PHY306 INTRODUCTION TO COSMOLOGY HOMEWORK 3 NOTES

1. Show that, for a universe which is not flat,

1 − Ω 1 − Ω = ,

where Ω = Ω r + Ω m + Ω Λ and the subscript 0 refers to the present time. [2]

This was generally quite well done, although rather lacking in explanation. You should really

define the Ωs, and you need to explain that you divide the expression for t by that for t0.

2. Suppose that inflation takes place around 10 −35 s after the Big Bang. Assume the following simplified properties of the universe:

● it is matter dominated from the end of inflation to the present day;

● at the present day 0.9 ≤ Ω ≤ 1.1;

● the universe is currently 10 10 years old;

● inflation is driven by a cosmological constant.

Using these assumptions, calculate the minimum amount of inflation needed to resolve the flatness problem, and hence determine the value of the cosmological constant required if the inflation period ends at t = 10 −34 s. [4]

This was not well done at all, although it is not really a complicated calculation. I think the principal problem was that people simply did not think it through—not only do you not explain what you’re doing to the marker, you frequently don’t explain it to yourself! Some people therefore got hopelessly confused between the two time periods involved—the matter-dominated epoch from 10 −34 s to 10 10 years, and the inflationary period from 10 −35 to 10 −34 s.

The most common error—in fact, considerably more common than the right answer—was simply assuming that during the post-inflationary epoch 1 − Ω ∝ . This is only true for a radiation dominated universe, whereas in this question you were told to assume matter domination. There’s really no excuse for this (and these people hadn’t simply misread the question, because they all started off by saying that the universe was matter dominated).

/ Most people failed to specify that the universe is nearly flat when taking = / . We saw in homework 2 that a matter dominated universe that isn’t flat has a more complicated form of a, which only tends to t2/3 as Ω tends to 1. Fortunately, we know that in this case Ω is close to 1, so it’s OK to take the matter-only solution—but you need to justify this.

Quite a lot of people seemed to be using a2 rather than a to calculate H (and hence Λ) at the end. Although the curvature depends on 1/ a2, the solution of the Friedmann equation is ∝ , not a2. I think this was another case of not thinking things through.

As usual, there were nowhere near enough words in the answers—people simply wrote down equations with no attempt to explain or justify them. This may in fact have contributed to the errors: if you’d tried to explain why you thought 1 − Ω ∝ , maybe you’d have seen that it wasn’t true!

3. For the matter-dominated, essentially flat universe described above, what would the horizon distance be at t = 300,000 years (the time of emission of the cosmic microwave background)? [1]

Most people got the right answer here, although almost everyone made life more difficult for themselves by converting to metres or parsecs (why? The expression is 3 ct , which naturally comes out in light-years). There was some dodgy rounding, with people quoting more than 4 sig. fig. despite having used 3×10 8 m/s for c (only correct to 3 sig. fig.), or rounding an intermediate result to 2 sig. fig. and yet quoting the final one to three.

Explain quantitatively why this presents a problem, given the uniformity of the cosmic microwave background. [1.5]

Many people did not take the ‘quantitatively’ seriously, even though it was in bold, did not provide numbers, and therefore did not get any marks. Many others simply copied the logic we used in the lectures, although it is not immediately obvious that the toy model we are using here has been set up to have the CMB emitted at redshift 1000 (in fact it has, but you have to calculate this: you can’t simply assume that it’s true). Quite a lot of people forgot to allow for the expansion of the universe, and therefore wound up with an angle too small by a factor of 1000. (The majority of these people still proceeded to assume that this corresponded to 2°, even though it’s plainly 0.002°: that cost you another half mark. If you’ve made a mistake, own up to it: don’t simply fudge the answer to get the number you expect. I will spot this, and then I’ll dock you more marks for whatever the fudge was!) Some people even got the expression for θ the wrong way up, which should have got them a huge angle, but they then proceeded to decide that this was in arc seconds, even though it plainly wasn’t (an angle calculated by dividing two lengths that are in the same units is always in radians: only dividing AU by parsecs gives you arc seconds), and divided by 206265 to get a number in the right ballpark (but no marks).

Briefly explain how the introduction of inflation solves this problem. [1.5]

Although I didn’t require a quantitative explanation here, I was looking for logical rigour. A lot of people failed to distinguish between the entire universe and the visible universe (the latter does expand from a causally connected pre-inflation region, but there’s no reason to believe that said region constitutes the entire universe), and I did want you to tell me that the inflationary expansion is faster than light, as this is critical to making the trick work (you have to have spacetime expanding faster than the horizon distance, and the horizon distance fairly obviously expands at the speed of light). There was also a particularly unfortunate spelling error: casual is not at all the same as causal (in fact it’s practically the opposite: if you were to say that there was a ‘casual’ relation between two things, you certainly would not be claiming that one caused the other!).