And with this we switch over to English. Some of you were not here yesterday because you thought we would do exercises and I have to apologize to the camera because no we did not do exercises but we did lectures on the subject.

So we're having a gap here but that is my fault not the fault of the camera because I forgot to tell the people that we were going to cancel them and when I realized that I forgot to say that we would cancel the exercises it was too late to do anything.

So the people who happen to watch this sometime in the future will not hear my pointed opinion about inflation and perhaps that's for the best.

But if you want the handouts and things like that it is possible to send me an email for the people who are here.

All the slides that we talked about yesterday are in the handout.

So what we did yesterday first was we talked about inflation which is an exponential growth of the universe over about 100 double times in length

starting about 10 to the minus 32 seconds after the Big Bang.

And this is a phase where people think it should be possible to have a phase transition in some particle that nobody understands that's known in flaton

that is a particle that's posited in many theories of quantum gravity things like that so there are enough ways to produce these inflatons if they exist.

For a time the cosmological constant which is the energy density of vacuum is being switched on to a higher value

which leads to an exponential increase of the volume of the universe where increase means

why don't you take this to the back

where exponential increase means roughly a size change from the size of a pea to the size of one astronomical unit.

And that effectively leads to a flattening of the universe if you want.

In the sense that if you take a pea and you make it to the size of one astronomical unit you are not going to see any curvature locally

and that explains why the omega value of our universe is close to unity.

So what this means then is we need to find what the real value of omega is for the universe and in order to do so

what we have to do is well we have to count what is the stuff that is gravitating and we have to count other things that lead to omega

and that is mainly well the energy density of vacuum.

So what we then did in the last 25 minutes of yesterday's lecture is we went through the different ways on how to determine gravitating material

and the conclusion of that is essentially we apply methods that we already talked about before Christmas.

We use the Virial Theorem if you don't believe it. We look at Bremsstrahlung emission from galaxy clusters

and if you don't believe that I briefly talked about gravitational lenses.

The methods aren't that important right now. The important thing is what comes out is a consistent value

that the amount of material that is gravitating and that is what we call omega m is roughly 0.25.

So 25% of the matter density that is required to close the universe.

Out of this 0.25 about 10% is in baryons and that is well constrained from Big Bang Nucleosynthesis.

So the rest is dark matter and we don't understand what it is. End of story here.

So if we believe in inflation then well we need to find an omega lambda of 0.75.

If not inflation is dead because it is very difficult to produce a universe that say has an omega of 0.8 or so.

That would be very difficult but asymptotically we require asymptotically flat.

So just to make a point about dark matter because that often leads to confusion.

So I just said omega m is 0.25 or here in this slide I say 0.3.

The details don't matter and I just forgot to update this.

So we have two types of dark matter. That is the important thing.

One thing is we know how many baryons there are. That is an omega of 0.02.

We know this from Nucleosynthesis.

We saw that the Nucleosynthesis calculations give us a very very narrow range of omega in baryons that is possible.

That is consistent with all that we see.

We find that both from Nucleosynthesis calculations and comparing those with measurements of primordial abundancies.

So abundancies of stars that haven't conventionally changed the elemental composition.

The composition of clouds between galaxy clusters that were never in stars and so on.

So the outcome of this is omega in baryons is 0.02.

But if you now count how many baryons we really see we find that the total amount of baryons that are observable is less than 0.02.

That is not too surprising because yesterday for example we saw that in galaxy clusters there is a large amount of hot material that is between the galaxies.

About the same amount of mass than what we have in galaxies is in the hot gas between galaxy clusters.

It is easy to hide material by just having a very very tenuous medium, a medium with very low density that is hot.

The reason for this is if you have a medium that has a low density it will not radiate a lot of its energy away.

The fact for this is just that the emissivity that material that an ionized plasma has will always be bremsstrahlung-like.

So it will be proportional to the square of the particle density.