The Hubble Diagram

As you examined galaxy clusters and spectra in the last two sections, you went through the same steps that astronomers like Edwin Hubble went through in 1929. Now, you have only one task left: you must make a Hubble diagram, pulling together your data to learn something about the universe.

The Hubble Diagram and the Big Bang

The key breakthrough that led astronomers to the big bang picture was the linear relationship between distance and redshift on the Hubble diagram. Two important observations led astronomers to this picture. First, the linear relationship between distance and redshift does not depend on direction in the sky - in one direction we see redshifts, as if galaxies are receding from us, and in the opposite direction we also see redshifts, not blueshifts. Everywhere it seems that galaxies are moving away from us, and the farther they are, the faster they appear to be moving. Second, counts of galaxies in various directions in the sky, and to various distances, suggest that space is uniformly filled with galaxies (averaging over their tendency to cluster).

From the second observation, we can infer that our region of space is not special in any way - we don't see an edge or other feature in any direction. While all galaxies appear to be moving away from us, this does not mean that we are at the center of the universe. All galaxies will see the same thing in a statistical sense - an observer on any galaxy who makes a Hubble diagram would see a linear relationship in all directions. This is exactly the picture you get if you assume that all of space is expanding uniformly, and that galaxies serve as markers of the expanding, underlying space. The expanding universe model would not have worked if astronomers had found anything except a linear relation between distance and redshift.

The term "big bang" implies an explosion at some location in space, with particles propelled through space. If this were true, then with respect to the site of the explosion, the fastest-moving particles will have traveled furthest, leading to a linear relationship between distance and velocity. But this is NOT the concept behind the big bang cosmological picture. The explosion model is actually more complex than the big bang cosmological model - you need to say why there was an explosion at that location and not some other location; what distinguishes the galaxies at the edge as opposed to closer to the center, etc. In the cosmological picture, all locations and galaxies are equivalent - everybody sees the same thing, and there is no center or edge.

Hubble did not measure the redshifts himself - those were already measured for a few dozen galaxies by Vesto Slipher. Hubble's key contribution was to estimate the distances to galaxies and clusters and to realize that the data in his diagram could be represented by a straight line.

The linear relation between redshift and distance is expressed as

c z = H₀ d ,

where c is the speed of light, z is the spectroscopic redshift, d is the distance, and H₀ is a constant of proportionality called the Hubble constant whose value depends on the units used to measure the distance d. The sub-naught tells us "evaluated at the present cosmic epoch," which suggests that its value may have been different at an earlier cosmic time. Note that as we observe galaxies at progressively greater distances, we are seeing them as they were progressively farther in the past, because it has taken the light from them longer to reach us. In other words, larger d means we are looking at things at earlier cosmic epochs. (For sufficiently large d, we might expect a departure from the simple linear relation, but that's another story.)

If you were to ask an astronomer what the distance to a particular galaxy was, most likely she or he would measure the redshift z and use the formula above to compute d. This is not what we are going to do: we are trying to establish that the formula itself is valid, which means that we must estimate d independently from our measurement of redshift.

Absolute and Relative Distances

A quasar found by SDSS at redshift 5.8, or about 2800 Mpc.

What you measured in the Distances section was relative, not absolute, distance. Having an absolute distance means we know the value of d in inches or meters or something - astronomers use a unit called the megaparsec (Mpc), where 1 Mpc = 3.1 x 10²² m. (To give you a sense of this distance, the Andromeda galaxy is a bit less than 1 Mpc away.) If we use such units, then H₀ has units of km per sec per Mpc. The currently favored value is H₀ = 70 km/sec/Mpc. The error associated with this number is about 10%, which reflects the uncertainty in measuring absolute distances to galaxies.

Because you could measure only relative distances in the Distances, you have no information on the value of H₀. But your work is still a significant accomplishment: the linear relationship is what motivated the big bang picture, not any particular value for H₀.

Putting it All Together

The data you have so far show a relationship between distance and redshift, and imply that the universe is expanding. This is an amazing result, but remember that you have only looked at a few galaxies in one tiny part of the sky. Scientists need a large amount of data to be convinced, and many of them would be skeptical of your conclusions. They might say that something strange was happening in that part of the sky, or that what you found was only a statistical coincidence.

In fact, Edwin Hubble and other astronomers also had difficulty convincing scientists of this discovery. After he announced his discovery in 1929, he teamed up with astronomer Milton Humason and embarked on a systematic program to look trace the diagram to larger distances and higher redshifts. They looked at thousands of galaxies, trying to prove that the linear characterization was really valid. They succeeded: by 1937, the redshift-distance relation was firmly established by these observations.

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