# Comoving redshift and area of sphere

1. Nov 19, 2013

### bowlbase

1. The problem statement, all variables and given/known data
Standard candles may be used to measure the "luminosity distance", using DL =
(L/4F)1/2, where L is the source's intrinsic luminosity, and F is the observed
flux. Inthis problem you will relate the luminosity distance to the previously discussed angular
diameter distance DA. Consider a source at redshift z, which emits light isotropically. In its rest frame, suppose it emits a pulse of total energy Ee over a short time interval Δte. Suppose that today, at z = 0, we have a telescope with collecting area ΔA. For simplicity, assume that the universe is spatially at, with $\Omega$curv = 0, so that the angular diameter distance is related to the comoving distance by DA = aDco.

The pulse of light isotropically emitted at redshift z is now, at z = 0, spread out
over the surface of a sphere. Assuming that this source is at comoving distance
Dco from us, what is the comoving area of that sphere today? What is the
proper physical area of that sphere today
? Assuming that there is no intervening
absorption or scattering, what fraction of the photons emitted by the source
impinge on our telescope with collecting area ΔA? Express your answer in terms
of Dco.

2. Relevant equations

3. The attempt at a solution

I think that I need to do an integral somewhere with z going from z to 0. However, at first I just thought that the the surface area (SA) would just be $4\pi(\frac{aD_{co}}{2})^2$ but Dco is just suppose to be the distance from me, the observer, to the source of the emitted energy.

Anyway, I know that SA=$4 \pi r^2$. I just need to find 'r' from z → 0. I know that $D_{co}=\int\frac{c dt}{a(t)}$. a has a time dependence so I can't just replace it with a-1=1+z. If I do I just get $\int_z^0c(1+z)dt$. Which would be $-cz(1+z)$
I don't believe a negative distance would properly describe this situation.

Also, the second question I bolded has me confused. Isn't comoving distances suppose to describe how things actually are? How are the two bolded questions different at all?