ASTRON 329/429, Fall 2015 – Problem Set 3

Due on Tuesday Nov. 3, in class. All students must complete all problems.

1. More on the deceleration parameter. In problem set 2, we defined the deceleration parameter q0 (see also eq. 6.14 in Liddle’s textbook). Show that for a universe containing only pressureless matter with a cosmological constant Ω q = 0 − Ω (t ), (1) 0 2 Λ 0 where Ω0 ≡ Ωm(t0) and t0 is the present time. Assuming further that the universe is spatially flat, express q0 in terms of Ω0 only (2 points).

2. No Big Bang? Observations of the cosmic microwave background and measurements of the abundances of light elements such as lithium believed to have been synthesized when the uni- verse was extremely hot and dense (Big Bang nucleosynthesis) provide strong evidence that the scale factor a → 0 in the past, i.e. of a “Big Bang.” The existence of a Big Bang puts constraints on the allowed combinations of cosmological parameters such Ωm and ΩΛ.

Consider for example a fictitious universe that contains only a cosmological constant with ΩΛ > 1 but no matter or radiation (Ωm = Ωrad = 0). Suppose that this universe has a positive Hubble constant H0 at the present time. Show that such a universe did not experience a Big Bang in the past and so violates the “Big Bang” constraint (2 points).

3. Can neutrinos be the dark matter? Thermal equilibrium in the early universe predicts that the number density of neutrinos for each neutrino flavor in the cosmic neutrino background, nν, is related to the number density of photons in the cosmic microwave background, nγ, through nν + nν¯ = (3/11)nγ. Here, nν¯ accounts for anti-neutrinos, which have the same mass as the corre- sponding neutrino. a) What is the expected average total number neutrinos and anti-neutrinos per cm3 in the cosmic neutrino background today? (1 point) b) Suppose that neutrinos in the cosmic neutrino background were to be dominant form of dark matter in our universe. What would the sum of the neutrino masses, Σmν ≡ m(νe)+m(νν)+m(ντ ), have to be to explain the dark matter density parameter ΩDM ≈ 0.27 inferred by the latest cosmo- logical parameter measurements? (1 point) c) In class, we noted that the sum of the neutrino masses is constrained to be Σmν < 0.23 eV. What is the maximum fraction of the total inferred dark matter density that can be explained by primordial neutrinos? (1 point)

4. Distances in an arbitrary EOS universe. Suppose that you are in a spatially flat universe containing a single component with a constant equation of state parameter w. – 2 – a) Derive expressions for the comoving, luminosity, and angular diameter distances as a function of redshift (dcom(z), dlum(z), and ddiam(z)). (2 points) b) At what redshift does ddiam(z) peak? (1 point)

5. Cosmological surface brightness dimming. The surface brightness SB of an astronomical object is defined as its flux S divided by the solid angle subtended on the sky. Thus, SB ∝ S/(δθ)2. Suppose that a class of objects are both standard candles and standard rulers, so that their in- trinsic luminosity and size are known and constant. Derive an expression for the apparent surface brightness of these objects as a function of redshift, SB(z). Your answer should only involve the present day surface brightness, SB0, of these objects and the redshift z. (2 point)

6. Age of the Galaxy. Our Galaxy’s age can be estimated using the radioactive decay of uranium. Uranium is produced through the “r-process” (involving rapid neutron captures) in supernovae. From our understanding of the r-process, the initial abundances of the two isotopes U 235 and U 238 are expected to be in the ratio

235 U 238 ≈ 1.65. (2) U initial The decay rates of the isotopes are

λ(U 235) = 0.97 × 10−9 yr−1 (3)

λ(U 238) = 0.15 × 10−9 yr−1. (4) The present abundance ratio is 235 U 238 ≈ 0.0072. (5) U final a) Use the decay law U(t) = U(0) exp (−λt) (6) to estimate the age of the Galaxy (1 point). b) Assuming that the Galaxy formed one billion years after the Big Bang, use this observation to derive an upper limit on the present-day Hubble constant H0 (in units of km/s/Mpc) assuming a critical-density universe containing only pressureless matter and a cosmological constant. You can use the result that in such a universe  √  2 1 1 + 1 − Ω0 H0t0 = √ ln √ (7) 3 1 − Ω0 Ω0 (2 points).