Skolon said:
What did we know about acceleration of Universe (a'(t)) and its future?
Well, it's difficult to predict the future with confidence. The difficulty is that we cannot say for certain what the causes of this acceleration are. Without knowing the causes, we can't be sure about the future history.
However, if the simplest model is correct, that the cause is a cosmological constant or something that behaves very much like one, then yes, \ddot{a} and \dot{a} will both remain positive forever.
What happens is that as the universe expands, the matter and radiation continue to dilute, but the cosmological constant remains at the same density no matter the expansion. Then we look at the first Friedmann equation:
H^2 = {8 \pi G \over 3} \rho
and we see that if the normal matter and radiation decay away, and all we are left with is an energy density that is constant and doesn't change in time, then we approach a situation where the Hubble parameter H(a) approaches a constant value. Now, the Hubble parameter is defined as:
H(a) = {\dot{a} \over a}
So what happens is that this ratio of the time derivative of the scale factor to the scale factor itself gets smaller and smaller as the radiation and matter dilute away, approaching a constant value, let's call it H_f (f for future). So in the distant future, we have:
H(a) = H_f
If we evaluate this in terms of the definition of the Hubble parameter, we have a very simple differential equation:
\dot{a} = H_f a
Which becomes:
a(t) = a(0)e^{H_f t}
So if the dark energy acts as a cosmological constant into the future, then the future is exponential expansion that never ends.
The difficulty is that we can't be certain what the cause of the accelerated expansion is just yet, so we can't say for sure one way or the other whether this is the future of our universe or not. By virtue of being the simplest explanation, it is perhaps the most likely, but that isn't very convincing to many, given how little we know about dark energy's nature.
