Blog #32: WS10.1, #3

3. Observable Universe. It is important to realize that in our Big Bang universe, at any given time, the size of the observable universe is finite. The limit of this observable part of the universe is called the horizon (or particle horizon to be precise, since there are other definitions of horizon). In this problem, we'll compute the horizon size is a matter dominated universe in co-moving coordinates. 
a) First of all, why do we use the light to figure out the horizon size? 

Since light has a finite speed and finite time has passed since the Big Bang, the part of the universe that is observable to us can be no larger than the distance that light can travel in the time interval since the Big Bang.

b) Light satisfies the statement that \(ds^2=0\). Using the FRW metric, write down the differential equation that describes the path light takes. We call this path the geodesic for a photon. Choosing convenient coordinates in which the light travels in the radial direction so that we can set \(d\theta=d\phi=0\), find the differential equation in terms of the coordinates t and r only. 

The FRW metric is:
\(ds^2=-c^2dt^2+a^2(t)[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]\)
Since we know that \(ds^2=d\theta=d\phi=0\), it becomes:
\(0=-c^2dt^2+a^2(t)[\frac{dr^2}{1-kr^2}]\)
\(c^2dt^2=a^2(t)[\frac{dr^2}{1-kr^2}]\)
\(\frac{dr}{dt}=\frac{c}{a(t)}(\frac{1}{1-kr^2})^{-1/2}\)

c) Suppose we consider a flat universe. Let's consider a matter dominated universe so that a(t) as a function of time is known. Find the radius of the horizon today (at \(t=t_0\)). 
Move all terms with variable r to the LHS and t to the RHS. Integrate both sides, namely r from 0 to \(r_{horizon}\) and t from 0 to \(t_0\). 

\(\int^{t_0}_0\frac{cdt}{a(t)}=\int^{r_{horizon}}_0(\frac{1}{1-kr^2})^{1/2}dr\)
We know that k = 0, since this is a flat universe. We also know that \(a\propto t^{2/3}\) in a matter-dominated universe.
\(c\int^{t_0}_0t^{-2/3}=\int^{r_{horizon}}_0(\frac{1}{1})^{1/2}dr\)
\(3ct_0^{1/2}\propto r_{horizon}\)