| By considering the path of a photon, show that the comoving proper distance between an object and the origin is given by | |
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| where te is the time of emission and to is the time of observation. | [2] |
| This is bookwork. The main issue with "Show that" questions is that it is not enough simply to write down the right equations in approximately the right order. You must explain the steps you are taking, any approximations you are making, and so on. Most people lose marks in this type of question through not doing this. For example, in this case you must explain that the spacetime interval for a photon is always zero (because photons travel at the speed of light) – it is not enough just to start with "c2dt2 = a(t)2dr2". | |
| Hence show that, in an expanding universe, the observed light will be redshifted such that | |
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| where λe is the emitted wavelength and λo is the observed wavelength. | [4] |
| Another set derivation, and the comment above applies here to an even greater extent. |
(2008 Q5(a).)