Notes on Cosmology


Modern cosmology is founded on three facts and a conjecture. The facts are: that on the largest scales, the universe looks the same in all directions (the isotropy of the universe); that the universe looks more and more different from our local neighborhood as we look further and further away; and that the further an object is from us, the faster it is moving away (Hubble's law). The conjecture, grandly called the Copernican principle, is that our local corner of the universe is nothing special. We know that the Earth is one of several planets orbiting the sun, the sun is one of billions like it in the Milky Way, and the Milky Way is a typical large galaxy of the sort we see in every direction when we look out into the far universe. The conjecture goes a step further and assumes that the Milky Way's neighbourhood in the realm of galaxies is quite ordinary, neither unusually crowded nor particularly deserted. Combined with the isotropy of the universe, this implies that, again on the largest scales, the universe looks and behaves the same everywhere; we call this the homogeneity of the universe.

When we look at anything, we see it, not as it is now, but as it was when the light left it. This time difference is called the look-back time, and it allows us to reconcile the changes we see in the universe at large distances with the idea of homogeneity: we are looking back into the history of the universe, and it was different from the present in those early days. For the most distant DRAGNs and galaxies, the look-back time can be up to 80 per cent of the age of the universe. In comparison, the DRAGNs in this Atlas are relatively nearby: the furthest of them are seen when the universe was a third younger than today, about 4 billion years ago (just about the time that the Earth was first forming).

The Expansion of the Universe

In 1929 Edwin Hubble showed that the further a galaxy is away from us, the faster it is moving away; this is now called Hubble's law, and it means that the universe is expanding.

What Hubble actually measured was the redshift, usually written z , of the spectrum of the galaxies: he found that the spectrum was `stretched out' so that a feature that should have been at a certain wavelength was actually detected at a wavelength longer by a factor (1 + z). The usual cause of such a redshift is the Doppler effect, and by assuming this it is possible to calculate the recession velocity of the galaxy. In the cosmological case, it turns out that the Doppler formula is only an approximation; the deeper meaning of the redshift is that the factor (1 + z) gives the ratio of the size of the universe at the time the light was emitted, to the size it is today.

This expansion is quite a subtle concept. It does not mean that everything (galaxies, planets, people, ants, atoms) is getting bigger; if that were the case, we would never know, because our tape measures and rulers would be growing at the same rate. The idea is clearest for a toy universe consisting of `dust' spread evenly through space, with individual particles not interacting at all with each other. Expansion means that the distance between any two particles is getting larger. Now think of a row of particles: since each is moving away from its immediate neighbours (at the same rate, from homogeneity), the relative motion must be proportionately larger for particles separated by larger distances: Hubble's law. Of course the real universe is lumpy on `small' scales; instead of being a smooth soup, matter is condensed into galaxies and stars and all the rest (just as well for us!). Within these objects the original expansion has been overcome by gravity. This is a runaway process. Start with a small region in the early universe with a slightly higher-than-average density. Its gravity will slow down both its own expansion and that of surrounding matter, so both the size of the region and its relative over-density increase. Eventually the expansion near the centre is stopped altogether. The size of the region affected continues to grows as outlying particles have time to move in. At present the largest regions in which the expansion has been stopped are the clusters of galaxies a few million light years across. For points with larger separations, the motions induced by local density peaks are small compared to the relative motion from the overall expansion.

The Age of the Universe

An expanding universe must have been smaller in the past, and in fact the distance between any two points approaches zero roughly 13 billion years ago. This moment of ultra-high density is called the Big Bang, and marks the birth of the universe (at least to all intents and purposes). The age of the universe is therefore about 13 billion years. For about half a million years after the Big Bang, the universe was opaque to electromagnetic radiation. This sets a maximum distance that we can see: radiation emitted just as the universe became transparent (the cosmic microwave background), that is reaching us now, has travelled 13 billion light years.

For more on the Big Bang and the cosmic microwave background, see the Jodrell Bank CMB Home page

How far away are DRAGNs?

DRAGNs are among the most distant objects know. They are so far away that some of our simple ideas about `distance' break down.

The trouble has two sources. First, the universe is expanding. So by distance, do we mean the distance now, or the distance when the light we are picking up was emitted, which is smaller by a factor (1 + z)? The distance actually travelled by the light is in between these two.

The second trouble with distance is that on the largest scales space may be 'curved', as predicted by general relativity. This curvature means that the simple geometry we use to calculate distances is no longer correct; for instance, the perimeter of a circle of very large radius R is not necessarily 2 pi R. At present we only have an upper limit on the curvature of the universe, and in the popular `inflationary' theory the curvature is zero (so 2 pi R is exactly right).

For practical purposes, we mainly use distances as a tool to convert from apparent (angular) size to actual size, and from apparent brightness (flux density) to actual luminosity. Unfortunately the angular size distance and luminosity distance needed to do this are different from each other and from the normal metric distance, which is what we'd get from a tape measure that happened (right now) to connect us with the DRAGN. The luminosity distance is bigger than the angular size distance by a factor of (1 + z)2. If the curvature is zero, the metric distance is (1 + z) times the angular size distance; in a curved universe the connection is more complicated.

How do we know how far away DRAGNs are?

Distances to nearly all DRAGNs are found through Hubble's law, R = v / H0 , where R is the current metric distance, H0 is Hubble's constant, and v is the current recession velocity. Unfortunately we don't know Hubble's constant very accurately, and we can only measure redshifts, not recession velocities.

To find Hubble's constant we have to apply Hubble's law to objects whose distances and recession velocities are already known. Over the last century, astronomers have constructed a long chain of arguments to find the distances to galaxies, called the cosmological distance ladder. Small uncertainties at each step of the argument build up so at the end we don't know the distances at all accurately. Fortunately, new results are changing this very rapidly, and by the end of the century we should know H0 to within 10 per cent. Some progress has already been made, and the erstwhile factor of two discrepancy has now narrowed down to the range

50 < H0 < 75 km/s/Mpc.
Because the uncertainty in H0 will scale all distances by the same amount, it does not affect relative distances, sizes, or luminosities.

For galaxies seen at look-back times a small fraction of the age of the universe, v = c z , where c is the speed of light and z is the redshift. For larger look-back times we need to find the relation between redshift and distance, which depends on the curvature of the universe and how the expansion rate has changed with time. Direct measurement depends on the top end of the distance ladder, which is not particularly good even for relative distances, but redshift definitely increases smoothly with (metric) distance out to z = 2 at least.

General relativity makes simple predictions for the luminosity distance-redshift relation for an isotropic homogeneous universe, which just depend on the deceleration constant, q0 , and the cosmological constant. These predictions are known as Friedmann universes (plural because you get a different universe for each pair of values for the constants). The cosmological constant is usually assumed to be zero because that makes the theory more elegant, and there are no strong observational reasons for having it. The straightness of the redshift-distance relation implies that, roughly,

-0.9 < q0 < 0.5.
Contrary to an incorrect statement on an earlier version of this page, negative values of q0 occur when there is a significant positive cosmological constant, which functions as a sort of anti-gravity. Recent observations surprisingly suggest that q0 is negative, with a value of around -0.5. However, this is still not certain so we have not yet updated our calculations from the standard assumptions given below. For redshifts less than 0.5, as in this Atlas, the resulting uncertainty in distance is less than about 20 per cent.

Standard Assumptions

For calculations of power and distance in this Atlas, we assume a Friedmann universe with
H0 = 75 km/s/Mpc, q0 = ½ , and zero cosmological constant.
This choice implies that the universe has zero curvature, consistent with the theory of inflation.

The two DRAGNs in the Virgo cluster are close enough their recession velocity is affected by gravitational motions, so it is not safe to use the Hubble law. Instead we assume a distance of 16 Mpc, based on recent Hubble Telescope measurements of distances to Virgo cluster galaxies. This corresponds to a cosmological redshift for the Virgo cluster of 0.0040 for our value of H0.

Cosmological Effects

We said earlier that angular size distance is smaller than the luminosity distance by a factor (1 + z)2 . The total power we receive is proportional to the inverse square of the luminosity distance, so for an object at high redshift, the surface brightness is dimmer by a factor of (1 + z)4, whereas normally it is independent of distance. This is a big effect: a factor of five even at a redshift of 0.5. Actually this formula applies only if we detect the radiation at all frequencies, whereas normally we only observe over a fixed and narrow range of frequencies, but this doesn't change the result much.

A curious relativistic effect is that time is dilated (i.e. appears to run slower) in distant objects, by a factor of (1 + z) .

Malmquist Bias

When we make observations in astronomy, our instruments set a limit on the faintest objects (stars, DRAGNs, whatever) that we can see. That is, in any observation there is a minimum flux density below which we will not detect an object. Now, the flux density is proportional to the luminosity divided by the square of the distance, so it is possible to see more luminous objects out to larger distances than intrinsically faint objects. This means that the relative numbers of intrinsically bright and faint objects that we see may be nothing like the relative numbers per unit volume of space; instead, bright objects are over-represented; and the average luminosity of the objects we see inevitably increases with distance. This is the Malmquist bias.

DRAGNs come in a wide range of radio power (i.e. luminosity), with the lowest power examples relatively common and more and more powerful ones becoming rarer and rarer. At a given distance (or redshift) there is a minimum power that is visible, from the Malmquist effect, but, also, only a small fraction of the objects within that distance will be much more powerful than the minimum, since they will be relatively rare. As a result, in a sample of DRAGNs chosen with a definite minimum flux density (as in the Atlas), there is a very strong relation between power and redshift, which is entirely due to the way the objects were selected; in reality there is a mix of radio powers at all redshifts. For DRAGNs, the relation is strengthened further by a real effect, cosmological evolution.

Cosmological Evolution

One of the most dramatic ways that the universe changes with look-back time is that powerful DRAGNs (and other active galactic nuclei, like quasars) were once much more common. There seems to have been a `quasar epoch' between redshifts 3 and 1 when the universe was lit up by these objects. Since then they have become hundreds of times more rare (this is not an effect of Malmquist bias). Even over the range of redshifts included in this Atlas (0 < z < 0.5), powerful DRAGNs have declined in density by a factor of ten or so. This change is known as cosmological evolution of the DRAGN population.

It is tempting but probably incorrect to associate cosmological evolution with a change in the properties of individual DRAGNs over the history of the universe. Because the number of DRAGNs declines so steeply with radio power, a long slow decrease in the power of each DRAGN would result in a dramatic decline in the numbers of DRAGNs at a given power. Unfortunately there are quite good reasons to believe that individual DRAGNs have lives which are very brief compared to the the age of the universe.

At present we do not even know what causes the birth or death of individual DRAGNs, or AGN in general, and so we are very far from understanding cosmological evolution. Many astronomers have pointed out that the quasar epoch is also the period in which galaxies themselves are expected to first have formed. It is all too easy to suggest reasons why this period should have spawned so many quasars; the hard part is finding ways to put such ideas to the test.

For more information on cosmology (with lots of pictures), see Ned Wright's Cosmology Tutorial and the Introduction to Cosmology Web Site at NASA's Goddard Space Flight Center. (The latter is an extended advert for their latest satellite project, but you can take that into account).

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Last modified: 1999 October 26
J. P. Leahy