91Ó°ÊÓ

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Suppose we detect radiation that was emitted by some galaxy far away. In the time the radiation traveled to reach us, its wave- length doubled. What happened to the scale factor of the universe in that time?

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). How much brighter would the sky be if it were uniformly filled with Suns, rather than the one we have? (Hint: think of the solid angle covered by the Sun relative to the whole sky.)

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Show that, if the universe were infinite in age and extent the cosmological redshift is not sufficient to get us out of Olbers's paradox.

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Show that the density of the universe is proportional to \(1 / R^{3}(t)\).

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). For the case \(k < 0,\) find an expression for \(R(t)\) valid for large \(R .\) What are the limits on \(R\) for your expression to be valid?

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Show that the density parameter \(\Omega\) is twice the deceleration parameter \(q\).

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). If the current density of the universe is \(1 \times 10^{-29} \mathrm{g} / \mathrm{cm}^{3},\) what value would be needed for the cosmological constant \(A\) in order for the universe to be static?

For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Suppose we observe an object that is \(10 \mathrm{Mpc}\) away. At what wavelength is the Ha line observed if (a) the object has no other motion, (b) the object has an additional motion away from us at \(1000 \mathrm{km} / \mathrm{s}\), (c) the object has an additional motion towards us at \(1000 \mathrm{km} / \mathrm{s}\).