SE4

We now consider the real meaning of the constant k which appears in the Friedma
equation
a ̇
a
2
=
8πG
k
ρ − 2 .
3
a
(4.1)
While the Newtonian derivation in the last chapter introduced this as a measure of the
energy per particle, the true interpretation, apparent in the context of general relativity, is
that it measures the curvature of space. General relativity tells us that gravity is due to the
curvature of four-dimensional space–time, and a full analysis can be found in any general
relativity textbook. Here I will be purely descriptive, and focus on the interpretation of
k as measuring the curvature of the three spatial dimensions. Further details of general
relativistic cosmology can be found in Advanced Topic 1.
We have demanded that our model universes be both homogeneous and isotropic. The
simplest type of geometry which can have this property is what is called a flat geometry,
in which the normal rules of Euclidean geometry apply. However, it turns out that the
assumption of isotropy is not enough to demand that as the only choice. Instead, there are
three possible geometries for the Universe, and they correspond to k being zero, positive
or negative.
4.1
Flat geometry
Euclidean geometry is based on a set of simple axioms (e.g. a straight line is the short-
est distance between two points), plus one more complex axiom which says that parallel
straight lines remain a fixed distance apart. These are the basis for the standard laws of
geometry, and lead to the following conclusions:
• The angles of a triangle add up to 180 ◦ .
• The circumference of a circle of radius r is 2πr.

Such a geometry might well apply to our own Universe. If that is the case, then the Universe
must be infinite in extent, because if it came to a definite edge then that would clearly violate
the principle that the Universe should look the same from all points. 1
A universe with this geometry is usually called a flat universe.
4.2 Spherical geometry
Euclid always hoped that the more artificial final axiom could be proven from the others.
It wasn’t until the 19th century that Riema demonstrated that Euclid’s final axiom was
an arbitrary choice, and that one could make other assumptions. In doing so, he founded
the subject of non-Euclidean geometry, which forms the mathematical foundation for Ein-
stein’s theory of general relativity.
The simplest kind of non-Euclidean geometry is actually very familiar to us; it is the
spherical geometry which we use, for instance, to navigate around the Earth. Before wor-
rying about the Universe having three dimensions, let’s examine the properties of the two-
dimensional surface of the Earth, shown in Figure 4.1.
First of all, we know that a perfect sphere looks the same from all points on its surface,
so the condition of isotropy is satisfied (e.g. if someone hands you a snooker ball and asks
which way up it is, you’re not going to be able to tell them). But, unlike the case of a flat
geometry, the spherical surface is perfectly finite in extent, its area being given by 4πr 2 .
Yet there is no boundary, no ‘edge’ to the surface of the Earth. So it is perfectly possible
to have a finite surface which nevertheless has no boundary.
If we draw parallel lines on the surface of the Earth, then they violate Euclid’s final
axiom. The definition of a straight line is the shortest distance between two points, which
means that the straight lines in a spherical geometry are segments of great circles, such as
the equator or the lines of longitude. 2 The lines of longitude are an excellent example of
the failure of Euclid’s axiom; as they cross the equator they are all parallel to one another,
but rather than remaining a constant distance apart they meet at both poles.
If we draw a triangle on a sphere, we find that the angles do not add up to 180 ◦ degrees
either. The easiest example to think about is to start at the North Pole. Draw two straight
lines down to the equator, ninety degrees apart, and then join them with a line on the equator.
You have drawn a triangle in which all three angles are 90 ◦ , shown in Figure 4.1.
The circumference of a circle also fails to obey the normal law. Suppose we draw a
circle at a radius r from the North Pole, and we’ll choose r so that our circle is the equator.
That radius, measured on the surface of the sphere, corresponds to a quarter of a complete
circle around the Earth, so r = πR/2 where R is the radius of the Earth. However, the
circumference c is given by 2πR, so instead of the usual relation one has c = 4r for a circle
drawn at the equator. The circumference is less than 2πr. Problem 4.1 looks at the general
case; you may find it helpful to glance at the figure on page 35 now.
Although I have only considered specific cases where the algebra is easy, it is true that
whatever triangle or circle is drawn, you’ll always find
• the angles of a triangle add up to more than 180 ◦ ;
1 Well,
2 Note
that’s almost true. See Advanced Topic 1.3 for a way to bypass that conclusion.
that, apart from the equator, lines of latitude are not straight lines; this is why aeroplanes do not follow
lines of latitude when flying, because they are not the shortest way to go!

Figure 4.1 A sketch of a spherical surface, representing positive k. A triangle is shown which
has three right angles!
• the circumference of a circle is less than 2πr.
If you make the triangles or circles much smaller than the size of the Earth, then the
Euclidean laws start to become a good approximation; certainly we don’t have to worry
about Euclidean laws being broken in our everyday existence (though the appreciation that
the Earth is spherical is vital for the plaing of long distance journeys). So a small triangle
drawn on a sphere will have the sum of its angles only marginally larger than 180 ◦ . This
property makes it rather hard to measure the geometry of our own Universe, because the
neighbouring region which we can measure accurately is only a small fraction of the size
of the Universe and so will obey nearly Euclidean laws whatever the overall geometry.
One of the most important conceptual points that you need to grasp is that our three-
dimensional Universe can have properties just like the two-dimensional surface of the
sphere. Unfortunately our brains are not conditioned to think of three dimensions as being
curved, so we must work by analogy with the two-dimensional situation I’ve just described.
When we think of the surface of a sphere being curved, we naturally imagine the sphere
as an object in our own three-dimensional Universe, and think of it as curved in that sense.
The important point is that the curvature is a property of the two-dimensional surface of
the sphere itself; the triangles and circles whose properties I’ve just described are drawn
on the surface. A classic application of this was the ancient Greeks’ use of these laws to
deduce that the Earth is spherical, and even to obtain a good estimate of its diameter. When
we discuss the spherical geometry, there is actually no need to think of it as existing within
our three-dimensional space at all, and we must remember when discussing the geometry
that we are assumed to be restricted to the surface of the sphere, and not allowed to move
off the surface either towards or away from the notional centre.
All this is an analogy to what might happen with our three-dimensional Universe. The
analogue of the two-dimensional sphere is called a three-sphere. In the rather unlikely
event of four-dimensional creatures existing, they would be able to visualize the curvature
of three-dimensional space in just the same way we can visualize that of two-dimensional

Figure 4.2 A sketch of a saddle surface, representing the hyperbolic geometry obtained when
k is negative. A rather exaggerated triangle is shown with its sum of angles well below 180 ◦ .
space. However, like the two-dimensional sphere, the possible curvature of three-space is
an intrinsic property and there is no actual need of a higher-dimensional space for it to live
in. Obtaining a correct mental picture of this is one of the big challenges in understanding
our Universe!
A universe with a spherical geometry, like the surface of the Earth, has a finite size but
no boundary. All points are equivalent. If we live in a spherical geometry, and travel in
a straight line, we would not go on for ever and ever. Rather, eventually we would come
back to where we had started from, from the opposite direction, exactly in the maer
that someone travelling outward from the North Pole on the Earth eventually returns there
from the opposite direction. Such a universe corresponds to choosing a positive value for
the quantity k appearing in the Friedma equation. Because the special properties of the
spherical geometry are due to its curvature, k is often therefore called the curvature term.
A universe with k > 0 is normally referred to as a closed universe, because of its finite
size.
4.3 Hyperbolic geometry
The final choice is k negative. The corresponding geometry is known as hyperbolic, and
is much less familiar than spherical geometry. It is normally represented by a saddle-like
surface as in Figure 4.2; it is hard to see that this is consistent with isotropy but in fact it
is. In a hyperbolic geometry, parallel lines never meet – in fact they break Euclid’s axiom
by diverging away from one another.
The behaviour of the hyperbolic geometry can be guessed from what has gone before;
it is the opposite of the spherical geometry. One finds that
• the angles of a triangle add up to less than 180 ◦ ;
• the circumference of a circle is greater than 2πr.
Because parallel lines never meet, such a universe must be infinite in extent, just as in the
flat case. The situation k < 0 is known as an open universe.

In Chapter 9 we’ll explore which of the three possible geometries, summarized in
Table 4.1, seems best suited to describe the real Universe.
4.4
Infinite and observable universes
What does it actually mean for the Universe to be infinite, as in the flat and open cases?
This property is nothing to do with the Universe lasting forever; what it means is that the
Universe is already infinite in size even at a finite time. It genuinely goes on forever in all
directions. Even despite this, it is still able to expand – the distances between objects can
still increase regardless of whether the total Universe is infinite in extent or not. (You might
consider the integers; these form an infinite set, but you can still multiply each by two to
get a new infinite set where the separation between the numbers is twice as large.)
Nevertheless, this description is only a model, and we have no way of discovering
whether the actual Universe does indeed go on forever. Cosmologists often talk about a
different concept to the entire, possibly infinite, Universe, namely the observable Uni-
verse. This corresponds to the portion of the Universe we can actually see, and is limited
by the finite speed of light. As the Universe gets older the observable Universe becomes
larger and larger, by a combination of two effects. Firstly, the Universe is expanding, and
secondly light has had longer to travel across the Universe. In practice, our knowledge
of the Universe is restricted to this portion and we have no way of telling whether it does
indeed continue into the infinite distance as required by the cosmological principle. It may
be, for example, that the Universe becomes highly irregular on extremely large scales, and
there are some theoretical models which predict that this might happen. Another possibility
is discussed in Advanced Topic 1.3.
4.5 Where did the Big Bang happen?
A common question is ‘where did the Big Bang happen?’, suggesting perhaps that one
could point in a specific direction and say ‘That way!’. After all, in a conventional explosion
that is a perfectly reasonable question to ask, as all the material flies outwards from the

ignition point. Unfortunately, for the Big Bang things aren’t so simple, and in a sense the
answer is ‘everywhere and nowhere’.
First of all, remember that our entire foundation is the cosmological principle, telling
us that no point in the Universe is special. If there were a particular point where the ‘Bang’
happened, that would clearly be a special point and violate the cosmological principle.
Rather, space and time themselves were created at the instant of the Big Bang (unlike a
conventional explosion where the material flies through pre-existing space). If we take any
point in the present Universe and trace back its history, it would start out at the explosion
point, and in that sense the Big Bang happened everywhere in space.
In another sense, the location of the Big Bang is nowhere, because space itself is evolv-
ing and expanding, and it has changed since the Big Bang took place. Imagine the Uni-
verse as an expanding sphere; at any instant ‘space’ is the surface of the sphere, which is
becoming bigger with time (again I’m thinking of a two-dimensional analogy to our real
three-dimensional space). The place where the ‘Bang’ happened is at the centre of the
sphere, but that’s no longer part of the space, the surface of the sphere, in which we live.
In particular, being constrained to the surface of the sphere means we are unable to ‘point’
to the place where the explosion is supposed to have happened. However, all the points in
our current space were once at the centre of the sphere, when the Big Bang took place.
4.6 Three values of k
Because there are only three distinct possibilities for the geometry, many people indicate
this explicitly by scaling their variables so that k takes on one of only three possible values,
namely k = −1, 0 or +1, corresponding to the open, flat and closed cases respectively.
This can be achieved by rescaling the scale factor by multiplying it by a fixed constant,
namely a
ˆ = a/ |k|, in cases where k is non-zero. This leaves H = a/a
̇ unchanged, and
removes the k from the final term in Equation (4.1), which now reads
a
ˆ ̇
a
ˆ
2
=
1
8πG
ρ ± 2 ,
3
a
ˆ
(4.2)
with ‘−’ for positive k, ‘+’ for negative k and the last term absent if k = 0.
If this rescaling is used, then (except in the k = 0 case) one loses the freedom to set
a
ˆ = 1 at the present time, as will be used in Section 5.3. In effect, what one is doing is
choosing to measure comoving distances in units of the so-called curvature scale, which is
the scale on which the effects of curved space must be included, rather than in astronomical
units such as megaparsecs.
In this book I will not rescale k to one of these three discrete values, though most of
the time I will be specializing to k = 0 anyway.