SE3

It is perfectly possible to discuss cosmology without having already learned general rel-
ativity. In fact, the most crucial equation, the Friedma equation which describes the
expansion of the Universe, turns out to be the same when derived from Newton’s theory
of gravity as it is when derived from the equations of general relativity. The Newtonian
derivation is, however, some way from being completely rigorous, and general relativity is
required to fully patch it up, a detail that need not concern us at this stage. Readers with
some familiarity with general relativity may wish to study the relativistic derivation given
in Advanced Topic 1.
In Newtonian gravity all matter attracts, with the force exerted by an object of mass M
on one of mass m given by the famous relationship
F =
GM m
,
r 2
(3.1)
where r is the distance between the objects and G is Newton’s gravitational constant. That
is, gravity obeys an inverse square law. Because a force on an object induces an acceleration
which is also proportional to its mass, via F = ma, the acceleration an object feels under
gravity is independent of its mass.
The force exerted means there is a gravitational potential energy
V = −
GM m
,
r
(3.2)
with the force being in the direction which decreases the potential energy the fastest. Like
the electric potential of two opposite charges, the gravitational potential is negative, favour-
ing the two objects being close together. But with gravity there is no analogue of the re-
pulsion of like charges. Gravity always attracts.
The derivation of the Friedma equation requires a famous result due originally to
Newton, which I won’t attempt to prove here. This result states that in a spherically-
symmetric distribution of matter, a particle feels no force at all from the material at greater
radii, and the material at smaller radii gives exactly the force which one would get if all the
An Introduction to Modern Cosmology, Third Edition. Andrew Liddle.
c 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.

Figure 3.1 The particle at radius r only feels gravitational attraction from the shaded region.
Any gravitational attraction from the material outside cancels out, according to Newton’s the-
orem.
material was concentrated at the central point. This property arises from the inverse square
law; the same results exist for electromagnetism. One example of its use is that the grav-
itational (or electromagnetic) force outside a spherical object of unknown density profile
depends only on the total mass (charge). Another is that an ‘astronaut’ inside a spherical
shell feels no gravitational force, not only if they are at the centre but if they are at any
position inside the shell.
3.1 The Friedma equation
The Friedma equation describes the expansion of the Universe, and is therefore the most
important equation in cosmology. One of the routine tasks for a working cosmologist is
solving this equation under different assumptions concerning the material content of the
Universe. To derive the Friedma equation, we need to compute the gravitational po-
tential energy and the kinetic energy of a test particle (it doesn’t matter which one, since
everywhere in the Universe is the same according to the cosmological principle), and then
use energy conservation.
Let’s consider an observer in a uniform expanding medium, with mass density ρ, the
mass density being the mass per unit volume. We begin by realizing that because the Uni-
verse looks the same from anywhere, we can consider any point to be its centre. Now
consider a particle a distance r away with mass m, as shown in Figure 3.1. (By ‘particle’,
I really mean a small volume containing the mass m.) Due to Newton’s theorem, this par-
ticle only feels a force from the material at smaller radii, shown as the shaded region. This

material has total mass given by M = 4πρr 3 /3, contributing a force
F =
GM m
4πGρ rm
,
=
2
r
3
(3.3)
and our particle has a gravitational potential energy
V = −
4πGρ r 2 m
GM m
= −
.
r
3
(3.4)
The kinetic energy is easy; the velocity of the particle is r ̇ (I’ll always use dots to mean
time derivatives) giving
T =
1 2
m r ̇ .
2
(3.5)
The equation describing how the separation r changes can now be derived from energy
conservation for that particle, namely
U = T + V ,
(3.6)
where U is a constant. Note that U need not be the same constant for particles separated
by different distances. Substituting gives
U =
1
4π
m r ̇ 2 −
Gρ r 2 m .
2
3
(3.7)
This equation gives the evolution of the separation r between the two particles.
We now make a crucial step in this derivation, which is to realize that this argument ap-
plies to any two particles, because the Universe is homogeneous. This allows us to change
to a different coordinate system, known as comoving coordinates. These are coordinates
which are carried along with the expansion. Because the expansion is uniform, the rela-
tionship between real distance r and the comoving distance, which we can call x, can be
written
r = a(t) x ,
(3.8)
where the homogeneity property has been used to ensure that a is a function of time alone.
Note that these distances have been written as vector distances. What you should think
of when studying this equation is a coordinate grid which expands with time, as shown in
Figure 3.2. The galaxies remain at fixed locations in the x coordinate system. The original
r coordinate system, which does not expand, is usually known as physical coordinates.
The quantity a(t) is a crucial one, and is known as the scale factor of the Universe.
It measures the universal expansion rate. It is a function of time alone, and it tells us
how physical separations are growing with time, since the coordinate distances x are by
definition fixed. For example, if, between times t 1 and t 2 , the scale factor doubles in value,
that tells us that the Universe has expanded in size by a factor two, and it will take us twice
as long to get from one galaxy to another.

Figure 3.2 The comoving coordinate system is carried along with the expansion, so that any
objects remain at fixed coordinate values.
We can use the scale factor to rewrite Equation (3.7) for the expansion. Substituting
Equation (3.8) into it, remembering x ̇ = 0 by definition as objects are fixed in comoving
coordinates, gives
U =
4π
1
m a ̇ 2 x 2 −
Gρ a 2 x 2 m .
2
3
(3.9)
Multiplying each side by 2/ma 2 x 2 and rearranging the terms then gives
a ̇
a
2
=
8πG
kc 2
ρ − 2 ,
3
a
(3.10)
where kc 2 = −2U/mx 2 . This is the standard form of the Friedma equation, and it
will appear frequently throughout this book. In this expression k must be independent of x
since all the other terms in the equation are, otherwise homogeneity will not be maintained.
So in fact we learn that homogeneity requires that the quantity U , while constant for a given
particle, does indeed change if we look at different separations x, with U ∝ x 2 .
Finally, since k = −2U/mc 2 x 2 which is time independent (as the total energy U
is conserved, and the comoving separation x is fixed), we learn that k is just a constant,
unchanging with either space or time. It has the units of [length] −2 . An expanding Universe
has a unique value of k, which it retains throughout its evolution. In Chapter 4 we will see
that k tells us about the geometry of the Universe, and it is often called the curvature.

3.2 On the meaning of the expansion
So what does the expansion of the Universe mean? Well, let’s start with what it does not
mean. It does not mean that your body is gradually going to get bigger with time (and
certainly isn’t an excuse if it does). It does not mean that the Earth’s orbit is going to get
further from the Sun. It doesn’t even mean that the stars within our galaxy are going to
become more widely separated with time.
But it does mean that distant galaxies are getting further apart.
The distinction is whether or not the motion of objects is governed by the cumulative
gravitational effect of a homogeneous distribution of matter between them, as shown in
Figure 3.1. The atoms in your body are not; their separation is dictated by the strength
of chemical bonds, with gravity playing no significant role. So molecular structures will
not be affected by the expansion. Likewise, the Earth’s motion in its orbit is completely
dominated by the gravitational attraction of the Sun (with a minor contribution from the
other planets). And even the stars in our galaxy are orbiting in the common gravitational
potential well which they themselves create, and are not moving apart relative to one an-
other. The common feature of these environments is that they are ones of considerable
excess density, very different from the smooth distribution of matter we used to derive the
Friedma equation.
But if we go to large enough scales, in practice tens of megaparsecs, the Universe does
become effectively homogeneous and isotropic, with the galaxies flying apart from one
another in accordance with the Friedma equation. It is on these large scales that the
expansion of the Universe is felt, and on which the cosmological principle applies.
3.3
Things that go faster than light
A common question that concerns people is whether faraway galaxies are receding from
us faster than the speed of light. That is to say, if velocity is proportional to distance, then
if we consider galaxies far enough away can we not make the velocity as large as we like,
in violation of special relativity?
The answer is that indeed in our theoretical predictions distant objects can appear to
move away faster than the speed of light. However, it is space itself which is expanding.
There is no violation of causality, because no signal can be sent between such galaxies.
Further, special relativity is not violated, because it refers to the relative speeds of objects
passing each other, and caot be used to compare the relative speeds of distant objects.
One way to think of this is to imagine a colony of ants on a balloon. Suppose that the
fastest the ants can move is a centimetre per second. If any two ants happen to pass each
other, their fastest relative speed would be two centimetres per second, if they happened
to be moving in opposite directions. Now start to blow the balloon up. Although the ants
wandering around the surface still caot exceed one centimetre per second, the balloon
is now expanding under them, and ants which are far apart on the balloon could easily
be moving apart at faster than two centimetres per second if the balloon is blown up fast
enough. But if they are, they will never get to tell each other about it, because the balloon is
pulling them apart faster than they can move together, even at full speed. Any ants that start
close enough to be able to pass one another must do so at no more than two centimetres
per second even if the Universe is expanding.

The expansion of space is just like that of the balloon, and pulls the galaxies along with
it.
3.4 The fluid equation
Fundamental though it is, the Friedma equation is of no use without an equation to de-
scribe how the density ρ of material in the Universe is evolving with time. This involves
the pressure p of the material, and is called the fluid equation. (Unfortunately the standard
symbol p for pressure is the same as for momentum, which we’ve already used. Almost
always in this book, p will be pressure.) As we’ll shortly see, the different types of material
which might exist in our Universe have different pressures, and lead to different evolution
of the density ρ.
We can derive the fluid equation by considering the first law of thermodynamics
dE + pdV = T dS ,
(3.11)
applied to an expanding volume V of unit comoving radius. 1 This is exactly the same as
applying thermodynamics to a gas in a cylinder. The volume has physical radius a, so the
energy is given, using E = mc 2 , by
E =
4π 3 2
a ρc .
3
(3.12)
The change of energy in a time dt, using the product rule, is
dE
da 4π 3 dρ 2
= 4πa 2 ρ c 2
+
a
c ,
dt
dt
3
dt
(3.13)
while the rate of change in volume is
dV
da
= 4πa 2
.
dt
dt
(3.14)
Assuming a reversible adiabatic expansion dS = 0, putting these into Equation (3.11) and
rearranging gives
ρ ̇ + 3
p
a ̇
ρ + 2
a
c
= 0 ,
(3.15)
where as always dots are shorthand for time derivatives. This is the fluid equation. As
we see, there are two terms contributing to the change in the density. The first term in
the brackets corresponds to the dilution in the density because the volume has increased,
while the second corresponds to the loss of energy because the pressure of the material has
done work as the Universe’s volume increased. This energy has not disappeared entirely
of course; energy is always conserved. The energy lost from the fluid via the work done
has gone into gravitational potential energy.
1 Don’t
confuse V for volume with V for gravitational potential energy.

Let me stress that there are no pressure forces in a homogeneous universe, because the
density and pressure are everywhere the same. A pressure gradient is required to supply
a force. So pressure does not contribute a force helping the expansion along; its effect is
solely through the work done as the Universe expands.
We are still not in a position to solve the equations, because now we only know what
ρ is doing if we know what the pressure p is. It is in specifying the pressure that we are
saying what kind of material our model universe is filled with. The usual assumption in
cosmology is that there is a unique pressure associated with each density, so that p ≡ p(ρ).
Such a relationship is known as the equation of state, and we’ll see two different examples
in Chapter 5. The simplest possibility is that there is no pressure at all, and that particular
case is known as (non-relativistic) matter.
Once the equation of state is specified, the Friedma and fluid equations are all we
need to describe the evolution of the Universe. However, before discussing this evolution,
I am going to spend some time exploring some general properties of the equations, as well
as devoting Chapter 4 to consideration of the meaning of the constant k. If you prefer
to immediately see how to solve these equations, feel free to jump straight away to Sec-
tions 5.3 to 5.5, and come back to the intervening material later. On the way, you might
want to glance at Section 3.6 to find out why a factor of c 2 mysteriously vanishes from the
Friedma equation between here and there.
3.5
The acceleration equation
The Friedma and fluid equations can be used to derive a third equation, not indepen-
dent of the first two of course, which describes the acceleration of the scale factor. By
differentiating Equation (3.10) with respect to time we obtain
2
kc 2 a ̇
a − a ̇ 2
a ̇ a ̈
8πG
ρ
̇
+
2
=
.
a
a 2
3
a 3
(3.16)
Substituting in for ρ ̇ from Equation (3.15) and cancelling the factor 2 a/a
̇ in each term gives
a
̈
−
a
a ̇
a
2
= −4πG ρ +
kc 2
p
+
,
c 2
a 2
(3.17)
and finally, using Equation (3.10) again, we arrive at an important equation known as the
acceleration equation
4πG
a
̈
= −
a
3
ρ +
3p
c 2
.
(3.18)
Notice that if the material has any pressure, this increases the gravitational force, and so
further decelerates the expansion. I remind you that there are no forces associated with
pressure in an isotropic universe, as there are no pressure gradients.
The acceleration equation does not feature the constant k which appears in the Fried-
ma equation; it cancelled out in the derivation.

3.6 On mass, energy and vanishing factors of c 2
You should be aware that cosmologists have a habit of using mass density ρ and energy
density interchangeably. They are related via Einstein’s most famous equation as = ρc 2 ,
and if one chooses so-called ‘natural units’ in which c is set equal to one, the two become
the same. For clarity, however, I will be careful to maintain the distinction. Note that the
phrase ‘mass density’ is used in Einstein’s sense – it includes the contributions to the mass
from the energy of the various particles, as well as any rest mass they might have.
The habit of setting c = 1 means that the Friedma equation is normally written
without the c 2 in the final term, so that it reads
a ̇
a
2
=
k
8πG
ρ − 2 .
3
a
(3.19)
The constant k then appears to have units [time] −2 – setting c = 1 makes time and length
units interchangeable. Since the practice of omitting the c 2 in the Friedma equation is
widely adopted in other cosmology textbooks, I will drop it for the remainder of this book
too. In practice, it is a rare situation indeed where one has to be careful about this.