or closed. We would like to greater than zero, so the range of q given by

define this parameter so that it is dimension-

06q612

less (just as the scale factor is dimensionless),

and is independent of the time t0 that we will produce an open universe.

choose for our reference epoch. The latter

The deceleration will depend on the density

requirement says that the parameter should

# $

of matter in the universe. We can define a critical

depend on quantities such as R R and R R. With

density, crit, such that the universe is closed if

these ideas in mind, we define the deceleration

crit and open if crit. If crit, we will

parameter as

have k 0, and the universe is on the boundary

$

R2

a ba # b

R between open and closed. This last point allows

q (20.23)

R R us to find crit, since it is the density for k 0, or

$

q 1/2. If we set q 1/2 in equation (20.24) and

RR

# solve for the density, , we have

R2

3 H2

$

Since the expansion is slowing down, R is neg- (20.26)

crit

G

8

ative, and q is positive. If we use equations (20.11),

$ #

(20.16) and (20.17) to eliminate R and R, this It is convenient to define a density parameter ,

becomes (see Problem 20.7) which is the ratio of the true density to the criti-

cal density. That is

4G

q (20.24)

3 H2 (20.27)

crit

We can look at ranges of q for the three cases We can easily show (see Problem 20.8) that

of k: 2q. If H 70 km/s/Mpc, then crit is

384 PART V THE UNIVERSE AT LARGE

20.4 Cosmology and general

relativity

Scale Factor

20.4.1 Geometry of the universe

When Einstein developed the general theory of

relativity, he realized that it should provide a cor-

rect description of the universe as a whole.

Einstein was immediately confronted with a

result equivalent to equation (20.17), which says

that if the density is not zero, the universe must

be expanding or collapsing. This was before

Time

Now

Hubble™s work, and most believed in a static

(steady-state) universe.

To get around this problem, Einstein intro-

Fig 20.5. Oscillating universe. In this picture t0 is the cur-

duced a constant, called the cosmological constant,

rent time.The universe is the single cycle that contains t0. If

the universe is oscillating, then, after all the material comes , into general relativity. It had no measurable

back together, the expansion starts again. effect on small scales, but altered results on cos-

mological scales. For example, in equation (20.17)

1 10 29 g/cm3. In the final section of this the effect of the cosmological constant would be

/8 G. This makes

to replace the density by

chapter, we will discuss observations that can

it possible # to have a non-zero density, but a zero

determine the actual value of .

value for R. Einstein withdrew the cosmological

There is a final point to consider if the uni-

constant when he heard of Hubble™s work, declar-

verse turns out to be closed. After the expansion

ing the cosmological constant to be his biggest

stops, a collapse will start. Eventually, all of the

mistake. However, theoreticians have tended to

matter will come together into a dense, hot

keep it in the theory, and then formally set it to

state for the first time since the big bang. Some

zero, or consider models with a non-zero . The

people have taken to calling this event the big

best determinations of various cosmological

crunch. It is natural to ask what will happen

parameters, discussed below and in the next

after the big crunch. It has been suggested that

chapter, suggest that may have a non-zero

the universe might reach a high density and

value.

then bounce back, starting a new expansion

Following Einstein™s work, a number of people

phase. If this can happen, then it might happen

worked out cosmological theories, using differ-

forever into the future, and might have hap-

ent simplifying assumptions. The models are gen-

pened for all of the past, as indicated in

erally named after the people who developed

Fig. 20.5. Such a universe is called an oscillating

them. The de Sitter models are characterized by

universe.

k 0 and a non-zero (positive) cosmological con-

If our universe turns out to be closed, can we

stant; the Friedmann models have a zero cosmo-

tell if it is oscillating? Some theoreticians have

logical constant and also zero pressure (a good

argued that the big crunch/big bang in an oscil-

approximation at low density); Lemaitre models

lating universe strips everything down to ele-

have non-zero density and a cosmological con-

mentary particles, and therefore destroys all

stant. As a result of his work, Lemaitre noted that

information on what has come before. Others

(independent of the value of ) there must have

have argued that there are certain thermody-

been a phase in its early history when the uni-

namic properties of the universe that might tell

verse was very hot and dense. This phase is called

us if it is oscillating. Others have taken a wait-

the big bang.

and-see attitude, pointing out that we will need a

Many of the general relativistic results are

quantum theory of gravity to understand the

similar to those we obtained in the previous

densest state that is reached.

20 COSMOLOGY 385

R(t)

section. This is because both depend on the fact

that, in a spherically symmetric mass distribu-

tion, matter outside a sphere has no effect on the

evolution of matter inside the sphere. One modi-

/8 G if

fication is the replacement of by

you want a non-zero cosmological constant.

t

However, the general relativistic approach

gives us a deeper insight by providing a geomet-

ric interpretation of the results. For example, the

space-time interval, in spherical coordinates,

becomes

1ds 2 2 1c dt 2 2

1dr2 2

R 1ct 2 c r2 3 1d 2 2 sin2 1d 2 2 4 d

2

kr2

1

(20.28)

(This is sometimes called the Robertson“Walker

metric.) In this equation, R(t) has the same mean-

Fig 20.6. Schematic representations showing the relation-

ing as before, and r, and are the usual spherical ship between cosmological model and the geometry of

coordinates of objects such as galaxies. In cosmol- space-time.The top graph shows an open model, with nega-

ogy, it is important to use what is known as a co- tive curvature (like a saddle).The middle graph shows a

moving coordinate system. This system expands universe that will expand forever, but is on the boundary, and

the geometry is ¬‚at The bottom graph shows a closed

with the universe. The form of this metric

model, and the geometry has a positive curvature.

ensures that the models are homogeneous and

[© Edward L.Wright, used with permission]

isotropic (the cosmological principle). (Note that

in books on general relativity, authors sometimes

use different symbols: the scale factor is written

as a(t), and k becomes 1/R2, where R is the “radius triangles are always 180 . In this case space is infi-

of curvature” of the universe. These books also nite. If k 1, space-time has a positive curva-

often take a system of units in which c 1, so it ture, like the surface of a sphere. The sums of the

does not appear explicitly in equations.) angles of triangles are always greater than 180 .

In general relativity, whether the universe is Space must be finite, just as the surface of a

open, closed or on the boundary tells us some- sphere is finite. Finally, if k 1, we say that

thing about the geometry of space-time. That space-time has a negative curvature. The sums of

information is contained in the constant, k, the angles of triangles are always less than 180 .

which has the value zero for the boundary, 1 for In this case, space is infinite. The relationship

a closed universe, and 1 for an open universe. between geometry and the type of cosmological

However, k now tells us something about the cur- model is summarized in Table 20.1.

vature of space-time (Fig. 20.6). If k 0, space- We can get a feel for the geometry of the uni-

time is flat (Euclidean). The sums of the angles of verse by considering a two-dimensional analogy.

Parameters of various cosmological models.

Table 20.1.

Type k q Curvature Extent

Boundary 0 1/2 1 ¬‚at in¬nite

Closed 1 1/2 1 positive ¬nite

Open 1 0q 1/2 0 q 1 negative in¬nite

386 PART V THE UNIVERSE AT LARGE

horizon is growing. We have seen that over small

distances even the surface of a sphere appears

flat. The curvature becomes apparent as you can

survey larger areas. This means that, as our hori-

zon grows, the curvature might become more

apparent.

As the universe expands, we would like to

keep track of the separation between any two co-

moving points. We start by considering nearby

points at r and r dr. The proper distance

between those points is

R1t 2 dr

11 kr 2 2 1>2

If the points are far apart (like a distant

galaxy to here), then we have to integrate that

expression.

We can also use our analogy to see that it is

meaningless to talk about the radius of the uni-

verse. In three dimensions, our sphere has a

radius, but in two dimensions we can only talk

about the surface. This is one reason why the

scale factor R(t) is a better way of keeping track

of the expansion. Even though we cannot talk

about the radius of the universe in a meaning-

ful way, we can talk about the curvature of our

surface. So we can talk about the radius of cur-

vature of the universe. The larger the radius of

curvature, the closer the geometry is to being

flat.

Our expanding sphere analogy also tells us

that it is not very meaningful to talk about the

Fig 20.7. The universe as an expanding sphere. As the

center of the universe. The sphere has a center,

sphere expands, the coordinate system expands, and the

but it is not in the universe, which is the surface

radius of curvature changes.The dark circle with the lighter

only. There is nothing special about any of the

interior represents the part of the universe that we can see,

points on the surface of the sphere. If we go back

with us at the center of the circle. A photon could have trav-

in time to very small times, our sphere will be

eled from the circle to the center in the age of the universe.

very small, and at t 0 all the points are

As the universe expands and ages, our horizon expands also.

Since our horizon expands at the speed of light, new objects together, at the center. So the proper way to talk

are always coming over the horizon. about the center is as a space-time event far in

our past.

20.4.2 Cosmological redshift

We consider the universe as being confined to the

surface of an expanding sphere, as shown in We can also see that the redshift (Hubble™s law)

Fig. 20.7. One concept that we can now visualize fits in as a natural consequence of the expan-

is that we have a horizon due to the finite age of sion (Fig. 20.8). As the universe expands, the

the universe t (as noted in our discussion of wavelengths of all photons expand by the same

Olbers™s paradox). We can only see light emitted proportion that cosmic distances expand. That

toward us within a distance equal to ct. This is, they expand in proportion to the scale factor.

20 COSMOLOGY 387

Fig 20.8. Cosmological redshift. As the universe expands,

represented by the expanding sphere, the wavelengths of all

photons increase in proportion to the scale increase. In this

¬gure the arrow traces the route of a photon, emitted in the

¬rst frame, through an expanding universe in the second

frame, and absorbed in the third frame.The solid part of the

arrow shows where the photon has already been.

We define the redshift, z, to be

¢

z

0> z. This gives

where ¢ , so 1

0

1

R1t 2

z

1 (20.29b)

Remember, since the radiation is emitted before

the reference epoch, R(t) 1, so z 0.

If radiation is emitted at wavelength 1 at epoch We can derive an approximate expression for

t1, and detected at wavelength 2 at epoch t2, the redshift for radiation emitted some time t

then in the recent past, where t V t0. Using a Taylor

R1t2 2 series, we have (see Problem 20.11)

2

R1t1 2

(20.29a)

R1t0 2 ¢tR 1t0 2

1

1 #

R1t0 ¢t 2

(20.30)

If we let t1 be some arbitrary time, and t2 be

¢tR 1t0 2

#

the reference epoch t0 (for which R 1), this

1

becomes

Combining this with equation (20.29b) gives

1

0

¢tR 1t0 2

R1t 2

#

z

1 1

388 PART V THE UNIVERSE AT LARGE

#

¶c2

R2 kc2

ab

Setting t t0 in equation (20.12) gives us, 8G

R 1t0 2

# (20.33a)

H0, so R2

R 3

z H0 ¢t (20.31)

and

If a photon takes t to reach us, it must have $

¶c2

c d

R 4G 3P

been emitted from a distance d c t . Using this

(20.33b)

c2

R 3 3

to eliminate t in equation (20.31) gives

where P is the gas pressure, usually taken to be

cz H0 d (20.32)

zero, except when the matter is hot and dense.

which is Hubble™s law. (Remember, this approxi-

Remember, the density at any time is 0/R3. So the

mation is for small t.)

mathematical effects of a non-zero density and a

It is important to note that equation (20.29)

non-zero are different, since the density term

tells how to interpret the redshifts of distant

will have an extra factor of the variable R. We will

galaxies. It is tempting to say that these galaxies

look at separate cases below.

are moving relative to us and their radiation is

Since H (1/R)(dR/dt), we can write Einstein™s

therefore Doppler-shifted. However, in comput-

equation as

ing a relative velocity for a Doppler shift, we take

the difference between the velocities of the two kc2

3cH d

2

¶c2

8G (20.34)

galaxies. These two velocities must be with R2

respect to a co-moving coordinate system.

From these, the deceleration parameter becomes

Therefore, apart from its peculiar motion, each

galaxy™s velocity is zero with respect to this coor-

c 4 Ga bd

1 ¶c2 P0

dinate system. Therefore, strictly speaking, there 0

q0 (20.35)

H2 3 3c2

3

is no Doppler shift due to the expansion of the 0

universe. The redshift arises as a result of the Note that this reduces to the classical case (equa-

increase in wavelengths of all photons moving P 0.

tion 20.24) when

through an expanding universe. We therefore The integration of these equations to give R(t) is

call it the cosmological redshift. Any additional generally quite difficult. To simplify the situation,

motions with respect to the co-moving coordi- it is useful to look at limiting cases, namely zero

nates would produce a Doppler shift (red or blue) cosmological constant and zero density. Zero cos-

in addition to the cosmological redshift. mological constant would also be an approximate

As a consequence of this, we should not description of a universe with small cosmological

directly interpret the redshift of a galaxy as giv- constant, where the matter term dominates.

ing a particular distance. The amount of redshift Likewise, zero density would also approximately

just tells us the amount by which the scale factor describe a case with low density where the cos-

has changed between the time the photon was mological constant dominates. In each case, we

emitted and the time it was detected. For this rea- must also look separately at zero, positive and

son, we often talk about the redshift of a particu- negative curvature. Below, we give results for var-

lar galaxy, and don™t bother to convert it to a dis- ious limiting cases without deriving them. You

tance. For example, we simply say that 3C273 is at can verify that they are solutions by plugging

z 0.15. To convert a redshift to a distance we them into the appropriate equations (see

need a particular model for how R(t) has evolved. Problem 20.18). The results for some of these

models are shown in Fig. 20.9.

20.4.3 Models of the universe Models with zero are called Friedmann

In general relativity, the solutions for R(t) are dif- models. For a flat universe (k 0), R(t) in these

ferent from the Newtonian case. The equation in models is given by

general relativity, analagous to equation (20.19),

t 2>3

R1t 2 1const. 2 a b

is called Einstein™s equation. There are two parts

(20.36a)

relevant to our discussion of cosmology: t1

20 COSMOLOGY 389

/8 G,

like those with a mass density, EFF

2

„¦Μ = 0 , „¦Λ = 1 constant in space and time. A positive value of

behaves like a negative effective mass density

„¦Μ = 0 , „¦Λ = 0

(repulsion), and a negative value of behaves like

„¦Μ = 1 , „¦Λ = 0

Scale Factor R(t)

a positive effective mass density (attraction). So,

„¦Μ = 2 , „¦Λ = 0

for positive , we would expect the expansion to

1

accelerate, and for negative we would expect

the expansion to stop and reverse.

For zero curvature (k 0), the result for R(t) is

R1t 2 1const.2 exp a tb

¶

(20.37a)

B3

0

-10 0 10

Note that for positive , this corresponds to an

Gyr from now

exponential growth in the expansion rate. The

Fig 20.9. Scale factor vs. time for various cosmological

flat model with zero density and a non-zero is

models. Models are all chosen to have R 1, now, and a

sometimes called the deSitter model.

Hubble constant 65 km/s/Mpc (so they all have the same

For positive curvature (k 1), there are

slope now). In terms of the density parameter (de¬ned in

3 12

solutions only for ¶ 7 0, R1t 2 a b , in which

equation 20.45), the models are (from top to bottom):

( M 0, 1), ( M 0, 0) ( M 1, 0) ¶

( M 2, 0). [© Edward L.Wright, used with

case

permission]

R1t 2 cosh a tb,

3 ¶

(20.37b)

B¶ B3

where t1 is any convenient constant by which to

where t is zero when R(t) has its minimum value.

scale the result. If we choose it to be t0, the current

For negative curvature (k 1)

age of the universe, taking the constant in front to

be equal to unity makes R 1 now. So, the flat

sin h a

3 ¶

model which describes our universe would be tb for ¶ 7 0

B¶ B3

R1t 2 µ (20.37c)

t 2>3

R1t 2 ab sin a

3 ¶

(20.36b) tb for ¶ 6 0

B¶ B3

t0

This result is the same as for the flat There are different ways in which we could

Newtonian case (equation 20.21). define the distance, since we are dealing with

For positive curvature (k 1) and negative objects whose separation changes between the

curvature (k 1), the results are mathemati- time a photon is emitted at one galaxy and

cally different from the Newtonian case, and are received in another. A convenient definition of

also complicated to express. They do have similar distance in this case is that which we would

characteristics to their Newtonian counterparts. associate with a distance modulus, m M. This

Namely, the positive curvature case produces a would tell us how to convert apparent bright-

universe that expands, reaches some maximum nesses (or magnitudes) into absolute brightnesses

R, and then contracts. The negative curvature (or magnitudes).

produces an expansion that lasts forever. As light travels from a distant source, the

Models with non-zero cosmological constant observed brightness decreases as the photons

are called Lemaitre models. As we have said, for from that source spread out on the surface of a

the sake of simplicity, we consider the empty sphere. Let the radius of that sphere be a. If the

Lemaitre models. One such model is also shown in geometry of space-time is flat, the surface area of

that sphere is 4 a2. So, the observed brightness

Fig. 20.9 these models are useful for any universe

falls as 1/a2, just as for light from nearby stars

dominated by the cosmological constant. As, we

said above, these universes will behave roughly (Chapter 2). If the geometry is not flat, then

390 PART V THE UNIVERSE AT LARGE

falloff can be greater or less than 1/a2. So the rela- 20.5 Is the universe open

tionship between observed brightness and dis-

or closed?

tance depends on the geometry of the universe. If

0, and q0 0, and if z is not too large, then

the relationship between distance modulus and In this section we look at evidence that might

redshift is given by allow us to decide whether the universe is open

or closed. It is impressive that we can even ask

m M q0)

25 5 log10(cz/H0) 1.086z (1 (20.38)

such a question and hope to achieve an answer.

where q0 is the current value of the deceleration The basic question is whether the actual den-

parameter. In this expression, cz/H0 is in mega- sity is less than or greater than the critical den-

parsecs, accounting for the factor of 25 in front sity. We could start by adding up the density of all

(see Problems 20.14 and 20.15). the matter we can see, to find out if it gives

The geometry of the universe also determines crit. However, we already know that there is

how the apparent angular size varies with dis- a problem with dark matter, so, if we only

tance. For flat geometry, and an object that is not include the visible matter, we will be missing a

too distant by cosmological standards, the angle significant amount. Of course, if the visible mat-

subtended by an object of length L at distance D is ter is sufficient to close the universe, then we

L/D (in radians) as long as L V D. For an object at don™t have to worry about the dark matter. It

a cosmologically significant distance, the angu- turns out that if the visible matter is insufficient

lar size is to close the universe, then we have to account for

z2 2

the dark matter. It turns out that the density of

L11

visible matter is about 1% of the critical density.

DL

If the universe is to be closed, the dark matter

q2 11 z22 must do it. From Table 20.2, we see that the

L 0

H0c 5q0z 1q0 12 3 11 2q0z 21 2 14 6

(20.39) amount of dark matter required to close the uni-

verse is greater than the dark matter in clusters

where DL is the luminosity distance. of galaxies. There also appears to be a trend

It is useful to have an estimate of the rela- towards more dark matter on larger scales.

tionship between the Hubble time, H0 1, and the Therefore, we would not be surprised if there is

true age of the universe, t0. Obviously, this enough dark matter to close the universe.

depends on which model we use to describe the However, in our attempt to see if the universe is

universe. open or closed, we can only include dark matter

For the Friedmann models ( 0) the results that we know is present by its gravitational

are as follows. effects on visible matter. We can therefore

For a flat universe (k 0), include the dark matter in galaxies, and clusters

of galaxies, since we can detect its gravitational

(2/3) H0 1

t0 (20.40a)

For positive curvature (k 1),

(2/3) H0 1 Table 20.2. Mass-to-light ratios on

t0

0 (20.40a)

different scales.

For negative curvature (k 1)

Scale M/L (solar units)

(2/3) H0 1 H0 1

t0 (20.40c)

Milky Way to Sun 3

It is also useful to have an estimate of the rela-

Spiral galaxy disk 10

tionship between the age of the universe when a

Elliptical galaxy 30

photon was emitted and the redshift. In one

Halo of giant elliptical 40

model it is given by

Rich cluster of galaxies 200

3/2 To close the universe 1200

t (Gyr) z)

10.5 (H0/65 km/s/Mpc) (1 (20.41)

20 COSMOLOGY 391

effects. This still leaves us a factor of five short of years ago. If we include near and distant objects

closing the universe. in a plot of Hubble™s law, we should be able to see

We have said that the best way to measure the deviations from a straight line as we look farther

mass of any object is to measure its gravitational back in time.

effect on something. If we want to determine the The difficulty comes in the methods used for

mass of the Earth, we measure the acceleration of measuring distances to distant objects. In our dis-

gravity near the surface. Therefore, instead of try- cussion of the extragalactic distance scale, we

ing to find all of the matter needed to close the saw that for the most distant galaxies we can see,

universe, we can look for its gravitational effects. we cannot look at individual stars, such as

We can try to measure the actual slowing down Cepheids, within a galaxy. Instead, we must look

$

of the expansion of the universe to see if R is at the total luminosity of a galaxy. We already

large enough to stop the expansion. When we do know that the luminosities of galaxies change as

this, we are determining the current value of the they evolve. Galactic cannibalism provides us

deceleration parameter from its original defini- with the most spectacular example of this, but

tion (equation 20.23). Using the fact that even normal galaxies change in luminosity with

R 1t0 2

#

H0 (equation 20.12), it becomes time. Therefore, if we calibrate the distance scale

R 1t0 2

using the luminosities of nearby galaxies, we can-

$

not apply this to more distant galaxies, precisely

q0 (20.42)

H2 because we are seeing them as they were in the

0

$

past. Before we can interpret observations of dis-

We don™t actually try to measure R(t0). What

tant objects, we must apply theoretical evolu-

we try to measure is the current rate of change of

#

tionary corrections. These corrections can be so

the Hubble constant H(t0). We would therefore

$ #

large that they can make an apparently closed

like to express R(t0) in terms of H0 and H(t0). We

universe appear open or an open universe appear

start with equation (20.11):

closed.

R 1t 2 H1t 2 R1t 2

#

When we discussed the extragalactic distance

scale, we mentioned that supernovae might pro-

Differentiating both sides with respect to t

vide very useful standard candles. In this case,

gives

the more reliable standards are those that arise

R 1t 2 H1t 2 R 1t 2 H 1t 2 R1t 2

$ # #

in close binary systems (type I). Nearby examples

# tell us how to relate the light curve to the peak

Setting t t0, and remembering that R (t0) H0,

luminosity. So if we can compare the observed

this becomes brightness with the known luminosity, we can

R 1t0 2 H 1t0 2

$ #

calculate the distance. HST has been particularly

H2 (20.43)

0

useful for studying these objects, and can detect

Substituting this into equation (20.42), we have faint ones at cosmologically significant dis-

H 1t0 2

# tances. These results suggest that the expansion

c 1d

q0 (20.44) of the universe may not be slowing very much.

H2

0

In fact, a few points are consistent with an accel-

Equation (20.44) tells us that if we can meas- eration in the expansion. This would be consis-

ure the rate of change of the Hubble parameter, tent with a non-zero cosmological constant.

we can determine q0. Unfortunately, measuring However, before these preliminary results are

#

H(t0) is not easy. This should not be a surprise accepted, many more such objects need to be

since measuring H0 is not easy. In principle, we observed, and we need to examine whether

#

can measure H(t0) by taking advantage of the fact these particular standard candles are not stan-

that we see more distant objects as they were a dard as thought.

long time ago. If we can determine H for objects An alternative approach is to measure the

that are five billion light years away, then we are curvature of space-time by surveying the universe

really determining the value of H five billion on a large scale. One way of carrying this out is

392 PART V THE UNIVERSE AT LARGE

with radio source counts. We divide the universe 2

into shells, such as for our discussion of Olbers™s

paradox earlier in this chapter. We then count

the number of radio sources in each shell. We use

Scale Factor R(t)

radio sources because we can see them far away.

(With large, sensitive optical telescopes, optical

1

counts are now being used also.) If the geometry

of space-time is flat (Euclidean) the number of

sources per shell will go up as r2. If the geometry

of space-time is curved, that curvature will

become more apparent as we survey larger

regions. Therefore, as we look farther away, we 0

-10 0 10

would expect to see deviations from the r2 Gyr from now

dependence. What is actually varying is the rela-

Fig 20.10. Scale factor vs. time for the cosmological model

tionship between r and surface area. Of course, as which best ¬ts current data.The model has a Hubble con-

we look far enough to see such deviations, we are stant of 65 km/s/Mpc and M 0.3, 0.7. [© Edward

also looking far back in time, and we are seeing L.Wright, used with permission]

radio sources as they were. Again, evolutionary

corrections are necessary. The results so far are now, it had to be very close to the critical density

consistent with a flat universe. in the past.

There are also more indirect methods that If there is a non-zero cosmological constant,

have proved fruitful. These involve an under- then it is possible to have a flat universe.

standing of the formation of elements in the big Remember, we defined the density parameter for

bang, and will be discussed in the next chapter. matter in equation (20.27) as

The results of these so far support a universe

M/ crit (20.45a)

which is open. In addition, they only give infor- M

mation on the density of material that can par- We saw that if there is a non-zero , then we can

ticipate in nuclear reactions, and may not define an effective density due to that , as

include the dark matter. /8 G, so we can define a density parameter asso-

One of the interesting aspects of this whole ciated with as

problem is that we should be so close to the

/8 G (20.45b)

boundary. Of all the possible values for the den- crit

sity of the universe, ranging over many orders of Then the total density parameter for the universe

magnitude, we seem to be tantalizingly close to would be

the critical density. Cosmologists have wondered

(20.45c)

TOT M

whether this is accidental, or whether it is telling

us something significant about the universe. Fig. 20.10 shows R vs. t for what is currently

They have noted that if is not exactly unity, the best estimate of the model universe with

then it evolves away from unity as the universe 1 ( M 0.3, 0.7). We will discuss the

TOT

becomes older. This means that for the actual future of the universe in the next chapter when

density to be pretty close to the critical density we look at the big bang.

Chapter summary

Cosmology is the study of the universe at the One of the fascinating things about cosmol-

largest scale. It asks about the large-scale struc- ogy is that we can do normal astronomical obser-

ture of the universe and how it has evolved. When vations to answer cosmological questions. The

we are talking about cosmological scales, the consideration of the simple question, “Why is the

smallest building blocks are the galaxies. sky dark at night?” leads us to the profound

20 COSMOLOGY 393

conclusion that the universe must have a finite As the universe expands, the wavelengths of all

size, a finite age, or both. the photons in the universe increase by the same

In explaining the large-scale structure of the amount as the scale factor increases. This is

universe, we start with the simplest assumptions, called the cosmological redshift. It is the redshift

that the universe is homogeneous and isotropic. that produces Hubble™s law. In treating the struc-

This means that (on cosmological scales), the aver- ture of the universe, we must be sure to account

age properties are the same from place to place, for the effects of general relativity. The geometry

and the appearance is the same in all directions. of the universe may behave differently from that

A class of theories that also include the assump- of a normal flat surface.

tion that the universe doesn™t change with time, Each particle in the universe feels the gravita-

so-called steady-state theories, are no longer sup- tional attraction of all the other particles in the

ported by observational evidence. Cosmological universe. Whether the universe will expand for-

models in which the universe is becoming less ever, or the expansion will eventually reverse,

dense as it expands have an era in the early uni- depends on the total density of material in the

verse when it was hot and dense. This era is called universe. This has turned out to be very hard to

the big bang, and models that include it are called measure, though we have been able to determine

big-bang cosmological models. that there is not nearly enough luminous matter

In keeping track of the expansion of the uni- to close the universe. So, if the universe is closed,

verse, it is useful to deal with the scale factor, then the dark matter must be responsible. The

which tells us how much distances between evolution of the universe may also be affected by

galaxies have changed from one time to another. a non-zero cosmological constant.

Questions

*20.9. What observations can we do to verify that

*20.1. How large a scale do we have to look at

the universe is isotropic?

before the cosmological principle can be

20.10. If we lived in a contracting universe,

applied? How does this scale compare with

would we still observe a cosmological

the distance over which light could have

redshift?

reached us in the age of the universe?

*20.11. How can we measure the curvature of the

20.2. Restate the argument in our discussion of

universe without getting outside it?

Olbers™s paradox using galaxies instead of

*20.12. What is the universe expanding into?

stars as the sources of light.

20.13. What are the advantages of using the scale

20.3. In our discussion of Olbers™s paradox, does

factor R(t) to keep track of the expansion of

it matter whether we talk about the appear-

the universe?

ance of the daytime or night time sky?

20.14. Where is the center of the universe?

*20.4. Suppose that we were trying to invoke inter-

20.15. What are the various interpretations of the

stellar dust as a way out of Olbers™s paradox

quantity k, discussed in this chapter?

by saying that it is the scattering by the

20.16. If the universe is closed, can we distinguish

dust that blocks out the distant light, not

a “one-time” universe from an oscillating

absorption. The dust will therefore not heat.

universe?

Why doesn™t this argument help?

20.17. Of the methods described for deciding

20.5. How does the universe having a finite size

whether the universe is open or closed,

or age save us from Olbers™s paradox?

which ones rely on measuring the gravita-

20.6. Does our motion towards the great attractor

tional effects of all of the matter in the

violate the part of the cosmological principle

universe?

that the universe should appear isotropic?

20.18. If the universe is expanding, how is it possi-

20.7. What is the observational evidence that the

ble for that expansion to reverse?

universe is expanding?

20.19. Why do we say that if the universe is closed,

20.8. Does Hubble™s law rule out the steady-state

then the dark matter must do it?

models?

394 PART V THE UNIVERSE AT LARGE

Problems

20.10. If the current density of the universe is

For all problems, unless otherwise stated, use H0

1 10 29 g/cm3, what value would be

70 km/s/Mpc.

needed for the cosmological constant in

20.1. Suppose we detect radiation that was emit-

order for the universe to be static?

ted by some galaxy far away. In the time the

20.11. Show that equation (20.30) can be obtained

radiation traveled to reach us, its wave-

by the appropriate use of a Taylor series.

length doubled. What happened to the scale #

20.12. What values of H(t0) would be needed to

factor of the universe in that time?

make q0 equal to (a) 0, (b) 1/2, (c) 1?

20.2. How much brighter would the sky be if it

20.13. Derive an expression for the critical density

were uniformly filled with Suns, rather than

(equation 20.26) without introducing the

the one we have? (Hint: think of the solid

deceleration parameter q0.

angle covered by the Sun relative to the

20.14. Show that if the distances are given by

whole sky.)

Hubble™s law, then the distance modulus is

20.3. Show that, if the universe were infinite in

given by equation (20.33) without the last

age and extent the cosmological redshift is

term on the right.

not sufficient to get us out of Olbers™s

20.15. Compare the distances obtained using

paradox.

Hubble™s law and equation (20.33) with

20.4. Estimate the rate of continuous creation

q0 1/2 for objects with z (a) 0.1, (b) 1.0,

required to keep the density constant (at

10 29 g/cm3). Express your answer in (c) 3.5, (4) 5.0.

protons/yr/km3. 20.16. Estimate the distance modulus, m M,

for objects with z (a) 0.1, (b) 1.0, (c) 3.5,

20.5. Show that the density of the universe is

proportional to 1/R3(t). (4) 5.0.

20.17. Estimate the age of the universe at the time

20.6. For the case k 0, find an expression for R(t)

when radiation was emitted from objects

valid for large R. What are the limits on R

with z (a) 0.1, (b) 1.0, (c) 3.5, (4) 5.0.

for your expression to be valid?

20.18. Suppose we observe an object that is 10 Mpc

20.7. Show that equation (20.24) follows from

away. At what wavelength is the H line

equations (20.11), (20.16) and (20.17).

observed if (a) the object has no other

20.8. Show that the density parameter is twice

motion, (b) the object has an additional

the deceleration parameter q.

motion away from us at 1000 km/s, (c) the

20.9. Rewrite equations (20.17) and (20.19) in

object has an additional motion towards us

terms of the density parameter, substituting

at 1000 km/s.

for the critical density from equation (20.26).

Computer problems

#

20.4. For k 0, plot a graph of the distance modulus

20.1. For the case k 0, find an expression for R vs. t,

(m M) vs. z.

and plot a graph of your result.

20.5. Using equation (20.41), plot a graph of t in

20.2. For k 0, what is the difference between the cur-

gigayears vs. z, for H0 50, 75 and 100 km/s/Mpc.

rent age of the universe and the current value of

the Hubble parameter?

20.3. For k 0, how different is the Hubble parameter

for objects with z 0.1, 1, 3, 5 and 103?

Chapter 21

The big bang

In the preceding chapter, we noted that Lemaitre detectable as a low intensity background of

first pointed out that if the universe is expanding, microwaves.

then there must have been an era in the past

21.1.1 Origin of the cosmic

when it was much denser than it is now. This hot,

background radiation

dense early era was named the big bang by Fred

Hoyle, a steady-state cosmologist, in an attempt to To help visualize the evolution of the early uni-

ridicule the theory. The theory survived the verse, we again rely on an analogy with an expand-

ridicule, the name remained, and we now refer to ing sphere, as shown in Fig. 21.1. Remember, in

all cosmological models with an evolving universe this analogy, the universe is the surface of the

as ˜big-bang cosmologies™. In this chapter, we will expanding sphere. All particles and radiation must

see what we can learn about conditions in the big move along the surface. If you have a balloon, you

bang, and what the relationship is between those can follow along with this analogy.

conditions and the current state of the universe. When the universe was young enough to have

its temperature higher than 3000 K, the atoms

were all ionized. The universe was a plasma of

21.1 The cosmic background

nuclei and electrons. The free electrons are par-

radiation ticularly efficient at scattering radiation. They

provided a continuum opacity for any radiation

Following the idea that the universe was very hot present. This means that radiation would not

and dense, George Gamow suggested, in 1946, that travel very far before getting scattered; the uni-

when the universe was less than about 200 sec- verse was opaque. The radiation therefore stayed

onds old, the temperature was greater than one in equilibrium with the matter. The spectrum of

billion kelvin, hot enough for nuclear reactions the radiation was that of a blackbody at the tem-

to take place rapidly. In 1948, Ralph Alpher, Hans perature of the matter. As the universe expanded,

Bethe and Gamow showed (in a paper often the density decreased, and the temperature

referred to as the alpha/beta/gamma (for the decreased. As the matter cooled, the radiation

names of the authors) paper) that these nuclear also cooled. Then the point was reached at which

reactions might be able to explain the current the temperature dropped below 3000 K. (Various

abundance of helium in the universe. (We will estimates place this at a time some 100 000 years

discuss the synthesis of the elements in the next after the expansion started.) At the lower temper-

section.) In a more thorough analysis of the prob- ature, the electrons and nuclei (mostly protons, or

lem, Alpher and Robert Herman, in a classic paper helium) combined to make atoms. This is called

the era of recombination. The neutral atoms are very

published in 1948, found that the early universe

should have been filled with radiation, and that inefficient at absorbing radiation, except at a few

the remnant of that radiation should still be narrow ranges of wavelengths corresponding to

396 PART V THE UNIVERSE AT LARGE

(a)

(d)

(b)

(e)

Fig 21.1. Diagram showing radiation and matter in an

expanding universe. (Remember, this is a two-dimensional

analogy.) (a) Before decoupling.The matter is dense and hot

and the matter and radiation are in equilibrium. (b) At decou-

pling the universe is transparent, and the radiation now moves

around without being absorbed. Photons are moving in all

(c) directions. (c) Protogalaxies are starting to form. Photons are

moving in all directions and are redshifted as the universe

spectral lines. For all practical purposes, the uni- expands. (d) Galaxies have formed, and photons are still mov-

verse became transparent to the radiation. Since ing in all directions.The redshift becomes larger as the uni-

the radiation and matter no longer interacted sig- verse expands. (e) Today, an inhabitant of any galaxy would see

the redshifted photons coming at them from any direction.

nificantly, we say that they were decoupled.

If we look at Fig. 21.1, we see that the last pho-

tons emitted by the plasma just before decoupling the material in the galaxies. Anyone in one of those

should still be running around the universe. A galaxies, looking around, should see radiation com-

relatively small fraction of those photons have ing at them from all directions. The radiation does

bumped into galaxies, and have been absorbed by undergo one change. As the universe expands, all

21 THE BIG BANG 397

of the radiation is redshifted, a result of the cos- “3

mological redshift. This is illustrated in Figs. 21.1(d)