1. Consider a flat universe with only matter.
(a) Prove that ε=εe,0a-3, where εe,0 is the critical density today.
(b) Prove that a(t) = (t/t0)^2/3.
(c) Find the current proper distance, comoving distance, luminosity distance, an angular diameter distance to a galaxy at redshift z.
(d) Find the horizon distance.
(e) Find the age of the Universe in terms of H0.
(f) Find the maximum angular diameter distance of the Universe.