Consider a flat universe containing only matter and cosmological constant.
(a). Suppose a signal is sent from some comoving coordinate $r$ at a time between $t_{i}$ and $t_{f}$ to a cosmological observer at $r=0$. Show that the largest possible value of $\mathrm{r}, r_{\max }$, is given by the formula
$$
r_{\max }=\int_{t_{i}}^{t_{f}} \frac{d t}{a(t)} .
$$
(a). Using this result, prove that if there is no dark energy, all events in spacetime will eventually become visible to the observer at $r=0$.
(c). Next show that if $\Omega_{\Lambda}>0$ that there are events which we can never see. The boundary of the future events we can see is the cosmological event horizon.
(d). In terms of the parameters of the universe now $\left(\Omega_{M}, \Omega_{R}, H_{0}\right)$, find an equation determining the maximum redshift $z_{\max }$ defined such that if an object at $z \leq z_{\max }$ emits a signal now we can see it at some point in the future. For the parameters describing our universe, what is the numerical value of $z_{\max }$ ?

Consider a relativistic gas of particles of mass $m$ in thermal equilibrium at temperature $T$. The number density of particles with momentum of magnitude $k$ is:
$$
n(k)=\frac{4 \pi g k^{2}}{(2 \pi \hbar)^{3}}\left(\frac{1}{\exp \left(\sqrt{k^{2}+m^{2}} / k T\right)-\sigma}\right),
$$
where $g$ is the number of spin states of the particle species and $\sigma=+1$ if the particles are bosons (Bosé-Einstein distribution) and $\sigma=-1$ if the particles are fermions (Fermi-Dirac distribution). (If the above formula for $n(k)$ is unfamiliar to you, review it!).
(a). Use the second law of thermodynamics (the formula for a differential change in entropy) to find $\rho(T)$ and $p(T)$ in terms of integrals involving the distribution $n(k)$. (You do not need to do the integrals.)
(b). Show that the results obtained for $\rho(T)$ and $p(T)$ imply that such particles obey an equation of state of the form
$$
p=w_{\sigma}(k T / m) \rho .
$$
In other words the equation of state is a linear relationship between pressure and energy density, where the coefficient depends only on the statistics of the particles $(\sigma=\pm 1)$ and the ratio of the temperature to the mass.
(c). By computing the integrals defining pressure and energy density, show that for cold matter, $k T \ll m, w_{\sigma}$ vanishes (for both bosons and fermions).
(d). By computing the integrals defining pressure and energy density, show that for hot matter, $k T \gg m, w_{\sigma}=1 / 3$ (for both bosons and fermions).
(e). Make a plot of $w_{\sigma}(T / m)$ and verify that it interpolates between the answers for cold and hot matter above.

Consider a gas of helium-4. Let the temperature $T$ be sufficiently small that the gas is nonrelativistic, but sufficiently hot that the helium may be ionized either singly, to He ${ }^{+}$, or doubly, to $\mathrm{He}^{++}$. We assume that the plasma is in a state of chemical and thermal equilibrium, and denote by $n$ the total number density of all helium nuclei (ionized or not).
(a). Let $X_{k}$ for $k=1,2$ be the fraction of helium atoms in the plasma that are $k$-ionized, and $X_{e}$ be the fraction of total electrons that are unbound. Derive the coupled Saha equations:
$\frac{1-X_{1}-X_{2}}{X_{1}\left(X_{1}+2 X_{2}\right)}=\frac{n}{4}\left(\frac{m_{e} k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left(\frac{B_{1}}{k T}\right), \quad \frac{X_{1}}{X_{2}\left(X_{1}+2 X_{2}\right)}=n\left(\frac{m_{e} k T}{2 \pi \hbar^{2}}\right)^{-3 / 2} \exp \left(\frac{B_{2}}{k T}\right)$ where $B_{1}$ and $B_{2}$ are the first and second binding energy of helium.
(b). In our universe, helium makes up about $24 \%$ of the baryonic matter density. Taking this as input, and using the results of part a) make a plot of the ionization fractions $X_{e}, X_{1}, X_{2}$ for temperatures between $5000^{\circ}$ Kelvin and $20000^{\circ}$ Kelvin, and briefly describe your findings. At about what temperature is $99 \%$ of the helium in the universe plasma in electrically neutral atoms? At what temperature is it $99 \%$ doubly ionized?
(c). Suppose the laws of physics were modified so that there was no such thing as hydrogen and all the baryonic matter in the universe was found in helium with negligible amounts of heavier elements. In this hypothetical universe, determine the temperature at last scattering of photons by charged matter. (Assume in this hypothetical universe that the total helium density is again $24 \%$ of the baryon density in our universe. Assume also that $\Omega_{M}$ takes the same value as in our universe, so the hypothetical universe contains a bit more dark matter to compensate for the missing hydrogen.)