Chapter 17: Cosmology

In the big bang model, the expansion of the universe is slowed down by gravity. If there is enough matter in the universe, then the expansion can be overcome and the universe will collapse in the future. The density of matter that is just sufficient to eventually halt the expansion is called the critical density. The equation for the critical density is:

ρ_crit = 3H₀² / 8πG

You can see that the critical density is proportional to the square of the Hubble constant — a faster expansion requires a higher density to overcome the expansion. How empty is space? Astronomers have learned that the mean density of matter in the universe is within a factor of a few of the critical density. We can calculate ρ_crit by inserting the gravitational constant, G = 6.67 × 10^-11 Nm² / kg², and adopting H₀ = 65 km/s/Mpc. We first convert the Hubble constant to metric units, H₀ = (65 × 1000) / 3.1 × 10²² = 2.1 × 10^-18s^-1. Now we can solve to get ρ_crit = 3 × (2.1 × 10^-18)² / 8 × 3.14 × 6.7 × 10^-11 = 7.9 × 10^-27 kg/m³ ≈ 10^-26 kg/m³. This amazingly low density is equal to about five hydrogen atoms per cubic meter, or a single hydrogen atom in a volume the size of the average TV set. However, in that same cubic meter, there will be 5 to 10 billion low-energy photons of microwave background radiation, the feeble relic of the big bang.

How well does this density match up to the density of large objects in the universe? The typical distance between large galaxies is about 2 Mpc, so we can see what the average density would be if a galaxy like the Milky Way were the only object in a box 2 Mpc across. The mass of the Milky Way, including dark matter, is about 10¹² M_Sun, or 10¹² × 2 × 10³⁰ = 2 × 10⁴² kg. The volume of a 2 Mpc cube is (2 x 3.1 x 10²²)³ = 2.4 x 10⁶⁸ m³. The typical density is therefore mass divided by volume, or 2 × 10⁴² / 2.4 x 10⁶⁸ ≈ 10^-26 kg/m³. Thus we can see that the typical spacing of galaxies corresponds to a density roughly equal to the critical density.

The density parameter is the ratio of the observed density to the critical density:

Ω₀ = ρ / ρ_crit

The density parameter is therefore a number with no units. If ρ > ρ_crit, giving us a value of Ω₀ > 1, then the universe is closed and it will eventually collapse. If ρ / ρ_crit, giving us a value of Ω₀ < 1, then the universe is open and it will expand forever. The special case where ρ = ρ_crit and Ω₀ = 1 represents a flat universe, which will expand forever at an ever-decreasing rate.

How do the actual measurements of matter density compare to the critical density? Mass measurements on the largest scales give Ω₀ = 0.2 to 0.3, or a few tenths of the critical density. However, most of this mass is in the form of dark matter! The density of normal matter is actually five times lower, corresponding to Ω₀ = 0.04 to 0.05. Therefore if the critical density is five hydrogen atoms per cubic meter, our universe only contains one hydrogen atom per 4-5 cubic meters. The density is a single hydrogen atom in a volume the size of a small bathroom. The rest of the mass is made up of ubiquitous dark matter — we do not yet know what kind of particles they are.

How does cosmic background radiation in the universe fit into calculations of critical density? We have seen that the microwave background radiation consists of 10⁹ photons per cubic meter. There are ten billion times more photons than hydrogen atoms in the universe. We can calculate the equivalent mass of all this radiation using the formula E = mc². The result is that all the microwave photons in space are equivalent to 5 × 10^-31 kg/m³, a density that is 20,000 times lower than the critical density. In other words, the density parameter from big bang radiation is only 1/20,000 = 0.00005, far lower than the density parameter from matter. Despite the huge number of microwave photons, the universe is truly dominated by matter.