### Friedmann's Acceleration Equation

The second Friedmann equation describes the universe's rate of expansion, how quickly the expansion is speeding up or slowing down. In short the equation tells us that if there is matter present in the universe then either its expansion rate is decreasing or its contraction rate is increasing. Friedmann originally solved Einstein's field equations with the Robertson-Walker metric and found two solutions, one was the Friedmann equation and the second was an equation which when subtracted by the first Friedmann equation gave the second Friedmann equation. However I want to explain how I first learned about the acceleration which I believe is he most easy and accessible way to derive it. We still need to use the first Friedmann equation that I spoke about previously.

Now multiply each term of the equation by the scale factor squared and you get

If you differentiate this equation with respect to time

Then divide by

Then the equation becomes

Finally substitute in the following term from the fluid equation

This gives us the equation in its final form

Although there is no force associated with pressure in a universe described by the Roberston-Walker metric, i.e., one that's isotropic, as you can see from the equation if we were to increase pressure we would further decelerate the expansion. On occasion you will see cosmologists who have mass density replaced with energy density. If you use natural units where c = 1 these become interchangeable. Though this need not concern us here.

The interesting thing about this equation is that if pressure p is positive then I can make the right hand side of the equation positive (if I also make the left hand side negative), then the equation tells me that if there is any matter in the universe the left hand side cannot be zero. So in standard cosmology with the cosmological constant omitted the expansion of the universe must be slowing down (or we equivalently say the contraction is spreading up). Perhaps, though we might speculate that I n the early universe the material content is such that it made the pressure negative in which the expansion would rapidly accelerate. This is known as "inflation" and is accepted by most cosmologists, with still a few reasonable sceptics but we'll discuss that in another post. For now I'll leave it at that.

1. P.S. I've talked about the first Friedmann equation and the fluid equation in previous posts. For anyone wondering what these are.

2. Under the first equation, it should read "scale factor squared."

### Three Things William Lane Craig Gets Wrong About the Hartle-Hawking No Boundary Proposal

Classical standard hot Big Bang cosmology represents the universe as beginning from a singular dense point, with no prior description or explanation of classical spacetime. Quantum cosmology is different in that it replaces the initial singularity with a description in accord with some law the "quantum mechanical wave function of the universe", different approaches to quantum cosmology differ in their appeal either to describe the origin of the material content of the universe e.g., Tyron 1973, Linde 1983a, Krauss 2012 or the origin of spacetime itself e.g., Vilenkin 1982, Linde 1983b, Hartle-Hawking 1983, Vilenkin 1984.

These last few proposals by Vilenkin, Hartle-Hawking and others are solutions to the Wheeler-DeWitt equation and exist in a category of proposals called "quantum gravity cosmologies" which make cosmic applications of an approach to quantum gravity called "closed dynamic triangulation" or CDT (also known as Euclidean quantum gravity). I&#…

### How Should Thatcherites Remember the '80s?

Every now and again, when I talk to people about the '80s I'm told that it was a time of unhinged selfishness, that somehow or other we learned the price of everything but the value of nothing. I can just remember that infamous line from Billy Elliot; 'Merry Christmas Maggie Thatcher. We all celebrate today because its one day closer to your death'. If it reflected the general mood of the time, one might wonder how it is she won, not one but three elections.

In an era when a woman couldn't be Prime Minister, her launch into power was accidental owing in part to Manchester liberals and the Winter of Discontent. Yet I'm convinced her election victory in '79 was the only one that ever truly mattered. Simply consider the calamity of what preceded it, the 1970s was a decade of double-digit inflation, power cuts, mass strikes, price and income controls, and the three day week. Britain was sick, it needed fundamental restructuring but no one seemed to quite under…

### Can inflation be eternal into the past?

Back in 2003 a paper appeared on the arXiv titled "Inflationary spacetimes are not past complete" that was published by Arvind Borde, Alan Guth and Alexander Vilenkin which has had considerable amounts of attention online. The theorem is rather uninteresting but simple and doesn't require a very complicated understanding of math. So I thought I'd explain the result here.

It's purpose is to demonstrate that inflationary models are geodesically incomplete into the past which they take as "synonymous to a beginning" but Vilenkin stresses that the theorem can be extended to non inflationary models so long as the condition of the theorem that the average rate of expansion is never below zero is met. These models too then are incomplete into the past. Consider the metric for an FRW universe with an exponential expansion

Where the scale factor is

Since the eternal inflation model is a "steady state cosmology" the mass density and the Hubble paramet…