Friedmann Equation Analysis, expansion of the universe?

[SmcWill]

Friedmann's Eq can be viewed here https://ned.ipac.caltech.edu/level5/March08/Frieman/Equations/paper1x.gif
What I don't get is that all the texts/analyses of Friedmann's equation say that if the right hand side is negative it means that the universe will expand reach a critical point and then contract. But if the right hand side is negative doesn't it mean that a dot, the rate of change of the scaling term, is imaginary? Also I don't understand how this squared term can be negative, or even how we can analyze something from it being negative. Thank you!

 

[PeterDonis]

if the right hand side is negative doesn't it mean that a dot, the rate of change of the scaling term, is imaginary?

They mean the RHS of the second Friedmann equation, the one with $\ddot{a} / a$ on the LHS.

Also, you may be misunderstanding the sources you're looking at (links to them would be helpful). The RHS of the second Friedmann equation being negative means the expansion of the universe is decelerating; but that alone doesn't tell you whether the expansion will eventually stop and then turn into contraction. You need more information to determine that.

 

[SmcWill]

I understand that if that equation is negative then the acceleration of the scale constant is concave down. But then doesn't that mean that we must know the sign of a double dot and a dot? For example it tells us that a(t) is concave down. but what if it were a function like this http://www.biology.arizona.edu/biomath/tutorials/functions/images/function_concave_down.gif (the left graph), and is there any evidence that it isn't this way? Also, thank you for your reply that cleared a lot of confusion up.

 

[PeterDonis]

doesn't that mean that we must know the sign of a double dot and a dot?

No. It is perfectly possible for $\ddot{a}$ to be negative but $\dot{a}$ positive.

is there any evidence that it isn't this way?

We know that $\dot{a}$ is positive in our universe now. That's what "the universe is expanding" means, and we have lots of evidence that the universe is expanding.

In our actual universe now, $\ddot{a}$ is also positive. That is what is meant by "the expansion of the universe is accelerating". But that has only been the case for the last few billion years; before that, $\ddot{a}$ was negative, but $\dot{a}$ was still positive (the universe was still expanding--and it has been since the Big Bang).