PX 389: Cosmology
Andrew J. Levan
A.J.Levan@warwick.ac.uk
May 4, 2016
Contents
1 Overview and key observations
1.1 Outline . . . . . . . . . . . . .
1.2 The night sky is dark . . . . . .
1.3 Redshift and Hubble Law . . .
1.4 Scale factor → redshift . . . . .
1.5 The cosmological principle . . .
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space-time
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fluid and acceleration equations
The Fluid Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Acceleration Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and finally ........ equations of state . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Describing the evolution of the universe
2.1 Some basic relativity . . . . . . . . . . .
2.1.1 Faster than the speed of light . .
2.1.2 The equivalence principle . . . .
2.2 Curvature . . . . . . . . . . . . . . . . .
2.3 The Friedman equation . . . . . . . . .
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basics of
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3 Measuring Curvature
3.1 The relativistic Freidman equation . . . . .
3.2 Critical density . . . . . . . . . . . . . . . .
3.2.1 Mass density – stars . . . . . . . . .
3.2.2 Dark Matter . . . . . . . . . . . . .
3.2.3 The Cosmic Microwave Background
3.2.4 The flat universe . . . . . . . . . . .
4 The
4.1
4.2
4.3
5 Model Universes
5.1 Matter Dominated . .
5.2 Radiation Dominated
5.3 Cosmological Constant
5.4 Curvature Dominated
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Dominated
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6 Multi-component universes
6.1 Different densities at different times . . . . . . . . . . . . . . .
6.2 Solving the Friedman equation for a multi-component Universe
6.3 Observationally: Measuring expansion and acceleration . . . . .
6.3.1 The Hubble parameter - H . . . . . . . . . . . . . . . .
6.3.2 The deceleration parameter q . . . . . . . . . . . . . . .
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7 Recent Observations of the Universe at large
7.1 Testing the Benchmark model . . . . . . . . . . . . . . . . . .
7.2 Future tests of benchmark cosmologies . . . . . . . . . . . . .
7.2.1 Large scale supernova searches . . . . . . . . . . . . .
7.2.2 High resolution Cosmic Microwave Background maps .
7.2.3 Baryon Acoustic Oscillations . . . . . . . . . . . . . .
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8 Dark Matter
8.0.4 Dark matter temperature . . . .
8.1 MACHOS . . . . . . . . . . . . . . . . .
8.2 WIMPS . . . . . . . . . . . . . . . . . .
8.2.1 Supersymmetry . . . . . . . . . .
8.2.2 Axions . . . . . . . . . . . . . . .
8.2.3 Searches for particle dark matter
8.3 Modified Newtonian Dynamics . . . . .
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9 Structure in the universe
9.1 The Jeans Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Formation in a radiation or matter dominated Universe . . . . . . . . . . . . . .
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10 Inflation
10.1 The horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 The flatness problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Acceleration expansion at early times – Inflation . . . . . . . . . . . . . . . . . .
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11 Baryogenesis and Nucleosynthesis
11.1 Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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12 The
12.1
12.2
12.3
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Big Bang
The Planck scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vacuum fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Branes and the Epkyrotic Universe . . . . . . . . . . . . . . . . . . . . . . . . . .
A The universe on a side of A4
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B Problems and Answers
B.1 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Problem Set with Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Health Warning
I have put this together with the aim of helping to clarify key points of the course, and to provide
a repository for the aspects of the cosmology which are directly covered. The idea is that this
will provide a basis for further reading, directing you to the areas which will aid your ultimate
performance in the exam. In the past I have primarily used this as a aide memoir for myself,
but am posting it online at the request of several people taking the course. It is inevitable that
some typos and errors will have slipped through (this has not had the level of editing which any
of your textbooks will have, not is it intended to be your lecture notes, which should be largely
free of errors). If, at any time the contents of this appear to contract what you read in the
textbooks (below) it is likely that these notes are in error. If you are worried about any possible
errors, or on careful examination cannot understand any differences, then please see/email me.
Also, please note that these notes are an overarching set of material that may be covered in
the course. The exact flavour of the course may vary from year to year somewhat, and so the
content of the lectures (including what is said and not written) should be viewed as the final
arbiter of what is required (and examinable) for the course.
Textbooks
The lectures are intended to be contained, and from 2014 onwards are also available via
Echo360. However, should you wish to consult textbooks then I can recommend the following.
B. Carrol & Ostlie – Introduction to Modern Astrophysics
This is not a formal textbook on cosmology but is the recommended text for the second year
stars and galaxies course and also contains some useful information for this course
M. Roos – Introduction to cosmology
B. Ryden – Introduction to cosmology
Andrew Levan - Vlissingen, the Netherlands (December 2014).
4
Chapter 1
Overview and key observations
Cosmology is principally concerned with the origin and evolution of the universe as a whole.
It is a relatively new subject, at least in the way which it is presented now, as a quantative
science. Nonetheless it roots reach back well into the past, since its fundamental objective –
understanding where the universe in which we live came from, and where it is going – have been
central questions since the dawn of civilization.
The earliest cosmologies came from the Ancient Greeks, who placed the earth at the centre
of the universe, with the sun, moon and other planets revolving around it in a series of different
“circles” (Ptolemy’s epicycles). Although contrived this model was understandable since the
conceptual leap from viewing stars merely as points on the sky, to viewing them as other suns,
perhaps complete with their own solar systems, is a huge one. Copernicus was the first to
suggest that the sun, and not the earth lay at the centre of the universe, but this was not
broadly accepted until some time after, Newton recast our vision of the universe by providing a
mathematical underpinning to the Copernican view.
The discovery of external galaxies again questioned our location in the universe, although,
the natural assumption then became that our own galaxy much lie the centre. Only the the
1950’s, when Baade finally showed that the Milky Way was really rather like other galaxies did
arguments assigning us a special place in the universe finally give way, allowing the picture of
modern cosmology, which we will study during this course, to emerge.
1.1
Outline
The questions and aims which will be addressed over the timescale of this course will be:
• Key observations - the birth of cosmology
• Describing the universe and space time
• Model universes - describing their evolution
• Measuring the key properties of our Universe
• The formation of structure
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• The early universe
Although much of this course will be describing the models which explain the above questions
there will also be a focus on the observations which provide evidence for these; why are they
important? what are their flaws? and what future directions will help us to better understand
cosmology?
At the end of the course you should have a quantative understanding of the current status
of cosmology, focussed on the breakthroughs in the 20th century which enable us to describe its
evolution, but sufficiently up to date to take into account recent research highlights, including
those which have happened as recently as 2007.
Modern cosmology is built on several key observations, and it important to understand these,
and their origin before moving on to more complex descriptions of the universe.
1.2
The night sky is dark
This, remarkably simple observation actually provides an important cosmological lesson, and
was first studied by Olbers in (1826). It essentially takes the Newtonian view, that the universe
is an infinite space occupied by an infinite number of ”other suns”, and demonstrates that,
despite its simplicity the fact that the night sky is dark has profound implications for the nature
of the universe.
Imagine that the universe consists of an infinite number of stars. The flux (F ) from each
star, as received at the earth is simply
L
,
(1.1)
4πr2
where L is the luminosity of the star. These stars are arranged in some configuration about
the sun (which we integrate from since it is our viewpoint, not because it occupies and special
place in the universe). Suppose we split up the sky into a series of spherical shells. The light
from each steradian of the shell, nr2 dr is simply,
F =
L
nL
× nr2 dr =
dr
2
4πr
4π
and therefore the brightness of the night sky per unit area is
dJ =
Z ∞
(1.2)
nL ∞
dr = ∞
(1.3)
4π 0
0
Thus, if the universe did consist of an infinite number of stars, then the sky would not be
dark, but blindingly bright. This is known as Olbers paradox. In fact, as we shall see later the
solution to this is simple.
J=
Z
dJ =
The universe is a finite age, Light from sources at distance greater than the speed of light time
the current age cannot yet have reached us.
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Figure 1.1: Olbers paradox. In a universe filled uniformly with stars or galaxies the sum of the
light from each star, integrated over the universe is infinite. The solution to this problem is that
the age of the universe is finite. In this systems objects over our horizon are invisible, since their
light has not had time to reach us since the big bang.
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1.3
Redshift and Hubble Law
Spectrographs, which spread the light from a given source over a range of wavelengths allow
precise measurements of the radial velocities of stars and galaxies from the us. This is possible
because of discreet spectral features in both stars and galaxies, such as absorption lines (in
stellar atmospheres) and emission lines (from excited gas). The relative location of these lines,
with respect to that measured in the laboratory is known as the redshift. In simple observer
terms
(1 + z) =
λobs
λrest
(1.4)
Locally this enables us to, for example, measure the orbits of stars around the Milky Way1 .
n a larger scale it also allows us to measure the velocities of extragalactic systems. As it happens
most galaxies (aside from our nearest neighbours) are moving away from us.
Edwin Hubble realised in 1929 that this isn’t simply a random motion, but that the redshift
is correlated with its distance, that is that more distant galaxies are moving away from us more
rapidly. He coined this into the now famous Hubble’s Law
v = H0 d
(1.5)
Where H0 is a number known as the Hubble Constant, and is currently constrained, principally by observations of Cephied variables to be H0 = 72 ± 2 km s−1 Mpc−1 . Thus a galaxy
moving away from us at 500 km s−1 would be a distance of ∼ 7 Mpc away.
In practice the determination of H0 has been one of the major challenges in astronomy since
it was first discovered by Hubble in 1929. Indeed, the Hubble Space Telescope is so named
because one of its chief goals was to improve the value of the Hubble constant. The principle
difficulty in this determination is not in measuring redshifts, which are straightforward, but in
measuring distances. All distance measurements are built from the distance ladder, but the
direct measurements which underpin this are only possible in our own galaxy (and even then
only in stars close to the sun). The value of the Hubble constant is now determined principally
from observations of Cepheid variable stars (whose period correlates with luminosity), allowing
a means of measuring the distances of galaxies of to ∼ 30 Mpc, and therefore placing much
better constraints on H0 than was possible in the era of Hubble.
Hubbles law states that there is a linear relationship between velocity and distance, with a
proportionality constant H0 = 72 km−1 Mpc−1 . This law holds out to z ∼ 0.2
One consequence of Hubble’s law is that it seems to refute the claim that we are not at a
special place in the universe. Surely if every other galaxy is moving away from us then we do
occupy some prime position. This may be true if the observed redshift was purely the result of
relative motions in a fixed space, however, the truth is that the space itself is expanding. The
common analogy is that of blowing up a balloon. Imagine sticking pictures on the side of a
1
Sometimes rather than plot z the velocity cz is plotted instead, note that this can be larger than c!
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Figure 1.2: A triangle of galaxies, expanding uniformly.
balloon and then inflating it. The pictures remain the same size, but each of them moves away
from each other one, much as each other galaxy appears to move away from us.
This can be shown with slightly more mathematical rigour.
Suppose the triangle shown in Figure 2.2 expands uniformly. In this case the relative shape
of the triangle is preserved in time. i.e.
r12 (t) = a(t)r12 (t0 )
r23 (t) = a(t)r23 (t0 )
r31 (t) = a(t)r31 (t0 )
(1.6)
Where a(t) is the so called scale factor which will be important later.
Imagine an observer in galaxy one. The relative velocities observed for galaxies 2 and 3 are
dr12
= ȧr12 (t0 ) =
dt
dr31
v31 (t) =
= ȧr31 (t0 ) =
dt
v12 (t) =
ȧ
r12 (t)
a
ȧ
r31 (t)
a
(1.7)
Where we have made the obvious substitution that r12 (t0 ) = r12 (t)/a(t).
The form of this equation is clearly that of Hubble’s law. In this case value of H0 can be
equated with ȧ/a, at the time t = 0 (i.e. the current time). More generally the value H is
known as the Hubble parameter, and is variable with time.
The Hubble parameter is simply the rate of change of the scale factor of the universe
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An important point here is that the expansion of the universe implies that at some point the
two galaxies were in contact with one another. Using this we can calculate the time at which the
two galaxies were in contact. Since the time the galaxies were in contact they have expanded to
r21 = v21 t0 , therefore:
t0 =
r12
r12
=
= H0−1
v12
H0 r12
(1.8)
This is known as the Hubble time, and for H0 = 72 km s−1 Mpc− 1 then t0 , also known as
tH = 14.0 ± 1.4 Gyr. This is the natural explanation of Olbers paradox.
Note that this also gives a natural distance for the universe, the distance which light can
travel in a Hubble time, simply c/H0 = 4300 ± 400 Mpc.
1.4
Scale factor → redshift
The above shows how we can write the evolution of an expanding Universe based on the change
of the scale factor. This in practice actually directly relates to the more measurable parameter,
the redshift, or velocity of the object. Because of this, if we can measure the redshift of a galaxy
we can directly infer the relative size of the Universe at the point that light was emitted.
Consider two adjacent galaxies which are moving apart from one another in the Hubble flow,
with a relative velocity dv. For this galaxy we can write Hubble Law as
ȧ
dr
a
= dv/c , we can substitute in above for dv
dv = Hdr =
now, from the doppler effect dλ/λemit
ȧ
da
dλ
ȧ dr
= dt =
=
λemit
a c
a
a
(1.9)
(1.10)
This can be integrated
Z
dλ
=
λemit
Z
da
a
(1.11)
so ln(λ) = ln(a) + c, or λ ∝ a
We now can return to our definition of redshift
λobs
aobs
=
(1.12)
λemit
aemit
since by definition aobs = 1 (i.e. the Universe is defined to have a scale factor of 1 at the
present time), the simplifies to
1+z =
1
(1.13)
a
In other words. If we observe a galaxy at z = 1 we observe it when the Universe was exactly
half of its present size. If we look at the Cosmic Microwave background at a redshift of z = 1100
we see the Universe when it was ∼ 1/1110th of its present size, and 109 times denser then it is
today.
1+z =
10
1.5
The cosmological principle
The discovery by Baade in the 1950’s that the Milky Way was a galaxy very much like any other
led to the acceptance of the view that we do not occupy a priveldged space in the universe. This
has in turn led to a stronger statement, known as the cosmological principle. This states that.
The universe is homogeneous – no preferred location
The universe is isotropic – no preferred direction
When looking around us day to day this is patently ridiculous. The world we inhabit is
clearly neither homogeneous or isotropic, there are clearly different (and distinct) locations, and
directions which can easily be determined. Indeed, not only does this not hold true on the
scale of humans it does not hold true on the scale of normally studied astronomical objects.
The solar system is clearly not the same on average – the earth is not the same as Jupiter, or
interplanetary space. Even the galaxy, or the set of galaxies which form the local group are not
especially homogeneous, and have obvious preferred directions (e.g. towards the Milky Way or
M31).
However, while this concept may seem odd, when viewed on the largest scales (which are,
of course the scales we are most interested in cosmology) the universe is both homogeneous and
isotropic – the relevant scales are > 200 Mpc.
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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 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
12
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 dH
(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,
GMg mg
r2
where mg is the gravitational mass, is equal to the force due to uniform acceleration
Fg =
F = mi a
(2.4)
(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
13
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).
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.
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 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.
14
Figure 2.2: Flat (left), negative (centre) and positive (right) curved spaces.
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, 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 =−
4πGρmr
GM m
=−
2
r
3
15
(2.10)
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.
16
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
As energy is conserved the total energy is just
(2.12)
1
4πGρmr2
U = T + V = mṙ2 −
(2.13)
2
3
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 co-moving co-ordinate and x is the physical (non-moving co-ordinate) 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
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.
17
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.
amax = −
GMS
3kc2
=
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.
ȧ
8πGρ
= H2 =
a
3
This can be re-arranged to give a critical density,
3H 2
8πG
If the universe is at this critical density then k = 0 and it will expand forever.
ρcrit =
18
(2.19)
(2.20)
Chapter 3
Measuring Curvature
In the last lecture we saw how mass/energy causes curvature in the universe, and set up (from
a Newtonian viewpoint), the Friedman equation, which describes the evolution of the size of
the universe. We also saw broadly how the curvature constant can impact the behaviour of the
universe. In this lecture we will take this a step further, looking at how we measure curvature,
and what observations today imply about the curvature of the universe.
3.1
The relativistic Freidman equation
The equation we derived in the last lecture is known as the Newtonian Freidman equation,
because it was derived from basic Newtonian physics. There also exists a fully relativistic
version (which we shall not derive). The form of this is:
2
ȧ
a
=
8πG
κc2
Λ
−
+ = H2
2
2
2
3c
3
R0 a
(3.1)
The three changes here are due to the replacement of the mass density with the energy
density, since
E 2 = m2 c2 + p2 c2
(3.2)
so that photons can in fact contribution to the curvature of the universe since they carry
momentum and energy. The second is to replace k with κ, and subsequently re-scale using R0
(currently observed scale factor). This means that rather than worry about a range of positive
or negative values for k, it is either 1,0,-1, when normalised with the current scale factor.
The final addition is the term Λ/3, this term was added by Einstein when it became clear that
the “original” Friedman equation required the universe to be either expanding or contracting.
As the viewpoint at the time favoured a steady state Universe he introduce the Λ term, which
he called the cosmological constant. Although this was thought to be his biggest blunder after
the discovery of universal expansion in recent years it has become clear that the universe does
have cosmological constant (which will be discussed in more detail later).
19
3.2
Critical density
Since the Universe is curved by the impact of its mass energy, it will be flat if it achieves a
critical density, which is derived by setting k = 0 in the non-relativistic Friedman equation.
3H 2
8πG
Commonly this critical density is re-written as
ρcrit =
Ω=
ρuniverse
ρcrit
(3.3)
(3.4)
and the contribution of individual components (e.g. wM ) is indicated separately. We can of
course calculate a numerical value of ρcrit now by substituting in the Hubble constant
ρcrit,0 =
3H02
= 1.4 × 1011 M M pc−3 = 9.2 × 10−27 kgm−3
8πG
(3.5)
Simply using E = mc2 this can also be written as an energy density.
crit,0 =
3c2 H02
= 8.3 × 10−10 Jm−3
8πG
(3.6)
So, an important component of the curvature of the universe is its mass/energy density. A
simple (at least in principle) way of measuring this is to “count-up” the obvious sources of mass
energy within the universe.
3.2.1
Mass density – stars
A first idea is simply to count up the mass. This isn’t actually as easy as it looks. We cannot
trivially measure the mass of each star in the universe, and add it up. What is much simpler is
to measure the integrated light, and then turn this into a mass. To do this, we need to define
something call a mass to light ratio. We can easily define this based on the sun, which has a
mass of one solar mass (M = 2 × 1030 kg) and a luminosity of one solar luminosity (L ), which
in the B-band equates to 4.7 × 1025 Watts s−1 . We can now measure the luminosity of galaxies
within several Mpc of our own and average to get the mean B-band luminosity
j = 1.2 × 108 L,B M pc−3
(3.7)
If we assume that all stars have the same mass to light ratio we can trivially convert this
into a mass. Of course, in truth stars do not have a uniform mass to light ratio. Low mass stars
put out much less energy than high mass stars, in general this can be approximated by a power
law
dL
∝ M3
dM
20
(3.8)
However, low mass stars greatly outnumber the higher mass stars, their initial mass function
is
dN
∝ M −2.35
(3.9)
dM
In practice this means that much of the light comes from high mass stars, while most of the
mass lies in the lower mass objects. Nonetheless the assumption of a mass to light ratio similar
to the sun is not a bad one, and hence assuming a mass to light ratio of one gives a mass density
of 1.2 × 108 M M pc−3 . This provides a fraction of the critical density of ∼ 0.0001. In practice
a slightly more complete estimate is that the mass of stars locally is ∼ 5 × 108 M , and thus
5 × 108
∼ 0.004ρcrit
(3.10)
1.4 × 1011
It is clear that stars alone do not provide the necessary mass-energy to close the universe.
Of course, not all the mass in the universe lies in stars, we know that there exists dust, while
there are also stellar remnants, such as black holes and neutron stars which can increase this
number, but only by a factor of a few, not sufficient to close the universe. The total estimated
density in this form of matter (known as the baryon density) is
ρstars =
Ωb = 0.04
(3.11)
an order of magnitude larger than our estimate of the stellar density, but not enough to
flatten the universe. If this were the true picture of the universe then it would be open, and
expand forever. However, we have not yet explored all the possible sources of mass-energy within
the universe.
The orbital velocity of an object around a mass is simply given by
a=
v2
GM (R)
=
R
R2
(3.12)
i.e.
s
v=
GM (R)
R
(3.13)
Therefore, the velocity in orbit around a point mass (or any object whose entire mass is
enclosed within radius R) should fall of as R−1/2 . Therefore, this is what we might expect to
observe for stars far out from the cores of galaxies, where there is little mass. A typical galaxy
light profile can be described as an exponential disk, where the intensity of light varies as
I(R) = I(0) exp
−R
,
RS
(3.14)
where RS is the scale length of the disc, and for typical galaxies is a few kpc. Since the light
falls off exponentially there is little light (and from our arguments above, little mass) outside
this point.
21
3.2.2
Dark Matter
As luminous matter apparently provides very little of the critical density of the universe we are
naturally led to the question of whether any other sources of matter exist. Indeed, evidence
for some form of dark matter can be found in the rotation curve of the Milky Way, and other
galaxies. The rotation curve of a given galaxy should be described simply by its orbital rotation
velocity
s
v=
GM
R
(3.15)
Where M is the mass of the galaxy and R the radius from its centre. If the galaxy were
dominated by starlight then beyond the visible stars M would be constant and the velocity
should fall as R−0.5 . However, in the majority of galaxies the rotation curves are flat out to
large radii indicating that the mass is continuing to increase with increasing R. This component
is known as dark matter. There are various possible explanations for the origin of dark matter,
and we will return to these later in the course. From the perspective of measuring curvature
the important factor is that dark matter is the dominant source of matter in the universe, but
nonetheless only provides around 25% of the critical density. Not sufficient to result in a spatially
flat universe.
3.2.3
The Cosmic Microwave Background
It turns out that possibly the best means of measuring the mass density of the universe is
the Cosmic Microwave Background. This is actually a very useful tool for several branches of
cosmology and we will come back to it later in the course. Here we investigate briefly how it
places limits on the curvature.
The Cosmic Microwave Background (CMB) is the imprint of recombination of electrons
and protons which occurred roughly 300,000 years after the big bang. The image essentially
contains a measure of the structure in the universe at that time, and indicates that the universe
was highly uniform. The scale of the inhomogenities in the CMB temperature map is only
< δT T >∼ 1 × 10−5 ). None-the-less there does exist structure within the CMB, and this
structure can place limits on the curvature of the universe.
The size of these imhomogenities is set by time which they have had to grow. Because of the
age of the universe at the time of CMB formation this sets a rather stringent limit on this size
of vp × t, where vp is the speed of sound in the plasma and t is simply the age of the universe at
CMB formation. vp is a significant fraction of the speed of light (∼ 0.6c), and implies that the
largest structures in the CMB can only have reached physical sizes of ∼ 200, 000ly. Thus, by
measuring the size of the largest angular structures in the CMB maps we can obtain a physical
measure of the angular size distance at the redshift of the CMB (z = 1100). This makes the
CMB anisotropies a standard ruler. The measured size of the CMB anisotropy can then be
compared to the expectations under different cosmological models. Since the light has travelled
through the entire universe it size is modified by the mass–energy of the universe. So, measuring
its apparent size, and comparing to the known size enables the total mass energy of the universe
to be measured.
22
Recent results from the Wilkinson Microwave Anisotropy Probe (WMAP), indicate that the
universe indeed does attain critical density with wtot = 1.02 ± 0.02.
3.2.4
The flat universe
When all the sources of mass energy within the universe are co-added it appears that the universe
does attain (within observational errors) the critical density. However, the visible light of the
universe is unable to explain this, since all the stars/dust/gas only create a few percent of the
critical density. Even allowing for a component of the universe to be made up of dark matter
∼ 25% of the total density is accounted for. The rest of the critical density is in some largely
unknown form, for which the term dark energy has been coined. This was first discovered from
supernova observations, and we will return to dark energy, and its impact on the evolution of
the universe in later lectures.
23
Chapter 4
The fluid and acceleration equations
We have now derived the fundamental equation for the evolution of the universe - the Friedman
equation - this, when solved allows us to compute the evolution of the universe under a myriad
of different physical models. However, we are still not in a position to solve it, largely since it
contains two unknowns the scale parameter (a(t)) and the density (ρ or ). We therefore need
to search for more fundamental laws which will allow us to solve the Friedman equation.
4.1
The Fluid Equation
The Friedman equation is basically a statement of energy conservation (total = potential +
kinetic = constant). Equally, the first law of thermodynamics is similar, linking the heat flow
in and out of a system (dQ) with a change in the internal energy (dE), the pressure p and the
change in volume dV .
dQ = dE + pdV
(4.1)
In a homogeneous universe dQ = 0 (it is an adiabatic expansion - adiabatic expansions do
not increase the entropy of the universe since dS = dQ/T .) Therefore
Ė + pV̇ = 0
(4.2)
The universe can essentially be considered an expanding sphere, with co-moving radius
r(t) = a(t)x, and therefore volume,
V (t) =
4π
a(t)3 x3
3
(4.3)
so
4π 3 2
V̇ =
x (3a ȧ) = V
3
ȧ
3
a
(4.4)
The internal energy is then
E(t) = V (t)(t)
24
(4.5)
The total rate of change of the internal energy of the universe is given by
Ė = V ˙ + V̇ = V
ȧ
˙ + 3 a
(4.6)
This can be written more simply, combining the above equations as
V
ȧ
ȧ
˙ + 3 + 3 P
a
a
=0
(4.7)
or even
ȧ
˙ + 3 ( + P ) = 0
a
(4.8)
and is known as the fluid equation.
4.2
The Acceleration Equation
The acceleration equation is a third equation, which can be derived from the fluid equation and
the Friedman equation. It tells us if the expansion of the universe is slowing down, or speeding
up.
The standard Friedman equation, multiplied by a2 takes the form
ȧ2 =
8πGa2 kc2 Λa2
− 2 +
3c2
3
R0
(4.9)
If we differentiate this we get
2ȧä =
8πG 2
2Λȧa
(a
˙ + 2aȧ) +
2
3c
3
(4.10)
Which, when divided by 2ȧa gives us
ä
4πG a
Λ
=
˙ + 2 +
2
a
3c
ȧ
3
(4.11)
ȧ
˙ + 3 ( + P ) = 0
a
(4.12)
But the fluid equation
Can be re-arranged to give
a
= −3( + P )
ȧ
Which can be substituted into equation 4.11 to give the acceleration equation
˙
ä
4πG
Λ
= − 2 ( + 3P ) +
a
3c
3
25
(4.13)
(4.14)
Note that any positive energy density contributes to a negative acceleration (the relative
velocity between two points is getting less). The pressure component is due to the thermal
motions of the particles which make up the universe. Normal baryonic matter exhibits a positive
pressure, and also contributes to the universe slowing down.
However, if the universe actually had a different pressure, say one which have
(4.15)
3
The it would in fact have a positive acceleration (a tension). This might sound odd, but it
is exactly what the cosmological constant provides (it has P = −).
P <
4.3
and finally ........ equations of state
We are now just one step away from having the mathematical machinery to describe (at least
at a basic level) the evolution of the universe. What we need now is an equation of state, which
enables us to solve our 3 equations (but only independent equations) in terms of only 2 unknowns,
rather than the current three (a(t), (t), P (t)). Thankfully this can be fairly straightforward.
Essentially we need to know a relation of the form
P = P ()
(4.16)
Generally in cosmology we write a simple form
P = w
(4.17)
where w is a dimensionless number. For example an ideal gas (which it turns out isn’t such
a bad approximation for the universe would have),
P =
ρ
kT
µ
(4.18)
Note that here k is the Boltzman constant, not the curvature constant.
Because of relativity we need to be thinking in terms of mass-energy, rather than just mass,
so that ∼ ρc2 , i.e. most of the mass energy comes from the mass of the particles, and not from
their motion (you can check this just by calculating mc2 for a proton, and comparing that with
its kinetic energy at a given velocity < c. Therefore
P =
kT
µc2
(4.19)
We can simplify this further by noting that for a non-relativistic gas the relation
D
3kT = µ v 2
E
(4.20)
Thus if we define an equation of state parameter, w
w=
2
v
3c2
26
<< 1
(4.21)
Then we can write the equation of state of a a non-relativistic gas (i.e. a matter dominated
universe) as
Pnonrel = wnonrel
(4.22)
In the relativistic case, which is for example where photons (or neutrinos) dominate the
universe the equation of state is
1
Prel = rel
3
(4.23)
i.e. w = 1/3.
In general cosmologists can then distinguish between different model universes simply by
differing values of the equation of state parameter. A matter dominated universe, to first order
has w = 0. Radiation dominated universes have w = 1/3, and mildly relativistic systems have
some value in between.
It is interesting to note that there is nothing to stop negative values of w, which give rise to
negative pressure (tension). For example any universe where
1
(4.24)
3
will provide an positive acceleration. Indeed the cosmological constant has w = −1, and is
thought to be responsible for the currently observed acceleration of the universe.
w<−
The last couple of lectures have been hard going, but this is the basic maths we need to
explain the universe. In the next few lectures we will apply these rules, and try to understand
how the universe at large evolves.
27
Chapter 5
Model Universes
5.1
Matter Dominated
For matter, the equation of state parameter w=0. Thus matter has no contribution to the
pressure. Therefore the fluid equation can be re-written as
3ȧ
3ȧ
( + P ) = ˙ + = 0
a
a
This can be rewritten, using some slight of hand as.
˙ +
(5.1)
1
1 d
(3a2 ȧ + a
˙ 3 ) = 3 (a3 ) = 0
(5.2)
3
a
a dt
In, other words, the energy density of a matter dominated universe, falls off as the cube of
its volume. When you think about it this really isn’t very surprising.
We can now simplify the problem somewhat. In the case where κ = 0, the form of the Friedman equations doesn’t change when they are multiplied by a constant, because in essence they
are normalised against the current scale factor. Therefore we are free to “scale” the Friedman
equation to make it easier to solve, the most obvious scaling being a0 = 1 (i.e. the universe
measured one universe across at the current time). In this case we can relate everything relative
to this time, so;
0
a3
Substituting this into the Friedman equation gives
=
(5.3)
8πG0 1
(5.4)
.
3c2 a
This can formally be separated, but is easiest to solve by substitution. We make a first guess
that the evolution of the universe is a power-law (as you may have already noticed, everything
in astronomy is a powerlaw). So we take the form a(t) ∝ tq . In this case the exponents on each
side of the equation are
ȧ2 =
28
LHS = t2q−2
RHS = t−q
(5.5)
The solution is when these two match. Namely when q = 2/3. So
a(t) =
5.2
t
t0
2/3
(5.6)
Radiation Dominated
We can perform essentially the same analysis above for a radiation dominate universe. In this
case w = 1/3 and the fluid equation becomes
3ȧ
4ȧ
( + P ) = ˙ + = 0
a
a
Which can be rewritten as for matter as
˙ +
1
1 d
(4a3 ȧ2 + a
˙ 4 ) = 4 (a4 ) = 0
4
a
a dt
(5.7)
(5.8)
So the density of a radiation dominated universe falls off as 1/a4 , faster than a matter
dominated universe. Since we know the universe contains both matter and radiation, the fact
that they behave differently with increasing scale factor immediately implies that at early times
in the universe the radiation component may be much more important than it is now. In any
case, we are currently concerned with a universe containing only one component. In this scenario
we can now solve the Friedman equation for a radiation dominated universe by assuming that
the evolution follows a powerlaw a ∝ tq , and substitute in as we did in the case of matter
domination.
LHS = t2q−2
RHS = t−2q
(5.9)
The solution to this being q = 1/2, so in a radiation dominated universe the scale factor
varies as
a(t) =
t
t0
29
1/2
(5.10)
5.3
Cosmological Constant Dominated
The case of a cosmological constant dominated universe is slightly different. In this case w = −1.
Since, P = − and ˙ = 0, the Fluid equation
3ȧ
( + P )
(5.11)
a
essentially reduces to zero. What this means is that the energy density from Λ is constant (this
is why it is called the cosmological constant). However, this is also slightly odd. Unlike a matter
or radiation dominated universe in which the energy density varies with scale factor as one might
expect, the actual contribution from the cosmological constant increases with the scale factor
since the energy density per unit volume is constant.
Considering a flat universe we have all the energy density provided by Λ so that
˙ +
3c2 H 2
8πG
(5.12)
8πGΛ 2
a
3c2
(5.13)
Λ,crit =
and the Friendman equation is simply
ȧ2 =
which can be re-written as
ȧ = H0 a
(5.14)
Which has the simple solution of
a(t) = exp H0 (t − t0 )
5.4
(5.15)
Curvature Dominated
This scenario sounds rather bizare; a completely empty universe consisting entirely of curvature.
It is also known as the Milne Universe. In a universe consisting only of curvature the Friedman
equation is
−κc2
ȧ2 =
(5.16)
R0
This can be solved for different values of κ. The simplest is of course κ = 0, there is nothing
(mathematically) wrong with this solution, but a static, flat and empty universe is not the most
exciting object to study.
If κ is positive then there is no solution allowed since once gets an imaginary ȧ. This makes
sense since it is not possible to fill the universe with sufficient mass energy to make it positively
curved, while also making it empty!
If κ is negative then the solution becomes
ȧ = ±
30
c
R0
(5.17)
and integrating gives
a(t) =
t
t0
(5.18)
For all of these scenarios we can also calculate the behaviour of the Hubble parameter
H = ȧ/a. This yields
Hmatter = 2/3t
(5.19)
Hradiation = 1/2t
(5.20)
HΛ = H0
(5.21)
This essentially covers all the bases for single component universes. The various different
possible evolutions of the universe in each of the cases described above is shown in Figure 5.1,
note that not all of them require a big bang (i.e the cosmological constant dominated universe is
infinitely old). Of course, the real universe is more complicated than these simple models, and
we consider that next.
31
Figure 5.1: The possible fate of the universe under the assumption that it consists entirely of
either mass, raditation, cosmological constant or curvature
32
Chapter 6
Multi-component universes
6.1
Different densities at different times
The differing dependencies on the energy density derived in the last lecture point to something
obvious, and yet highly important.
At different times in the Universe’s history, it is dominated by different components
When the Universe is very small, radiation, which is of very little importance today, is
dominant, simply because r ∝ 1/a4 , at intermediate times matter is dominant, while as the
Universe gets progressively bigger, the contribution from Λ, which doesn’t fall off as the Universe
is expands, becomes the driving force in its evolution.
This is interesting, but also problematic. It implies that the single component model Universe’s that we considered last lecture aren’t necessarily representative of the real Universe.
Again we know this isn’t surprising, its obvious from the world around us that the Universe contains matter and radiation. It also seems likely from various experiments that it also contains
Λ.
Today we believe that the relative contributions of radiation, matter and cosmological constant can be summed up as part of our benchmark cosmological model. In particular:
ΩM,0 = 0.23,
ΩR,0 = 1 × 10−4 ,
ΩΛ = 0.73
(6.1)
From this, and the relative dependencies on density with scale factor we can estimate the
points at which the Universe switches between different periods of dominance. Since there is a
flat Universe
ΩM + Ω R + Ω Λ = 1
(6.2)
In practice only two of these are ever important in our Universe (i.e. at the point at which
matter and Λ are equivalent the contribution from radiation is negligible. The same is true
33
of Λ when radiation and matter are contribute equally). Hence, we can calculate the point of
equality by
where ΩM,0 = M /c and ΩR,0
ΩR,0
ΩM,0
= 4 ,
3
a
a
= R /c . Hence,
ΩR,0
= aM R ≈ 10−4 .
ΩM,0
(6.3)
(6.4)
This is an epoch well before the formation of the Cosmic Microwave background, which
occurred at a ∼ 10−3 .
Similarly, we can calculate the epoch when matter and Λ were equal contributors to the
Universal density
ΩM,0
= ΩΛ,0 ,
a3
(6.5)
so
ΩM,0
ΩΛ,0
!1/3
= aM Λ ≈ 0.72.
(6.6)
The Universe can then be broken down into three distant eras, radiation dominance, matter
dominance and Λ dominance. In most cases then a single solution may be effective, although
there are switching points at which the scenario may be less straightforward.
6.2
Solving the Friedman equation for a multi-component Universe
If we want to solve the Friedman equation for these multiple component Universe’s the it is
useful to re-write the Friedman equation in terms of Ω, so that the classic Friedman equation,
2
ȧ
a
=
8πG
κc2
Λ
−
+ = H2
2
2
2
3c
3
R0 a
(6.7)
divided by H 2 looks like.
1=
and since
8πG
3c2 H 2
8πG
κc2
−
3c2 H 2
R02 a2 H 2
(6.8)
= 1/c , this can be written as
1=
κc2
− 2 2 2
c R0 a H
(6.9)
κc2
R02 a2 H 2
(6.10)
or
1−Ω=−
34
An important point here is that the RHS cannot change sign. κ = −1, 0, 1, and do so always,
for the entire life of the Universe. This leads to an important conclusion to start with.
Changing the dominant components of the Universal energy density does NOT change its
curvature
Now, we can evaluate this at the current time, when a0 = 1.
κ
H02
(Ω0 − 1)
=
c2
R02
(6.11)
Substituting this back into the Friedman equation gives
H2 =
8πG
H02
−
(Ω0 − 1)
3c2
a2
(6.12)
which, upon division by H02 is
H2
1 − Ω0
+
=
2
c,0
a2
H0
(6.13)
Separating this into separate energy components gives.
H2
(1 − Ω0 )
Ωm,0 Ωr,0
= 3 + 4 + ΩΛ +
2
a
a
a2
H0
(6.14)
This is a general form of the Friedman equation, that can be integrated to obtain the solution
for the dependence of a on t, for any arbitrary combination of energy densities (one can even
easily add more exotic equations of state if needed). Multiplying by a2 and taking the square
root gives an equation that can be integrated.
Z t
Z a
H0 dt =
0
0
da
h
Ωm,0
aa
+
Ωr,0
a2
+ ΩΛ a2 + (1 − Ω0 )
(6.15)
i1/2
The main problem with this is that in most cases it must be integrated numerically. Analytical solutions can only be obtained in cases where there are one or two terms. For example,
in the current Universe, where the contributions from matter and Λ are of the same order of
magnitude, we have ΩΛ,0 = 1 − Ωm,0 , and the analytical solution is

a
2
log 
H0 t = p
a
3 1 − Ωm,0
mΛ
s
3/2
+
1+
a
amΛ
3


(6.16)
where
amΛ =
Ωm,0
ΩΛ,0
35
!1/3
(6.17)
Feel free to try and re-produce this as an exercise, but in practice you won’t be asked to solve
the Friedman equation for more than a single component. Fortunately, this actually provides a
pretty good estimate most of the time. After all, for most of the time, either a/amΛ >> 1 or
a/amΛ << 1. In other words the solution reduces to that for either a matter dominated, or Λ
dominated Universe. i.e. the above equations reduce to
a(t) ≈ amΛ exp
q
(1 − Ωm,0 )H0 t
(6.18)
for a/amΛ >> 1, or
3q
Ωm,0 H0 t)2/3 .
(6.19)
2
So, while the behaviour at the transition is more complex, for most of the Universe’s history
it behaves rather simply. This is shown graphically in Figure 6.1, and in many ways is the
punchline of this lecture.
a(t) ≈ (
The Universe consists of many components, but for the majority of its history a single one is
dominant, and it expands or contracts based on that component alone.
6.3
Observationally: Measuring expansion and acceleration
6.3.1
The Hubble parameter - H
6.3.2
The deceleration parameter q
In a sense, what we want to look at with the deceleration parameter is for an explanation of the
expansion or contraction of the universe which is physics free, in other words, in an observable
which is not determined purely by the physics driving the evolution, which we may or may not
understand. In practice it is something that we can determine if we can measure a change in
the Hubble parameter as we move away from the local Universe.
To get a handle on this we can expand the form of a(t) around t = t0 . In other words,
a(t) = a(t0 ) +
da
dt
1 da2
(t − t0 ) +
2 d2 t
t=t0
!
(t − t0 )2 + .........
(6.20)
t=t0
dividing this by a(t0 ) gives us
1
a(t) = 1 + H0 (t − t0 ) + q0 H02 (t − t0 )2
2
(6.21)
where
äa
ä
q0 = − 2 = −
ȧ
aH 2
q0 is potentially an observable, but is also a link to the make up of the Universe.
36
(6.22)
Figure 6.1: The evolution of different components of the energy density with scale factor. As
can be seen Λ = m at a ∼ 0.7), while m = r at a much earlier epoch a ∼ 3 × 10−4 .
37
The acceleration equation is
4πG
ä
= − 2 ( + 3P )
a
3c
(6.23)
This can be re-arranged into a form for q0 by dividing by H 2 and multiplying by −1 to give
−ä
1 8πG
=
( + 3P )
2
aH
2 3c2 H 2
(6.24)
Note that the form 3c8πG
2 H 2 is equal to 1/c , where c is the critical density of the universe.
This allows us to re-write the acceleration equation in terms of the critical density parameter
Ω = w /c , where w is the energy density in a component of the universe with equation of state
parameter w. Summing over the possible components we get
−ä
1X 1
( + 3P )
= q0 =
2
aH
2 w c
(6.25)
Since the equation of state links P and , we can substitute in for P = w to get
−ä
1X
Ωw (1 + 3w)
= q0 =
2
aH
2 w
(6.26)
Now, summing over the possible values of w for matter (0), radiation (1/3) and cosmological
constant (−1) gives
1
q0 = Ωr,0 + Ωm,0 − ΩΛ,0
2
38
(6.27)
Chapter 7
Recent Observations of the Universe
at large
7.1
Testing the Benchmark model
Over the past few lectures we have mainly been concerned with developing a model for understanding how the Universe expands or contracts based on the different components which
contribute to its energy density at any given time. The natural question to ask is then if observations provide a handle to directly measure this, and if they support this model. We have seen in
an earlier lecture that the benchmark Universe can be written on the basis of the contributions
of the different components today
• Ω0 = 1
• Ωm,0 = 0.27
• ΩΛ,0 = 0.73
• Ωr,0 ∼ 10−4
We are now interested in measuring how the Universe evolves from this position, and if the
models we derived in the previous lectures are a good fit to the data. You can probably guess
that they are, unless I have been maliciously wasting your time, but its still relevant to consider
how we actually make those measurements.
In previous lectures we have seen that different components of the density of the Universe
have dominated at different times in the past. Radiation was dominant at early epochs, while
matter dominated for the next 10 billion years. The cosmological constant (or whatever it is that
is causing the Universe to accelerate) has only come to dominate in rather recent history. This
means that the scale factor evolution has not always been the same, it has changed according
to the dominant component of energy density in the Universe. We have also seen that we
can estimate the point at which the scale factor will change by determining the tipping fro
e.g. acceleration to deceleration. This final change happened at around z ∼ 0.7, and is quite
abrupt. In comparison, the change from matter to radiation domination occurred very early in
39
the Universe, and was only associated with a small change in the way the scale factor evolves
(t1/2 → t2/3 ). Given that this change occurred at an epoch we can never directly measure
because it takes place before the formation of the CMB, we will have to content ourselves with
measuring the change for matter to cosmological constant dominance.
7.2
Future tests of benchmark cosmologies
The benchmark cosmology which we have described above represents a cornerstone of physics
over the past century. However, it has by no means been a fixed model over that time. A hundred
years ago the static model of the universe was the only game in town, until the discovery of
Hubble’s law in the late 1920’s. Even after this we remained woefully short of the model
we currently assume. Most notably because of the absence of a cosmological constant from
observational models prior to the early-mid 1990’s.
We are now in a period which some like to call the era of precision cosmology, but precision
in cosmology is not necessarily what we would call precision in other fields. Indeed, the errors
we currently have on cosmological parameters are broadly comparable to those which would
be considered just about acceptable in a school laboratory were we talking about measuring
electrical current etc. Therefore, despite our success in creating a self consistent picture of the
universe there remains much to do if we are to truly understand it.
Furthermore, despite having described the benchmark model in the previous chapters, we
still know little about its actual constituents. In describing our benchmark model we make the
assumption that the universe is 23% dark matter and 73% dark energy. That is to say that
about 4% of the universe is made up of things that we broadly understand, the rest we do
not. In fact, even most of the 4% which is made of baryons is actually invisible to us. We
can understand how dark matter and dark energy impact the evolution of the universe since we
can see this from observations, but we have only concepts of what they may be. This situation
clearly needs to be resolved, and it is this which leads to the drive to improve our knowledge
of cosmological models. Indeed, this is a cornerstone of NASA’s beyond Einstein programme
(its current vogue), and will be a chief goal of at least one (and possibly) more future space
observatories.
The basic concept here is simple, to find a means of pinning down the cosmological parameters
with greater accuracy than has been possible to date. We can potentially do this a number of
ways, so I’ll discuss only a few possibilities here.
7.2.1
Large scale supernova searches
One of the most successful tool for pinning down dark energy, and for general cosmographic use,
has been type Ia supernovae. It has proved possible to calibrate these to reasonable accuracy
(∼ 7% errors in distance) and this has made them the tool of choice for both current, and
next generation endeavours. The aim of most supernova searches now is simply to locate large,
homogeneously selected samples of type Ia supernovae which can be followed up. The work
can be split into two main categories. The first is studies of relatively local supernovae. These
studies aim to calibrate the standard candle relationships which are used to measure distances.
There is a well known relation between the peak brightness of a SN Ia and its decay rate, known
40
as the Philips relation. In this the brighter the SN, the slower its decline. Correcting for this
difference in lightcurve shape is what enables. SN Ia to be accurate distance estimators. Much
work is continuing now using local samples of SN Ia to further refine this relationship, and also
to test how well SNIa behave as standard candles at a variety of wavelengths. For example,
when we look at optical wavelengths at z ∼ 1, we see light which has been redshifted by a factor
of (1 + z), that is, visible light at z ∼ 1 actually originated in the ultraviolet in the rest frame
of the supernovae, so we cannot simply assume that the behaviour observed at low redshift in
the optical will be the same as at high redshift in the ultraviolet.
7.2.2
High resolution Cosmic Microwave Background maps
Measuring the first peak of the Cosmic Microwave background provides one of the strongest
constraints on the total density of the Universe, and in turn the fraction contributed by dark
energy. There has been a significant investment in resources towards this aim over the past
decade. The Wilkinson Microwave Anisotropy Probe (WMAP) is a NASA mission launched
in 2002. Over the course of a 9 year mission it accrued detailed temperature and polarization
data acros the CMB, pinning down the detailed cosmological parameters with great detail (the
WMAP papers are amongst the most highly cited in all of science over the past decade). The
cummulative 9 year data provides the tightest constraints on the total density parameter of
Ω =. More recently, the European Space Agencies Planck satellite has provided an even more
detailed and high resolution view, although only the initial temperature data has been provided
to date.
7.2.3
Baryon Acoustic Oscillations
Baryon acoustic oscillations are echoes of the cosmic microwave background imprinted on the
large scale disribution of galaxies. Recall that the largest structures in the CMB have a size
which is determined by the age of the Universe at the epoch of CMB formation, and the speed
of sound in the plasma which dominated the early Universe. At decoupling the matter and
radiation separate (i.e. the Universe ceases to be a plasma), so the photons free stream to us
as the CMB, only impacted by the gravitational pull of the Universe as a whole. However,
the baryons continues to evolve and interact, and proceeded to create large scale structure in
the Univesrse. However, this large scale is frozen into the galaxy distribution. Large scale sky
surveys such as the Sloan Digital Sky Survey have measured the galaxy distribution on these
very large scales, and indeed seek a peak in the clustering power at large radii (much larger than
typical clustering length of galaxies) – these are known as baryon acoustic oscillations (BAOs).
Since this started as a standard ruler at the time of CMB formation, its measured size as a
function of redshift provides us with a direct measurement of how the scale factor has evolved
between the CMB and the epoch where the BAO is observed. Hence measuring the evolution
of BAOs provides a powerful route to measuring the scale factor as a function of cosmic time,
and hence of directly testing model cosmologies. However, it should be borne in mind that
making such a measurement is extremely challenging, since it requires a detailed 3D map of the
positions of galaxies in space. This is far from trivial even with modern technology. Secondly, it
is limited by the number of oscillations that fit within our horizon at a given epoch, thus there
41
is a statistical noise limit which cannot be beaten even with perfect observations.
42
Chapter 8
Dark Matter
As we have already discussed dynamical evidence for dark matter in the universe is widespread.
The most notable evidence comes from the rotations curves of normal galaxies (like our own)
which are flat at large radii, far beyond the extent of the visible disk. The expected orbital
rotations scales as
q
vorb = GM (r)/r
(8.1)
and should fall off as r−1/2 beyond the visible light if M (r) is not continuing to increase. This
indicates the presence of invisible matter beyond the radius of the majority of the stars, and a
primary motivator for the presence of dark matter.
However, this only works in systems which have ordered rotation, and this only comprises
a few astrophysical objects (mostly spiral galaxies). Many others have stars on essentially
randomised orbits, and so this approach cannot be used. Instead, for a system in equilibrium
(which most can be assumed to be) we can make use of the Viral theorem. In doing this there
are several observational considerations that should be considered (if doing the experiment
properly), there are intentionally glossed over here for the sake of clarity.
2T + V = 0
(8.2)
where T is the kinetic energy and V the gravitational potential energy. For a system of stars
P1
2
1
2
T =
2 mv = 2 M hvi , where M is the total system mass and hvi is the velocity dispersion.
The gravitational potential energy is given by
3GM 2
(8.3)
5R
In practice this comes from the assumption of a spherically symmetric distribution of mass,
which only feels the force internal to it (M < r), that in turn acts like a point source. In this
system dm = ρ4πr2 dr and hence
V =−
dU = −
GM (< r)
ρ4πr2 dr
r
Since M (< r) = 34 πr3 ρ this becomes
43
(8.4)
U =−
16 2 2
π ρ
3
Z R
r4 dr = −
0
16 2 2 5 −3GM 2
π ρ R =
15
5R
(8.5)
Hence using the virial theorem gives
M=
5 hvi2 R
3G
(8.6)
So, in principle if we measure the velocity dispersion (note that the measured and true
velocity dispersions need some adjustment since we only measure the radial component), and
the size of the object then we can determine its mass. In principle this should work for many
systems e.g., star clusters, globular clusters, elliptical galaxies, galaxy clusters. Since
we can also measure the luminosity of these objects we can then determine their mass to light
ratio. This provides one of the few diagnostics we can make of the nature of dark matter, since
we can determine how the dark matter distribution depends on the mass or size of a
given object.
As we saw in lecture 3, it is common in astronomy we define the Sun as our ultimate standard
and define its mass to light ratio as unity M /L = 1. Table 8.1 shows the mass to light ratios
seen in different types of astronomical systems.
In general more massive systems (in terms of the mass of stars) exhibit relatively more dark
matter, as evidenced by higher mass to light ratios. This suggests that the gravitational pull of
larger systems is sufficient to ”trap” more dark matter. It also shows that dark matter is not
uniformly distributed across space.
An exception to this rule comes from the local population of dwarf spheroidal galaxies. These
galaxies have stellar masses (i.e. the masses of the stars in the system) which are comparable
to those of globular clusters, but are typically a factor of 10 larger. Like globular clusters they
are mostly ancient systems, consisting of old stars. However, unlike globular clusters, which
have no evidence for dark matter, the dwarf spheroidal galaxies are dominated by dark matter,
with mass to light ratio’s of 100 or more at times. That systems of similar stellar mass but
different size can have such different dark matter contents is suggestive that dark matter has
some minimum scale size or tens of parsecs and masses of 107 M . Systems small than this do
not have sufficient gravitational pull to hold onto dark matter.
8.0.4
Dark matter temperature
This leads to the concept of a dark matter temperature, which really refers to the speed
at which dark matter particles are travelling (in analogy to the speed that particles in a gas
move as the gas is heated). The two types of dark matter that are generally discussed are
hot dark matter (HDM) and cold dark matter CDM. In practice there can be a wide
range of properties in between and warm dark matter, tepid dark matter and others have been
proposed. Hot dark matter moves relativistically, and the classic example would be the massive
neutrino. Because of its velocity it moves too rapidly to be caught up in collapsing peaks
in the baryon density during structure formation. This results in hot dark matter particles
being scattered broadly throughout the Universe, and not clustering in massive structures as is
44
Table 8.1: Masses an M/L ratios of common objects
Object
Globular Cluster
Spiral (disc)
Spiral (halo)
Elliptical
Galaxy Cluster
Coma Cluster
Dwarf Spheroidal
M(stars)(M )
106
1011
1011 +
109 − 1012
1010 − 1014
106 − 10 − 8
M/L
1
2-10
5-10
10-100
100+
250
10-100
observed. Further, since the mass of haloes in galaxy formation models is strongly dependent on
dark matter (since it is the dominant) source of mass. Small scale structure is effectively wiped
out in hot dark matter models. In practice, a strong bound has been put on the contribution of
hot dark matter, in particular the neutrino as Ων < 0.0072.
In contrast cold dark matter readily falls into initially shallow potential wells and has a much
larger influence on the process impacting galaxy formation. It can create the low mass haloes
that are necessary for matching simulations to the observed Universe. However, the problem it
faces is that it overpredicts the abundance of these halos, something known as the substructure
crisis. There are numerous solutions to this problem that have been proposed, including altering
the prescription for dark matter (e.g. self interacting dark matter), observational biases, or the
presence of some haloes which simply contain few (or even no) stars. It is likely that one or
more of these explanations are correct, and so the substructure crisis causes less concern that
previously.
Having accepted that cold dark matter is the likely explanation for the observed properties of
the Universe at large scales, we are now faced with the question of what the dark matter actually
is. There are two broad categories of models, those which arise from particle physics, and invoke
Weakly Interacting Massive Particles (WIMPS) and those which come from astronomy,
and use the (intentionally invented) name Massive Compact Halo Object (MACHO).
8.1
MACHOS
MACHOS are essentially compact stars, which are sufficiently cool that their luminosity is insufficient to be observed, even with todays large ground based telescopes. The most likely MACHO
candidates are:
• Cool white dwarfs - ∼ 0.6 M
• Brown dwarfs - ∼ 0.08 M
• Neutron stars - ∼ 1.4 M
• Black Holes - ∼ 3 − 104 M
45
• Other wierd objects, quark stars etc
Although these objects are too faint for direct detection with telescopes it is possible to
test what contribution they may make to dark matter by observing there effect on light from
background stars as it passes close to the MACHO. This is a phenomena known as microlensing,
which also has applications to extra solar planet research.
The basic principle of microlensing is that as a MACHO moves across in front of a background
star the light from the star is bent and magnified by the gravitational field. The light is deflected
by an angle
4GM
α= 2
(8.7)
c b
where b is the so called impact parameter and is the separation of the star and MACHO. This
deflection can create a so called Einstein Ring, a perfect ring around the lensing object (although
it is typically very small and not resolved for MACHO microlensing events). The size of this
ring is
4GM 1 − x 1/2
θE =
(8.8)
c2 d
x
where x = dm /d, the ratio of the distance to the star d and the lens dm . θE is thus maximized
for x = 0.5. This means that the maximum magnification and optical search is for MACHOs
towards the Magellanic Clouds, where the background stars are roughly twice as far away as
any candidate MACHOS. For typical parameters for the Magellanic Clouds
−4
θE = 4 × 10
M
arcsec
M( )
!1/2 d
50kpc
(8.9)
In addition to the bending of the light into a Einstein Ring there is also amplification, while
the background star is within the Einstein ring. Setting
u = b/θE
(8.10)
the amplification is then
A=
u2 + 2
u(u2 + 4)1/2
(8.11)
The total timescale of the lensing event (again for a typical event towards the Magellanic Cloud)
is
−1
dθE
M 1/2
v
∆t =
≈ 90days
(8.12)
2v
M
200km/s
where v is the velocity of the MACHO. Thus the length of the lensing event is dictated by the
mass of the MACHO. The rate of MACHOS observed essentially sets their space density, and
so the product of mass per MACHO and density yields the total mass in MACHOS, and can be
compared with the measured dark matter density in galaxy halos.
The punchline is that although some microlensing events have been observed, typically with
lenses being low mass stars (brown dwarfs or white dwarfs) the total mass within is not enough
to be responsible for the observed value of ΩM , but can probably make up only 20% of it.
46
8.2
WIMPS
WIMPS are thought to be elementry particles are are predicted under certain variations of the
standard model of particle physics. They must be electrically neutral in order to interact only via
gravitational and potentially weak forces. They must be particles that are beyond the standard
model since no standard model particle has the required properties. There are various attempts
to extend the standard model which arise due to concerns within it. In particular about how one
might include gravity, or about various fine tuning issues. In particular there is the Hierarchy
problem – this refers to the fact that the Higgs vacuum expectation (vev - the average value of
the field in a vaccum) lies at around 250 GeV, while the Planck scale, where quantum effects
of gravity are needed is at 1019 GeV – it may well be expected that there is new physics in the
massive energy range. The second problem is a fine tuning problem associated with the Higgs
mass lying in the electroweak regime (125 GeV). Since it is expected that the electroweak and
strong forces become a “Grand Unified Force” at energies of above 1016 GeV the cancellation
to get a Higgs mass of only ∼ 100 GeV must be better than one part in 1014 , implausibly small.
8.2.1
Supersymmetry
Supersymmetry is an appealing solution that allows the interchange of bosons and fermions. It
means that each standard model particle has an supersymmetric counterpart. For example the
sneutrino is a boson that is the superpartner of the neutrino. The most likely candidate for
dark matter is the neutralino, which is a fermion. Rather complexly there are actually a family
of neutralios which are superpositions of the superpartners of gauge bosons of the standard
model (e.g. photinos, zibo (Z-boson super partners) with the super partner of the Higgs Boson
(Higgsino). The lightest neutralino is expected to be stable, and may be present in large numbers
within the Universe. Since it does not decay it is extremely difficult to detect. It is clear that
the symmetry is not a perfect one. Superpartners do not have the same mass as their normal
cousins (and only a difference in spin), but some mass hierarchy, otherwise they would have
been discovered in large numbers via particle accelerators already. Their mass must therefore
be in the range 100 - 1000 GeV.
A final feature of supersymmetric partners is that they provide a new quantum number, R,
that is conserved in multiplication, and that they a majorama particles, which means they can
act as their own antiparticle. This is potentially valuable when it comes to searching for these
objects, as we see below.
8.2.2
Axions
Axions are created by extensions to the standard model containing the strong force. The physics
behind the strong force actually allows for a strong electric dipole to be associated with the
neutron. Clearly one has not been observed, and so a (broken) symmetry was developed that
forced the neutron to be neutral (a similar symmetry forces the photon to be massless). If this
symmetry is broken it allows for a low mass (µeV) particle to be produced, this is known as the
axion.
47
8.2.3
Searches for particle dark matter
Cryogenic detectors
There are numerous ongoing searches for particle dark matter, which may manifest itself in
many ways. The traditional route of undertaking such searches has been to use deep detectors,
isolated from significant particle backgrounds on the Earth surface (e.g. cosmic rays etc). These
detectors then search for dark matter, relying on the direct (weak) interaction of a dark matter
particle with an atomic nucleus. This is possible since the normal interactions between atoms and
incoming particles are mediated by electromagnetic interactions between the incoming particle
and the electrons in the atom, such that nucleon interactions will be rare. When they do occur
they can be detected via several general routes;
• Phonon interactions – when a dark matter particle interacts with a nucleon in a crystal
lattice, it rattles the lattice, creating vibrations which travel as phonons through the lattice
and are potentially detectable.
• Charge changes – The interaction that creates vibrations will also move charges away
from their equilibrium positions since the nucleus will draw electrons with it. This means
that dark matter created currents through the detector that can again be observed.
• Temperature changes – The interactions within the sample necessitate the transfer of
kinetic energy from the WIMP into the sample. This in turn will create a measurable
change in temperature (for a sufficiently stable initial temperature, and accurate enough
thermometer), and may be measured.
• Cherenkov radiation – When nucleons are hit by the relatively heavy WIMP particles
they receive a kick which imparts velocity to them. This velocity may be larger than the
speed of light in the medium in which they reside. If this is the case as they slow they will
give off Cherenkov radiation.
Accelerator detectors
In principle, particle accelerators whose collision energy exceeds the rest mass energy of a dark
matter particle should be capable of producing particle dark matter. Once detected the detailed
analysis of the resulting particle tracks should allow its existence to be inferred. In practice the
weak interaction of dark matter particles means that they are unlikely to interact within the
detector. However, they would still carry energy and momentum away from the instrument.
Given this, a smoking gun for dark matter particles would be collision events which appear not
to conserve energy and momentum. By reconstructing this missing energy and momentum it
would be possible to constrain the properties of the particles themselves.
Astrophysical detection
In addition to Earth-bound searches there is also the possibility of conducting direct astrophysical searches for dark matter particles. While most naturally astrophysical observations are
48
sensitive to larger scale dark matter (such as MACHOs), various possibilities may allow the
detection of dark matter association with astronomical sources.
In particular, since neutralinos are majorana particles, interactions between neutralinos can
cause annihilation and that the result must be a normal particle. e.g.
χ+χ→γ+γ
R = −1 × −1 = 1 × 1
(8.13)
These particles may be photons or could equally be electron – positron pairs. If γ-rays are
produced then their energy will be equal to Eγ = mχ c2 . If a γ-ray telescope observing the sky
was sensitive at this energy then a line would be observed whose energy would directly provide
the rest mass energy of the neutralino.
There have been some suggestions by orbiting satellites (most notably PAMELA) that there
is an excess of positrons (positrons are much rarer than electrons so easier to detect an excess
or). This excess could be produced by dark matter annihilations, although other explanations
such as position accelerations in pulsar winds also remain plausible.
Similarly, axions may be produced by nuclear reactions inside the sun, and there is a recent
suggestion that they may have been discovered based on a varying X-ray background observed
by the XMM-Newton experiment, although such analysis is understandably complex, and awaits
confirmation.
8.3
Modified Newtonian Dynamics
F = mµ
a
a
a0
(8.14)
We do not know what µ is, but it is an empirical function that for low a becomes a/a0 and
for high a tends to 1. a0 is a new fundamental constant of nature, the acceleration constant
a0 = 1.2 × 10−10 m s−2 . This means in the high acceleration regime >> a0 , F = ma as usual,
but at low accelerations
F =
ma2
a0
(8.15)
With this functional form in place we can ask ourselves how the rotation curve at large radii
in a galaxy may work. In this case a = v 2 /r and so
ma2
GM m
=
a0
R2
and since a = v 2 /R in a circular orbit
mv 4
GM m
=
2
R a0
R2
49
(8.16)
v 4 = GM a0
(8.17)
In other words, the rotation at large radii will naturally result in a constant rotation velocity
as observed.
This does have problems though. Firstly, it is not a relativistic theory. The concept of an
acceleration constant requires a frame against which the acceleration can be measured. It is
therefore directly at odds with a relativistic theory. People have attempted to adapt this by
creating TeVeS (Tensor Vector Scalar Gravity)
50
Chapter 9
Structure in the universe
9.1
The Jeans Mass
At early times (i.e. viewing the Cosmic Microwave Background 300000 years after the big bang)
the universe was very smooth, with variations of order 1 part in 105 , yet today the universe
consists of individual stars, galaxies and planets. How is it possible to go between the two
extremes? How does structure in the universe form? This is the question we want to address
this lecture.
In essence we are initially concerned with conditions to form gravitationally bound structures.
Any overdensity in the Universe will see to collapse on a dynamical timescale.
tdyn
1
=
=
(4πGρ)1/2
c2
4πG
!1/2
(9.1)
This is essentially a free-fall time scale, e.g. if you consider gravitational acceleration
GM
R2
a=
(9.2)
and a time to fall distance R = 1/2at2 , you get
R∼
GM 2
t
R2
(9.3)
and for a spherical mass M = ρR3 , this equates to
t∼
1
(Gρ)1/2
(9.4)
For the earth’s atmosphere, this dynamical time is approximately 9 hours. Clearly the
atmosphere is not a puddle at our feet, and so something opposes this collapse. This is pressure.
For a matter dominated Universe (matter is after all the thing collapsing) the equation of state
parameter is given by
w=
kT
µc2
51
(9.5)
and signals will travel through the matter at the soundspeed (cs ).
tpre =
R
cs
where
dP
cs = c
d
(9.6)
1/2
(9.7)
p
i.e. since P = w this means the sound speed is cs = c (w).
To avoid collapse a perturbation must respond to a gravitationally induced change more
rapidly than the change can propagate. In other words tpre < tdyn . Setting these two equal
gives a Jeans length (R = λJ ).
c2
4πG
λJ = cs
!1/2
(9.8)
This is a slightly simplified derivation. Gathering all the relevant factors of π etc. together
yields the correct result.
λJ
=
cs
πc2
G
!1/2
= 2πcs tdyn
(9.9)
Structure forms on scales that are larger than λJ , or its correspond Jeans Mass, MJ ,
4
MJ = πρλ3J
3
Smaller structures are pressure supported.
9.2
(9.10)
Formation in a radiation or matter dominated Universe
Clearly, if we want to understand the evolution of the Universe we want to know the Jeans
mass/length at any given time. Here we can note that the energy density (in a flat universe) is
given by
3c2 H 2
8πG
and that the age of the Universe t = 1/H, so
=
tH
1
=
=
H
3c2
8πG
!1/2
(9.11)
1/2
3
2
=
tdyn
(9.12)
substituting in we get
λJ = 2πcs tdyn = 2π
52
1/2
3
cs
2
H
(9.13)
Now we can essentially work out the Jean’s mass or length as a function of the sound space
√
c w.
Radiation Dominated: w = 1/3, cs = 0.58c In this case
λJ = 2π
1/2 1/2
3
1
c
2
3
H
≈
3c
H
(9.14)
Note, this is significantly larger than the horizon, (c/H). The corresponding mass is MJ ∼
7 × 1018 kg, more massive than the largest galaxy clusters. In other words, structure doesn’t
form in a radiation dominated Universe.
This isn’t strictly true, since the crucial point isn’t the radiation domination, but the speed
that the material is moving at. in practice this applies at any time while the matter and the
radiation are coupled (i.e. before formation of the CMB). At the point when decoupling occurs,
we move to a matter dominated Universe. Then
cs (baryon) =
kT
µc2
1/2
c
(9.15)
We know kT = 0.26 eV directly from the CMB, so cs ∼ 10−5 c. At this point the density of
baryons ρbaryon = 5 × 10−19 kg m−3 , and the Jeans Mass is MJ ∼ 105 M , roughly the mass of
a dwarf galaxy.
In other words, structure formation begins immediately after decoupling.
In practice, the situation is slightly more complex than this. Most of the matter in the
Universe is dark, it does not interact electromagnetically, and so isn’t coupled to the radiation
in the early Universe, this material is extremely important, since it collapse from the beginning,
and so the haloes that have become somewhat overdense in dark matter during the pre-CMB
epoch go on to host galaxies and galaxy clusters.
53
Chapter 10
Inflation
So far we have outlined a benchmark model for the universe, which provides a good description
of the observations from the cosmic microwave background until the present day. However, while
this description is powerful and very successful there remain some puzzles within it, and these
require a novel explanation in the form of inflation.
10.1
The horizon problem
When we view the cosmic microwave background we observe that it has essentially the same
temperature across the entirety of the visible sky. That is, points opposite eachother on the
sky have essentially the same temperature. The similarity of temperatures across the sky is
indicative of thermal equilibrium, that is to say that the size of the universe at the time of the
CMB shows the signs of being in thermal equilibrium. Yet we know that at this stage the hasn’t
been time for signals to travel across the universe and thus for it to have come into thermal
equilibrium. This is shown schematically in Figure ??. Our horizon is defined as the surface ct0
from our location in any direction, but two points opposite eachother on the horizon thus have
a separation of 2ct0 , i.e. the light travel time from the two points is greater than the age of the
universe. they have never been in causal contact, so how are they in thermal equilibrium? This
is known as the horizon problem.
10.2
The flatness problem
The flatness problem refers to the statement that the universe is flat. Again measurements from
the CMB imply that Ω0 = 1.02 ± 0.02, but why is this the case? There is no special reason to
believe that the universe should be flat, it could have taken any value, say 105 0, or 10−50 , unity
is a very special value, and it is odd that the universe could (randomly) take that value.
A common argument is that we don’t know for sure that the universe does have the critical
density 1.02 ± 0.02 is not certainly 1. Perhaps it is just chance that the value is close to 1.
However, this can be dismissed by considering how the density of the universe evolves with time
if it is dominated by different components. We saw in Lecture 6 that the evolution of the density
of the universe with time can be written as
54
1 − Ω(t) =
−kc2
R02 a(t)2 H(t)2
(10.1)
evaluated at the present moment this is simply
1 − Ω0 =
−kc2
R02 H02
(10.2)
where we have made the common assumption a0 = 1, diving the two equations gives
1 − Ω(t)
H02
=
1 − Ω0
H(t)2 a(t)2
(10.3)
We also know from earlier lectures that the Friedman equation for multicomponent universes
can be expressed as
H2
Ωm,0 Ωr,0
(10.4)
= 3 + 4
2
a
a
H0
Ignoring the ΩΛ term since we are looking back towards the early universe where Λ is less
important. Substituting this into 13.3 above gives
1 − Ω(t) =
(1 − Ω0 )a2
Ωr,0 + aΩm,0
(10.5)
where Ω0 = 1.02 ± 0.2, Ωm = 0.27, Ωr = 10−5 . We can now extrapolate this back in time. For
example when a = 10−4 , and matter and radiation contributed equally then 1 − Ω(t) ∼ 10−5 . At
the Planck time of 10−44 then 1 − Ω(t) ∼ 10−60 . In other words, any difference that we observe
now get progressively small as time goes backward, so that if the universe is approximately
flat now then immediately after the big bang it is flat to 1 part in 1060 . This means that it is
extremely like universe is flat, not just nearly flat, since a universe which is flat remains so for
ever.
These two problems essentially related to apparent fine tuning of the universe under the
standard model. Somehow it must have had the same temperature throughout, and have always
been flat. This may be the case, but the probability of it is very low, and it can appear contrived
if we cannot explain why the universe appears as it does.
10.3
Acceleration expansion at early times – Inflation
The solution to these problems comes from a theory by Alan Guth, called inflation. It basically
suggests that at very early times in the universe ∼ 10−36 s the universe underwent a brief period
of exponential expansion, greatly enlarging our horizon and flattening the universe. Here is how
it works.
First, take the acceleration equation
ä
4πG
Λ
= − 2 ( + 3P ) +
a
3c
3
55
(10.6)
and an equation of state P = we. Any value of w ¡ -1/3 gives an accelerating universe. So, if at
early times for some reason the equation of state parameter of the universe became < −1/3, the
universe would accelerate. For simplicity lets consider the case of a cosmological constant-like
equation of state.
The Friedman equation for a Λ dominated universe is
2
Λi
3
(10.7)
a(t) ∝ eHi t
(10.8)
ȧ
a
=
The solution to this is exponential expansion
where Hi is the Hubble parameter at inflation, and is a constant for Λ dominance. So, if inflation
kicks in at time ti and ends at time tf it dramatically impact the scale factor of the universe,
so that
a(t) = ai (t/ti )1/2
a(t) = ai exp (HI (t − ti ))
t < ti
(10.9)
ti < t < t f
(10.10)
1/2
a(t) = ai exp (Hi(tf − ti ))(t/tf )
t > tf
(10.11)
So the scale factor of the universe increases during inflation by a factor of
a(tf )
= eN
a(ti )
(10.12)
If inflation begins at 10−36 s and ends at 1034 s then N = 100, an important point to note here
is that while this is a very short time (10−34 s) it is actually 100 Hubble Times, so if the same
were to happen today inflation would not end until the universe was 100 times its current age
(roughly 1400 billion years old).
The increase in scale factor is thus e100 ∼ 1043 , and so rapidly the scale factor has been
stretched. This means that a region of space the size of our horizon prior to inflation becomes
1043 times the size of our horizon afterwards. In other words, regions which are in causal
contact (and thus thermal equilibrium) are increased in size so much that they are now much
larger than our horizon. Under this scenario it is entirely unsurprising that the universe has the
same temperature. This solves the horizon problem.
Now let us consider the flatness problem. As we had before
1 − Ω(t) =
−kc2
R02 a(t)2 H(t)2
(10.13)
During inflation H = Hi always, and so this can be simplified to
1 − Ω(t) ∝
1
∝ exp −2Hi t
a(t)2
56
(10.14)
Figure 10.1: A cartoon diagram of how inflation works. The universe may initially be significantly non-flat, and be evolving further away from flatness. When inflation kicks in the universe
rapidly flattens and then remains flat for a long period (though it may not look so in log space!),
eventually, depending on the components of the universe which dominate the universe may
evolve away from flatness.
so
|1 − Ω(tf )| = exp (−2N )|1 − Ω(ti )|
(10.15)
If N = 100 then this means that during inflation the universe is flattened by a factor of 10−87 !
If the universe was anywhere within 87 orders of magnitude of being flat before inflation then it
will be flat afterwards. This is not fine tuning, the universe really could have chosen any value
to start with. This solves the flatness problem.
There is another rather remarkable fact about inflation when it is considered. If the scale
factor really increases by a factor of 1043 then tiny quantum fluctuations are enlarged phenomenally, these tiny variations suddenly become the size of galaxy clusters, and are, perhaps the
source of the structure we see in the universe today.
57
Chapter 11
Baryogenesis and Nucleosynthesis
Another key question is the origin of the particles (mainly atoms, electrons) that the universe
is made up from. If we extrapolate the Universe back to early times, t ∼ 10−12 s, when the
temperature was T ∼ 1 TeV (note that these energies are similar to those being created in the
LHC, which is why people talk about it creating energies similar to those just after the big
bang). In this scenario the Universe is radiation dominated and so
t −1/2
T = 10 K
1s
−1/2
t
kT = 1MeV
1s
10
(11.1)
11.1
Baryogenesis
At some (poorly constrained) point in this early Universe we set a first fundamental property
of matter. Namely that the Universe is made of matter and not antimatter. We know this
from observations today. Clearly the solar system, and indeed the Milky Way galaxy are made
of matter. We have good reasons to believe that all of the rest of the visible Universe is also
made of matter, since we do not see sufficiently large scale voids to explain regions of matter
and antimatter, nor do we see obvious spatial clustering in photons which may originate from
matter – antimatter annihilation.
However, there is evidence that much of the Universe was at some point in the form of
antimatter. This evidence primarily comes from the observed photon to baryon ratio. In the
current Universe np /nb ∼ 109 (note that this is the number density of photons to baryons, not
the energy density, these are low energy photons). If we go to a point in the early universe
where kT > mb c2 , where mb is the mass of a baryon then the baryons are in equilibrium with
the photons, and we might expect there to be somewhat similar numbers.
γ+γ *
) b + b̄
58
(11.2)
This is a good time to set about making asymmetry, because all that you need to do is make
a tiny fraction of additional baryons (1 part in 109 ). In other words we can define an efficiency
η=
nb − nb̄
nγ
(11.3)
However, the problem arises because we know very little about the physics that might introduce such a change, and very little of it has been physically tested (although this is one of the
aims of the LHC). However, there are some conditions which can be imposed which may explain
it. These are the so-called Sacharov conditions, which state that
1) Baryon number is not conserved.
2) CP violation
3) Interactions out of thermal equilibrium, probably because of the rapid expansion of the
Universe.
While the physics underlying these issues remains opaque, one thing that is clear is that
from this early time onwards the Universe was dominated by matter rather than antimatter,
and this is what we must consider when moving further.
11.2
Nucleosynthesis
Most of the heavy elements in the universe have been formed in the core’s of stars, and subsequently ejected when the stars reach the end of their lives, either in the winds of red giants, or
more explosively as supernovae ejecta.
However, in the early Universe, the first generations of stars were clearly formed out of
material that had not previously been processed by stellar nuclear fusion, and the bulk atomic
properties of the Universe were actually set in the first few minutes. This is the subject of big
bang nucleosynthesis, and is what we shall consider in this lecture.
As the temperature passes through 1 MeV, the thermal energy of the Universe is rather
comparable to nuclear binding energies, and so nucleosynthesis can be begin.
At first, neutrons and protons are in equilibrium
n + νe *
) p + e−
n + e+ *
) p + ν̄e
(11.4)
Since p and n have different masses this can create/require energy
Qn = mn c2 − mp c2 = 1.29Mev
(11.5)
n → p + e + ν¯e
(11.6)
Also, neutrons decay
the halflife for this reaction is 900 seconds.
59
In equilibrium the number of protons and neutrons is given by the Maxwell Boltzman distribution
nn = gn
mn KT
2πh̄2
3/2
−mn c2
exp
KT
!
mp KT
2πh̄2
3/2
−mp c2
exp
KT
!
np = gp
(11.7)
where gn = gp = 2. We can now take the ratio of these to give
nn
=
np
mn
mp
!3/2
−(mn − mp )c2
exp
KT
!
(11.8)
This can be simplified since mn /mp ∼ 1, to
nn
−Qn
= exp
np
KT
(11.9)
This proves an exponential decline in the number of neutrons with decreasing temperature.
If this equilibrium holds the after 5 minutes the nn /np ratio is 10−6 . Clearly this is not correct,
since there are a reasonable number of neutrons present in the Universe today.
The solution to this apparent contradiction is that the reactions which govern this equilibrium
are weak interactions, the cross section of these is
−47
σw = 10
2
m
kT
1M eV
2
(11.10)
This can be compared to the Thomson cross section of 7 × 10−29 m2 , which makes it clear
that the weak interaction will only provide a high cross section for interactions at very high
temperatures.
The rate of interactions is then given by
Γ = nσv
Γ = nν σ w c
(11.11)
Since
nν ∝ a−3 ∝ t−3/2
(11.12)
σw ∝ T 2 ∝ a−2 ∝ t−1
(11.13)
and
The total rate of interactions is proportional to
60
Γ ∝ t−5/2 ∝ a−5
(11.14)
In other words the reaction rate falls rapidly with time, and eventually Γ = H, so that there
is less than one interaction per Hubble time. At this point we get something called Freezeout
Tf reeze = 9 × 109 K
tf reeze = 1 s
At this time the neutron to proton ratio is approximately 0.2.
At this point nucleosynthesis can proceed, and various reactions begin
p + p → D + e+ + νe WEAK
n + n → D + e− + ν¯e WEAK
p + n → D + γSTRONG
(11.15)
This later reaction, as a strong reaction, dominates the observed rate. This builds up Deuterium, which then reacts further
D + p →3 He + γ
D + n →3 H + γ
(11.16)
and eventually Helium is made
3
3
He + n →4 He + γ
3
3
H + p →4 He + γ
H + D →4 He + n
He + D →4 He + p
(11.17)
Fusion continues until all of the neutrons are locked up in Helium.
From the predicted ratio of nn /np we can then calculate the expected mass fraction in Helium.
For each neutron we have 5 protons, and hence for each He nucleus we have two protons and
two neutrons and hence an additional 8 unbound protons. This suggests that the mass fraction
in He (Y) is equal to 4/12 = 1/3.
In practice the actual number is Y=0.24. The reason for this is twofold.
• Some neutrons are locked up in heavier elements
• Some neutrons decay before they are locked up into nuclei
61
The heavier element reactions are
4
4
He + D →6 Li + γ
He +3 He →7 Be + γ
(11.18)
One might wonder, given that the most stable nucleus is 56 F e, why does the reaction not
proceed to give us a universe made of Fe and unbound protons. The reasons for this are twofold.
Firstly, there are not enough neutrons to realistically do this, but more importantly there is not
enough time to build these more complex elements. The reaction
4
He +4 He →8 Be + γ
(11.19)
leads to an unstable element, which decays in 10−16 s, it is very difficult to get beyond
this in straight nucleosynthesis. In stars, these heavier elements are built up through chains of
reactions known as the triple−α processes. However, this process is slow. Since the window for
nucleosynthesis in the early Universe is short (15 minutes) there simply isn’t enough time to
build up the heavier elements. Hence at the end of nucleosynthesis, the Universe is dominated
by Hydrogen and Helium, with only traces of heavier elements.
62
Chapter 12
The Big Bang
Throughout this course we have discussed cosmology entirely in the context of the big bang
model for the universe. Although in many ways it was inflation which put the ”bang” in the
beginning of the universe it is not inflation which brought the universe into existence. Despite
the very strong evidence requiring a big bang, and the great success of the model there remains
little insight into what happened at the time of the big bang itself.
One of the big questions, both in physics and philosophy is therefore what caused the big
bang? This is a question which is very difficult to address for a variety of reasons, nonetheless
there do exist a number of possible mechanisms for creating the big bang, and we will briefly
discuss them here.
12.1
The Planck scale
An important scale in the early universe is known as the Planck scale, defined as
Λp =
Gh̄
c3
1/2
Gh̄
c5
1/2
∼ 1.6 × 10−35 m
(12.1)
∼ 5.4 × 10−44 s
(12.2)
and the associated Planck time
tp =
Once the universe shrinks below the Planck scale the particle wavefunctions overlap one
another and it becomes impossible to consider gravity in a classical (or even general relativistic
way). This means, that without a theory of quantum gravity (one of the great unsolved problems
in physics) it is difficult, if not impossible to extrapolate beyond this time. In essence this means
that models for the big bang are by necessity uncertain.
12.2
Vacuum fluctuations
The most commonly discussed concept for the creation of the big bang is the Vacuum fluctuations. This basically states that it is possible to borrow energy from a vacuum to create a
63
particle – antiparticle pair, which they continues to create further particles, until ultimately it
creates the universe we see today. This is allowed because of the uncertainty principle which
states that
h̄
∆E∆t >
(12.3)
2
and provides a viable (if far from certain) possibility for the big bang.
12.3
Branes and the Epkyrotic Universe
Alternative models for the big bang come from recent attempts to derive so called Theories of
Everything (TOEs). These include multidimensional theories such as String Theory and MTheory which envisage each elementary particle as a sting or membrane oscillating in 10 or
11 dimensional space. If the universe does contain a greater number of dimensions than the
classical 4 (3 spatial plus one time) then it creates a new field of String Cosmology, and a the
possibility of new explanations for the Big Bang. One of these models, postulated by Neil Turok
and collaborations at Cambridge is that the Big Bang was caused by the collision of two 11
dimensional branes. The ripples in the space time spreading out in our 4-dimensional space
time then create our universe. In this model the Friedman equation is modified to
H2 =
3c2
1+
8πG
2λ
(12.4)
where λ is the so called brane tension, which can provide a source on energy in the universe,
but can also introduce degeneracy.
One interesting consequence of this model is that the universe itself may be cyclic, with the
big bang happening multiple times, with the clock restarting after every brane collision. This
has also been coined as the ”Big Splat” scenario.
These different models for the big bang remain very uncertain, although future space missions
which might detect gravitational waves can distinguish between the two possible models briefly
illustrated above.
64
Appendix A
The universe on a side of A4
65
Distance
The universe on a side of A4
MACHO and
WIMP
searches microlensing
+ gammaray
D
H0
v=
Hubble’s law obeyed in
local universe
velocity
a α eHt
ω = -1
Figure A.1: The
ΩΛ,0 = 0.73, Ωm,o = 0.27, Ωr,0 = 10-5
H0 = 72 km s-1 Mpc-1
Spatially flat Ω0 =1
Type Ia supernova searches
Λ
66
Jeans mass ~105 Msol.
Stars/Galaxies form
Radial distance
10 ~Gyr - Matter- Cosmological constant equality
Observational challenges/facts
Key points/ideas
Key periods in Universe history
Rotation curves flat:
Dark Matter Dominant:
Probably Cold
On large scales universe is
homogeneous and isotropic
(Cosmological Principle)
Acceleration
Horizon size
Nucleosynthesis
N+p => D
D+p => 3He
D+D => 4He
First stars/galaxies
Time (total age = 13.5 Gyr)
Matter
a α t2/3
ω=0
velocity
Inflation
~300,000 years Cosmic Microwave background (e+p => H + γ)
Fluid
Peak of star formation
Jeans mass >1018 Msol.
Galaxies can’t collapse
Big bang
Vacuum fluctuations?
Brane collisions?
Friedman
a α t1/2
ω = 1/3
10-36s - increase scale
factor by ~1040
Solves flatness problem horizon problem
a α eHt
ω = -1
Radiation
Energy densities
εΛ = const
εm α 1/a3
εr α 1/a4
Matter-radiation equality
Appendix B
Problems and Answers
B.1
Problem Set
1a) A galaxy has a measured recession velocity of 5000 km s−1 , what is its;
i) redshift
ii) distance
b) Show that, in any uniformly expanding space the Hubble constant can be attributed to
the current rate of change of the scale factor ȧ/a.
c) Explain what is meant by the terms standard candle and standard ruler.
2) a) Show that in a universe consisting purely of matter, that the universe will be flat if the
matter density is
c =
3c2 H02
8πG
(B.1)
b) Using the fluid equation show that the energy density of matter varies as 1/a3 (you may
assume P = ω, where ω = 0).
c) Hence, solve the Friedman equation, assuming that the scale factor has a power law dependence of a(t) ∝ tq . What is the value of q?
d) How does H evolve? What does this mean for the evolution of the critical density of the
universe?
3) The Friedman equation, in terms of multiple component universes can be re-written as
67
H2
Ωm,0 Ωr,0
(1 − Ω0 )
= 3 + 4 + ΩΛ +
2
a
a
a2
H0
(B.2)
a) Explain the origin of each term.
b) Given that the universe is spatially flat, with Ωm,0 = 0.27 and ΩΛ,0 = 0.73, at what scale
factor were the two equal?
c) What redshift does this correspond to?
d) Perform the same exercise for matter and radiation, assuming Ωr,0 = 8.4 × 10−5 .
e) At very early times the universe was dominated by radiation only. Using the equation above
(or otherwise), show that in this phase its expansion was governed by a(t) ∝ t1/2 .
4) a) Explain what is meant my the terms horizon problem and flatness problem.
b) Given that the Friedman equation can be written as
1 − Ω(t) = −
κc2
R02 a(t)2 H(t)2
(B.3)
Show how the density of the universe (1 − Ω(t)) evolves for universes dominated by matter,
radiation and cosmological constant.
c) How does a brief period of acceleration in the early universe solve the horizon and flatness problems?
d) If the energy density and pressure of the inflaton field are given by
1
φ = φ̇2 + V (φ)
2
1 2
Pφ = φ̇ − V (φ)
2
(B.4)
Under what conditions will the inflaton field mimic a cosmological constant with equation of
state parameter ω ∼ −1.
e) Sketch a potential which may satisfy this criteria.
5) a) For a given component of the universe, with equation of state parameter ω, show that the
fluid equation can be re-written as
d
da
= −3(1 + ω) ,
a
68
(B.5)
and hence that
= 0 a−3(1+ω) .
(B.6)
b) By solving the Friedman equation for this generic equation of state, assuming a power-law
dependence (tq ), show that the scale factor evolves as
a(t) ∝ t2/(3+3ω) ,
(B.7)
assuming ω 6= −1.
6) a) Outline the evidence for dark matter in galaxies and clusters.
b) Show that the rotation curve of a galaxy should follow the form
s
vorb =
GM
.
R
(B.8)
c) Two possible origins of dark matter are WIMPS and MACHOS, what does each term mean?
Give an example of a possible WIMP and MACHO candidate.
d) Modified Newtonian Dynamics (MOND) states that Newton’s law should be modified to
the form,
a
F = ma.
(B.9)
a0
in very low accelerations. Show that in this case MOND predicts flat rotation curves in galaxies.
B.2
Problem Set with Answers
1a) A galaxy has a measured recession velocity of 5000 km s−1 , what is its;
i) redshift
Redshift is simply z = v/c = 0.017
ii) distance
Hubbles law state v = H0 D, or D = v/H0 = 70 Mpc
b) Show that, in any uniformly expanding space the Hubble constant can be attributed to the
current rate of change of the scale factor ȧ/a.
Suppose the triangle shown in Figure 1 expands uniformly. In this case the relative shape of
the triangle is preserved in time. i.e.
r12 (t) = a(t)r12 (t0 )
r23 (t) = a(t)r23 (t0 )
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Figure B.1: A triangle of galaxies, expanding uniformly.
r31 (t) = a(t)r31 (t0 )
(B.10)
Where a(t) is the so called scale factor which will be important later.
Imagine an observer in galaxy one. The relative velocities observed for galaxies 2 and 3 are
dr12
= ȧr12 (t0 ) =
dt
dr31
v31 (t) =
= ȧr31 (t0 ) =
dt
v12 (t) =
ȧ
r12 (t)
a
ȧ
r31 (t)
a
(B.11)
Where we have made the obvious substitution that r12 (t0 ) = r12 (t)/a(t).
The form of this equation is clearly that of Hubble’s law. In this case value of H0 can be
equated with ȧ/a, at the time t = 0 (i.e. the current time). More generally the value H is known
as the Hubble parameter, and is variable with time.
c) Explain what is meant by the terms standard candle and standard ruler.
A standard candle is simply an object of known luminosity, from its flux we can therefore measure its distance, or, more formally its luminosity distance. A standard ruler is an object with a
known length, therefore by measuring its apparent dimensions its true distance can be calculated.
Both of these are functions of the assumed cosmological model.
2) a) Show that in a universe consisting purely of matter, that the universe will be flat if the
matter density is
c =
3c2 H 2
8πG
70
(B.12)
Take the Friedman equation with zero cosmological constant. Spatially flat implies that k = 0.
So, if this is the case the Friedman equation reduces to
2
ȧ
8πG
=
a
3c2
Which can be re-arranged to give the required form above
(B.13)
b) Using the fluid equation show that the energy density of matter varies as 1/a3 (you may
assume P = ω, where ω = 0).
The fluid equation simply states
ȧ
˙ + 3 ( + P ) = 0
a
(B.14)
ȧ
˙ + 3 = 0
a
(B.15)
1 2
2
3a
ȧ
+
a
˙
=0
a3
(B.16)
since ω = 0, this reduces to
This can be rewritten as
which is equal to
1 d
(a3 ) = 0
(B.17)
a3 dt
Since a > 0 then the solution for this equation is a3 =const, or ∝ 1/a3 .
c) Hence, solve the Friedman equation, assuming that the scale factor has a power law dependence of a(t) ∝ tq . What is the value of q?
The Friedman equation reduces to
2
ȧ
a
=
8πG
3c2
(B.18)
If we assume that a(t) ∝ tq and ∝ 1/a3 , then we can equate each side of the equation in terms
of dependence on q, so:
8πG 1
(ȧ)2 =
(B.19)
3c2 a
The LHS is proportional to t2q−2 and the RHS is proportional to t−q . This can be solved in
q = 2/3.
d) How does H evolve? What does this mean for the evolution of the critical density of the
universe?
71
H = ȧ/a, = 2/3t
3) The Friedman equation, in terms of multiple component universes can be re-written as
H2
(1 − Ω0 )
Ωm,0 Ωr,0
= 3 + 4 + ΩΛ +
2
a
a
a2
H0
(B.20)
a) Explain the origin of each term.
The first term refers to the expansion of the universe, and refers to the Hubble parameter at
an arbitary time (t) compared to its value at the current time (t0 ).
The second term describes the evolution of the component of the universe which consists of matter ∝ 1/a3
The third term describes the evolution due to radiation ( ∝ 1/a4 )
The fourth term describes the evolution due to the cosmological constant ( = const)
The final terms describes the total density of the universe, or its curvature.
b) Given that the universe is spatially flat, with Ωm,0 = 0.27 and ΩΛ,0 = 0.73, at what scale
factor were the two equal?
The two are equal at scale factor a = Ωm,0 /ΩLambda,0 )1/3 ∼ 0.7
c) What redshift does this correspond to?
Since a = 1/(1 + z), z ∼ 0.4
d) Perform the same exercise for matter and radiation, assuming Ωr,0 = 8.4 × 10−5 .
e) At very early times the universe was dominated by radiation only. Using the equation above
(or otherwise), show that in this phase its expansion was governed by a(t) ∝ t1/2 .
4) a) Explain what is meant my the terms horizon problem and flatness problem.
The horizon problem refers to the fact that all points along our horizon in the cosmic microwave
background have very similar temperatures, indicating they were once in thermal equilibrium.
However their distance from eachother on opposite sides of the sky is twice the horizon distance
(2c/H0 ), and so they will never be in contact.
The flatness problem refers to the measurement of the universe as flat. Since the Universe
could have taken any value. Further, differences from universal flatness as a function of time
mean that if the universe is flat now (ΩT = 1.02 ± 0.02) then it was even closer to being flat at
early times. Why?
b) Given that the Friedman equation can be written as
1 − Ω(t) = −
κc2
R02 a(t)2 H(t)2
(B.21)
Show how the density of the universe (1 − Ω(t)) evolves for universes dominated by matter,
72
radiation and cosmological constant.
c) How does a brief period of acceleration in the early universe solve the horizon and flatness
problems?
d) If the energy density and pressure of the inflaton field are given by
A brief period of superluminal expansion solves this problem since, as you have shown above
for matter and radiation differences from flatness increase over time, but for the cosmological
constant they decrease. Therefore, a brief phase where the cosmological constant was dominant flattens the universe. Further, since the expansion at this time was vastly superluminal
is stretches regions which were in causal contact prior to the accelerated expansion, and leaves
them out of causal contact at later times.
1
φ = φ̇2 + V (φ)
2
1 2
Pφ = φ̇ − V (φ)
2
(B.22)
Under what conditions will the inflaton field mimic a cosmological constant with equation of
state parameter w ∼ −1.
Since P = w, and for a cosmological constant P = − (w = −1) this can be achieved if
2
˙
phi << V (φ)
e) Sketch a potential which may satisfy this criteria.
See lecture notes for lecture 13
5) a) For a given component of the universe, with equation of state parameter ω, show that the
fluid equation can be re-written as
d
da
= −3(1 + ω) ,
a
(B.23)
= 0 a−3(1+ω) .
(B.24)
ȧ
˙ + 3 ( + 3P ) = 0
a
(B.25)
and hence that
The Fluid equation states that
73
and can be rewritten using P = we to give
ȧ
d
da 1
˙ + 3 (1 + w) =
+ 3 . (1 + w) = 0
a
dt
dt a
(B.26)
d
da
= −3 (1 + w)
a
(B.27)
which can be simplified to
Integrating this gives
da
a
(B.28)
log = −3(1 + w) log a
(B.29)
= a−3(1+w)
(B.30)
Z
d
= −3(1 + w)
e
Z
or
which can be further simplified to
b) By solving the Friedman equation for this generic equation of state, assuming a power-law
dependence (tq ), show that the scale factor evolves as
a(t) ∝ t2/(3+3ω) ,
(B.31)
assuming ω 6= −1.
The Friedman equation for a flat universe is
2
ȧ
a
=
8πG
3c2
(B.32)
Substituting in from the above question
8πG (−1+3w)
a
(B.33)
3c2
Now if we assume a powerlaw dependence of a ∝ tq , we can get the value of q by equation the
LH and RH sides, so:
2q − 2 = −q − 3wq
(B.34)
ȧ2 =
Which can be trivially solved for q to give
q=
2
3 + 3w
(B.35)
6) a) Outline the evidence for dark matter in galaxies and clusters.
Dark matter is primarily known to exist because of its dynamical (gravitational) impact on
galaxies. Principally the rotation curves of galaxies at large radii are flat, white the velocity
dispersions in clusters are much larger than expected based on the observed luminous matter,
suggesting a large dark component
74
b) Show that the rotation curve of a galaxy should follow the form
s
vorb =
GM
.
R
(B.36)
equating the centripetal acceleration with gravitational attraction
GM
v2
= 2
R
R
(B.37)
which can be re-arranged
s
vorb =
GM
.
R
(B.38)
c) Two possible origins of dark matter are WIMPS and MACHOS, what does each term
mean? Give an example of a possible WIMP and MACHO candidate.
WIMP stands for Weakly Interacting Massive Particle, and an example of this is the neutralino,
MACHO stands for Massive Compact Halo Object, an example is a black hole.
d) Modified Newtonian Dynamics (MOND) states that Newton’s law should be modified to the
form,
a
F = ma.
(B.39)
a0
in very low accelerations. Show that in this case MOND predicts flat rotation curves in galaxies.
Again equal centripetal acceleration, but this time allow for the additional term included via
MOND.
or
a2
GM
= 2
a0
R
√
GM a0
a=
R
(B.40)
(B.41)
If this is centripetal acceleration = v 2 /r then
v2
=
r
√
GM a0
R
(B.42)
v = (GM a0 )1/4
(B.43)
or
Which is a constant value.
75