. 19
( 28)


we will see that the relative abundance of hydro-
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
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

spherical shells, centered on the Earth, each of
thickness dr. The volume of each shell is then
4 r2 dr

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
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
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

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)
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
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
r0 1dR dt 2 H1t 2R1t 2 r0

Dividing by r0 gives
H1t 2 R1t 2
dR dt (20.11)
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

(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

R 1t 2

20.3 Cosmology and Newtonian Substituting this into equation (20.15), we have
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
3 R2
for a particle of mass m on the surface of the
sphere at position r is Noting that
GM1r2 m ˆ
d1R2 2
mr (20.13) #$
r 2RR
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
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
that 11 R2 2 1dR dt 2
M1r2 r d11 R2 dt, we have

Substituting into equation (20.13) gives
cR d
8 G0
d #2
  0 (20.18)
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
R GR (20.15) 8 G0
3 R2 k
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
1t 2

R3 1t 2
in Fig. 20.4(a).

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

We integrate the left side from zero to R and the
right side from zero to t to give
a bt
2 3>2 0
R (20.21)
3 3
This means that R is proportional to t2/3. The uni-

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
3 Rmax

Solving for Rmax:

8 G0
Rmax   (20.22)
Age of Universe
After Rmax is reached, the universe starts to col-
lapse. We say that the universe is closed.
We next look at the case k 0. If k is negative
then k is positive, and the right-hand side of equa-

No Gravity
tion (20.19) is always positive. As R gets very large,
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
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.

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  
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
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-
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
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
# solve for the density, , we have
3 H2
Since the expansion is slowing down, R is neg-   (20.26)
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
q   (20.24)
3 H2 (20.27)

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

20.4 Cosmology and general
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

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
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
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.

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.
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,
1ds 2 2 1c dt 2 2

1dr2 2
R 1ct 2 c r2 3 1d 2 2 sin2 1d 2 2 4 d

(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

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
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
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
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.

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

0> z. This gives
where ¢ , so 1

R1t 2
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)
R1t1 2
R1t0 2 ¢tR 1t0 2
1 #
R1t0 ¢t 2
If we let t1 be some arbitrary time, and t2 be
¢tR 1t0 2
the reference epoch t0 (for which R 1), this
Combining this with equation (20.29b) gives
¢tR 1t0 2
R1t 2
1 1

R2 kc2
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)
If a photon takes t to reach us, it must have $
c d
R 4G 3P
been emitted from a distance d c t . Using this
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
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
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
relevant to our discussion of cosmology: t1

/8 G,
like those with a mass density, EFF
„¦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
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

-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

R1t 2   cosh a tb,
3 ¶
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

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

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
visible matter is about 1% of the critical density.
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
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)

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
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):
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,
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)
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.
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

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
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
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
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

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.

*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
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.
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
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?

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
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




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

of the radiation is redshifted, a result of the cos- “3
mological redshift. This is illustrated in Figs. 21.1(d)


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