Reading:

Roos sections 1.4 -- 1.6. (attached)
Weinberg Mathematical Supplements 2, 3. (attached)

Problems:

1. (6 points) In class we discussed the flatness problem, claiming that the quantity $1 -\Omega$ grows with the expansion of the universe, in our quasiNewtonian approximation. For $\Omega_o$ as close to one as measured now, this means that it must have been fine-tuned extremely close to one in earlier epochs. Here you will demonstrate the problem.
   1. In class we showed that a galaxy's energy
      
      $$E = \frac{1}{2}\ mr^2 (t)\, H^2(t)\ \left(\ 1 - \frac{\rho(t)}{\rho_c(t)}\ \right)$$
      
      
      is constant in time, where
      
      $$\rho_c (t) = \frac{3 H^2(t)}{8\pi G}\ \ .$$
      
      
      Show this implies that the quantity
      
      $$r^2(t)\ \ (\ \rho_c(t) - \rho(t)\ )$$
      
      
      is also constant.
   2. Assume $\rho \propto r^{-n}$, as in the cases discussed in class, and define $\Omega (t)$ as the ratio $\rho(t)/\rho_c(t)$. Using part (a), express the ratio
      
      $$\frac{1 - \Omega(t_1)}{ 1- \Omega(t_2)} \$$
      
      
      in terms of the ratio $r(t_1)/ r(t_2)$.
   3. The universe, which is now $10^{18}$ s old, became matter-dominated at $10^{12}$ s. If $(1- \Omega)$ is $0.9$ now, what must it have been at $t = 10^{12}$ s, when the universe first became matter-dominated? (Neglect our very recent departure from matter domination.)
   4. Before $10^{12}$ s, the universe was in a radiation-dominated era. Given your value for $(1- \Omega)$ at $10^{12}$ s, what must it have been at $10^2$ s, when big bang nucleosynthesis occurred? (Note the universe was radiation-dominated during this entire time interval.)