How do we know the radius of the universe is 47 billion light years? Hubble deep field image at maximum resolution shows a few of the furthest galaxies very small &amp; faint, only just visible, but suggests that there is more to see beyond if only the resolution were high enough. I know the age of the universe is 'only' about 14 billion years, but if Hubble can only show galaxies 13.2 billion light years away, how do we know the radius of the universe is 47 billion light years?
The observable universe's edge, by definition, is the farthest part of the Universe that light could have reached us from. Let's say light from the observable universe's edge has just reached us. The longest it could have possibly travelled for would be 13.8 billion years, since the Universe is only 13.8 billion years old.

As such, one might assume that the light came from 13.8 billion light years away, and hence the observable universe's radius should be 13.8 billion light years. The problem is that this overlooks a key phenomenon: the expansion of the universe. 

The location from where the light had been emitted has grown increasingly distant from us. In other words, by the time the photon has reached us and we are able to see the object emitting it, the object has moved much farther than when it emitted the photon. This is why we are able to see objects of distances far beyond 13.8 billion light years.

So to work out the actual numbers, let's suppose an object emitted a photon at the beginning of the universe $t=0$, and it just reached us today ($t_0 = 13.8 \ Gyr$). Evidently, this object lies at the edge of the observable universe, so finding its distance allows us to find the observable universe's radius.

Let's consider a given time $t$ in the past. Suppose our photon travelled for a small interval of time $dt$; it follows that the distance it just travelled away from its source would be $dx = c \ dt$. Due to the expansion of the universe, that $dx$ will expand over time; compared to today, it was a factor $a(t) = \frac{dx(t)}{dx(t_0)}$ smaller (this is called the scale factor at time $t$). Hence, the distance it gained from its source grew to be $dx(t_0) = \frac{dx(t)}{a(t)} = \frac{c \ dt}{a(t)}$ by today.

At every time $t$, it increased by an amount $dx$. So if the photon reached us at time $t_0 = 13.8 Gyr$ after it was emitted, then the total distance it gained from its source is the sum of these changes:
$$r=\int_0^{t_0} dx(t_0) = \int_0^{t_0} \frac{c}{a(t)} dt \approx 46 \ Gly$$
Which is therefore the distance of the object from us.