Accelerating expansion of the universe: so? Please help me out, I’m missing something.
We know that, right now, space is expanding at roughly 73km/s/Mpc.
This means: two points in space 1Mpc away from each other “move” 73 km farther away every second. Of course they are not actually moving, it’s the space between them that’s expanding: that’s why two objects can drift apart faster that the speed of light if they are far away enough).
Now: scientists tell us that the universe is not just expanding, but that it’s expansion is accelerating.
Initially my understanding of the accelerating expansion was: if now it’s 73km/s/Mpc, at some point in the future it will be 74, 75, ….
But it turns out it’s not like that. 
It turns out that that 73 is actually DECREASING (which is why the Hubble sphere is expanding) and that “accelerating expansion of space” means something different.
Apparently it means “accelerating growth of the scale factor” $a(t)=\frac{d(t)}{d0}$ (where $d(t)$ and $d0$ are the proper distance between two points at time t and time 0).
So I understand: “accelerating universe” just means that $a’’(t)>0$ (i.e. the proper distance between two points changes over time at in increasing rate), while the Hubble parameter $H(t)=a'(t)/a(t)$. decreases over time  (so at some point in the future two points in space will “move” away from each other at, say, 65km/s/Mpc instead of 73).
In other words: the proper distance between the two objects increase faster and faster as the distance between them increases, BUT every single Mpc between them actually increases (slightly) SLOWER.
BUT: if everything I’ve said is correct (and please tell me if it’s not), I don’t get this:
If “accelerating universe” does NOT mean “increasing Hubble parameter”, but it just means that the proper distance between two objects increase at an increasing rate WHAT’S THE BIG DEAL?
I mean: the very basic fact that space is expanding everywhere implies that as time pass two objects will drift away faster and faster, since their recession velocity increase with their distance.
What am I missing?
 A: The Hubble constant is defined by:
$$ H = \frac{\dot{a}}{a} \tag{1} $$
where $a(t)$ is the scale factor as you describe. Acceleration means $\dot{a}$ is increasing, but then of course $a$ is also increasing so how $H$ behaves with time depends on which of the two increases faster. We can calculate how $H$ changes with time simply differentiating equation (1) with respect to time. Using the quotient rule we get:
$$\begin{align}
\frac{dH}{dt} &= \frac{a\ddot{a} - \dot{a}\dot{a}}{a^2} \\
              &= \frac{\ddot{a}}{a} - \left( \frac{\dot{a}}{a} \right)^2 \\
              &= \frac{\ddot{a}}{a} - H^2 \tag{2}
\end{align}$$
So $H$ increases if $\ddot{a}/a > H^2$ and decreases otherwise. From the FLRW metric, assuming that pressure is negligable, we know that:
$$ \frac{\ddot{a}}{a} = -\frac{4\pi G \rho}{3} + \frac{\Lambda c^2}{3} \tag{3} $$
I actually estimated the current value of the two terms in equation (3) in the question If we were able to double the mass of the ordinary matter in the universe it will recollapse?. Inserting these values we get:
$$ \frac{\ddot{a}}{a} = 2.25 \times 10^{-36} \text{s}^{-2} $$
The current value of $H$ was calculated by the Planck expeiment to be $67 \text{km} \space \text{s}^{-1}/\text{Mpc}$, and converting to time units I get:
$$ H^2 = 4.72 \times 10^{-36} \text{s}^{-2} $$
And putting these values into equation (2) I get:
$$ \frac{dH}{dt} = -2.47 \times 10^{-36} \text{s}^{-2} $$
and that's why the Hubble constant is decreasing even though the expansion is accelerating. Basically the increase in $a$ is outstripping the increase in $\dot{a}$. You would need the comological constant to be a factor of two and a bit bigger for $H$ to grow with time.
A: One consequence is that if the distance between objects is forever increasing, it rules out the big crunch hypothesis and certain cyclic models of the universe whereby eventually galaxies stop moving apart and instead converge.
Instead a big rip fate of the universe is implied if expansion is endlessly accelerating.
See Phanton Energy and Cosmic Doomsday for more information.
A: The idea that the proper distance is accelerating is a proposed explanation for certain data about the brightness of supernovae. The cosmological models discovered before these observations couldn't account for such acceleration. So those models have to be revised, which is important.
Another implication of the purported acceleration is that if general relativity is true the matter or fields producing the acceleration must have negative energy density:
http://arxiv.org/abs/0803.0982.
Such matter or fields would have to be rather different from the matter or fields we are familiar with, so lots of physicists have been trying to come up with candidates. One way to do this is to propose some entirely new idea, possibly from a theory of quantum gravity. Another is to look for candidates in fields described by particle physics. So this development could produce new ways to test ideas in particle physics or quantum gravity. There are also unsolved problems about what this would imply for the future of the universe in any given scenario.
Another less exotic possibility is that the dimming of the supernovae has nothing to do with acceleration of the proper distance. David Wiltshire has proposed that it might be a result of differences in spatial curvature between bound systems within the cosmic expansion and the curvature due to the expansion itself:
http://arxiv.org/abs/0712.3982
http://arxiv.org/abs/1311.3787.
I think the big deal is that it is a large unsolved problem whose solution will involve discovering new physics.
A: I would like to remark that for an accelerating universe, i.e. $\ddot{a}>0$, we require $\Lambda >0$ (or something like quintessence). This follows from the second Friedmann equation. If there is a nonzero cosmological constant, then the Hubble-rate will converge to $H_0 \sqrt{\Omega_{\Lambda}} \simeq 62 ~ km/(sMpc)$. This follows from the first Friedmann equation.
And indeed, the big deal is how to interpret and understand the origin of the cosmological constant lambda and why it is of order $\Lambda \sim 10^{-120} M_{pl}^4$. Or if the acceleration is due to quintessence, how does it arise exactly? 
Other attempts are to study modifications of gravity or relevant deviations from the cosmological principle (which might mimic the measured acceleration). 
psm.
