Right now, across the universe, distant galaxies are slowly accelerating away from each other, rather than receding at a constant speed or decelerating in their recession. The actual rate of expansion, the increase in the average distance between highly separated galaxies, is accelerating.
Interestingly, whilst the expansion is accelerating, our Hubble sphere still increases in size and can overtake photons coming from a certain distance beyond, even "today". That distance is our cosmological event horizon and it only coincides with our Hubble distance at the end of time. Only in a universe where the rate of expansion is always exponential is the Hubble horizon an event horizon.
I was just trying to find out how the various illustrations of expansion, which usually describe an expansion that is accelerating exponentially (the raisin bread applet above or any model where distances keep "doubling" over regular time intervals) relate to the conditions of expansion in our universe at various times in its history. The answer is, according to Lineweaver and Davis, that it is an illustration of the inflationary epoch, or a possible fate in an accelerating FRW universe, which means the universe doesn't normally expand quite like that. Whilst recession speed always increases with distance, this is all about whether the recession speed of a particular galaxy decreases (deceleration), increases (acceleration) or remains constant over time. With constant expansion, our Hubble distance recedes at c and a galaxy at that distance remains on the edge of our Hubble sphere, constantly receding at c.
In the absence of a cosmological constant, for instance, objects can only separate at a constant speed in an "empty universe" model. There is no extra space pushing things apart at ever increasing speeds. Sans cosmological constant but with a universe containing matter like ours, objects can only decelerate in their relative recessions. With a cosmological constant like dark energy might be, the rate of deceleration would be less than without the dark energy, but only once it has levelled out and turned into an acceleration (~4.79 Gyr ago) would galaxies be accelerating away from each other.
As I understand it.