7.2 The temperature of matter and radiation
The different reactions by which neutrons and protons came together soon after the instant of the big bang to produce heavier nuclei will have proceeded at different rates according to the energies of the particles involved. The first step in calculating nuclear abundances is therefore to make some assumption about these energies.
The particles at any instant have a wide range of energies; this obviously complicates matters. Fortunately, however, it is possible to make one very important simplification. This is based on the concept of equilibrium. We have already touched on this topic in the context of our discussion of the origin of the 3 K radiation. Let us now look into it in a little more detail:
We use an analogy. Suppose that a large number of molecules of an ordinary gas are introduced into an insulated container. To all external appearances, the state of the gas does not change with time. It is at a fixed pressure and a fixed temperature and is said to be in equilibrium. Of course, on a microscopic scale, a great deal is happening as the molecules collide with one another. Nevertheless, the measurable properties of the gas remain effectively constant. The reason for this is that the distribution of energy among the molecules fluctuates to only a very small extent about a well-defined average pattern. Thus, although the energy of any given molecule continually changes as it collides with others, details like the most probable energy for a molecule, or the probability of a molecule having an energy greater than, say, 10−20 J are preserved. Now it turns out that, once the temperature of the gas has been specified, it is possible to predict the average pattern of energy distribution in the gas at equilibrium – the temperature of matter serves as a label for the average mixture of energies of the gas molecules. For example, the most probable energy for a molecule is proportional to the temperature, and could, in the absence of a thermometer, be used to define the temperature scale.
If we now have a mixture of gas and radiation, and these are allowed to come into equilibrium with each other, not only will the particles of the gas adopt a characteristic energy distribution, but so also will the photons of the radiation. As we discussed earlier in connection with the microwave radiation, this distribution of photon energies is known as a thermal spectrum, and is rather different from the Maxwell-Boltzmann distribution for molecules. Nevertheless, the distribution can again be labelled by a single number – the temperature of radiation. This is defined to be proportional to the most probable energy (or frequency) of a photon. Our two temperature scales for matter and for radiation are consistent in that, whenever radiation is in equilibrium with matter, the values of the two temperatures agree.
Returning now to the early Universe, it is important to realise that the very high density of matter and radiation at that time gave rise to a frequency of collisions between protons, neutrons, electrons, and photons sufficiently high to ensure that the various components of the Universe were in almost perfect equilibrium. Thus, their respective distributions of energies could be labelled by a single common temperature. The electrons had the same temperature as the protons and the neutrons and the photons, because the collisions were so frequent that no part of the Universe could get out of step with the rest. This is very important because it allows us to replace the great complexity of the possible energy distributions by a single parameter – the temperature.
Strictly speaking, the various equilibria – between particles, and between particles and radiation – can never have been quite perfect because the Universe was expanding all the time, and therefore cooling rapidly. There was therefore never a steady approach to a final condition such as we associate with the behaviour of, say, a cup of coffee cooling to equilibrium with its surroundings. Indeed, as we have pointed out before, the Universe has still not reached equilibrium. Nevertheless, there was near-equilibrium in the early Universe because of the high rate of collisions.