There is something I don't understand about the Hubble parameter $H$, as it seems to clump two concepts together that I can't quite unify in my head. On the one side, we have
$$V = D H$$
which means that for a given distance $D$, there is a certain amount of new space created over time – and $H$ is simply the factor that makes this relationship work. So, for example, say we have two points 1 Mpc apart, this would mean they recede at about 70 km/s away from each other (given our current approximation of $H$).
Now the thing I can't wrap my head around is that
$$T =\frac{1}{ H}$$
is also the age of the universe. Contrary to claims made on, say, Wikipedia this means that $H$ cannot possibly have been a constant throughout the past 13 billion years, because mathematically $1/H$ means that $H$ must be continually shrinking as the universe ages.
So if $H$ did start out as some huge value and is now shrinking over time, doesn't this mean the expansion of the universe is slowing down? Because if $H$ is shrinking, I'll get a lower value of $V$ today than I'll get tomorrow. Shouldn't the notation then be more like
$$V = DH(t)$$
So which one is it? If $1/H$ is simply the solution for $D=0$, how can we use it as the expansion-velocity-per-unit-of-distance at the same time? What's worse, how can literature say $H$ has probably been more or less constant forever and simultaneously assert that $1/H$ is the current age of the universe? What am I missing?
Best Answer
I think what fundamentally needs to be explained here is this:
(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).