# If the Universe is only 14 billion years old, how can we see objects that are now 47 billion light years away?

When talking about the distance of a moving object, we mean the spatial separation NOW, with the positions of both objects specified at the current time. In an expanding Universe this distance NOW is larger than the speed of light times the light travel time due to the increase of separations between objects as the Universe expands. This is not due to any change in the units of space and time, but just caused by things being farther apart now than they used to be.

 Milky Way Image source- Apple Mac OS X Mountain Lion(10.8) theme wallpaper

What is the distance NOW to the most distant thing we can see? Let’s take the age of the Universe to be 14 billion years. In that time light travels 14 billion light years, and some people stop here. But the distance has grown since the light traveled. The average time when the light was traveling was 7 billion years ago. For the critical density[the density of the Universe necessary so the expansion rate of the Universe is just barely sufficient to prevent a recollapse. Numerically the critical density is 3Ho2/[8*pi*G] or 19*[Ho/100]2*10-30 gm/cc] case, the scale factor for the Universe goes like the 2/3 power of the time since the Big Bang, so the Universe has grown by a factor of 22/3 = 1.59 since the midpoint of the light’s trip. But the size of the Universe changes continuously, so we should divide the light’s trip into short intervals. First take two intervals: 7 billion years at an average time 10.5 billion years after the Big Bang, which gives 7 billion light years that have grown by a factor of 1/(0.75)2/3 = 1.21, plus another 7 billion light years at an average time 3.5 billion years after the Big Bang, which has grown by a factor of 42/3 = 2.52. Thus with 1 interval we got 1.59*14 = 22.3 billion light years, while with two intervals we get 7*(1.21+2.52) = 26.1 billion light years. With 8192 intervals we get 41 billion light years. In the limit of very many time intervals we get 42 billion light years. With calculus this whole paragraph reduces to this–

Another way of seeing this is to consider a photon and a galaxy 42 billion light years away from us now, 14 billion years after the Big Bang. The distance of this photon satisfies D = 3ct. If we wait for 0.1 billion years, the Universe will grow by a factor of (14.1/14)2/3 = 1.0048, so the galaxy will be 1.0048*42 = 42.2 billion light years away. But the light will have traveled 0.1 billion light years further than the galaxy because it moves at the speed of light relative to the matter in its vicinity and will thus be at D = 42.3 billion light years, so D = 3ct is still satisfied.
If the Universe does not have the critical density then the distance is different, and for the low densities that are more likely the distance NOW to the most distant object we can see is bigger than 3 times the speed of light times the age of the Universe. The current best fit model which has an accelerating expansion gives a maximum distance we can see of 47 billion light years.