# Section 5: Beyond Hubble's Law

Figure 11: Dark energy is now the dominant factor pushing the universe to expand.

The supernovae results by themselves show that the universe is accelerating, but they don't say exactly what is causing it. Nor, by themselves, do they tell us how much of the universe is matter and how much is the agent causing the acceleration. The supernovae measure the acceleration in cosmic expansion, which stems from the difference between the component of the universe that makes things speed up () and the component that makes them slow down (). Apparently, now has the upper hand in the cosmic tug-of-war, but we'd like to know how much of the universe is in each form. We can obtain a much more complete idea of the contents of the universe by including other strands of evidence.

To determine the cosmology of the universe we live in, the two most effective pieces of information are the geometry of the universe, which tells us about the sum of the amount of matter and the cosmological constant (or whatever it truly is) driving the acceleration, and direct measurements of the amount of matter. In Unit 10, we learned that the cosmic microwave background (CMB) gives us direct information on the total amount of matter in the universe. It turns out that the CMB also gives us an excellent measurement of the geometry.

## Density, geometry, and the fate of the universe

Einstein's theory of general relativity describes the connection between the density of the universe and its geometry. A particular value of the cosmic density, called the critical density, corresponds to flat space—the geometry of Euclid that is taught in schools. For cosmic expansion, this term depends on the current rate of expansion—the Hubble constant. More precisely, the critical density is given by , where G is the gravitational constant that we first met in Unit 3.

The arithmetic shows that, for a Hubble constant of 70 km/sec/Mpc, = 9 x 10-27 kg/m3. This is a significantly small number compared with the emptiest spaces we can contrive in a laboratory on Earth. It corresponds to about five hydrogen atoms in a cubic meter. Modern evidence, especially from the analysis of the cosmic microwave background that we encountered in Unit 10, shows that our universe has the geometry of flat space, but the sum of all the forms of gravitational matter is too small to supply the needed density.

Figure 12: The geometry of the universe depends on its density.

Astronomers usually compare any density they are measuring to the critical density . We call this ratio omega () after the last letter in the Greek alphabet. (We should use the last letter to describe something that tells us how the world will end.) So = —a pure number with no units. The total density of the universe is simply the sum of the densities of all its constituents, so is equal to the matter density that we discussed in unit 10 plus the density of anything else out there, including the energy density associated with the cosmological constant, . A value of less than one means that the universe has an "open" geometry, and space is negatively curved like the surface of a saddle. If is greater than one, the universe has a "closed" geometry, and space is positively curved like the surface of a sphere. And if equals one, the geometry of the universe is that of flat space.

Modern evidence, especially from the analysis of the cosmic microwave background, shows that our universe has the geometry of flat space, but the sum of all the forms of gravitating matter is too small to supply the needed density. The energy density associated with the cosmological constant, we believe, makes up the difference.

## The anisotropic glow of the cosmic microwave background

The measurement of cosmic geometry comes from observations of the glow of the Big Bang, the cosmic microwave background (CMB). As we saw in Unit 10, Bell Labs scientists Arno Penzias and Robert Wilson observed this cosmic glow to be nearly the same brightness in all directions (isotropic), with a temperature that we now measure to be 2.7 Kelvin.

Although the CMB is isotropic on large scales, theory predicts that it should have some subtle texture from point to point, like the skin of an orange rather than a plum. The size of those patches would correspond to the size of the universe at a redshift of 1,000 when the universe turned transparent, and the radiant glow we see today was released. We know the distance to these patches, and we know their size: The angle they cover depends on how the universe is curved. By measuring the angular scale of this roughness in the CMB, we can infer the geometry of the universe.

Figure 13: The cosmic microwave background, shown here, has subtle texture from point to point.

The technical difficulty of this measurement is impressive: The variations in temperature that we must measure correspond to about 0.001 percent of the signal. However, in 2000, astrophysicists measured the angular scale well from Earth and better from the WMAP satellite three years later. The angular scale is just about 1 degree, which corresponds with amazing precision to the angle we would measure if the geometry of the universe as a whole were Euclidean. To an uncertainty of just a few percent, the universe is flat, and + is 1.

The results from Unit 10 are consistent with of about 1/3. If the total of and is 1, as the angular scale of the temperature fluctuations in the CMB strongly suggest, this suggests that 2/3 of the energy density in the universe is made up of . Not only is there something driving cosmic acceleration, as the supernovae show, but the CMB observations also require a lot of it.

## Summing up

Figure 14: Combining evidence from supernova and the CMB makes a strong case for dark energy.