PHY228 Introduction to Cosmology Homework 3 Answers

Historical Understanding 1. Explain the basic features of the Steady State model of cosmology. [3] Basic principle: universe should look the same (on large enough scales) at any point in space and at any point in time. [1]

Therefore: • since the universe is expanding, new matter must constantly be created to maintain the same average density at all times; [0.5] • since H must be the same at all times, expansion must be exponential, and the universe is infinitely old; [0.5] • the general appearance and frequency of different types of object must not change with cosmic epoch (no evolution); [0.5] • galaxies can have a range of ages, though the typical age will be 1/H, and there will be no trend in age with redshift. [0.5]

The mark scheme here is fairly flexible: more detail can be traded off against fewer points. The basic “Perfect Cosmological Principle” must be included for full marks.

The Steady State model was introduced in 1948, and died for all practical purposes in the mid 1960s. If you were a young cosmologist in 1968, what observations would have led you to prefer the Big Bang model over the Steady State model? For each observation you select, explain carefully why you feel that it disfavours the Steady State model. [4] Redshift distribution of quasars and radio galaxies: from last two points above, quasars should be spread evenly through space, so no peak in redshift distribution. However, observations show that they are much more common at redshift ~2 than in the local universe. Identification of radio sources with optical counterparts with measurable redshift disproved the option that faint radio sources were a local, intrinsically faint, class of objects. (The first quasar redshift was measured in 1962.) The argument that quasar redshifts are not cosmological is tenable, because we do not know what quasars are at this point, but contrived. Existence and spectrum of CMB: existence is favourable for the Big Bang, which naturally expects a thermal relic radiation from original hot dense plasma, but does not of itself disprove Steady State (not unreasonable to suggest that radiation could be created along with matter, and not even unreasonable to suggest that it could have a blackbody spectrum, cf. Hawking radiation; could also suggest that it is thermalised background starlight from extremely distant galaxies). However, as blackbody spectra with different temperatures (i.e. different redshifts) do not sum to a blackbody, one would expect deviations from the blackbody spectrum. In 1968, observations are consistent with a blackbody spectrum, but only the long wavelength tail has been measured. Helium abundance: seems excessively high compared to the abundances of the heavier elements. This would be expected in the Big Bang model, where it is made in the early universe as well as in stars (cf. Peebles 1966). It is harder to explain in the Steady State, where all elements are made using currently operating processes: one has to invent some kind of helium factory (Hoyle suggested supermassive stars). Note that the deuterium abundance in not well known at this time, so cannot be used: this weakens the force of the Big Bang predictions, since we cannot now use the consistency of the calculations as a point in favour. In all three cases, the observation can be explained in the Steady State, but only by invoking extra hypotheses which are not an integral part of the theory. On the other hand, they arise naturally in the Big Bang. Therefore, the argument that one should not introduce more new ideas than necessary (Occam’s Razor) suggests that the Big Bang model is preferable.

Mathematical and numerical problem-solving 2. Show that if the universe expands from a1 to a2, a blackbody spectrum with temperature T1 remains a blackbody spectrum, but its temperature is reduced to T1a1/a2. [2] 2hν 3 1 [Note that the Planck function is Bν (T) = .] c2 ehν kT −1

The Planck intensity B
(T) is the energy per unit volume. If the Universe expands from a1 to a2, the frequency of a given photon decreases from
to
' = a1
/a2, and “unit volume” at a1 3 becomes volume (a2/a1) at a2. Since energy is conserved, the amount of energy that was 3 carried by photons of frequency
in unit volume at a1 is now spread out over volume (a2/a1) (and is being carried by photons of frequency
'). [1]

Therefore, the energy per unit volume at a2 carried by photons of frequency
' is 3 3 3 3 3 3 2hν 1 a1 2hν ′ a2 1 a1 2hν ′ 1 Bν ′ = = = 2 hν kT 3 2 3 hν ′a2 kTa1 3 2 hν ′ kT′ c e −1 a2 c a1 e −1 a2 c e −1 if we define T' = a1T/a2. This has the same form as the original B
(T), so it is still a Planck function, but now corresponding to T' instead of T. [1] 3. If we consider a “tired light” model, in which light loses energy as it traverses space, what happens to a blackbody spectrum when the energy E1 of each photon has been reduced to E2 = E1a1/a2 (i.e. the equivalent of an expansion from a1 to a2)? [1]

We still change
to
' = a1
/a2, but we no longer change the unit volume, because space is not expanding in a tired light model. Therefore the Planck function becomes 2hν ′3 1 a3 ′ 2 Bν ′ (T ) = 2 hν ′ kT′ 3 . c e −1 a1 This is not a blackbody spectrum, even though the shape is right, because the normalisation is wrong. [1] 4. Suppose that inflation takes place around 10−35 s after the Big Bang. Assume the following simplified properties of the universe:

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

 at the present day 0.8 ≤  ≤ 1.2;

 the universe is currently 1010 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. [5] In a radiation dominated universe, |1 –
| ∝ a2, and a ∝ t1/2, so |1 –
| ∝ t. Therefore, if |1 –
| ≤ 0.2 at t = 1010 years = 3 × 1017 s, at t = 10−35 s we must have |1 –
| ≤ 6.5 × 10−54. [1.5] In an inflationary epoch, if we assume this is equivalent to a large cosmological constant, we 2 have |1 –
| ∝ 1/a . If we assume we start at ai with |1 –
| ~ 1, then to end with |1 –
| ≤ 6.5 × 10−54, we must increase a by a factor of (6.5 × 10−54)−1/2 = 3.9 × 1026. This is the amount of inflation needed, though since in the inflationary epoch a = ai exp(H(t – ti)) it is more usual to quote ln af/ai = 61, and say we need 61 e-foldings of inflation. [2] 26 −35 −34 We know af/ai = exp(H(tf – ti)) = 3.9 × 10 , ti = 10 s, and tf = 10 s. Put in these numbers to get H = 6.8 × 1035 s−1. Since  = 3H2, it follows that  = 1.4 × 1072 s−2. [1.5]