November 3, 2003

The Angular Power Spectrum

The patterns in the cosmic microwave background are due to random processes, set up in the very early universe. As if at some early time, the universe became agitated, like a cauldron of boiling water and started to slosh around. But unlike with the boiling cauldron, we are not entirely sure about the origin of this randomness. Most cosmologists now believe that it is an imprint of quantum noise from the very beginning of time. It is a clean, elegant hypothesis but has not been completely confirmed. To describe random processes we need to use a branch of mathematics known as statistics. Instead of describing a random process in terms of the exact values that it takes at any point in space or time, we talk about the probability of taking a given value. Let us think of a simple example, the surface of an ocean or a lake. Suppose the water is very rough with waves and a lot of activity. We may be able to say that the average height of the waves is, for example one metre. This means that if we look at all the waves and measure their heights, we will find that there are a few very small ones and a few very big ones, but most of the waves have a height of about one metre. The average height of the waves is a statistical measure of the roughness of the surface of the water. But we might also like to ask how big the waves are, or how much of the surface of the water does a typical wave take up? This is different from the height of the wave. For example one might have very small, but high waves which look choppy. Or one might have very long waves and high waves. Or even very long and shallow waves. We then need to characterize the average spatial extent of the waves. Or even better, the average heights of waves of a given spatial extent. For example we may find that long waves that cover an appreciable part of the surface of the water are normally very shallow. Or that small waves that pop in and out of existence normally do so with gusto and are very high. The average height of the waves as a function of spatial extent will contain this information. Or to be more accurate the range in heights of waves as a function of spatial extent will tell us what are the typical sizes we may extent when we look at the surface of the sea.

Posted by Pedro Ferreira at November 3, 2003 11:34 PM