# Accelerated expansion of the universe caused by dark energy

Maybe I am misunderstanding this concept but my question is the following: would we state that the universe is accelerating its expansion if we measure the speed of an astrophysical object when it reaches a distance 'd' from Earth and the corespondent velocity 'v' due to Hubble flow and after that we measure the velocity of another object more later in time when it also reaches the same distance 'd' of the first object from Earth and the measurament shows these two velocities are equal? Should we then state the universe 'baloon' is expanding but the rate of expansion is constant?

We describe the expansion of the universe with a scale factor that we conventionally call $$a$$. We take $$a=1$$ right now, so if in the future the universe has doubled in size that means $$a=2$$, or if we look back to a time when in the past when the universe was half the size $$a = \tfrac12$$.
To understand what this scale factor means we can relate it to any convenient property of the universe. For example we could consider the average spacing between galaxies, or possibly the average spacing between galaxy clusters. If we say the universe has doubled in size, i.e. $$a = 2$$, we mean this average spacing has doubled. Alternatively consider the average density of matter in the universe. If the size of the universe doubles we expect the average density of matter to fall by a factor eight, so the density is related to $$a$$ by $$\rho \propto 1/a^3$$.
Now we understand what we mean by the scale factor, the acceleration or otherwise of the expansion is just a statement of the way $$a$$ changes with time. If $$a$$ were a constant that would mean the universe was static i.e. the average spacing wasn't changing and neither was the average density. Alternatively if $$a$$ is increasing linearly with time then the expansion rate is constant.
So an accelerated expansion just means the scale factor $$a(t)$$ is increasing at a rate that is faster than linearly in time. In fact for a dark energy dominated universe we expect $$a \propto e^{kt}$$ for some constant $$k$$.
It seems natural to beginners to associate $$H$$ with the "expansion rate", however this is not totally accurate and it often leads to confusion like this. The Hubble parameter is really given by the expansion rate normalized by the scale factor: $$H=\frac{\dot{a}}{a}.$$ For galaxies at proper distance $$d$$ the recession velocity is given by Hubble's law as $$v=Hd=\frac{\dot{a}}{a}d.$$ The situation you're describing would correspond to $$v=\text{constant}$$ at $$d=\text{constant}$$, so $$H=\dot{a}/a$$ must also be constant. The solution to the resulting differential equation $$\dot{a}=Ha$$ is $$a(t)\propto\exp(Ht),$$ which is certainly accelerating (in fact this is what our universe will tend to in the far future, assuming of course there's no crazy phase transition or anything). Constant expansion on the other hand would mean an object with recession velocity $$v$$ maintains that velocity indefinitely. In other words, at fixed proper distance the velocity would go as $$1/t$$ (since $$a(t)\propto t$$).