Look-Back Times

The velocity of light plays a central role is astronomy and in physics because it places an upper limit on speeds in our Universe.

A Cosmic Speed Limit
According to the Einstein's Theory of Relativity, nothing in our Universe can exceed the velocity of light; thus, it is a kind of cosmic speed limit against which all other velocities may be measured. More generally, light is part of the electromagnetic spectrum, which includes infrared radiation, radio waves, gamma rays, X-rays, ultraviolet radiation, and so on.

All of these are a form of light; they just have energies that differ from the visible light that our eyes can see. Thus, these forms of electromagnetic radiation all travel at the speed of light too. Furthermore, contrary to normal intuition, the theory of relativity tells us that light always travels at the same speed relative to some observer, no matter what the relative motion of the observer. Although this may seem strange, it has been confirmed in many experiments: it is the way the Universe works.

Looking Back in Time
Because light travels at a large but finite speed, it takes time for light to cover large distances. Thus, when we see the light of very distant objects in the Universe, we are actually seeing light emitted from them a long time ago: we see them literally as they were in the distant past.

For example, when Supernova 1987A occurred in the "nearby" galaxy called the Large Magellanic Cloud (above figure), its light was observed on Earth in 1987. But the distance to the Large Magellanic Cloud is about 170,000 light years. Thus, we normally say that Supernova 1987A occurred in 1987, but it really happened about 170,000 years earlier. Only in 1987 did the light of the explosion reach the Earth! If we want to know what the Large Magellanic Cloud looks like "now", we will have to wait 170,000 years. In comparison, the Sun is only about 8 light-minutes away. So the light we see from the Sun represents what the Sun looked like 8 minutes ago, and we must wait another 8 minutes to see what it looks like "now".

Because of this property of light coming from distant objects, astronomers often define a quantity called the look-back time. The look-back time is just the time since the light that we see from an object was actually emitted. The speed of light is so high that for nearby objects the look-back time is essentially zero, but for Supernova 1987A it was about 170,000 years and for very distant objects the look-back time could be 10 billion years or more.

Time Machines
Such look-back times become critical when we look at the largest distances because they literally allow us to peer into the early Universe. The most distant objects observed may now allow us to see what the Universe looked like when it was only about 1/10 of its present age. Large telescopes are not just devices for gathering faint light. They are time machines! The above left figure shows how the look-back time varies with redshift for three different assumptions concerning a parameter Ω0 that measures the average density of the Universe and that we shall discuss in Chapter 18 (the look-back time is expressed in the plot as a fraction of the Hubble time for each case). The following table gives recessional velocities, distances, and look-back times as a function of redshifts.

Redshifts, Distances, and Look-Back Times
Redshift v/c Distance
Time (yr)*
Age of the
0.00 0.00 0 0 0 100%
0.05 0.049 222 7.26 x 108 7.08 x 108 93%
0.10 0.095 430 1.40 x 109 1.336 x 109 87%
0.50 0.385 1694 5.52 x 109 4.57 x 109 54%
1.0 0.600 2704 8.82 x 109 6.48 x 109 35%
2.0 0.800 3901 1.27 x 1010 8.10 x 109 19%
3.0 0.882 4616 1.51 x 1010 8.78 x 109 12%
4.0 0.923 5103 1.66 x 1010 9.13 x 109 9%
5.0 0.946 5462 1.78 x 1010 9.35 x 109 7%
6.0 0.960 5742 1.87 x 1010 9.51 x 109 6%
10 0.984 6448 2.10 x 1010 9.75 x 109 3%
100 1.00 8312 2.71 x 1010 1.00 x 1010 ~0%
1000 1.00 8939 2.92 x 1010 1.00 x 1010 ~0%
infinity 1.00 9231 3.01 x 1010 1.00 x 1010 0%
*Assuming flat geometry   ** Percentage of age for present Universe when light emitted (flat geometry)

The look-back time in the above table is expressed both in years and as a percentage of the present age of the Universe when the light was emitted. For example, at a redshift of z = 2 we are seeing light that was emitted 8.1 billion years ago when the Universe was only 19% of its present age. Here is a Java calculator that allows you to calculate the recessional velocity, look-back times, and distances from the redshift for arbitrary values of the Hubble constant and the deceleration parameter. (The preceding table can be reproduced with this calculator by setting the Hubble constant to 65/km/s/Mpc and the deceleration parameter to 0.5, corresponding to a flat Universe with no cosmological constant.)

The Hubble Constant and Age

The age of 10 billion years assumed for the Universe in the preceding table depends on the Hubble parameter, which we chose as 65 km/s/Mpc. If, for example we had chosen H = 50 km/s/Mpc (as some researchers favor), the Hubble time would have been 19.6 billion years and the age of the Universe 13.1 billion years with our other assumptions. If instead we had chosen H = 80 km/s/Mpc (as some other researchers favor) the Hubble time would have been 12.3 billion years and the age of the Universe 8.2 billion years. As we have noted, a time as short as the latter would be difficult to reconcile with the minimum age of globular clusters.

Large Redshift and Look-Back Times
With the assumptions used to calculate the look-back times in the preceding table, the Hubble time is 15.1 billion years and the age of the Universe is 2/3 of that or a little more than 10 billion years (this depends on choice of Hubble constant, as described in the box). We will explain the factor of 2/3 in Chapter 18, but it is associated with slowing of the expansion by gravity, which makes the Hubble time an overestimate of the age for the Universe.

Note that the look-back times are approaching a constant value of 10 billion years for large redshift. This is because the look-back time cannot be larger than the age of the Universe, which is 10 billion years for the assumptions used in the table. For infinite redshift, the look-back time is exactly the age of the Universe. However, we cannot look back all the way to infinite redshift because the corresponding photons are redshifted to zero energy and so cannot interact with our detectors. Furthermore, as we shall see in Chapter 18, the Universe becomes highly opaque to light for redshifts larger than about 1000, so it would be difficult to observe higher redshifts than that.

Distances and Grains of Salt
As noted in the right panel, relating distances to large redshifts requires assumptions about the nature of the Universe. The distances in the preceding table would change somewhat if we made a different set of assumptions in relating them to the redshifts. We shall often quote a distance in later discussion of high redshift objects, but one should always take these distances with a small grain of salt. They are meant to convey a qualitative sense of distance, but the directly measurable quantity is redshift.