Introduction to Cosmology Example Exam Question: Answer

Lecture 16: Growth of Structure, part 2

CMB power spectrum
The diagram shows the CMB power spectrum using data from WMAP, CBI and ACBAR.

Briefly explain exactly what the "CMB power spectrum" actually is and what the symbols l and Cl represent. 		[3]
The steps you need to include are the following:
  1. Quantify the map by expressing it as a series of spherical harmonics: ΔT/T = ∑l ∑m alm Ylm(θ,φ).
  2. Convert the 2D map to a 1D distribution by constructing a correlation function: C(θ) = ⟨(ΔT/T)n(ΔT/T)n'⟩, where the average is taken over all pairs of directions n, n' separated by an angle θ.
  3. Then substitute the spherical harmonic series from (1) into the correlation expression from (2). It turns out that this gives 4πC(θ) = ∑l (2l + 1) Cl Pl(cos θ), where the Pl are the Legendre polynomials. In an ideal world, the sum goes from 0 to infinity, but in practice l is cut off at the low end (large angles) by the sky coverage of your experiment and at the high end (small angles) by your angular resolution.
[2]
Therefore l is the index of your spherical-harmonic expansion (therefore, the larger the value of l, the smaller the angle you are talking about), and Cl is the coefficient of that component in your Legendre polynomial expansion, and therefore represents the relative strength of that particular angular size in the overall pattern. [1]
Compare and contrast, carefully but without detailed mathematics, the growth of structure in a universe dominated by (i) non-baryonic cold dark matter, (ii) non-baryonic hot dark matter and (iii) baryonic matter. [5]
As with most descriptive questions, the mark scheme for this is fairly flexible, and a variety of answers are possible. The one below covers all the material and is properly structured – I have highlighted the elements of comparison and contrast that are needed to answer the question – but that doesn't mean that you would have to reproduce this word for word to get full marks!

In all three cases (compare), density fluctuations cannot possibly start to grow until the universe becomes matter dominated. However, in cases (ii) and (iii), (compare/contrast), galaxy-sized density fluctuations will still not start to grow, because they are prevented from doing so by radiation pressure – in case (iii), pressure from photons, because they couple to the baryons by electromagnetic interactions, but (contrast) in case (ii), pressure from the dark matter itself, which has a "radiation" equation of state because it is relativistic at this time. In case (iii), galaxy-sized density fluctuations will start to grow once the radiation decouples from the matter at recombination, whereas (contrast) in case (ii), the size of fluctuations that can grow will decrease gradually as the universe expands and the temperature, and therefore the velocity, of the dark matter particles decreases. 		[2]
The size of fluctuations that can grow is determined by the Jeans mass. For a non-relativistic gas such as case (i) or case (iii) after recombination (compare), this turns out to be about the size that presently corresponds to a dwarf galaxy, whereas (contrast) for case (ii) and case (iii) before recombination (compare), it is approximately the horizon size. [2]
Therefore, in summary:
  • for dark-matter dominated models (i) and (ii), structure growth starts at the epoch of matter-radiation equality, but in hot-dark-matter models only very large structures can form then;
  • in baryon-dominated models structures don't start to develop until recombination;
  • both cold dark matter and baryonic models form structure bottom up (small structures form first), whereas hot dark matter models form structure top down (large structures form first).
 [1]

(2005 Resit, part of Q8.)

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