- Introduction
- The expansion of the universe
- The age of the universe
- How far away are DRAGNs?
- How do we know how far away DRAGNs are?
- Standard assumptions
- Cosmological Effects
- Malmquist Bias
- Cosmological evolution

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).

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.

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

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

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 *H _{0}* 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

Because the uncertainty in50 < Hkm/s/Mpc._{0}< 75

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, *q _{0}* , 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,

Contrary to an incorrect statement on an earlier version of this page, negative values of-0.9 < q._{0}< 0.5

This choice implies that the universe has zero curvature, consistent with the theory of inflation.H= 75 km/s/Mpc,_{0}q, and zero cosmological constant._{0}= ½

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 H_{0}.

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

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.

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

jpl@jb.man.ac.uk