# Does the scale factor evolution in a matter dominated universe allow for acceleration?

One probably very trivial issue which is bothering me. Standard cosmology says that in a FRW metric, in a matter dominated flat universe (Pressure $$P = 0$$ and $$K = 0$$), the scale factor, $$a \propto t^{2/3}$$.

This gives $$\dot{a} \propto \frac{2}{3}t^{-1/3}$$ and $$\ddot{a} \propto \frac{-2}{9}t^{-4/3}$$.
In other words, $$\ddot{a} < 0$$, always.

Now, if current observations indicate that the universe is accelerating i.e. $$\ddot{a} > 0$$, does it imply that the good old result $$a \propto t^{2/3}$$ is incorrect?

Or am I interpreting something incorrectly?

• Yes, all you say is perfectly correct. The scale factor $a(t) \propto t^{2/3}$ doesn't apply to our universe.
– Cham
Commented Apr 3, 2019 at 10:01
• For a flat universe with both matter and dark energy, $a\propto\sinh^{2/3}(t/t_\Lambda)$. (This neglects radiation, but radiation was important only early, not now.) You can see how this transitions from $t^{2/3}$ at small $t$ when matter dominates to exponential in $t$ at large $t$ when dark energy dominates.. Commented Apr 3, 2019 at 16:40
• Thanks everybody for all the explanation Commented Apr 3, 2019 at 20:59

## 1 Answer

The result you quote is for a matter dominated universe, but the universe is dark energy dominated in the accelerating epoch. You could say that matter domination tried to halt expansion, but failed: dark energy started to dominate before expansion halted.