My calculation of the radius of the visible universe: where am I going wrong?

I searched Physics Stack Exchange and google and could only find wordy articles on this, but what I am after is the actual mathematical calculation. I took General Relativity in Physics, and I tried calculating the radius of the visible universe myself, but my calculation is not quite right. What am I missing? My calculation is as follows:

Take a photon emitted 13.8 billion years go. The distance it must have been from us, at the moment it should be

$R_{0}=ctimes t_{now}=3times 10^{8}m/stimes13.8 mbox{ billion years}=13.6 mbox{ billion lightyears}$

We of course need to account for the expansion of space over the intercal of 13.8 billion year. The Hubble constant is roughly equal to

$H_{0}=73.8 mbox{km/s/Mpc}=2.3917462 times10^{-18} mbox{/s}$

The metric for the expansion of the universe is:

$ds^{2}=-c^{2}dt^{2}+a(t)^{2}dr^2$

with approximately

$a(t)=a_{0}e^{frac{t}{t_{H}}}$

where $t_{H}=frac{1}{H_{0}}=4.181046 times 10^{17}s$

For convenience I choose $a_{0}=1$ so that the coordinate $r$ of the location where the photon was emitted is given by:

$r=R_{0}=13.6 mbox{ billion lightyears}$

we next have that:

$a_{now}=1 times e^{frac{13.8 mbox{billion years}}{13.249217 mbox{billion years}}}=2.834$

Now consider, again the the point in space that the photon was emitted from. It will be at the same coordinate $r=13.6 mbox{ billion lightyears}$, but because of the expansion of space, its distance from us now should be

$R=a_{now} times r = 2.834 times 13.8 mbox{ billion lightyears} = 39.1 mbox{ billion lightyears} $

Obviously though, the actual radius of the visible universe is believed to be about 46.5 billion lightyears, whereas I am calculating 39.1 billion lightyears, so I am under-calculating by 6.4 billion lightyears.

My questions are therefore: Where am I going wrong? Is there a paper that presents the actual detailed calculation? I tried googling this like crazy, but could not find the actual calculation.