Astro 321: Problem Set 1

Due April 7

1 Problem 1: Units

Convert the following quantitites by inserting the appropriate factors of c, h̄, k_{B} and unit conversions. You may find Peacock chapter 9 helpful.

• H_{0} = 100hkm s^{−1} Mpc^{−1} into (a) eV, (b) Mpc^{−1}, (c) Gyr^{−1}. [Corresponds to upper limit on the mass of a dark energy particle, the inverse Hubble length, inverse approximate age.]

• ρ_{crit} = 3H_{0}^{2}/8πG into (a) g cm^{−3}, (b) GeV^{4}, (c) eV cm^{−3}, (d) protons cm^{−3}, (e) M_{⊙} Mpc^{−3}. If the cosmological constant, has ρ_{Λ} = 2ρ_{crit}/3, what is its energy scale in eV (i.e. ρ_{Λ}^{1/4}). Compare that to the Planck mass.

• T_{CMB} = 2.728K to (a) eV. Assuming a black body distribution, convert this to number density n_{γ} in photons cm^{−3} and energy density ρ_{γ} in (a) eV cm^{−3} (b) g cm^{−3}, and Ω_{γ} = ρ_{γ}/ρ_{crit}.

• T_{ν} = (4/11)^{1/3}T_{CMB}. Use this to express n_{ν}, ρ_{ν} and Ω_{ν} in the above units assuming that the neutrinos are relativistic (and fermions and have three species!).

• With the above relic number density, now consider the case where one out of three neutrino species has a mass of 1 eV and the rest are massless. What is the density of relic neutrinos in units of the critical density Ω_{ν,massive}. For what mass is the density at the critical value.

2 Problem 2: Age

• Assume the universe today is flat with both matter (Ω_{m}) and a cosmological constant (Ω_{Λ}). Compute the age of the universe and plot your result for t_{0} in h^{−1} Gyr as a function of Ω_{m}.

• Assume instead that the universe is open with matter (Ω_{m}) but no cosmological constant. Compute the age of the universe and plot your result for t_{0} in h^{−1} Gyr as a function of Ω_{m}.

• Assume that there is only matter and radiation in the universe (no cosmological constant) and that the universe is flat. Integrate the age equation to determine the time at which the cosmic temperature was 10^{9}K and 1/3 eV.