SE5

In Chapter 3 we derived the equations satisfied by an expanding isotropic gas. They are the
Friedman equation 1
a ̇
a
2
=
8πG
k
ρ − 2 ,
3
a
(5.1)
which governs the time evolution of the scale factor a(t), and the fluid equation
ρ ̇ + 3
a ̇
p
ρ + 2
a
c
= 0 ,
(5.2)
which gives us the evolution of the mass density ρ(t). I stress that, despite their Newtonian
derivation, these are the real equations used by cosmologists, more traditionally derived
via the equations of general relativity as described in Advanced Topic 1.
This chapter finds and discusses some simple solutions to these equations, which will
be used extensively during the book. However, while they have wide applicability during
the early stages of the Universe, it will turn out that these are not sufficient to describe the
present state of the Universe, for which an extra ingredient, the cosmological constant, will
be needed. It is introduced in Chapter 7.
Before finding solutions to these equations, we can study two of their implications.
5.1
Hubble’s law
The Friedma equation allows us to explain Hubble’s discovery that recession velocity is
proportional to the distance. The velocity of recession is given by v = dr/dt and is in the
1 In accord with the discussion of Section 3.6, the c 2 on the final term has been dropped, so that its appearance
matches that of other cosmology textbooks.

same direction as r, allowing us to write
v =
| r ̇ |
a ̇
r = r.
|r |
a
(5.3)
The last step used r = ax, remembering that the comoving position x is a constant by
definition. Consequently, Hubble’s law v = Hr tells us that the proportionality constant,
the Hubble parameter, should be identified as
H =
a ̇
,
a
(5.4)
and the value as measured today can be denoted with a subscript ‘0’ as H 0 . Because we
measure Hubble’s constant to be positive rather than negative, we know that the Universe
is expanding rather than contracting.
We notice from this that the phrase Hubble’s constant is a bit misleading. Although
certainly it is constant in space due to the cosmological principle, there is no reason for
it to be constant in time. In fact, using it as a more compact notation, we can write the
Friedma equation as an evolution equation for H(t), as
H 2 =
8πG
k
ρ − 2 .
3
a
(5.5)
It is best to use the phrase ‘Hubble parameter’ for this quantity as a function of time, reserv-
ing ‘Hubble constant’ for its present value. Normally the Hubble parameter decreases with
time, for instance as the expansion is slowed by the gravitational attraction of the matter in
the Universe.
5.2 Expansion and redshift
The redshift of spectral lines that we used to justify the assumption of an expanding universe
can also be related to the scale factor. In this derivation I’ll make the simplifying assumption
that light is passed between two objects which are very close together, separated by a small
distance dr, as shown in Figure 5.1. I’ve drawn the objects as galaxies, but I really mean
two nearby points. According to Hubble’s law, their relative velocity dv will be
dv = H dr =
a ̇
dr .
a
(5.6)
As the points are nearby we can directly apply the Doppler law to say that the change in
wavelength between emission and reception, dλ ≡ λ r − λ e , is
dλ
dv
=
,
λ e
c
(5.7)
where dλ is going to be positive since the wavelength is increased. The time between
emission and reception is given by the light travel time dt = dr/c, and putting all that

Figure 5.1
A photon travels a distance dr between two galaxies A and B.
together gives
a ̇
da
a ̇ dr
dλ
= dt =
.
=
λ e
a c
a
a
(5.8)
Integrate and we find that ln λ = ln a + constant, i.e.
λ ∝ a,
(5.9)
where λ is now the instantaneous wavelength measured at any given time.
Although as I’ve derived it this result only applies to objects which are very close to each
other, it turns out that it is completely general (a rigorous treatment is given in Advanced
Topic 2). It tells us that as space expands, wavelengths become longer in direct proportion.
One can think of the wavelength as being stretched by the expansion of the Universe, and
its change therefore tells us how much the Universe has expanded since the light began its
travels. For example, if the wavelength has doubled, the Universe must have been half its
present size when the light was emitted.
The redshift as defined in Equation (2.1) is related to the scale factor by
1+ z =
a(t r )
λ r
.
=
λ e
a(t e )
(5.10)
and is normally only used to refer to light received by us at the present epoch.
5.3 Solving the equations
In order to discover how the Universe might evolve, we need some idea of what is in it. In a
cosmological context, this is done by specifying the relationship between the mass density
ρ and the pressure p. This relationship is known as the equation of state. At this point, we
shall only consider two possibilities.
Matter: In this context, the term ‘matter’ is used by cosmologists as shorthand for ‘non-
relativistic matter’, and refers to any type of material which exerts negligible pres-
sure, p = 0. Occasionally care is needed to avoid confusion between ‘matter’ used
in this sense, and used to indicate all types of matter whether non-relativistic or not.
A pressureless universe is the simplest assumption that can be made. It is a good

approximation to use for the atoms in the Universe once it has cooled down, as they
are quite well separated and seldom interact, and it is also a good description of a
collection of galaxies in the Universe, as they have no interactions other than gravi-
tational ones. Occasionally the term ‘dust’ is used instead of ‘matter’.
Radiation: Particles of light move, naturally enough, at the speed of light. Their kinetic
energy leads to a pressure force, the radiation pressure, which using the standard
theory of radiation can be shown to be p = ρc 2 /3. Problem 5.2 gives a rather hand-
waving derivation of this result. More generally, any particles moving at highly-
relativistic speeds have this equation of state, neutrinos being an obvious example.
I will concentrate on the case where the constant k in the Friedma equation is set equal
to zero, corresponding to a flat geometry.
5.3.1 Matter
We start by solving the fluid equation, having set p = 0 for matter. One way to solve it is
to notice a clever way of rewriting it, as follows
a ̇
ρ ̇ + 3 ρ = 0
a
=⇒
1 d
ρa 3 = 0
a 3 dt
=⇒
d
ρa 3 = 0 ,
dt
(5.11)
though one could also solve it more formally by noting that it is a separable equation.
Integrating tells us that ρa 3 equals a constant, i.e.
ρ ∝
1
.
a 3
(5.12)
This is not a surprising result. It says that the density falls off in proportion to the volume
of the Universe. It is very natural that if the volume of the Universe increases by a factor
of say two, then the density of the matter must fall by the same factor. After all, material
caot come from nowhere, and there is no pressure to do any work.
The equations we are solving (with k = 0) have one very useful symmetry; their form
is unchanged if we multiply the scale factor a by a constant, since only the combination
a/a
̇ appears. This means that we are free to rescale a(t) as we choose, and the normal
convention is to choose a = 1 at the present time. With this choice physical and comoving
coordinate systems coincide at the present, since r = a x. Throughout this book I will use
the subscript ‘0’ to indicate the present value of quantities. Denoting the present density
by ρ 0 fixes the proportionality constant
ρ =
ρ 0
.
a 3
(5.13)
Having solved for the evolution of the density in terms of a, we must now find how a
varies with time by using the Friedma equation. Substituting in for ρ, and remembering
we are assuming k = 0, gives
a ̇ 2 =
8πGρ 0 1
.
3 a
(5.14)

Faced with an equation like this, one can use formal techniques to solve it (this equation is
separable, allowing it to be integrated), or alternatively make an educated guess as to the
solution and confirm it by substitution. In cosmology, a good educated guess is normally a
power-law a ∝ t q . Substituting this in, the left-hand side has time dependence t 2q−2 and
the right-hand side t −q . This can only be a solution if these match, which requires q = 2/3,
and so the solution is a ∝ t 2/3 . As we have fixed a = 1 at the present time t = t 0 , the full
solution is therefore
a(t) =
t
t 0
2/3
;
ρ(t) =
ρ 0
ρ 0 t 20
=
.
a 3
t 2
(5.15)
In this solution, the Universe expands forever, but the rate of expansion H(t) decreases
with time
a ̇
2
H ≡ =
,
(5.16)
a
3t
becoming infinitely slow as the Universe becomes infinitely old. Notice that despite the
pull of gravity, the material in the Universe does not recollapse but rather expands forever.
This is one of the classic cosmological solutions, and will be much used throughout
this book.
5.3.2 Radiation
Radiation obeys p = ρc 2 /3. Consequently the fluid equation is changed from the matter-
dominated case, now reading
a ̇
ρ ̇ + 4 ρ = 0 .
a
(5.17)
This is amenable to the same trick as before, with the a 3 replaced by a 4 in Equation (5.11),
giving
ρ ∝
1
.
a 4
(5.18)
Carrying out the same analysis we did in the matter-dominated case gives
a(t) =
t
t 0
1/2
;
ρ(t) =
ρ 0
ρ 0 t 2
= 2 0 .
4
a
t
(5.19)
This is the second classic cosmological solution.
Notice that the Universe expands more slowly if radiation dominated than if matter
dominated, a consequence of the extra deceleration that the pressure supplies – see Equa-
tion (3.18). So it is definitely wrong to think of pressure as somehow ‘blowing’ the Universe
apart. However, in each case the density of material falls off as t 2 .
We’d better examine the fall off of the radiation density with volume more carefully.
It drops as the fourth power of the scale factor. Three of those powers we have already
identified as the increase in volume, leading naturally to a drop in the density. The final

stretching is proportional to a, and the energy of radiation proportional to its frequency via
E = hf , this results in a further drop in energy by the remaining power of a. This lowering
of energy is exactly the redshifting effect we use to measure distances.
The rate of decrease of the radiation density also has an explanation in terms of thermo-
dynamics, which is macroscopic rather than microscopic. Since the Universe in this case
has a pressure, when it expands there is work done which is given by p dV , in exactly the
same way as work is done on a piston when the gas is allowed to expand and cool. This
work done corresponds to the extra diminution of the radiation density by the final factor
of a.
5.3.3
Mixtures
A more general situation is when one has a mixture of both matter and radiation. Then
there are two separate fluid equations, one for each of the two components. The trick
which allows us to write ρ as a function of a still works, so we still have
ρ mat ∝
1
a 3
;
ρ rad ∝
1
.
a 4
(5.20)
However, there is still only a single Friedma equation (after all, there is only one Uni-
verse!), which now has
ρ = ρ mat + ρ rad .
(5.21)
This means that the scale factor will have a more complicated behaviour, and so to convert
ρ(a) into ρ(t) is much harder. It is possible to obtain exact solutions for this situation,
but they are quite messy so I won’t include them here. Instead, I’ll consider the simpler
situation where one or other of the densities is by far the larger.
In that case, we can say that the Friedma equation is accurately solved by just in-
cluding the dominant component. That is, we can use the expansion rates we have already
found. For example, suppose radiation is much more important. Then one would have
a(t) ∝ t 1/2
;
ρ rad ∝
1
t 2
;
ρ mat ∝
1
1
∝ 3/2 .
3
a
t
(5.22)
Notice that the density in matter falls off more slowly than that in radiation. So the situation
of radiation dominating caot last forever; however small the matter component might be
originally it will eventually come to dominate. We can say that domination of the Universe
by radiation is an unstable situation.
In the opposite situation, where it is the matter which is dominant, we obtain the solu-
tion
1
1
1
(5.23)
a(t) ∝ t 2/3 ; ρ mat ∝ 2 ; ρ rad ∝ 4 ∝ 8/3 .
t
a
t
Matter domination is a stable situation, the matter becoming increasingly dominant over
the radiation as time goes by.

Figure 5.2 A schematic illustration of the evolution of a universe containing radiation and
matter. Once matter comes to dominate the expansion rate speeds up, so the densities fall more
quickly with time.
Figure 5.2 shows the evolution of a universe containing matter and radiation, with the
radiation initially dominating. Eventually the matter comes to dominate, and as it does so
the expansion rate speeds up from a(t) ∝ t 1/2 to the a(t) ∝ t 2/3 law. It is likely that this
is the situation that applied in our Universe until fairly recently, as we’ll see in Chapter 11.
5.4
Particle number densities
An important alternative view of the evolution of particles, which will be much used later
in the book, is that of the number density n of particles rather than of their mass or energy
density.
The number density is simply the number of particles in a given volume. If the mean
energy per particle (including any mass–energy) is E, then the number density is related
to the energy density by
= n × E.
(5.24)
The number density is useful because in most circumstances particle number is conserved.
For example, if particle interactions are negligible, you wouldn’t expect an electron to
suddenly vanish into oblivion, and the same is true of a photon of light. The particle number
can change through interactions, for example an electron and positron could aihilate and
create two photons. However, if the interaction rate is high we expect the Universe to be

in a state of thermal equilibrium. If so, then particle number is conserved even in a highly-
interacting state, since by definition thermal equilibrium means that any interaction, which
may change the number density of a particular type of particle, must proceed at the same
rate in both forward and backward directions so that any change cancels out.
So, barring brief periods where thermal equilibrium does not hold, we expect the num-
ber of particles to be conserved. The only thing that changes the number density, therefore,
is that the volume is getting bigger, so that these particles are spread out in a larger volume.
This implies
n ∝
1
.
a 3
(5.25)
This looks encouragingly like the behaviour we have already seen for matter, but it’s also
true for radiation as well!
How does this relate to our earlier results? The energy of non-relativistic particles is
dominated by their rest mass–energy which is constant, so
ρ mat ∝
mat
∝ n mat × E mat ∝
1
1
× const ∝ 3 .
a 3
a
(5.26)
But photons lose energy as the Universe expands and their wavelength is stretched, so their
energy is E rad ∝ 1/a as we have already seen. So
ρ rad ∝
rad
∝ n rad × E rad ∝
1
1
1
× ∝ 4 .
a 3
a
a
(5.27)
These are exactly the results we saw before, equations (5.12) and (5.18).
Although the energy densities of matter and radiation evolve in different ways, their
particle numbers evolve in the same way. So, apart from epochs during which the assump-
tion of thermal equilibrium fails, the relative number densities of the different particles
(e.g. electrons and photons) do not change as the Universe expands.
5.5 Evolution including curvature
We can now re-introduce the possibility that the constant k is non-zero, corresponding to
spherical or hyperbolic geometry. Rather than seeking precise solutions, I will concentrate
on the qualitative properties of the solutions. These are actually of rather limited use in
describing our own Universe, because as we will see the cosmological models discussed
so far are not general enough and we will need to consider a cosmological constant (see
Chapter 7). Nevertheless, studying the possible behaviours of these simple models is a
useful exercise, even if one should be cautious about drawing general conclusions.
In analysing the possible dynamics, I will assume that the Universe is dominated by
non-relativistic matter always, which in practice is not a restrictive assumption. We have
already seen that if we assume that the constant k in the Friedma equation is zero, then
the Universe expands for ever, a ∝ t 2/3 , but slows down arbitrarily at late times. So we
know the fate of the Universe in that case. But what happens if k = 0?

Figure 5.3 Three possible evolutions for the Universe, corresponding to the different signs
of k. The middle line corresponds to the k = 0 case where the expansion rate approaches zero
in the infinite future. During the early phases of the expansion the lines are very close and so
observationally it can be difficult to distinguish which path the actual Universe will follow.
The principal question to ask is whether it is possible for the expansion of the Universe
to stop, which since H = a/a
̇ corresponds to H = 0. Looking at the Friedma equation
H 2 =
8πG
k
ρ − 2 ,
3
a
(5.28)
it is immediately apparent that this is not possible if k is negative, for then both the terms on
the right-hand side of the Friedma equation are positive. Consequently, such a universe
must expand forever. That enables us to study the late-time behaviour, because we can see
that the term k/a 2 falls off more slowly with the expansion than does ρ mat ∝ 1/a 3 . Since
a becomes arbitrarily large for the matter-dominated solution for negligible k, the k/a 2
term must eventually come to dominate. When it does, the Friedma equation becomes
a ̇
a
2
= −
k
.
a 2
(5.29)
Cancel off the a 2 terms and you’ll find the solution is a ∝ t. So when the last term comes
to dominate, the expansion of the Universe becomes yet faster. In this case, the velocity
does not tend to zero at late times, but instead becomes constant. This is sometimes known
as free expansion.
Things are very different if k is positive. It then becomes possible for H to be zero, by
the two terms on the right-hand side of the Friedma equation cancelling each other out.
Indeed, this is inevitable, because the negative influence of the k/a 2 term will become more

and more important relative to the ρ mat term as time goes by. In such a universe therefore,
the expansion must come to an end after a finite amount of time. As gravitational attraction
persists, the recollapse of the Universe becomes inevitable.
The collapse of the Universe is fairly easy to describe, because the equations governing
the evolution are time reversible. That is, if one substitutes −t for t, they remain the same.
The collapse phase is therefore just like the expansion in reverse, and so after a finite time
the Universe will come to an end in a Big Crunch. Problem 5.5 investigates this in more
detail.
These three behaviours, illustrated in Figure 5.3, can be related to the particle energy U
in our Newtonian derivation of the Friedma equation. If the particle energy is positive,
then it can escape to infinity, with a final kinetic energy given by U . If the total energy
is zero, then the particle can just escape but with zero velocity. Finally, if the energy is
negative, it caot escape the gravitational attraction and is destined to recollapse inwards.
There is a fairly precise analogy with escape velocity from the Earth (or the Moon, if
you want to worry about the atmosphere). If you throw a rock up in the air hard enough,
gravity will be unable to stop it and eventually it will sail off into space at a constant ve-
locity. If your throw is too puny, it will rapidly fall back. And in between is the escape
velocity, where the rock is just able to escape the gravitational field and no more.