# A singular problem

*The Cosmic Microwave Background temperature fluctuations measured by the WMAP mission. The fluctuations correspond to about 0.0002 degrees. Image: NASA / WMAP Science Team*

## Mathematics in the search for the origins of the universe

When astronomers were first able to measure the velocities of stars and galaxies relative to the Earth, they noticed a strange thing. Almost all of them were moving away from us and the further they were from us, the faster they receded. You might be tempted to think that this is because the Earth lies at some special central point in the universe, but in reality it doesn’t. The universe is expanding everywhere and an observer at any point would see the exact same thing. The same is true when you look at the distribution of stars and galaxies and the distribution of the cosmic microwave background left over from the big bang. Aside from local structure, the universe looks remarkably similar in all directions. This is a property scientists call isotropy.

Why this should be so is one of the most intriguing questions in cosmology and a topic of great interest to Professor Susan Scott of The Australian National University. “There’s really nothing special about where the Earth happens to be,” Professor Scott explains, “so why we have this apparent isotropy is a really fascinating question. To answer it we have to better understand how the universe has evolved to the present time.”

Unravelling that puzzle is the job of cosmologists. Because light travels at a finite speed, when we look at distant objects we are essentially looking into the past. This is a major reason why astronomers are always looking for larger and more powerful telescopes; distant galaxies are very faint, but the further away they are, the closer to the beginning of the universe we see them. And of course looking at how galaxies formed over time helps us to understand the formation of the universe in general.

However observational astronomy alone, can’t solve the big questions of cosmology. To do that requires highly complex mathematical models into which scientists can plug data from astronomical observation and particle physics. There are a number of competing cosmological models but they all incorporate the idea of a big bang: a point at which the universe rapidly expanded from an infinitesimally small singularity.

A simple example of a mathematical singularity is f(x) = 1/x. When x = 0 the function becomes one divided by zero which is an undefined quantity. Although the mathematics of cosmology is far more complex, the essential problem is the same.

“Solving the mathematics of the big bang as well as the final state of the universe is complicated by the initial and possibly final singularities.” Professor Scott explains, “You can’t do much when things go singular.”

To get around this, Professor Scott and her collaborators have been working on nice regular cosmologies that don’t have singularities. Although these don’t directly represent the physical universe we live in, they can be designed to have what’s known as conformal relationships with the physical universe. What this means is that you can solve the maths in one universe and extract results that are meaningful in another.

“An early contender for modelling the universe was Chaotic Cosmology.” Professor Scott says, “This is a theory in which the universe began with a big bang and entered an exceedingly hot and highly chaotic phase before organising itself in the way we now see. The problem is that whilst this is a nice picture, it’s not really compatible with either observation or thermodynamics which requires that entropy increases with time.”

The more recent Quiescent cosmological model still incorporates the notion of a big bang but the key idea is that the gravitational field also has an entropy associated with it. So in effect you have an early universe in which the gravitational field is very smooth and has very low entropy whilst the first matter to condense had a very high entropy. As time progresses the situation slowly reverses, with the gravitational field increasing in entropy as the matter cools. Of course the combined overall entropy has to increase in accordance with thermodynamics.

“Our work enables us to solve some difficult problems in the Quiescent cosmological model.” Professor Scott says, “A while back we had been able to incorporate the initial singularity of the big bang. Now we’ve recently expanded that work to model the final singularity that might be seen in a big crunch scenario, where the matter of the universe contracts back under gravity to a single point again. However this work isn’t about predicting whether the universe will continue to expand or collapse into a big crunch. The maths works equally well for both scenarios so we have to leave it to the observational cosmologists to provide some numbers to plug in.”

The most recent observational work suggests that the expansion of the universe, far from slowing down with time as one might expect in a big crunch scenario, is in fact accelerating. Astronomers have dubbed this phenomenon dark energy in reference to the unknown force that may be causing it and it’s a hot topic in modern astronomy.

“I think there are three really big unsolved questions in physics at the moment.” Professor Scott says, “The nature of dark energy, the unification of fundamental forces and of course us, by which I mean are we unique in the universe or not?”

It may be some time before we’re able to answer any of those questions but it’s certain that when we do, mathematics will play a central role.

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