| © CAMBRIDGE UNIVERSITY PRESS 1999 |

**3.3 Common Big Bang Misconceptions**

Although the concept of an isotropically expanding universe is straightforward enough to understand locally, there are a number of conceptual traps lurking in wait for those who try to make global sense of the expansion. The most common sources of error in thinking about the basics of cosmology are discussed below. Some of these may seem rather elementary when written down, but they are all fallacies to be encountered quite commonly in the minds of even the most able students.

THE INITIAL SINGULARITY
*into*.
The very terminology of the ``big bang'' suggests an explosion,
which flings debris out into some void. Such a picture is
strongly suggested by many semi-popular descriptions,
which commonly include a description of the initial
instant as one ``where all the matter in the universe is
gathered at a single point'', or something
to that effect. This phrase can probably
be traced back to Lemaître's unfortunate term
``the primaeval atom''. Describing the origin of the
expansion as an explosion is probably not a good idea
in any case; it suggests some input of energy
that moves matter from an initial state of rest.
Classically, this is false: the expansion merely
appears as an initial condition. This might
reasonably seem to be evading the point, and
it is one of the advantages of inflationary cosmology
that it supplies an explicit mechanism for starting the
expansion: the repulsive effect of vacuum energy.
However, if the big bang is to be thought of
explosively, then it is really many explosions that happen everywhere
at once; it is not possible to be outside the
explosion, since it fills all of space.
Because the density rises without limit as *t* -> 0,
the mass within any sphere today (even the size of
our present horizon) was once packed into an arbitrarily
small volume. Nevertheless, this does not justify the
``primaeval atom'' terminology unless the universe is
closed. The mass of an open universe is infinite;
however far back we run the clock, there is infinitely
more mass outside a given volume than inside it.

THE ORIGIN OF THE REDSHIFT
*z* = *v / c*) is quite clear. What is
not so clear is what to do when the redshift becomes
large. A common but incorrect approach is to use the
special-relativistic Doppler formula
and write

This would be appropriate in the case of a model
with = 0 (see below), but is
wrong in general. Nevertheless, it is all too common to read
of the latest high-redshift quasar as ``receding at 99.999 . . .
per cent of the speed of light''. The reason the
redshift cannot be interpreted in this way is because
a non-zero mass density must cause gravitational
redshifts [problem 3.4]; even this interpretation is hard
to apply rigorously when redshifts of order unity are attained.
One way of looking at this issue
is to take the rigid point of view that 1 + *z* tells us
nothing more than how much the universe has expanded
since the emission of the photons we now receive.
Perhaps more illuminating, however, is to realize that,
although the redshift cannot be thought of as a *global*
Doppler shift, it is correct to think of the effect as
an accumulation of the infinitesimal Doppler
shifts caused by photons passing between fundamental
observers separated by a small distance:

(where is a radial increment of proper
distance). This expression may be verified by substitution of the
standard expressions for *H (z)* and *d* / *dz*.
The nice thing about this way of looking at the result
is that it emphasizes that it is momentum that gets
redshifted; particle
de Broglie wavelengths thus scale with the expansion,
a result that is independent of whether their rest mass is non-zero.
To see this last point, replace *d* by *v dt* in general,
and consider the effect of an infinitesimal Lorentz
transformation: *p* =
(*E / c*^{2})
.

THE NATURE OF THE EXPANSION
*inward* relative acceleration). In the common
elementary demonstration of the expansion by means of
inflating a balloon, galaxies should be represented
by glued-on coins, not ink drawings (which will spuriously
expand with the universe).

THE EMPTY UNIVERSE
**Milne model** after
E. Milne, who attempted to establish a *kinematical*
theory of cosmology (e.g. Milne 1948).
We no longer believe that his model
is relevant to reality, but it has excellent pedagogical value.
The reason is that, since the mass density is zero,
there is no spatial curvature and the metric has to be the familiar
Minkowski spacetime of special relativity. Our normal
intuition concerning such matters as length and time can
then be brought into play in cases of confusion.

We therefore start with the normal Minkowski metric,

and consider how this is viewed by a set of fundamental
observers, in the form of particles that are ejected
from *r* = 0 at *t* = 0, and which proceed with constant velocity.
The velocity of particles seen at radius *r*
at time *t* is therefore a function of radius: *v* = *r /
t* (*t* = *H*_{0}^{-1},
as required); particles do not exist beyond the radius
*r* = *ct*, at which point they are receding from the origin
at the speed of light. If all clocks are synchronized
at *t* = 0, then the cosmological time *t'* is just related to
the background special relativity time via time dilation:

If we also define *d*
to be the radial separation between events measured by fundamental observers at
fixed *t'*, the metric can be rewritten as

To complete the transition from Minkowski to
fundamental-observer coordinates, we need to eliminate
*r*. To do this, define the velocity variable :

Now, the time-dilation equation gives *r* in terms of
*t* and *t'* as

and the radial increment of proper length is related to
*dr* via length contraction (since *d* is at constant *t'*):

The metric therefore becomes

This is the *k* = -1 Robertson-Walker form, and it demonstrates
several important points. First, since any model with < 1
will eventually go over to the empty model as *R* ->
,
this allows us to demonstrate that the constant of integration
in the Friedmann equation really does correspond to -*kc*^{2}
(see earlier) in the *k* = -1 case without using Einstein's field
equations. Second, it shows that it is more useful to
think of the comoving distance *S _{k}* ()
as representing the distance to a galaxy than
itself. In the empty model, the reason for this is that the
proper length elements