Surely it must mean that every second each Mpc expands by that much. Well what else, could it be.
So if the expansion is accelerating, then how can H0 be decreasing!? Let's look at Hubble's law: v=H0D, so
| D | v: |
|---|---|
| 1 | 70, |
| 2 | 140, |
| 3 | 210; |
| Mpc | km/s. |
How does accelerating expansion change this? (Yes I know, that our local group, let alone galaxy, is not actually expanding).
Am I missing something?
The Hubble parameter is not a direct measure of how fast the universe is expanding; it is the ratio of the rate of expansion of the scale factor $a$ (can think of this as the distance between two galaxies) to the scale factor itself, $$H = \frac{\dot{a}}{a}\ .$$ $H$ is not a constant (in cosmic time) at all. The label $H_0$ means the value of $H$ now.
An accelerating expansion means $\dot{a}$ is increasing. But of course $a$ is increasing at the same time as the universe expands. Thus it is quite possible (as indeed seems to be the case in our universe) for the Hubble parameter to be decreasing with time whilst $\dot{a}$ is becoming larger. It just means $a$ is increasing faster than $\dot{a}$.
If the Hubble parameter were constant then this would means that the expansion was exponential and the scale factor would grow exponentially with time - an extreme case of an accelerating expansion, which our universe may approach asymptotically in the distant future. See here and here.
An accelerating universe does not change Hubble's law at all. $v = H_0 d$ is perfectly correct as long as you correctly interpret the meanings of $d$ and $v$. They are the "proper distance" (the length of a ruler stretched between us and the distant object now) and the rate of change of that distance. In particular, $d$ is not the light travel time distance and $v$ is not linearly proportional to redshift. For these reasons, a plot (or table) of redshift against light travel time distance would not be a straight line (except at smallish distances, where it is approximately linear).
Accelerated expansion means that a given galaxy recedes from us faster in the future than it does today.
A decreasing Hubble rate $H$ means that for a set distance $d$, the recession rate of objects at distance $d$ in the future is lower than the recession rate of objects at distance $d$ today.
In the former case, we fix the object that we are thinking about. Since it recedes, this means that we are considering an ever-increasing distance as time goes on. In the latter case, we fix the distance, which means that we think about different objects at different times.
