- #1

- 1,238

- 10

## 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