# S16. Distance and Magnitude - SOLUTIONS

- Page ID
- 7852

### Exercise 16.1.1

**Answer**-
\(d_L = d_A (1+z)\)

### Exercise 16.2.1

**Answer**-
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

**Answer**-
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

**Answer**-
I have not produced a written solution for this yet.

### Exercise 16.4.2

**Answer**-
I have not produced a written solution for this yet.

### Exercise 16.4.3

**Answer**-
I have not produced a written solution for this yet.