# 17.6 Universal Expansion

The Hubble relation is our primary evidence for the expansion of the universe. When Einstein, Lemaitre, and others solved the equations of general relativity, they were able to describe how the size of the universe has changed with time. Astronomers use the symbol R to represent the scale or size of the universe at any time. You can think of R as the size of the universe, but more accurately it represents the distance between any two well-separated places. The cosmological principle says that any two points are moving apart at the same rate. Thus, the entire history of the universe is described by the way that R varies with time. Since the universe is expanding, R has been continuously increasing for billions of years. Remember that R describes the expansion of space that carries galaxies apart — the galaxies themselves are not expanding. Galaxies are just markers of expanding space.

Determining the Hubble constant using supernova type 1a. Click here for original source URL

The expansion of space provides a good way to define the cosmological red shift. If R is the scale of the universe at any time and R_{0} is the scale of the universe now, then the red shift is the ratio of the present scale to the previous scale, minus one. In equation form, z = (R_{0}/R)^{-1}. We see regions near us in space as they are now, so R = R_{0} and z = 0. Remember that looking out in space corresponds to looking back in time. However the universe was smaller in the past, so a distant region of space has R < R_{0} and z > 0. Conceptually, light left distant objects when the universe was smaller and the waves have since been stretched by the expansion of space. We can now relate the expansion of the space to the red shifts of distant galaxies and quasars. The light from a distant galaxy might have been emitted when the universe was half its present size. Using the definition of red shift given above, R_{0} = 2R and z = 1. The light from the most distant quasar was emitted when the universe was one eighth of its present size. In this case, R_{0} = 8R and z = 7. Ancient light is light that has been reddened due to the expansion of space.

Doppler Shift. Click here for original source URL.

How are cosmological red shifts related to the Doppler effect? A red shift caused by the Doppler effect is defined as z = Δλ/λ, where Δλ is the difference between the wavelength observed and the wavelength emitted. A cosmological red shift is z = (R_{0} - R)/R = ΔR/R, where ΔR is the difference between the scale of the universe observed now and the scale of the universe when the light was emitted. However, there are crucial distinctions between Doppler red shifts and cosmological red shifts. The Doppler effect applies to waves of any kind traveling through a medium. The cosmological red shift is caused by the expansion of the medium itself. The Doppler effect relates the red shift to the speed of the waves; at low red shifts z = v/c. By contrast, the cosmological red shift is not related to the speed of light at all. The limitation imposed by special relativity — that nothing can go faster than the speed of light — does not apply on the global scale of the universe, which is governed by general relativity. As a result, there can be remote regions of the universe that are moving apart faster than the speed of light!

Until the mid 1990s, it took only two numbers to describe all the possible models for the expansion. The first is the current expansion rate, given by the Hubble constant. The second is the mean density of matter in the local universe. In expanding universes, the rate of expansion decelerates (slows down) with time, because galaxies are pulling on all other galaxies. The strength of the deceleration depends on the mean density of matter, where most of it is dark matter. The deceleration is also related to the curvature of space, which determines the fate of the universe. It is a fundamental consequence of general relativity that the structure, or curvature, of space is related to the amount of matter in the universe. This simple view had to be amended with the discovery of the acceleration of the universe and the implication of a component called dark energy that worked counter to gravity and was speeding up the expansion rate. For the rest of this article, we will just describe the behavior of universes containing only matter.

The expansion rate currently is the Hubble constant, H_{0}. When the universe was denser and smaller, it had a higher expansion rate. If the universe had nothing in it, there would be no gravity to slow down the expansion. The result would be expansion at a constant rate. In an almost empty (i.e. low-density) universe, the deceleration is small, and R increases almost linearly with time. A low-density universe has negative space curvature in general relativity; it is called an open universe and it expands. At a certain critical density, the universe continues to expand to some maximum size at an ever-decelerating rate, taking an infinite amount of time to come to a halt. This special case has zero space curvature; it is called a flat universe. A universe in which the mean density is above the critical density has a positive curvature and is a closed universe. The mutual attraction of matter in such a universe is eventually enough to overcome the Hubble expansion. After R reaches a maximum value, the universe will collapse. As space begins to contract, all the galaxies will begin to rush toward each other and show blue shifts.

The critical density divides universes that will expand forever from universes that will eventually collapse. So astronomers characterize the big bang model by a density parameter (Ω_{0}), which is the ratio of the observed density to the critical density. The density parameter is a dimensionless number. If Ω_{0} < 1, the universe is open and will expand forever. If Ω_{0} > 1, the universe is closed and will collapse. Remember that there are many possible values of the mean density that correspond to an open universe. Equally, there are many possible values of the mean density that correspond to a closed universe. However, the special case of a flat universe only occurs if the mean density equals the critical density, Ω_{0} = 1.

Consider the analogy of a rocket launched from the surface of the Earth. The escape velocity is about 11 km/s. A rocket launched with an initial velocity of less than 11 km/s will decelerate as it rises. The Earth's gravitational attraction will eventually overcome the upward velocity and force the rocket back to the planet's surface. On the other hand, a rocket with an initial velocity above 11 km/s will escape from the Earth forever. Of course, the rocket will continue to slow down since the planet's gravity has a long reach, but it will never reverse its direction and fall back to Earth. In our universe, the Hubble constant is analogous to the launch velocity of the rocket, and the mean density to the mass of the planet. For each possible launch velocity of a rocket, there is a mass of planet where that velocity will just equal the escape velocity. Conversely, for every planet there is a single velocity that will allow the rocket to just escape gravity's pull. Larger planets have larger escape velocities. By analogy, for every possible value of the Hubble constant, there is a mean density that will just be enough to stop the universal expansion and close the universe. A larger Hubble constant requires a larger mean density to close the universe.

The following shows how the density parameter relates to the curvature of space and the future of the universe:

• Flat universe: zero curvature, Ω_{0} = 1, fate is to expand to a maximum size

• Closed universe: positive curvature, Ω_{0} > 1, fate is to reach a maximum size and collapse

• Open universe: negative curvature, Ω_{0} < 1, fate is endless, decelerating expansion

• Empty universe: negative curvature, Ω_{0} = 0, fate is endless and constant expansion

An estimate of the age of the universe comes from tracing the expansion back to the time when all galaxies were on top of each other (R = 0). Space had zero size when the universe formed. A linear Hubble expansion leads to an age estimate of 1/H_{0}. For the current best estimate of H_{0} = 71 km/s/Mpc, this age is roughly 13.7 billion years. Any expansion will have a deceleration, so the age estimate of 1/H_{0} is correct only in the artificial case of an empty universe. That is, 13.7 billion years is an upper limit to the age of the universe in the big bang model. Mathematically, it turns out that a flat universe has an age of 2/3(1/H_{0}) or about 9 billion years. Therefore, the big bang model predicts that the universe will have an age between 9 and 13.7 billion years if the geometry is open, and less than 9 billion years if the geometry is closed. The inclusion of dark energy alters this calculation and leads to a slight increase in the calculated age of the universe, to 13.8 billion years.