$$\require{cancel}$$

# Exercise 16.1.1

$$d_L = d_A (1+z)$$

# Exercise 16.2.1

The path is at constant $$\theta$$ and $$\phi$$ so only $$r$$ is varying. Therefore $$\sqrt{ds^2} = a(t) dr/\sqrt{1-kr^2}$$. The comoving length is given by integrating this up, while setting the scale factor equal to unity so $$\ell = \int_0^r dr/\sqrt{1-kr^2}$$. For $$kr^2 << 1$$, $$\ell \simeq \int_0^r dr(1+k r^2/2) = r + kr^3/6$$.

# Exercise 16.3.1

Curvature affects the history of the expansion rate, $$H(a)$$, via the Friedmann equation. It also affects the time a photon has to travel to come from coordinate distance $$r$$ due to the $$dr^2/(1-kr^2)$$ term in the invariant distance equation.

# Exercise 16.4.1

I have not produced a written solution for this yet.