| Define the terms proper distance, luminosity distance and angular diameter distance. | [1.5] |
| Proper distance: distance obtained by integrating the Robertson-Walker metric with dt = 0 (i.e. distance at some specific time t; the comoving proper distance is the proper distance evaluated at a=1). You could also quote the formula dP = ∫dt/a evaluated from the time of emission of the light being observed, te, to the time of observation, to. | [0.5] |
| Luminosity distance: distance obtained by comparing the observed flux (or apparent magnitude) with the emitted luminosity (or absolute magnitude): f = L/(4πdL2) or m – M = 5 log(dL/10). | [0.5] |
| Anguiar size distance: distance obtained by comparing the angle subtended by the object to its linear dimensions: θ = D/dA where D is the linear diameter of the object and θ is its angular diameter. | [0.5] |
| Show that the angular diameter distance dA and the luminosity distance dL are related by dA = dL/(1 + z)2 for an object at redshift z. | [3] |
| For the luminosity distance, we first need to calculate the area of a sphere of radius r (where r is the comoving proper distance) in the RW metric. This is straightforward: we set dr = dt = 0 and integrate over dΩ, getting 4πx2 where x is defined on the exam cover. Therefore, in a non-expanding universe, we would have f = L/(4πx2). | [0.5] |
|
In an expanding universe, however, each photon we receive has had its energy reduced
by a factor of (1 + z), from the redshift of its frequency. In addition, the
rate of reception of photons per unit time is reduced by another factor of
(1 + z). Therefore in fact
f = L/(4πx2(1 + z)2). Comparing this with the above definition, we see that dL = x(1 + z), |
[1] |
| For the angular size distance, we calculate the proper distance between the two edges of the object, assuming it is aligned perpendicular to us. Since it's bound to be a small angle, we don't really need to integrate – we can just write D = a(te)xθ [that's a letter x not a × sign!]. Comparing this with the definition, and writing a(te) = 1/(1 + z), we find that dA = x/(1 + z). | [1] |
| Comparing the expressions for dL and dA gives dL = dA(1 + z)2 as reuired. | [0.5] |
| Does this result rely on assuming that the universe has a flat geometry? | [0.5] |
|
No. Nothing we did in the derivation required setting x = r (which is true only
in a flat geometry); in fact, the factors of x cancelled out when we considered
the ratio. Note, however, that the expressions dL = dP(1 + z) and dA = dP/(1 + z) do require the universe to have a flat geometry, because in these cases we have set x = r. |
[0.5] |
(2006 Q4.)