(2) Geometry in Curved Spaces :
(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?
(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
(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
(5) Proper Distances
(6) Concordance Model
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.
(7) The Age Problem
(8) Vacuum Energy's Accelerating Expansion
(9) The Flatness Problem
This is an example of how a Universe that is slightly curved (unbound) at early times evolves very differently from a perfectly flat Universe. In this case, the expansion is much faster and reaches the same size (scale factor) and temperature very quickly. [Question 13.3 from Liddle].
(10) The Monopole Problem
(11) Big Bang Nucleosynthesis
(12) Origin of the CMB
n_{x} = g_{x} (m_{x} k T / 2 π ℏ^{2})^{3/2} exp[(μ_{x} − m_{x} c^{2}) / kT]
Consider the "chemical" recombination reaction: p + e = H + γ. Another fundamental result from statistical mechanics is that when this reaction is in equilibrium, μ_{p} + μ_{e} = μ_{H} + μ_{γ} where the general definition of chemical potential is μ/T = (∂S/∂N)_{U,T} (this relation arises from demanding that at a given temperature and energy, the total entropy, S, is at a maximum w.r.t. changing the particle numbers, N, of each species).
Note: Although this treatment seems to follow Ryden (pp 155-159), her MB relation is wrong -- by excluding μ she obscures the true logic behind the derivation of the Saha equation. The correct MB relation, and logic, is taken from Peebles (Principles of Physical Cosmology, pp 165 - 167).
n_{H} / n_{p}n_{e} = (g_{H} / g_{p}g_{e}) (m_{e}kT / 2πℏ^{2})^{-3/2} exp(Q / kT)
where Q = 13.6 eV is the binding energy of the electron in hydrogen.
Re-express the LHS of the Saha equation in terms of X, η and n_{γ}, and after bringing η and n_{γ} to the RHS, recast n_{γ} explicitly in terms of kT and now simplify the RHS. You should have a quadratic relation for X of the form: (1-X)/X^{2} = f(η,kT). Hence, show that the ionization fraction, X, is given by:
X = [√(1 + 4f) − 1 ]/2f where f(η,kT) = 3.84 η (kT/m_{e}c^{2})^{3/2} exp(Q/kT)
In general, optical depth is given by: τ = n σ L, where n is the particle density, σ is the particle scattering cross section, and L is the path length. In our case, σ = σ_{T} = 6.6 × 10^{-25} cm^{2} is the Thompson cross section; n = n_{e} = X η n_{γ} is the electron density, and L = cΔt is proper path length corresponding to a difference in epoch of Δt. Thus the full expression for optical depth, as a function of redshift is:
τ(z) = X(z) η n_{γ} σ_{T} cdt/dz' dz' where the integral is from z' = 0 to z, and we've transformed our path length, cdt, into a redshift interval.
Now, although many of these quantities are functions of redshift (e.g. n_{γ} and cdt/dz') the recombination transition occurs over such a narrow window in redshift that we can effectively set these to constants. Thus we have:
τ(E) = σ_{T} η n_{γ} cdt/dz' dz'/dT dT/dE X(E) dE, where E = kT in units of eV.
Pick a redshift, z, that is in the middle of recombination, and use relations: T = 2.725 (1+z) and T = 11,600 E to obtain values for dz'/dT and dT/dE; and the cosmological relation (see section 6ci): cdt = r_{H,0} dz / (1+z)E(z) to obtain a value for cdt/dz' (you'll need to use the concordance parameters for E(z), and don't confuse Peeble's E(z) with our energy variable E). Combine all these to get the pre-factor A in the relation τ(E) = A X(E') dE', with limits of E' from 0 to E.
To find the redshift of the cosmic photosphere, where τ(E) = 1, find the value of E such that the integral = 1/A (you will need to do the integral numerically). You should find that the photosphere occurs where X 0.1, so the gas that we see in the CMB is in fact pretty neutral.
(13) Growth of Structure