ASTR/PHYS 328/428
Cosmology
Problem Set 5
Due Tuesday 26 November 2024

  1. High Redshift Quasars

    The existence of quasars powered by supermassive black holes with
    MBH ≈ 109 M☉ at z = 7.5
    has been reported by both Banados et al (2018) and Yang et al (2020). The growth of such objects is constrained by the Eddington limit, which is the point at which an object becomes so bright that it blows itself apart via photon pressure.
    a) Assuming these beasts grow exponentially at the Eddington-limited rate with e-folding time τ = 50 Myr, what was the initial `seed' mass at early times?
    (By "early" you may assume t = 0, but think about what difference this makes.)
    [Hint: recall the age-redshift relation from the third homework. Assume the same vanilla ΛCDM parameters: Ωm = 0.3, ΩΛ = 0.7, and H0 = 70.]
    b) The early universe, as seen in the CMB, is extremely homogeneous. Structure forms from small initial perturbations that grow as δ ∼ a. Is this consistent with the result from (a)?

  2. Cosmic Microwave Background Radiation
    The microwave background is the relic radiation field of the Big Bang. It has a perfect thermal distribution with
    Trad = 2.7255 K. Remembering that thermal radiation obeys Wien's Law, Planck's Law, and the Stefan-Boltzmann relation, calculate these CMB basics:
    a) What is the peak wavelength of the relic radiation?
    b) What is its energy density?
    c) What is the number density of photons?
    d) If the mean mass density of baryons in the universe is ρb = Ωb ρcrit, what is the number density of baryons?
    [You may assume that the average baryon mass is that of a proton. Choose a value for Ωb h2 from the literature and cite your source.]
    e) What is the baryon-to-photon ratio?

  3. Acoustic Power Spectrum
    Max Tegmark built movies of what happens to the acoustic power spectrum of the CMB as cosmic parameters change. You can switch parameter by clicking its name at right under "combo movies". Watch these to answer the following:
    What happens as
    a) curvature increases?
    b) the density of cold dark matter increases?
    c) the baryon fraction increases?
    Think about why each parameter has the effect that it does.

  4. Recombination History of the Early Universe
    The ionization fraction, expressed as the ratio of free electrons to nuclei, Xe, can be determined by numerical solution of the Saha equation as the universe cools using a code like RECFAST. Use this RECFAST output run using Planck parameters to answer this question.
    [It may help to refer to Chapter 8 of the book, particularly section 8.3. (That'd be chapter 9 in the first edition.)]

    Plot Xe as a function of redshift.
    It may help plot redshift as log(1+z).
    a) Does this depiction imply that recombination is a sudden event?
    b) At what redshift does Xe = 1/2?
    c) How does this compare to the value given in Table 8.1?
    d) How can it be that Xe > 1 at very high redshift?
    [Hint: what is happening around z = 6000?]

    ASTR/PHYS 428 only:
    Matter remains entangled with the radiation field and no structure can form until well after recombination.
    Plot the matter and radiation (CMB) temperature against redshift.
    e) Find the redshift at which Tmatter = 1/2 Trad.
    f) What is the optical depth to this redshift? (see eq 8.45 in the book).
    g) The observed optical depth is at least τ = 0.06. How does this compare to you answer in (f)? What causes this difference?

  5. WIMPs in you
    It is widely assumed that the cosmic dark matter is made of Weakly Interacting Massive Particles (WIMPs). In order to explain the Galactic rotation curve, these need to have a local density of 0.25 GeV cm-3. If each WIMP has a mass of 100 GeV, how many are passing through your body at any given moment?
    [Hint: your volume follows easily from assuming that you have the density of water.]