| In the context of the standard (pre-inflation) Big Bang model, explain what is meant by the following catch-phrases, and why they are described as "problems": | |
| (i) the horizon problem; | [2] |
| In a matter-dominated universe, the horizon expands ∝ t, whereas a ∝ t2/3 – therefore the horizon at earlier times was relatively as well as absolutely smaller than it is now. Given this, one would assume that widely separated regions of the CMB would not have been in causal contact at the time of CMB emission, and should not therefore have nearly identical temperatures. The "horizon problem" is therefore the observed fact that widely separated regions of the CMB do have nearly identical temperatures. In the standard Big Bang, the only way to do this is to insert it as an initial condition – which then presents the problem of why the temperatures are not precisely equal. | |
| (ii) the flatness problem; | [2] |
| In a matter or radiation dominated universe which is not precisely flat, the departure from flatness increases with time. Therefore one would expect the universe now to be strongly curved (assuming it hadn't already collapsed in a Big Crunch!). The "problem" is that it is observed to be flat (to within ±0.01 according to WMAP). The solution in the standard Big Bang is to insert an initial condition that it be exactly flat, but there is no good theoretical motivation for this. | |
| (iii) the monopole problem. | [2] |
| In many Grand Unified Theories, the phase transition out of the high energy unified state is expected to produce point defects, which are very massive and behave as magnetic monopoles. Since the basic cause is lack of long range order, one expects about one per horizon volume, but they are produced very early (10-35 s or so), when the horizon volume is extremely small. One therefore expects that the density of the universe today should be heavily dominated by monopoles (again, assuming they haven't pushed it into a Big Crunch), which it manifestly is not. This one is probably escapable: since we do not know the correct Grand Unified Theory, we don't know whether to expect point defects or not. But it was a major driver during the initial theoretical explorations of inflation in the early 1980s. | |
| Briefly explain the term "inflation", and explain why introducing a period of inflation solves the above problems. | [2] |
| Inflation is a postulated brief period of extremely rapid (usually assumed exponential) expansion in the very early universe (around 10-35 s). | [1] |
| It solves the above problems because [(i) and (iii)] the entire currently visible universe expands from a single causally connected region (i.e. <<10-35 light-seconds across) in the pre-inflation universe (to solve (iii), we have to assume that inflation begins after the monopoles are produced, therefore "1 per horizon volume" translates to "at most 1 in the visible universe"!) and [(ii)] during an exponential expansion the curvature is diluted rather than enhanced, resulting in a very nearly flat post-inflation universe regardless of the pre-inflation curvature. | [1] |
| Inflation is generally modelled as a large positive cosmological constant. Show that in this model the expansion factor of the universe during inflation is given by a(t) = a(ti) exp[H(t – ti)], where ti is the time that inflation starts. If we assume that inflation starts at 10-35 s, ends at 10-33 s, and involves 100 e-foldings of expansion, what is the implied value of the cosmological constant Λ? | [3] |
| The Friedmann equation becomes ȧ2 = Λa2/3 = H2a2; therefore H in this model is constant (since H2 = Λ/3 and Λ is constant). Take square root and rearrange to get da/a = H dt. The limits of integration are ti and a(ti) at the lower end, and t and a(t) at the upper end (i.e. we're integrating from the start of inflation to time t during inflation). Therefore the integral gives ln a – ln a(ti) = H(t – ti). Take antilogs to get the required answer. | [2] |
| If we have 100 e-foldings, then we know that ln a – ln a(ti) = 100. We also know t – ti = 99×10-35 s. Therefore H = 1.01×1035 s-1, and Λ = 3H2 = 3.1×1070 s-2. | [1] |
(2005 Resit Q5(a) and (c).)