Redshifts

Until well into the twentieth century, it was not understood whether the great groupings of stars that were seen through telescopes were part of our own galaxy or distant galaxies in their own right. This puzzle was finally resolved by using Cepheid variables to establish a distance to the objects like the "spiral nebula" in the constellation Andromeda and to determine the size of our own galaxy. By around 1925 Hubble, building on the work of Leavitt, Hertzsprung, Shapley, and others, had established conclusively that objects like the Andromeda "Nebula" were in fact much further away than objects in our own galaxy. Thus, they were themselves independent galaxies.
The Expansion of the Universe
Then, in the late 1920s, Hubble (utilizing results obtained earlier by Slipher), combined Doppler shift measurements of radial velocities with distance measurements to conclude that almost all galaxies were flying away from the Milky Way. In addition, Hubble found that the velocity of recession for these galaxies was proportional to the distance from us: the further away the galaxy, the faster it was receding from us. It turned out later that there were systematic errors in the earliest measurements (see the right panel). However, although corrections for these mistakes changed the distance scales by significant amounts, they did not alter the fundamental conclusions:

1. There are many galaxies outside of our own.
2. These galaxies are all receding from us if we go to large-enough distances.
3. The velocity of recession is proportional to the distance from us.

Thus, this work early in this century altered forever our perception of the Universe. It established both that the Universe was a much larger place than previously thought, and that it was not static but was expanding.

The Distant Galaxies Are Flying Away
The galaxies are all flying away from each other (on very large distance scales), with the velocity of recession proportional to the distance between them. The adjacent image, taken by the Hubble Space Telescope, shows many galaxies billions of light years away. Most of the fuzzy patches are galaxies containing billions of stars. Most of the galaxies in this image are receding from us at high velocities because they are at large distances.

Space Itself Is Expanding
As we shall discuss further in Chapters 26 and 27, the expansion discovered by Hubble implies that space itself is expanding. That is, the Hubble expansion is not a result of the galaxies all moving away from the Milky Way in a fixed space. It is that the space itself, containing all the galaxies, is growing larger. Likewise, the redshift of spectral lines for a receding distant galaxy is not because the galaxy is moving away from us in space, but rather because the expansion of space stretches the wavelength of electromagnetic radiation traveling between the galaxies to longer wavelenths. Nevertheless, astronomers speak of a redshift for a galaxy and often relate it to a velocity of recession exactly as if the galaxy were moving away from us in space. This animation illustrates a more easily grasped two-dimensional model for the expansion of the Universe and this animation illustrates the redshift of light caused by expansion of the Universe.
Redshift Parameter
To measure the effect of radial velocity, astronomers commonly define a redshift parameter z. If the observed wavelength of a spectral line is L and the unshifted wavelength is L0

z = (L - L0) / L0

Because of this relation, we shall commonly refer to the redshift of a distant object rather than its distance or its recessional velocity. The recessional velocity is related to the redshift by the relativistic formula

z = [(1 + v / c) / (1 - v / c)]1/2 - 1       (valid for all velocities)

It also is often useful to solve this equation for the velocity in terms of the redshift. The result is

v / c = [(z+1)2 -1] / [(z+1)2 +1]       (valid for all velocities)

These formulas are called "relativistic" because they were derived using the special theory of relativity so they are valid for all velocities, including very large ones approaching the speed of light. These expressions reduce to the much simpler nonrelativistic form

z ~ v / c       (valid if v is much less than c)

but only for low velocities (that is, ones that are small compared with the speed of light; they may still be large by ordinary standards!). If velocities are more than a few percent of the speed of light, the more complicated relativistic formula must be used. Otherwise we would make a serious error for large redshifts. Once the velocity is related to the redshift by the above formulas, it can then be related approximately to a distance using the Hubble Law, as we describe in the next section.

Resolution of Olbers' Paradox

That the Universe is not static but expanding helps solve a paradox. This paradox has been known at least since the 1500s, but it was popularized by Heinrich Olbers in 1826 and is called Olbers' Paradox. Briefly, if the Universe is static, of infinite age and extent, and uniformly filled with stars and galaxies, the night sky should be as bright as the surface of a star. The essential argument is that, in this case, no matter in what direction we look our line of sight should eventually intersect a star. It follows that the sky should be glowing as if it were completely covered by stars with no dark sky showing through.

Obviously the night sky is dark, not glowing like the surface of the Sun, which leads to the paradox. The expansion of the Universe resolves the paradox, basically for two reasons. First, the Universe is not infinitely old, so light coming to us at finite velocity from the most distant stars has yet to reach us. Furthermore, stars live only a finite time. Therefore, at any one time we see only part of the starlight produced in the history of the Universe. Second, light from the most distant stars that we can see is dimmed because it is strongly redshifted by the expansion of the Universe. Detailed considerations indicate that the first reason is by far the most important one in resolving Olber's Paradox. Thus, at least two assumptions leading to Olbers' Paradox are not valid: the Universe isn't static and it isn't of infinite age. (We don't know yet whether it is of infinite extent.)