Selection of Homework Questions


Topic 16: Cosmological Context

(1) Energy in the CMB

  1. A spherical human (!) of radius 0.5m floats in the IGM, exposed only to the CMB. Their density and heat capacity are those of water (1 gm cm-3; 4.2 J gm-1 K-1). Starting at normal body temperature (36oC), if they absorb all radiation and emit none, how long is it before they boil: (make sure you get the right factors of for the absorbed energy!)

  2. For a thermal gas of number density n, and mean particle velocity v, how many particles strike unit area per second? Using the current baryon density b = 0.04 and Hubble Constant Ho = 72 km s-1 Mpc-1, what was the proton and electron density at the time of the CMB (assume the gas is fully ionized hydrogen). How much heat energy from the gas enters our unlucky subject at the time of the CMB (when Tgas = Tphotons) from (a) protons, and (b) electrons. How does this compare to the energy coming from the photons?

(2) Geometry in Curved Spaces :

  1. Derive the metric for a spherical 2-D surface of radius R using the coordinates r, (For example using the Earth, r is from the pole along the surface, is a change in longitude)

    (a) What is ds along a "line of latitude" at constant r from the pole for a small sector d of a circle. Hence, what is the circumference of a circle radius r.

    (b) If you can measure lengths to 1 km accuracy (e.g. a car odometer over a long journey), how big must r be to detect the curvature of the Earth by driving around a line of constant latitude (R 6000 km; assume you know r exactly and your uncertainty is in the circumference).

    (c) What's the relationship between the area of a spherical triangle and the sum of its interior angles (you do not need to derive this relation)? If you can measure angles to 1 arcmin, how big (side length) must an equilateral triangle be to detect the curvature of the Earth?

  2. 3-D spatial curvature radius within a region of mass density is roughly   Rc2 ~ 3 c2 / (8G). Show that this can be re-expressed as Rc = c × Porb / 8, where Porb is the period of a circular orbit about the system. [Roughly: Rc c × dynamical time].

    (a) Hence estimate the local spatial radii of curvature (i) near the Earth's surface, (ii) within the solar system (e.g. near the Earth's orbit), (iii) within the galaxy (e.g. near the sun's orbit), (iv) within the universe.

    (b) Does this curvature affect metrology with the following levels of accuracy: (a) two satellite GPS triangulation to 1 cm on earth (GPS altitude 20,000 km); (b) wide angle planetary separations to 1 arcsec; (c) wide angle globular cluster separations to 1 arcsec (GC radii out to 50 kpc); (d) angular separations within the local supercluster (50 Mpc) to within 1 arcmin.

    (Hint: use the triangle area relation from part c above).

(3) Equations of State

  1. Write down the equation for the total (relativistic) energy, E, of a particle of rest mass mo and momentum p. The de Broglie wavelength of this particle is given by = h / p, which increases with the scale factor, just like light: a. Derive an expression for the equation of state parameter, w, for a gas of these particles, assuming they all have the same mo and p, and that the total energy density is given by u = nE for n such particles per unit volume. Show that in the relativistic limit w 1/3 and in the non-relativistic limit w 0. (Recall: pressure P = w u = wc2 where u = is the total energy density).   [Ryden: Q 4.5].

  2. Imagine if, for some bizarre reason, all the matter in the Universe (both baryonic and dark matter) suddenly annihilated to become photons. A stupendously spectacular, but energy conserving, conversion. How would this change the future evolution of the scale factor, both in the short term, and the long term?

  3. Compare the histories of the scale factor for two slight variants on our actual universe:

    (a) Perfect matter-antimatter symmetry led to full annihilation in the first second.
    (b) Something suppressed annihilation, so all CMB photons are instead proton/electron pairs.

(4) Observing De/Acceleration

  1. Can we measure de/acceleration directly by watching for a gradual change in z over time for an object? Start with the fundamental definition of redshift: 1 + z = a(to)/a(te). Now, as time passes, both scale factors change, since obviously a(to) is increasing, but so is a(te) since the light we see sets out a little later. Now find dz/dt by differentiating (1 + z) w.r.t. time. What are da(to)/dt and da(te)/dt ? Substitute in for these, using E(z), to find an expression for dz/dt.

  2. Currently, the highest accuracy with which spectral features can be measured is ~0.01 A at 8000A. For an Einstein de-Sitter universe, how long must we wait between observations to detect a change in redshift of a spectral feature at a redshift of (a) 1, (b) 4, (c) 8. Assume that in each case, a spectral feature is identified near 8000A. Compare your answer for z = 1 with the concordance model.

  3. Of course, galaxies also experience "peculiar accelerations" -- i.e. their peculiar velocities build up/change over time. Consider a typical peculiar velocity of 800 km/s for a galaxy in a cluster (Porb ~ 1 Gyr) and in the field (tfall ~ tHubble). Will the peculiar accelerations of these two galaxies undermine the measurement of cosmic de/acceleration?

  4. Discuss, briefly, the advantages of using Lyman-alpha forest absorption lines (rather than galaxies) in this experiment.

(5) Proper Distances

(6) Concordance Model

  1. Use the concordance model parameters (m = 0.27,   v = 0.73,   rel = 8.4 × 10-5,   Ho = 72), to plot the following as a function of redshift. Use three separates graphs for a, b, c. Plot linear z ranges of 0 - 5 for a and b and log z from -1.0 to 5.0 for c. Mark on each plot the times of matter/vacuum equality (and for c, the time of relativistic/matter equality).

    1. The comoving (toady's) distance, r(to);   the emission distance, r(te);   the light-travel distance, c(to - te).

    2. Angular diameter distance, DA;
      Luminosity distance, DL.

    3. Hubble parameter H(z);
      The velocity history, v(z), of a galaxy which is currently at 1 Mpc   [ie a × H(z)].

    You will need to write a routine to evaluate E(z) and its integral. I suggest you make use of the integrator qromb (which also calls trapzd and polint) in Numerical Recipes.

  2. You observe a magnitude 28.0 supernova at z = 3, located 1.2 arcsec from its host galaxy nucleus. Spectra show emission line widths implying an expansion speed of 104 km/s. What's the absolute magnitude of the supernova; its projected distance from the galaxy nucleus; and what angular velocity (in mas/yr) does the expansion velocity correspond to? (Ignore K corrections in your calculation of the absolute magnitude).

(7) The Age Problem

(8) Vacuum Energy's Accelerating Expansion

(9) The Flatness Problem

(10) The Monopole Problem

(11) Big Bang Nucleosynthesis

(12) Origin of the CMB

(13) Growth of Structure

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