Friedmann equation - show Big Bang happened given conditions

[binbagsss]

Homework Statement

Use Friedmann equations to show that if $\dot{a} > 0$, $k<0$ and $\rho>0$ then there exists a $t*$ in the past where $a(t*)=0$

Homework Equations

[/B]
Friedmann :

$(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}$

The Attempt at a Solution

[/B]
re-arrange as:

$\dot{a}^2=1+\frac{8\pi G}{3}a^2\rho$

where I have used $k<0$ can be/is standard to set to $k=-1$

Then since $\rho>0$, and $\dot{a}$>0 implies I should take the positive square root of this , this implies that $\dot{a}>1$ for all time.

Now I do not follow the next part of my solution which says:

Thus $a(t)>0$ is decreasing at a rate that is always greater than $1$ and there was necessarily a finite time $t*$ in the past where $a(t*)=0$

How have we concluded a decrease, do we not need $\ddot{a}$ to make this conclusion? Can someone please explain where this comes from?

Many thanks in advance

 

[mfb]

I don't think you can just set k=-1, although it doesn't change the conclusion here. It is not necessary to do that, all you need is k<0 in the following argument.

If the first derivative is positive, then the original function decreases if we go back in time. This does not depend on the second derivative.