| The present universe has Ωm0 = 0.23 and Ωr0 = 8.4 × 10-5. Calculate the redshift z at which Ωm = Ωr (the epoch of matter-radiation equality). | [2] |
|
Since
H2Ωr = H02Ωr0/a4
for radiation and
H2Ωm = H02Ωm0/a3
for matter, we have
Ωr/Ωm = Ωr0/(Ωm0 a).
Therefore the value of a at which Ωr/Ωm = 1 is
a = 8.4 × 10-5/0.23 = 3.7 × 10-4. Hence z = 2740. |
[2] |
| The differential equation for structure growth shows that density perturbations should grow ∝ a in a matter-dominated universe. Explain why, despite this, one might expect that structures would not start to grow until the epoch of recombination at z ∼ 1100. | [2] |
| High-density regions can be prevented from collapsing by their internal pressure. At radiation-matter equality, the radiation pressure (Pr = εr/3) is great enough to prevent collapse of structures below the horizon size, although the gas pressure is not. While the baryonic matter is still fully ionised, the photons and baryons interact frequently, forming a single photon-baryon fluid; therefore, if radiation pressure prevents the photons from collapsing, the baryons won't either. | [1.5] |
| At recombination, protons and electrons combine to form neutral hydrogen. Neutral atoms interact far less readily with photons, so the photons decouple from the baryonic matter. This means that the radiation pressure no longer acts to prevent baryonic structures from collapsing, so structures can start to grow. | [0.5] |
| How does the recognition that most of the matter in the universe is non-baryonic cold dark matter affect the growth of structure? | [3] |
| Non-baryonic dark matter is electrically neutral and therefore does not interact with photons. Therefore the radiation pressure has no effect on it, and collapse can start at matter-radiation equality. | [1] |
| However, if the dark matter is relativistic at this time, it itself will have a "radiation" equation of state, and therefore its own pressure will prevent it from collapsing until its velocity has dropped below relativistic levels. | [1] |
| We define dark matter to be "hot" if it is relativistic (v ∼ c) at z ∼ 3000 (i.e. T ∼ 104 K), and "cold" if it is non-relativistic (v << c) at this temperature [dark matter which would be mildly relativistic, with v a significant fraction of c, is commonly called "warm"]. Therefore, if the dark matter is cold, it does not have a relativistic equation of state at matter-radiation equality, and so neither photon pressure nor its own pressure will prevent collapse. | [1] |
(2006 Resit Q7(b).)