Problems

#06

Estimate an age from $t=1/H_0$. We note that 1 Mpc has $3\times 10^{19}$ km and that 1 year has 31 million seconds. If $H_0=1$ km/s/Mpc, the age would be $3\times 10^{19}/3.1\times 10^7 \approx 10^{12}$ years. For $H_0 = 60, 70, 80$, the maximum age is $17, 14, 12.5$ billion years.

#12

The spectrum of the microwave background is a 3 K blackbody. Using Wien's Law, it has peak emission at $\lambda = 0.27 / T \approx 1$ mm. At redshift $z=6$, the background radiation would have been hotter (because the universe has expanded more since then, and so the temperature has been dropping). The peak wavelength for the blackbody would be shorter by a factor $1+z=7$, so the temperature would be 7 times great, which would place it around 21 K.