© 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 Beginning at the beginning, a common question asked by laymen and some physicists is what the universe expands 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 For small redshifts, the interpretation of the redshift as a Doppler shift (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

(3.66)

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:

(3.67)

(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 / c²) .

THE NATURE OF THE EXPANSION An inability to see that the expansion is locally just kinematical also lies at the root of perhaps the worst misconception about the big bang. Many semi-popular accounts of cosmology contain statements to the effect that ``space itself is swelling up'' in causing the galaxies to separate. This seems to imply that all objects are being stretched by some mysterious force: are we to infer that humans who survived for a Hubble time would find themselves to be roughly four metres tall? Certainly not. Apart from anything else, this would be a profoundly anti-relativistic notion, since relativity teaches us that properties of objects in local inertial frames are independent of the global properties of spacetime. If we understand that objects separate now only because they have done so in the past, there need be no confusion. A pair of massless objects set up at rest with respect to each other in a uniform model will show no tendency to separate (in fact, the gravitational force of the mass lying between them will cause an 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 A very helpful tool for regaining one's bearings in cases of confusion is a universe containing no matter at all, but only non-gravitating test particles in uniform expansion. This is often called the 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,

(3.68)

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 = H₀^-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:

(3.69)

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

(3.70)

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

(3.71)

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

(3.72)

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

(3.73)

The metric therefore becomes

(3.74)

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 -kc² (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 dsrc="../GIFS/omega.gif" alt="omega"> become progressively length-contracted at high redshift, giving a spuriously low value for the distance.