6 BIG BANG COSMOLOGY – THE EVOLVING UNIVERSE 6.2.43A radiation-dominated model of the Universe R We have just seen that in the early Universe, the dominant energy density is that due to the radiation within the Universe. The Friedmann equation that was described in Chapter 5 (Box 5.4) can be solved for such conditions and the way in which the scale factor varies with time for such a model is shown in Figure 6.7. One important feature of such a model is that the scale factor varies in the following way: R(t) ∝ t1/2 (6.17) 0 t −4 Because the energy density of radiation is dominant for times when R(t)/R(t0) < 10 , all cosmological models which start at t = 0 with R(0) = 0, will go through a phase Figure 6.73The evolution of the that is well described by this radiation-dominated model. Thus we are in the rather scale factor with time in a remarkable position that regardless of which type of cosmological model best cosmological model in which the dominant contribution to the describes the Universe at the present, we can be reasonably confident that we energy density arises from the know how the scale factor varied with time in the first few tens of thousands of radiation within the Universe years of the big bang. (i.e. during the radiation-dominated However, the temperature of the background radiation varies with scale factor era). according to T(t) ∝ 1/R(t) (Equation 6.6). It follows that during the radiation- dominated era the temperature of the background radiation varies with time according to T(t) ∝ t −1/2 (6.18) This describes how temperature changes with time in an expanding universe where the energy density of radiation is the dominant component. Of course, to use Equation 6.18 to predict the temperature, it is necessary to know the constant of proportionality between T and t −1/2. In fact, this can be derived from the Friedmann equation and Equation 6.18 becomes (T/K) ≈ 1.5 × 1010 × (t/s)−1/2 (6.19) (Note that T is measured in kelvin and t in seconds.) Equation 6.19 is an approximate relationship. As you will see later, other physical processes can change the temperature of the radiation in the real Universe during the radiation-dominated phase of its expansion. What is the temperature when the age of the Universe is one second? By substituting a value of t = 11s in Equation 6.19 the temperature is 1.5 × 10101K. Thus the temperature of the Universe in the first few seconds of the big bang was higher than the highest temperatures that are found in the cores of the most massive stars (where temperatures may reach about 1091K). This immediately suggests that nuclear reactions may have occurred in any matter that was present at this time. We will look into such processes in more detail in Section 6.4, but in the next section we will discuss even more extreme conditions: we will consider the processes that occurred when the Universe was less than 1 second old. 267 AN INTRODUCTION TO GALAXIES AND COSMOLOGY © The Open University QUESTION 6.3 Rearrange Equation 6.19 to find an expression for the age of the Universe for a given temperature. What was the age of the Universe when the temperature was 1061K? Express your answer in terms of years. 6.33The early Universe We have seen that when considered together, the cosmic microwave background and the expansion of the Universe imply that there was an early phase of the history of the Universe which was characterized by high temperatures and high densities. In particular, the temperature at any time t can be estimated using Equation 6.19. A natural question to ask then is how far back towards t = 0 can we go in understanding processes in the Universe? This will be the first major question that we shall consider in this section, and we will find that there is a limit to our knowledge of the evolution of the Universe. The remainder of the section will be concerned with understanding the processes that occurred before the Universe was about 1 second old. 6.3.13Cosmology and the limits of physical theory We have seen that the Friedmann equation gives a model for a radiation-dominated Universe that is characterized by the scale factor having a value of zero at the instant of t = 0. As we said in Section 6.2.2, the naive interpretation of this is that the Universe came into existence with an infinitely high temperature; the truth of the matter is that we don’t really understand the physical processes in the very early Universe. So, how early in the history of the Universe can we be confident that our physical theories really do apply? There are essentially two answers to this question, which reflect two levels of certainty in physical theory. The first approach is to say that theories are only well-tested for the ranges of physical conditions that can be explored by experiments. Thus, we may have a good deal of confidence in describing the Universe at times when the particle energies were similar to the highest values that can be imparted in large accelerator experiments. At present, this limit corresponds to being able to describe physical processes in the Universe that occurred after the temperature fell to below 10151K, which corresponds to a time of t ~ 10−91s. An alternative approach is to apply physical theories to conditions that never have been, and probably never will be, tested in the Earth-bound laboratory and to look for observable consequences in nature. Clearly, this is a somewhat more speculative approach than having to rely on ‘tried-and-tested’ physical theory. However, it is one way in which physical theories can be explored and developed, and is a very exciting field for cosmologists. While it might be expected that physical theories could be extrapolated to describe processes at ever increasing temperatures, it turns out that there is a well recognized limit to our theoretical understanding of the processes of nature. This limit arises because of a surprising incompatibility between the physical theory that is used to describe the interactions of subatomic particles and the theory that describes gravity. The interactions of subatomic particles are described by a branch of quantum physics called the standard model of elementary particles. The gravitational interaction is described, as you saw in Chapter 5, by Einstein’s general theory of relativity. 268.