What is $\Omega$?

Here's the little bit that you need to know about cosmology to understand what follows.

One of the numbers we'd most like to know about the Universe is its density. This number, when measured in appropriate units (to be specific, when measured in terms of the ``critical density'') is called $\Omega$.

[Technical note: In this document, I will use the symbol $\Omega$ to stand for the total density of all constituents of the Universe, including any cosmological constant or ``dark energy.'' If you don't know what I mean by this, don't worry about it for now.]

One reason that $\Omega$ is an important number to know is that it's related to the geometry of the Universe: if $\Omega =1$, then the Universe is spatially flat, which essentially means it obeys the usual rules of Euclidean geometry. If $\Omega \ne 1$, then it doesn't. For this reason and others, lots of effort has gone into measuring $\Omega$ over the years.

For quite some time now, people have harbored a theoretical prejudice in favor of the value $\Omega =1$. That is, many people thought it was likely that the Universe was flat. In recent years, the chief argument people have made in support of this is that a certain family of theoretical models of the early Universe, specifically the ``inflationary'' models, generically predict that the Universe should be flat.¹ Inflationary models do a good job at explaining a number of otherwise puzzling observations in cosmology, so lots of people believe that inflation has a high probability of being right. If you believe that inflation is probably right, then your prior probability density on $\Omega$ will say that $\Omega$ is probably equal to 1.