Chapter 2
Describing the evolution of the
universe - basics of space-time
In the last lecture we set up some of the basics in the universe. From this we now know.
• The universe is both isotropic and homogeneous
• The universe is expanding, and its current rate of expansion is 72 km s−1 Mpc−1
We now want to know how we can describe this expansion in more general terms. In
particular, in the first lecture we looked at describing the universe in terms of observational
parameters, such as distance and velocity. We now want to know how it evolves on a more
theoretical level, based on force, mass and energy.
The first thing to consider is that the only force of consequence in the evolution of the
universe (at this stage of its life) is gravity. The strong and weak nuclear forces act only
at short distances, and despite the fact that the electromagnetic force is 1036 stronger than
gravity on a pair of individual protons, the universe, on large scales is neutral.
In order to understand gravity, we need to understand, at least to first order something
about general relativity.
2.1
Some basic relativity
Before we can set about understanding the universe we need to understand some of the basics
of space time, so that we can set up the models that we need to describe the evolution of
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the universe as a whole. This course will, in general, not worry about relativistic effects,
but we do need to understand at least some of the basics of general relativity, and to set
up the necessary maths to enable us to describe the universe. This is not the most exciting
section of the course, but it is necessary to understand it before we can move on.
2.1.1
Faster than the speed of light
Firstly, before moving on to some basic of general relativity, a note on special relativity.
Galaxies can be moving apart from one another at faster than the speed of light. In the
last lecture we saw that the Hubble time t0 is defined as
1
H0
(2.1)
dH = ct0
(2.2)
t0 =
and a corresponding Hubble distance
Essentially (and this is simplified) it means that as
v = c = H0 d H
(2.3)
The velocity of these galaxies is the speed of light, anything beyond them is moving
at v > c. This does not violate special relativity since it is due to the space between the
galaxies expanding. Two galaxies, moving apart from one another at faster than the speed
of light are not causally connected.
2.1.2
The equivalence principle
This statement is central to how Einstein created General Relativity. At its most simple,
it say that the force due to gravity,
Fg =
GMg mg
r2
(2.4)
where mg is the gravitational mass, is equal to the force due to uniform acceleration
F = mi a
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(2.5)
where mi is the inertial mass.
The equivalence principle states simply that
mi = mg
(2.6)
That is to say that how strongly an object is pulled on my gravity is equal to its resistance
to acceleration by any other force – it is not a natural assumption to make.
This statement essentially means that it is impossible to tell between a uniformly accelerating frame, and one in a gravitational field. The classical thought experiment to prove
this is that if one were in a spacecraft uniformly accelerating then it should be impossible
for an astronaut within that craft to perform a local experiment (i.e. one inside the ship)
to determine if he or she was uniformly accelerating, or in a gravitational field. The natural
experiment would be to shine a light at the opposite wall. Since light travels in a straight
line, in the accelerating case the light would strike the wall at a lower point than it was
started from, due to the motion of the craft. However, in the Newtonian view at least,
the light, in the gravitational frame would still hit the same point on the wall. Since this
would provide a means of determining the difference between acceleration and gravity, it
would disobey the equivalence principle. Rather than take the easy option, and abandon
this, Einstein to the drastic, but apparently correct, view that in fact the light would still
be bent down by the gravity, because of the curvature of space-time itself.
Under this model, all masses bend space-time, and light follows geodesic lines through
this.
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Figure 2.1: The basic equivalence principle. There are no local experiments that observers
in either a uniformly accelerating frame, or a frame sitting in a constant gravitational field
can do which will determine which frame they are in. As a consequence light must be bent
by the presence of the gravitational field (or more generally, by mass energy).
This is well put in Barbara Ryden’s introductory cosmology book:
Newton
Mass tells gravity how to exert a force (F = −GM m/r2 )
Force tells mass how to accelerate (F = ma)
Einstein
Mass–energy tells space time how to curve
Curved space time tells mass energy how to move
2.2
Curvature
The effect that mass can have on the universe, and its curvature from this, can at crudely be
understood in terms of some simple geometries. Flat, is the standard Euclidean geometry
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Figure 2.2: Flat (left), negative (centre) and positive (right) curved spaces.
which we are familiar with. A simple way of understanding this is to imagine a triangle
drawn on a surface. In a flat systems the angles add up to π e.g. in Figure 3.1.
α+β+γ =π
(2.7)
In a positively curved surface - a spherical one
α + β + γ = π + A/R2
(2.8)
Where A is the area of the triangle, and R the radius of the sphere. In a negatively
curved space, which is hyperbolic
α + β + γ = π − A/R2
(2.9)
Geometrically many different possibilities exist, however because the universe is homogeneous and istotropic (yes we come back to this again). There are only three options, the
universe has constant positive curvature, constant negative curvature, or is flat.
2.3
The Friedman equation
We are interested in how a particle behaves under the influence of gravity in an expanding
universe. This is essentially described by how its potential and kinetic energy evolve. Note,
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Figure 2.3: A uniform density sphere contracting or expanding under its own gravity.
Figure 2.4: As co-moving co-ordinate system, which expands with the universe.
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that because of the cosmological principle we can use any particle we like, since there is no
difference between them.
It can be shown (and probably has been in earlier dynamics lectures) that in a sphere
mass outside the observer does not contribute to the gravitational force. This means that
we are only concerned about matter inside the radius of the particle. Assuming uniform
density (again an assumption we can make from the cosmological principle), then its mass
is simply given by, M = 4/3πρr3 , so
F =−
GM m
4πGρmr
=−
2
r
3
(2.10)
and the gravitational potential energy is given by
V = −GM m/r = −
4πGρmr2
3
(2.11)
Kinetic energy is just just
1
T = mṙ2
2
(2.12)
As energy is conserved the total energy is just
1
4πGρmr2
U = T + V = mṙ2 −
2
3
(2.13)
Now we can make a clever switch of co-ordinates, which makes understanding this simple.
These are called co-moving co-ordinates. This is just (thinking back to lecture 1)
r(t) = a(t)x
(2.14)
Where r(t) is the radius co-ordinate and x is the co-moving co-ordinate, (sometimes you
will also see this written as rs ) so that ẋ = 0. Substituting this in above gives:
1
4π
U = mȧ2 x2 −
Gρa2 x2 m
2
3
(2.15)
We can now multiply each side of this equation by 2/ma2 x2 to get
! "2
ȧ
a
=
8πG
2U
ρ+
3
mx2 a2
(2.16)
Commonly a substitution kc2 = −2U/mx2 is made, which reduces the equation to the
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standard Newtonian Freedman equation:
! "2
ȧ
a
=
8πG
kc2
ρ − 2 = H2
3
a
(2.17)
Note that the form (ȧ/a)2 is that of the Hubble parameter (this is not the Hubble
constant H0 , which we examined in the previous lecture as the current value of ȧ/a, but is
a more general Hubble parameter (H), describing this ratio at different times.
In this case, k is called the curvature constant, and defines one of three possibilities
for the curvature of the universe. Since the universe is homogeneous (yes, we come back
to this yet again) the curvature has to be constant, and cannot vary across the universe.
Therefore there are essentially only three possibilities for the curvature; uniform positive,
uniform negative, and flat.
We can examine the consequences of each in turn. Let us start by imagining a expanding
universe, where ȧ > 0. In cases where k is negative the right had side of the equation is
always positive, and so the universe will continue to expand forever. If k is positive then
the equation begins positive, but with time a(t) reaches a maximum.
amax = −
3kc2
GMS
=
Ux
8πGρ
(2.18)
At this point the expansion will stop and the second derivative of a will be negative,
and so the universe will contract.
Finally we can consider the case where k = 0. Since the universe is shaped by its mass
energy this only happens at a certain mass/energy density.
8πGρ
ȧ
= H2 =
a
3
(2.19)
This can be re-arranged to give a critical density,
ρcrit =
3H 2
8πG
If the universe is at this critical density then k = 0 and it will expand forever.
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(2.20)
k
k>0
k<0
k=0
Geometry
Spherical
Hyperbolic
Flat
Type
Closed
Open
Flat
Fate
Collapse
Expansion
Expansion
Table 2.1: The different types of universe depending on the value of the curvature constant.
Figure 2.5: The evolution of the universe with time for different values of the curvature.
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