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