Alright everyone.
Like all of you this is my first time here and conferring with my colleagues as I have, I think I am starting off
with a relatively technical talk compared to the other 2 of this morning.
But I know we're bright eyed and bushy tailed as we say.
And you'll be ready to go through and figure out how cosmology works.
So my goal here is to present normally an 18 lecture series in 29 minutes and 56 seconds. So without further ado.
The first observations from my perspective done on cosmology were made by Vesto Slipher.
A cosmologist almost none of you have heard of but you should know because he was the person who really started it all.
He went out and looked at galaxies and he saw on average their spectra were red shifted, they were stretched to the red.
Now followed on by this Hubble who had access to the largest telescope of his day went through
and looked at Slipher's galaxies and looked at the stars in them and used the brightness of the stars
to measure relative distances. The fainter the stars the further away.
And what he found in 1929 was that the faster the galaxy was moving away or the more its light had been stretched,
then the further the galaxy was away.
And so from that observation Hubble, it turns out, had a pretty good PR machine, he went through with this data
and said this means that the universe is expanding.
Now George Lemaître in 1927 had done more or less the same work and had done all the theoretical underpinnings of it as well.
George showed Einstein his work in 1927 and Einstein said to Lemaître that...
"Your mathematics looks just fine but your physics is abominable." ...because he predicted that the universe was expanding.
He also predicted that the universe started in what he called a primeval atom or what we would now call the Big Bang.
So, science is sometimes subject to revision and, you know, if this ever happens to you,
you just have to accept the fact that you're doing good work and move on. So, the theoretical underpinning comes from Einstein.
In 1907 Einstein had what he described as his greatest thought, the thought that acceleration due to gravity
and acceleration due to motion are equivalent.
Now it took Einstein 81/2 years working on essentially nothing else, to reconcile that and the basic ideas of special relativity.
And what emerged was his equations of general relativity.
And from those of course he was able to predict curved space, he was able to go through
and solve cosmology for the first time, something that Newton was never able to do with his laws.
The solutions to Newton's equations are that the universe has nothing in it.
That's the only viable solution if you try to understand a universe that goes on and on.
So, the problem Einstein had in 1917 when he tried to solve cosmology was the following.
It showed that the universe should be in motion and in 1917 it was 12 years before Hubble
or 10 years before Lemaître's measurements and so he invented something, the cosmological constant.
And this was a term which could offset matter in the universe and Einstein thought stabilised the universe
against collapse or expansion. It turns out it doesn't work mathematically. The equations still remain unstable.
And so it was a mistake. But of course in 1929 when it was realised that the universe really was expanding,
then it really was a mistake because it was something where Einstein could have, amongst everything else he did,
predicted the expansion of the universe. So, we'll come back to that.
So, from a mathematical point of view the work really got going when a person by the name of Friedman
in 1923 looked at Einstein's equations. And he started off with general relativity. And he added an assumption.
And this assumption was important because it made Einstein's equations, which are notoriously difficult to solve, quite tractable.
And that assumption was that the universe was homogenous and isotropic on large scales.
So that means that every part of the universe is on average like every other part of the universe.
And there's no preferred direction to space.
And with that you can take Einstein's equations and essentially bring them down to 2 nice simple differential equations.
And so these are known as the Friedman equations and these are the ones that Lemaître actually re-derived from scratch in 1927
as part of his PhD thesis. And all of Einstein's equations can break down for my purpose to this single equation today,
a nice relatively simple differential equation.
Now, in this equation you will see a few things which we in cosmology refer to as the scale factor, this term A.
So that tracks how big a piece of space is over time.
If the universe is curved as we'll talk about, it gives the radius of curvature of the universe.
Of course, we know what the gravitational constant is, we have density. And then we have this term called K.
Now before I go on to K I will say that the scale factor is observationally linked to the red shift that Slipher saw.
Indeed they really correspond exactly to each other.
As long as you get far enough away, you don't have to worry about the local effects of gravitational attraction.
And so I will interchangeably use scale factor and red shift just remembering that they're essentially the inverse of each other.
Alright, so let's talk about curvature. Of course, space can be curved and within general relativity you have a heavy universe.
That is one that has a lot of stuff in it. Space curves onto itself in sort of a spherical geometry and that's the K=+1 term.
On the other hand a light universe: that K has a value of -1.
And then there's the just right universe, that's the one that has K=0.
Alright, so we put these things together and we have to add a couple of other things
that we talk about all the time within cosmology. One is the Hubble parameter.
So, this is what we would refer to as the expansion rate of the universe.
So it's the time derivative, the normalised time derivative of the scale factor.
And over the last 20 years we've sort of finally settled down to what the value of the Hubble constant is.
And it has units for astronomers of 71 +/- 4 km/sec/Mpc.
And if you take, as you will see, the inverse or 1 over the Hubble constant, that has units of roughly 1 over,
is 14 billion years. The other thing we need to worry about is something known as the critical density.
The critical density provides how much material in the universe you need to make the universe transition from being open,
K=-1 to closed K=+1. If you have this amount you're obviously at K=0. And finally...
And if I calculate that from the Hubble constant and it turns out you can derive this critical density as the Hubble constant
and the few other factors including the gravitational constant.
And you can see that the critical density of the universe for the value of the Hubble constant is roughly 9*10^-27 kg/m^3.
So that's a very low number.
And it tells you in this room where the density is more like 1 is going to be a very different or much,
much denser part of the universe than normal.
And finally we describe the actual density of the universe compared to this critical density using a term called Omega.
So Omega we call the density parameter.
And essentially this sort of tells you how much stuff there is in the universe.
So that's most of the vernacular we need to understand what's going on in the universe.
So, if you go through and solve Friedmann's equations, then...
And we only assume that the universe has got stuff in it, like atoms or something else that has attractive gravity,
then you get parametric solutions. So the equations, although it's quite a simple differential equation.
The solutions are a little messy. And I show them here for you, for the K =-1 and 1 you get these parametric solutions.
The flat universe, you get a nice simple solution. So graphically let's look at what these things look like.
You have a universe which has nothing in it, empty universe. The solution is essentially just uniform expansion.
Universe is in motion and nothing changes, keeps on getting bigger and bigger at the same constant rate.
You will notice that the time between now and when things, when the scale factor was 0 is 1 over the Hubble constant.
I've told you the Hubble constant has a value such that 1 over the Hubble constant is about 14 billion years old.
So when you measure the Hubble constant you are effectively measuring something related to the scale of the age of the universe.
On the other hand if you have a universe that has less than the critical density, so a mega matter between 0 and 1,
then the universe is going to have a curved trajectory.
It is going to continue to get bigger and bigger and it will expand forever. It has that open geometry.
So it is infinite in time and infinite in space. It has a characteristic age which is a little shorter than 1 over H0.
So, for example, if you go to a mega matter equal 1 it turns out then the age of the universe is 2/3 times the inverse
of the Hubble constant. Finally, you have a universe which is heavy.
So this is a universe which is finite in time, a universe which expands and then begins to contract.
It is also finite in space, it wraps around on itself. It has a finite volume.
And so will this universe begin in a Big Bang, this universe of course ends in a "Gnab Gib" which is the Big Bang backwards.
So this is a universe that always appealed to most people. When I was younger I certainly liked this universe.
This was the universe I wanted to live in. So it's a good thing to go out and measure and see what the universe does.
Because the one thing I have learned in life is that one should not prejudge the universe.
The universe does what it wants and our job as scientists are to figure it out rather than to assume
it should be the way we want it to be. Now it turns out the universe may have stuff in it other than normal matter.
And we describe how the different forms of matter interact with gravity by their equation of state.
So the equation of state describes how the density of material reacts as space expands.
That's the... What we call its gravitational pressure is more or less what you're referring to in this case.
So, for example, normal matter has an equation of state. We say it's pressureless.
And in this case its density goes inversely of the volume of a piece of space.
If you want to think of it another way you can say that its density goes as the inverse cube of the scale factor.
So, fine. But what about a photon? Well, a photon is different.
If you have a photon in a box you have a single photon in the box so its density, its number density behaves just like an atom.
But the photon gets stretched by the box. And so as you stretch the photon its wavelength decreases and its energy decreases.
And so its energy density drops more quickly as the universe expands compared to a photon. It has a different equation of state.
And so its density goes as the volume to the -4/3 or conversely it goes as the inverse of the 4th power of the scale factor.
And finally you could have something like Einstein's cosmological constant.
So rather than thinking of the cosmological constant like Einstein described it,
we now think of the cosmological constant as being energy that is part of the fabric of space itself.
And so in this case, if I change the size of the box, its density remains constant.
And so it has an equation of state such that it's invariant with the scale factor.
And that means in the lingo of this W, it has an equation state of -1. Alright, the other thing we need to worry about is...
We talked about how density and geometry related.
Well it turns out all forms of matter combine to give you the total density of the universe.
And it doesn't matter if it's a photon, if it's an atom or it's a cosmological constant.
So when I want to look at the geometry of space, I have to add together everything that is in the universe.
The other thing to know is, if the universe is born flat, it stays flat.
If the universe is open, you cannot make something finite in the infinite and vice versa.
So it makes sense, if the universe was more open it will stay open. If it's born closed, finite, it's going to stay closed.
Now it may it turns out approach, asymptotically approach being infinite if it's finite.
But in general you stay what you were born as. So, to think about how material interacts with Friedmann's equations...
Let's take Friedmann's equation and make a simplification. We're only going to worry about the universe which is flat.
If I do that, I can get rid of this K term and I can rewrite replacing the critical density here.
And I can make a nice simple differential equation.
This is a differential equation that is easy to solve without resorting to looking up things in mathematical handbooks
like the previous solution. So let's take this as our basic Friedmann equation for the universe.
If you have a universe that is full of normal pressureless matter like atoms and you take the Friedmann equation
and I make the substitution that the scale factor and the density are inversely proportionate to the third power,
then I can substitute that into this equation and I get a differential equation which I can simply integrate.
If I integrate that differential equation, I find that the scale factor goes as the time to the 2/3 power.
The universe gets bigger and bigger and bigger as the T to the 2/3 power.
So it essentially goes on forever asymptotically reaches a maximum size.
So, that is a solution if the universe is full of what we're made out of.
On the other hand, if the universe were primarily made up of photons, then I have a different substitution.
I have a different differential equation and I get a different behaviour.
The universe expands as T to the 1/2 power. If I add a cosmological constant, again a different substitution.
A different differential equation and in this case rather than an asymptotic behaviour
I get an exponential growth of the universe. A universe full of the cosmological constant exponentially expands.
So it expands faster and faster, you notice the other ones slow down over time.
Alright, the other thing we need to worry about is the constituents of the universe.
If the universe is, for example, full of photons and atoms, you note that the density of photons is increasing
as the fourth power of the scale factor. Because when I scrunch the photons they become more energetic.
V, atoms, increases the third power of the scale factor.
And the cosmological constant of course doesn't increase at all back in time.
And so what is going on in the universe today may be different to what is going on in the past.
Because the ratio of what is dominating the universe, there's more than one thing in it, will change back in time.
And so you can see that we have different...
The ratio, for example, radiation to matter will scale as the red shift or the scale factor, inverse scale factor.
Another set of rules when you do these solutions is that accelerating models which I'll talk about in just a second tend
towards flatness over time.
That is if you have something like a cosmological constant in it and the universe is born open or closed,
it will asymptotically tend to becoming flat. That's the way the equations work.
If you have a universe full of stuff with equations of state of greater than -1/3 such as atoms and photons,
those asymptotically drive things away from flatness.
You always end up with an infinitely dense universe or an Omega=0 universe in those universes.
So, as you guys know, we think the universe has matter, radiation and apparently a cosmological constant.
And so right now we appear to be in a time as we'll discuss right here where the universe is beginning to exponentially expand.
But not too long ago in the past it would have been dominated by matter and expanding as T^2/3.
And in the more distant past radiation would eventually have dominated the universe.
Because it becomes more and more dense relative to everything else back in time.
So the universe in the distant past would have been expanding as T^1/2. Alright, the deceleration parameter.
One of the things we talk a lot about in my work of course is that the universe seems to be accelerating.
And so we have a parameter that describes, essentially it is a normalised version of the second derivative of the scale factor.
So it tells you how the scale factor, the rate of change of the scale factor, is changing over time.
And it turns out that this unitless measurement has a value of 0 if the universe is coasting, if the universe is empty.
It also turns out that if the universe has something with an equation of state of -1/3 that causes the universe
also not to accelerate or decelerate. Finally...
And something that would have an equation of state of -1/3 it turns out would be like a cosmic string.
It has energy per unit length and that has an equation of state, it turns out, of about -1/3.
If you have a universe which is radiation dominated, it slows down very quickly. It has a high rate of deceleration.
A universe that has got a cosmological constant accelerates, has a Q0 negative.
And the reason there's a negative sign out here, of course, is that when this parameter was defined,
we were very sure that the universe was decelerating. We wanted to call it the deceleration parameter.
And so they added a minus sign so that we were always working in positive units.
Which adds kind of the ridiculous situation now that we have a negative sign in front of something
that we call the deceleration parameter.
So we should probably redefine it, just call it the acceleration parameter, and get rid of that minus sign.
But history is hard to change. Alright. So, let's talk about some observations that constrain our theory.
Here is the cosmic microwave background. The sky is filled with the 2.73 degree black body.
It's uniformed to 1 part in 10^5. So the temperature of the universe is going to scale as this temperature times (1+z).
Now I haven't proven that but you can think that photon energy density is rising as the fourth power of the red shift.
And a black body's energy density rises to the fourth power of the temperature.
And so you get this nice little relationship which you can easily prove.
The density of the universe is of course rising as the red shift cubed or the inverse scale factor cubed.
And so you're going to have, if you look back in time, a time when the universe is both very hot and dense.
And hot and dense is like the centre of the sun, it's a nuclear reactor.
So, one of the first things we could do with this model is go through and predict
how much stuff should have been synthesised in the early universe.
You get hydrogen, helium, deuterium, some forms of helium which aren't easy to see, lithium, beryllium, also hard to see.
And so when you can go through and compare this to data that we have and I show here
the Big Bang nucleus synthesis calculations done as these lines for lithium, deuterium and helium fractions
and then some bounds on what the measurements are in our universe for the primordial abundances of those.
They're not easy to make. But you can see there's a nice consistent answer right there in the middle.
And the thing that we have to worry about is essentially this thing we call Omega baryon
or effectively atomic matter times the Hubble constant squared. That tells you essentially the density of...
ratio of density of photons to atoms in the universe.
So, if we want to connect to other observations we have to add something else. And that's geometry.
And to talk about geometry in astronomy, cosmology we use something called the Robertson-Walker line element.
And this essentially connects information in the universe. So, we want to find the distance between 2 points.
We have to worry about how far apart they're separated in time.
We have to worry about the coordinates in this space that's expanding.
We have to worry about the curvature and we have to worry about the dynamics of the universe,
how fast the universe itself is expanding.
And so it's not an easy piece of geometry because if I want to connect from those 2 points,
I got to worry about the curvature and how fast things are expanding and when those points are separated in time.
The Robertson-Walker line element takes care of that. It's only geometry. Has nothing to do with general relativity at all.
So, this allows us to go through and then make other observations that we might be able to connect to as an astronomer.
So, I like to measure how bright things are. This is what Hubble did.
We have something defined as how bright an object is as a function of red shift.
So it turns out if you go through and use the Friedmann equation and the Robertson-Walker line element.
You put it all together; you get a fairly nasty looking integral equation that tells you the answer.
It's not as bad as it looks, partially because it's easy to integrate functions now on a computer.
But it really is quite is simple in that.
How bright an object is depends exclusively on what is in the universe, its equation of state and how much.
So you'll see you have Omega. So what fraction of the critical density every form of matter is.
Accompanied with that form of matter's equation of state you have to worry about the curvature of the universe
which is just the sum of all the different density parameters.
And then everything is scaled out in front by this Hubble constant, the current expansion rate of the universe.
So, if you go through and graphically plot what these equations look like, you can see that universes have...
Things get fainter as you go further and further back in time.
And you can also go through and prove that the size that a ruler looks is the same answer as the luminosity distance.
You just have to scale it by a factor of 1+z^2.
And so, if you look at how big a ruler looks, as you look back in time things get smaller and smaller up to a point.
At about a red shift of 1 to 2 it turns out that the ruler starts looking bigger.
And that's effectively because the photons are travelling through a universe which was smaller in the past.
And so you're effectively getting this bending of light looking back into the past
that makes suddenly start appearing bigger as you go further and further back in the past. So we can go through and look at that.
Now, one of the first things when you go through and look at how big things look back in time is you can make a diagram here
where this is sort of like a... Well, this is a diagram showing us how big a horizon is, how big an area we can see right now.
So this tells us a co-moving distance. So that's a distance right now in current units of today's universe.
And then we go through and take out effectively the expansion here and so we get something called conformal time.
And this allows lines on this diagram to be straight.
It's a useful diagram because right now when we look back, we're right here, we look back,
we see objects that are separated by a huge distance and time.
If we go back to the cosmic microwave background, it turns out that is at a red shift of roughly 1,000, its 1,097.
And if you put that on this diagram and look at what the cosmic microwave background would see,
you would see that the light cones, or the cosmic microwave background only see a tiny little area because that's a long ways.
And this part of the universe we see looking out that way. And this part of the universe looking out that way can't see.
They don't share any information. They can't see the same parts of the universe.
There is no way for information from this part of the universe to this part of the universe to transfer.
And yet there's this amazing effect that the universe has the same temperature that direction as that direction.
Alright, so that was a bit of a mystery.
And the way it's been solved is by having the universe do something very funny early on, which we call inflation.
The idea is if you stretch out and you have exponential expansion in the very early times of the universe,
then you can go through and have those light cones overlap in the future.
And essentially the idea is you have a nice universe expanding as T^0.5.
And before that you suddenly have a time when the universe exponentially expanded. Why? We're not exactly sure.
We can guess that for some reason there was something that mimicked a cosmological constant in the early universe.
And for some reason it disappeared. The whole theory of that is known as inflation.
I'm not going to talk too much more about it except for it does do a couple of other nice things about the universe.
Here is a map of the galaxies in the nearby universe.
And this is what the nearby universe should look like if there were no primordial seeds to our universe.
There would be no large scale structure in the universe.
There would just be random fluctuations and we would live in a much less interesting universe.
Well, quantum fluctuations can provide the seeds of structure if the universe went through this period of inflation.
So the idea here is that quantum fluctuations produce real fluctuations if you have virtual particle pairs,
find themselves separated by more than... well beyond the horizon.
And so that suddenly allows these quantum fluctuations to become real and populate our universe
when the universe is expanding very, very quickly and give us the seeds of structures.
So, in this case if the universe is expanding very rapidly...
This can happen quite a lot where these particles find themselves separated.
And of course this exponential growth will drive the universe to flatness as I told you earlier.
That's a characteristic of a universe that is accelerating.
So during the '70's and '80's we had a model that came out known as inflation and cold dark matter.
It has the nice thing that it explains the uniformity of the cosmic microwave background,
provides the seeds of structure formation, predicts that the universe should be flat.
Now, it turns out we also had this issue that trying to make galaxies form we couldn't do it just with atoms,
we needed something else. And we called that cold dark matter or dark matter.
We need this stuff not to interact with atoms. It really needs to only interact by the weak force to make it work.
It gives the structure formation of the universe be it gravity and is consistent with the idea that when we look at galaxies
we don't see material to explain the gravity we see at large, large distances.
So, in 1994 there were different ways of looking at the universe.
Theorists really liked the idea of inflation and cold dark matter, which predicted that the universe should be flat.
On the other hand observers kept on measuring things and we found that the universe didn't seem to have enough matter in it
to be flat. And in some sense the difference was really that we were happy to get rid of inflation and cold dark matter
where the theorists weren't.
So, I'm going to skip that and say that with type 1A supernovae, objects which explode and act as candles,
standard candles if they are modified, and this modification was figured out by a team in Chile,
you can go through and measure distances with these, quite accurately in a relative sense.
And this was demonstrated by this Chilean team. And here is a Hubble diagram.
You can see it looking much better than Hubble's Hubble diagram.
And these objects provided the basis of measuring, essentially the expansion history of the universe.
And so in 1998 we were able to take these observations and look at the expansion history of the universe.
And we found that the only thing that would make sense was a universe that was accelerating back in time.
And so since I am running out of time, I'm going to skip ahead to the very end and say:
In the end when we put everything that we have together and talk about this more in our extra sessions.
You're left with a universe with general relativity, isotropy and inflation which provides the initial conditions
to make the universe sane, which gives us flatness, structure seeds and a uniform cosmic microwave background.
We need to have 41/2 % of the universe be atoms, 22.5 % of it to be cold dark matter and 73% of it to be dark energy.
The Hubble constant is 71 km/sec/Mpc plus or minus about 4.
And it predicts the expansion and history of the universe correctly, the growth of the structure of the universe correctly,
the geometry of the universe, the age of the universe, something I haven't talked about but the acoustic structure
and the cosmic microwave background and big bang nucleus synthesis.
It turns out there's one problem in that which I won't talk about now.
The principle issue that we have with this model is it works beautifully but it does require us to invent 95.5 % of the universe.
And I will end there.