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?
Cosmology – Understanding Hubble Time, the Age of the Universe, and Expansion Rate
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Actually, you're right: the Hubble constant is not really constant. At least, it's not constant in time.
The reason it's called a constant is that, when Edwin Hubble originally compared the recession velocities of galaxies with their distances in 1929, there was no reason to expect any particular pattern. After all, just a few years prior, people had thought there were no other galaxies. But what Hubble found was that, except for a small amount of random variation, the velocities of galaxies were proportional to their distances; in other words, the ratio $v/d$ was roughly the same for all galaxies he observed. The value of this ratio came to be known as Hubble's constant, $H_0 \equiv \frac{v}{d}$, because it was constant from one galaxy to the next, rather than varying randomly as one might have guessed at the time.
Of course, it wasn't long before people realized that if the recessional velocity of each galaxy was proportional to its distance, you could extrapolate back to some point in the past at which $d = 0$: all the galaxies would have started out in the same place. This gives you an effective age for the universe. If you use a simple linear extrapolation, from basic kinematics you get
$$t = \frac{d}{v} = \frac{1}{H_0}$$
So the Hubble "constant" is not constant in time, but rather is inversely related to the age of the universe. As the universe gets older, the Hubble constant gets smaller, as you would expect. This happens because the distance $d$ to any given galaxy increases with time.
However, the fact that the universe has an age doesn't create an absolute time. Sure, different observers at different points in spacetime will measure different values for the age of the universe. And sure, you could define a time coordinate system by specifying that the time coordinate for any observer is the age of the universe as measured by that observer. This is called the comoving time, and it is a useful and sensible way to set up a coordinate system in time. But it is not the only possibility, and there is certainly nothing so special about it that it deserves to be called "absolute." Any observer who is moving with respect to the universe as a whole (i.e. relative to the Hubble flow) would not measure time at the same rate as this comoving time.
I think what fundamentally needs to be explained here is this:
The physical interpretation of the Hubble time is that it gives the time for the Universe to run backwards to the Big Bang if the expansion rate (the Hubble "constant") were constant. Thus, it is a measure of the age of the Universe. The Hubble "constant" actually isn't constant, so the Hubble time is really only a rough estimate of the age of the Universe.
(source, emphasis added) You can verify this mathematically: if the Hubble time $1/H$ really did track the age of the universe (ignoring the general relativistic complications of what "the age of the universe" really means), then it must be the case that $H(t) = 1/t$. Given the definition of the Hubble parameter as $\dot{a}(t)/a(t)$, you can write
$$\frac{\dot{a}(t)}{a(t)} = \frac{1}{t}$$
This differential equation for $a$ has the solution $a(t) = ct$, which indicates that the universe would be expanding linearly in this case.
In reality, of course, the universe does not expand linearly, at least not always. But the available evidence suggests that it's been expanding pretty nearly linearly for a long time, $\dot{a}(t) \approx \text{const.}$ for the last 10-12 billion years, which is why the Hubble time is so close to the age of the universe as estimated from other methods (well, method - WMAP data).
Best Answer
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.
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