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# 17.13 Mean Cosmic Density

General relativity predicts that the geometry or curvature of space is determined by the mass content of the universe. This suggests that if we measure the mean density of matter in the local universe, we can determine whether it is above or below the critical density needed to overcome the universal expansion. Matter in the universe is lumpy — atoms coalesce into stars, and stars are bound by gravity into galaxies. The key to the concept of mean density is to imagine all the matter in the universe broken up into atoms and smoothly distributed through space.

The value of the critical density depends on the Hubble constant, since a faster expansion requires a larger density to overcome it. For an assumed value of H0 = 71 km/s/Mpc, the critical density is roughly 10-26 kg/m3. This tiny number is equivalent to 4 or 5 hydrogen atoms in a cubic meter of space, or, analogously, to the density of a grain of sand distributed over the volume of the Earth. The universe is fantastically empty! So all we have to do is measure the average density in a large volume and divide it by the critical density. This number is the density parameter, Ω0. It should be easy to determine & mdash; just count galaxies, add up their masses, and divide by the volume containing them to get the mean density. Then see if the mean density is larger or smaller than the critical density that is calculated using the best estimate of the Hubble constant. You might suspect that nothing in cosmology is as simple as it seems, and you are right!

What region of space represents a "fair sample" of the universe? Suppose you were conducting a census of a large country. If you counted the people in a 100 square mile area, you might get a very low density of people if the area was in the remote countryside. On the other hand, if your 100 square miles included a large city, you would calculate a very high density of people. Neither result would give a good estimate of the average density of people. You would need to take a number of samples covering the full range of living environments. It is the same with galaxies. Suppose astronomers survey a volume of only a few cubic mega parsecs. They might happen to choose a region of the universe with lower than average density and get a biased result. In fact, since the average distance between large galaxies is about 1 Mpc, they might even be unlucky and not find any galaxy in their survey volume. This does not mean that the density of the universe is zero!

Location of galaxies as a function of redshift from the 2df galaxy survey. Click here for original source URL.

The universe becomes homogeneous on scales of 300 Mpc and higher — that is, the number of galaxies does not fluctuate very much. Notice that 300 Mpc corresponds to a look-back time of about one billion years, which is less than 10 percent of the age of the universe. Therefore, a survey to this distance is still considered a "local" measurement and the issue of galaxy evolution is not important. Do we have to count galaxies across the entire sky to this distance? No, because the universe is isotropic and we will get the same census in any direction we look. Astronomers use surveys of slices, or "pencil beams," to measure the mean density of galaxies. A typical survey might have a depth of 200 Mpc and dimensions on the plane of the sky of 5 Mpc in each direction. The total volume is therefore 200 × 5 × 5, or 5000 Mpc3. The resulting sample will contain several thousand galaxies.

Astronomers have added up the luminosity of galaxies in a large region of space. The mass-to-light ratios of stellar populations are well determined, so the summed luminosity can be converted into a summed mass. Next, the total mass is divided by the volume to derive the average density. Finally, the average density is then divided by the critical density to get the density parameter. The result is a very low number, Ω0 ≈ 0.0002 to 0.003. In other words, stars make up only a few tenths of a percent of the critical density. We can use the cosmological principle to argue that our local volume is typical of every region of space. As remarkable as it sounds, the totality of stars in the universe — around 1021 of them contained in some 100 billion galaxies — does not come close to matching the critical density.

Temperature map of the universe, as measured by WMAP. Click here for original source URL.

Astronomers gain some insights by looking at the estimates of matter density on scales ranging from the nuclei of individual galaxies to galaxy super clusters. From models of cosmic nucleosynthesis, there is a narrow range of density for which the big bang predicts the correct abundance of light elements. The big bang model predicts this range of density from calculations that include all the conventional subatomic particles: protons, neutrons, electrons, and neutrinos. The density range predicted by nucleosynthesis calculations is Ω0 = 0.04 to 0.05, or 4% to 5% of the critical density. This leads us to two conclusions. First, the sum of all the stars in 100 billion galaxies only corresponds to Ω0 = 0.01, so the visible parts of galaxies only account for 15 to 25 percent of the matter predicted by nucleosynthesis calculations. In other words, most of the normal matter in the universe has not yet been detected! Astronomers speculate that this mass will be found as a thin gas lying between the galaxies. They predict that it will be heated to 100,000 K by ultraviolet radiation from young stars and rare but powerful quasars. Second, the density parameter from normal matter is far below the critical density. Normal matter is completely insufficient to close the universe and cause it to collapse in the future.

Although a census of visible matter indicates that the universe is open, we need to account for dark matter. At least 80-90% of the mass of most galaxies is dark. (Remember that dark matter particles are known to interact very weakly with normal particles and so they do not enter into the nucleosynthesis calculations.) Including this mass raises the mean density derived from stars by a factor of 6-7, implying that Ω0 = 0.06 to 0.07. This is still less than a tenth of the mass needed to close the universe. In addition, a lot of dark matter is found on scales larger than individual galaxies. Astronomers weigh dark matter on scales up to 100 Mpc by observing how galaxies are tugged around in response to the way dark matter is distributed. It is as if the galaxies are shiny marbles rolling on an undulating surface of black velvet. The undulations represent the high- and low-density regions of dark matter. The marbles will roll toward concentrations of dark matter and away from regions where dark matter is sparse. A larger excess of density will generate a larger motion. The map of these motions is used to weigh the dark matter. Including this mass raises the mean density by another factor of four, giving Ω0 = 0.25 to 0.30.

In 1998, measurements of the expansion rate of the universe made using supernovae indicated that some mysterious force is acting as a pressure to push the universe apart. The Hubble expansion is accelerating. Termed dark energy, it is now believed that the another 0.7 of the mass-energy density of the universe is made of dark energy. The sum of all these mass and energy components is close to the critical density, Ω0 = 1. This yields a flat geometry for space and time.