Welcome to the new decade. I hope you've all had some relaxing time and you probably have forgotten

everything we did in the first half of the semester. That's good. Thanks everybody for those of you who

did send me these bullet points and so on for the proposal round. I'll be getting back to you with

comments by the end of this week. If you haven't submitted me something about these proposals, you

can still do so. This was not a mandatory thing, but I would strongly suggest that if you plan to

participate in this whole proposal writing that you do so. Just because this forces you to think

about this stuff and that's probably a good start. Okay, so last time we saw each other we were

talking about how to build world models. We started with the metric, then decided that plugging in

metric into the Einstein equation that's shown here is a bit too difficult, so we chose the simple

Newtonian way to derive how the scale parameter which describes the evolution of the universe given

that the metric is fixed and doesn't change and can be parameterized only by the world time and the

scale parameter. And in the end what we did was we derived these equations for the temporal evolution

of the scale parameter and we ended up by taking a look at the different components in the Friedmann

equations written down like this and we saw that effectively what dominates the evolution here would

be the time dependent density, would be the pressure which if we write it in this form we can see as

another contribution to the density and the cosmological constant which if one looks at this

and we discussed this also scales with the scale parameter in the same way as these other two terms

and therefore can also be interpreted as being in relation to being a pressure that only depends on

nothing and therefore it's related to the pressure of the vacuum. We then briefly ended looking at the

Friedmann equations in the limit of a cosmological parameter of zero and we'll be doing this for the

rest of today but we'll be seeing later that this assumption is wrong but it makes it much easier to

understand what happens in the equations if we set lambda to zero for the moment. If we just take a

look at the Friedmann equation here and we write the Friedmann equation in terms of the Hubble

parameters so in terms of the ratio between the temporal change of the scale parameter r dot and

its value and then we saw that we got this equation here which we could solve for the curvature of

space and what we saw there is that the whole for a given value of the Hubble parameter the sine of

the curvature and therefore the question on whether the universe is open has an infinite volume or is

closed has a finite volume only depends on the density parameter here. And writing this in a

slightly different way there is a critical density therefore and the critical density is the one where

the left hand side of this equation equals zero so where space is flat and that is given by 3h squared

over 8 pi g and if the density of the universe is less than that then gravity is not strong enough

to stop the expansion of the universe if the density is higher than this critical parameter then

gravity is strong enough to fold the universe into a finite volume so to have curved space and because

this ratio of the proper density to the critical density is so important it has its own abbreviation

and this density parameter capital omega is one of the things that we will have to measure and we

also saw that this parameter is very small the critical density depending on the detailed value of

the Hubble parameter is a few hydrogen atoms per cubic meter this is not a typo there this is really

a few hydrogen atoms per cubic meter so the question on whether the universe is open or closed is

effectively the question on whether the universe is an ultra ultra ultra ultra ultra high vacuum or

only an ultra ultra ultra ultra high vacuum assuming that this was different numbers of ultras.

Okay so now this is something that we just see from the equation and the question now is how does

the universe really evolve right the boundary condition effectively is we know the current rate

of expansion that's something that we measure right so today we have a certain scale parameter

and we have a certain rate of expansion right so we know what our dot is today and the question is

what will the scale parameter be relative to today or what was the scale parameter in the past and

that only depends on the density so it depends again on the value of omega.

We'll first look at two simple cases and then we combine these two simple cases to have a more

generic feeling for what's going on so we will see also today or later that there are effectively

two cases of universes that are important and all others are mixtures of these two cases.

There's one case where the universe is dominated by material where it's matter universe where it's