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The Hubble time is about 14 billion years. The estimated current age of the Universe is about 13.7 billion years. Is the reason these two time are so close (a) a coincidence, or (b) a reflection that for much of its history the Universe has been expanding at a constant rate?

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It is in fact a reflection of the fact that the rate of expansion has been nearly constant for a long time.

Mathematically, the expansion of the universe is described by a scale factor $a(t)$, which can be interpreted as the size of the universe at a time $t$, but relative to some reference size (typically chosen to be the current size). The Hubble parameter is defined as

$$H = \frac{\dot{a}}{a}$$

and the Hubble time is the reciprocal of the Hubble parameter,

$$t_H = \frac{a}{\dot{a}}$$

Now suppose the universe has been expanding at a constant rate for its entire history. That means $a(t) = ct$. If you calculate the Hubble time in this model, you get

$$t_H = \frac{ct}{c} = t$$

which means that in a linear expansion model, the Hubble time is nothing but the current age of the universe.

In reality, the best cosmological theories suggest that the universe has not been expanding linearly since the beginning. So we would expect that the age of the universe is not exactly equal to the Hubble time. But hopefully it makes sense that if any nonlinear expansion lasted for only a short period, then the Hubble time should still be close to the age of the universe. That is the situation we see today.

For more information on this, I'd suggest you check out these additional questions

and others like them.

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thanks very much. –  Peter4075 Apr 8 '12 at 19:03
Thanks-- I've been randomly thinking about the meaning of the Hubble Time and although I could readily understand that it is the reciprocal of Hubble's Constant, I didn't see why this should be synonymous with the age of the Universe. In thinking about your explanation I still don't have a deep physical intuition about it (I am not sure why I am slightly embarrassed by that), although I am satisfied with the more abstract mathematical explanation. –  davecoulter Aug 3 '12 at 5:46

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