Last month’s multiverse musings delineated some philosophical objections to the existence of actual infinities. However, an extremely large but spatially finite universe could still negate the significance of the fine-tuning arguments used in Christian apologetics. Today, I want to address an email question particularly pertinent to this issue. The question is:

How do scientists measure the total density of the universe, and also the split between conventional matter, dark matter, and dark energy?

The importance of this question relates to the fact that the only hard experimental evidence currently weighing in on the size of the universe is the geometry of the universe. The WMAP image of the cosmic microwave background (CMB) radiation provides the most potent tool for measuring the geometry although the large-scale structure also contributes significantly. In a nutshell, a closed universe cannot be spatially infinite.

The total density of the universe, Ω_{total}, defines the universe’s geometry. For an open universe, Ω_{total} < 1 whereas a closed universe has Ω_{total} > 1. Ω_{total} = 1 means the universe is flat. Additionally, a measured value of Ω_{total} > 1 also constrains the size of the universe (not just our observable universe). The best measurements of Ω_{total} indicate a closed universe but are consistent with a flat or open universe within the error bars.

To get from the all-sky WMAP temperature measurements to a measurement of the cosmological parameters involves three steps:

- Transform the CMB sky map into a set of numbers that characterize the fluctuations.
- Run simulations to see how changing the various cosmological parameters affects the numbers characterizing the fluctuations.
- Find the parameters that best fit the data.

The first step utilizes the common mathematical tool of spherical harmonics. Basically, you convert this into a sum of these , then you determine the amount of power at each harmonic and make a graph like this.

The next step is where all the physics enters. The CMB photons were all emitted at a common time (when the universe cooled enough for nuclei and electrons to combine and form atoms) roughly 380,000 years after the big bang. The temperature fluctuations measured by WMAP record a snapshot of the physical scale and gravitational distribution of the universe’s density. Multiple factors affect this distribution, such as the total density, the number of baryons, the relative amounts of the components making up the total density, and the spectrum of the initial density fluctuations. Using simulations, cosmologists can form pictures of the CMB fluctuations assuming different cosmological parameters. For a more detailed description of how these processes interact, see Wayne Hu’s tutorial.

The final step involves comparing the simulated CMB distribution of harmonics with the distribution calculated from the WMAP data. Max Tegmark’s CMB movies provide nice illustrations of how changing the various cosmological parameters would affect the measured distribution of harmonics. Once the best fit to the data is found, you have your measurement of Ω_{total} (and a number of other parameters). The current best values for some relevant parameters are:

- Ω
_{total}= 1.02 +/- 0.02 - Normal matter = 4%, Exotic dark matter = 22%, and Dark energy = 74%.
- Spectral index n
_{s}= 0.95 +/- 0.17 (inflation predicts a value less than one).

So now you know how to convert the WMAP data into cosmological parameters.