I defer to Kuroneko too, but I thought it odd that you, Durandal, would've come up with a radius of the observable universe with a close mantissa but ten times smaller than I and wilfulton came up with, so I reexamined your analysis.
Let's go back in time where the Hubble law was linear to the distance calibrators availible at the time. The Hubble law has the form
v = H D (1)
Now consider: the lefthand side of equation (1) has dimensions of [L/T], so H must have dimensions of [1/T] in order to match dimensions. Let's express H in units of only inverse time, specifically, in inverse years. There are 31,557,600 seconds in a year, and there are 3.08568e19 kilometers in a megaparsec, so we apply these conversions to the Hubble constant,
H = 71 km/s/Mpc * 1 Mpc/3.08568e19 km * 31,557,600 s/1 y
= 7.2914290488e-14 1/y
Now, 1/H = 13.714 Gy. If we solve for D in (1) and substitute the maximum speed physically availible (c) for v, we get
D = c/H = c (13.714 Gy) = 13.714 Gly
And voilá! A figure close to ours. You must have lost a decimal place somewhere in your calculation.
Also, the Hubble constant is
not a rate at which the universe expands, because it's the wrong units. A rate of expansion would be a change in length per unit time, and the Hubble constant has dimensions [1/T]. Therefore, it is not a rate of expansion. Indeed, to first approximation, the Hubble constant is the inverse of the age of the universe.
To see why, rearrange the Hubble law, and rename 1/H = T, we get for all galaxies the following relation:
vT = D (2)
Ie, the familiar time-distance-constant speed relationship you learned in high school. Since T is constant with respect to the individual velocities and distances, at some time back in the past, all of the galaxies were confined to a small volume. Then they flew out in various speeds to spread across the universe, and some time T later, their velocies and distances display the above relation. In other words, T is the time since the beginning of this expansion (in other words, the Big Bang).
You can also think about it this way: Imagine the universe was half the age, but the velocities of the galaxies are the same as they are now. Then the distances would be half, because of (2). This implies a Hubble constant of twice that of today, yet the universe is expanding at the same rate as it does now, because the velocities of all those little parts are constant (as per assumption), but crammed into a sphere of half the present radius.
Now, if you hadn't messed up your figures, you would've been in the same ballpark as wilfulton and me, because the only thing we did was skip the step where we went from the Hubble constant to the age of the universe.
Of course, I've left out a lot of details Kuroneko would not, but this is the basic idea.
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