Cosmology lecture 2: the Cosmological EquationsSorry for leaving last week's lecture notes until this week

Well we started off by getting a 60 page handout which is not available online -

here is just some of it!

There is not much I can really say about the cosmological equations at this stage. Mathematics not being my strong point, I am not going to try to explain the equations because I am not ready. Clearly they're going to need a lot of work. I will just reproduce the interesting snippets from my notebook. (In fact, if my maths was stronger, I would have applied to do this course long ago. I reasoned myself into applying by telling myself again and again that public speaking used to be one of the things I'm worst at, and now it's one of the things I'm best at, so hopefully I can do the same for maths with enough work . . .)

None of the cosmological equations are simple to solve at local levels. But at the level of the Cosmological Principle - at a scale of 100 Mpc - the Universe is homogenous, so we can neglect density gradients and indeed all gradients. At that point, things become much simpler.

In the past, we had the perfect cosmological principle. That added a fourth dimension to the homogeneity of the Universe: time. In other words, the Universe would be the same no matter where we looked

*and when* we look (past, present, future). Now we know the Universe is expanding, we have had to jettison this one.

100 Mpc is a huge area. But it is still only a tiny, tiny fraction of the observable Universe, whose radius is roughly 6 gigaparsecs! So we can still do good statistics with the Universe using a "cell" of 100 Mpc.

We started by looking at equations involving spheres and potential energy - taking into account radius, mass, etc. A sphere acts on such and such a mass with gravitational energy. This is very "primitive" - we get very different results by including all the spheres whose gravity is felt, etc.

A thought experiment: Imagine the Universe is infinite. Start integrating gravitational points across the whole thing. For example, right and left. If you add up everything in every direction, ultimately we get the same force acting on us from all directions, so we get stretched equally everywhere.

To do any simple calculations about a body on a sphere we need to assume there are no net forces from outside.

Someone once asked Einstein how he understood an infinite Universe. Was it a sphere with a centre? The centre is everywhere and nowhere.

Incidentally, at this point, we were told, there was no one more skeptical of Newtonian theories (I take it this meant Newtonian gravity) than Newton himself. (This is not the first time I have heard that a scientist didn't believe his own answer.) He found a Newtonian universe unsatisfactory. One of his laws states that force changes with distance, which made him unhappy, but he knew of nothing better.

Einstein's universe changed this a lot, as I expect most of you know. Newton assumed absolutes of space and time. He separated them. Einstein realised you couldn't do that: if you separate space from time, you get different speeds of light. For example, half of Earth's year would see light coming from such and such a direction as a different speed from the other half of the year. This effect was looked for and never found. We can now measure the speed of light to a 10

^{-12} accuracy.

We heard a lot about conservation of energy (kinetic and potential) - I think this had to do with the Friedmann equations - but let's put it this way, things were going too fast for me to write anything comprehensible here.

But anyway, even while a sphere (eg the Universe) expands, the law of conservation of energy applies.

The internal energy of an object is its pressure (the opposite of pressure is tension). Thermodynamics comes into this. If the Universe has the same temperature all over (to a 100 Mpc scale), there should be no heat transfer. Heat transfer occurs when there is a gradient. The energy inside a volume is, you guessed it, E=mc

^{2} - can't get away from relativity! We were told to imagine the

second law of thermodynamics as a consequence of relativity.

By this time the lecturer must have noticed we were sitting there looking a bit stunned (I certainly was). He asked us if we knew how Einstein had arrived at E=mc

^{2}. Well, he said, he had a look at the blackboard and he wrote E =ma

^{2}. That didn't look right. So he tried E=mb

^{2} . . .

Then it got serious again and we were informed that the scale factor describes the Universe as a whole, so we need . . . . another equation! (We were going through the equations but as I have said I am not writing any of them out, because I can't yet explain them. But it's the conservation equation - 2.16 on the handout if you want to look.)

When starting from the first law of thermodynamics, we get the conservation equation. Scale factor, pressure, and density are all relevant. If we can't solve one equation from two variables, can we solve two equations from three variables? Not just like that (apparently) - but we know what to do now: we need an equation of state, which relates pressure and temperature.

Now we can start looking at what's in the Universe: what are their pressure, temperature, density etc? Cosmic streams, black holes, dust, dark matter, plasmas . . . all these things have different properties.

Dust is very simple: the pressure is 0. If we have a Universe filled with dust, the pressure is negligible. Structure and stability can form in dust. (We get equation 2.19 for that. I am also wondering what this was about only 10 equations

)

In dark energy, we were looking at a constant called gamma which won't as far as I know paste here and which is not quite the same as pressure . . . but anyway, dark energy seems to have negative pressure. Anything that makes a certain part of an equation - pressure and gamma - end up negative will be dark energy, and that's what makes whatever it is unique.

In this phase transition, substances are interacting. We can neglect this to give each substance its own equation of state.

We can differentiate the Friedmann equation with respect to time.

Acceleration of the Universe is proportional to the scale factor.

Starting from Newtonian law, we have now taken conservation of energy into account, and also found Newtonian law neglects pressure. Integration still obtains the Friedmann equations, and makes pressure appear to disappear. For the record, I have not the faintest idea as yet what this means. "It is cheating," we were told, "but very clever cheating, because it obtains something very useful."

If the pressure is negative but the density is positive (and of course we cannot have negative density . . . I wonder what the density of a black hole is?), this is called tension. It is a commonplace thing, for example surface tension on water which allows insects to skip around on a pond. Pressure is proportional to kinetic energy, including heat, a collision, etc. (

Van der Waals forces are important here.)

On the border between energy and dark energy are

Cosmic Strings. These cause zero acceleration of the expansion of the Universe.

All in all, I am exceedingly confused, and think I will have to work through those equations quite a few times - which will be hard as I don't even have any values to put into them. Oh well. Fortunately I am not the only one who is confused, even the maths graduates looked how I felt . . . and besides, why go on a course where you already understand everything?