$$\require{cancel}$$

# Exercise 7.1.1

Constant $$t$$, $$r$$, and $$\phi$$ so we have

\begin{equation*} \begin{aligned} \int ds = \int_{0}^{d\theta} a(t)r \,d\theta ' = a(t)r\,d\theta \end{aligned} \end{equation*}

# Exercise 7.1.2

Similar to 7.1.1 above, we now have constant $$t$$, $$r$$, and $$\theta$$, which gives

\begin{equation*} \begin{aligned} \int ds = \int_{\phi}^{\phi+d\phi} a(t)r\sin\theta \,d\phi ' = a(t)r\sin\theta \,d\phi \end{aligned} \end{equation*}

# Exercise 7.1.3

This is simply $$a^2(t)r^2 \sin \theta \, d\theta d\phi$$.

# Exercise 7.1.4

Using the result from 7.1.3 above, we now just integrate over $$\theta$$ and $$\phi$$, so the area is

\begin{equation*} \begin{aligned} A &= \int_{0}^{2\pi}\int_{0}^{\pi} a(t)rsin\theta d\theta d\phi \\ \\ &= \int_{0}^{2\pi} 2a^2(t)r^2d\phi \\ \\ &= 4\pi a^2(t)r^2 \end{aligned} \end{equation*}

# Exercise 7.1.5

We know that Luminosity = (Flux)$$\times$$(Surface Area). Making the appropriate substitutions we do indeed find that $$F = L/(4\pi d^2 a^2)$$.

# Exercise 7.2.1

As we found in Chapter 6, the rate of arrival of the wave crests will be slower than the rate of emission by the factor of $$a(t_r)/a(t_e)$$. The same argument applies to the rate of arrival of photons. We also saw that wavelength would be stretched out by a factor $$1+z \equiv \frac{\lambda_r}{\lambda_e} = a(t_r)/a(t_e)$$. Therefore the rate of arrival of photons will be slowed down by a factor of $$1+z$$.

# Exercise 7.2.2

The relationship between photon energy and wavelength is $$E = \frac{hc}{\lambda}$$. Substituting this into the definition of redshift, we find that

\begin{equation*} \begin{aligned} E_r/E_e = \frac{1}{1 + z} \end{aligned} \end{equation*}

# Exercise 7.3.1

Solving Equation 7.4 for $$d_{\rm lum}$$ we get
\begin{equation*} \begin{aligned} d_{\rm lum} = \sqrt{\frac{L}{4\pi F}} \end{aligned} \end{equation*}
\begin{equation*} \begin{aligned} d^2_{\rm lum} = \frac{L}{4\pi}\frac{4\pi d^2(1 + z)^2}{L} \quad \Longrightarrow \quad d_{\rm lum} = d(1 + z) \end{aligned} \end{equation*}