This is the question:
By considering both the Friedmann and acceleration equations, and assuming a pressureless Universe, demonstrate that in order to have a static Universe we must have a closed Universe with a positive vacuum energy. Using either physical arguments or mathematics, demonstrate that this solution must be unstable.
I started off with the friedmann equation and equated H^2=0 and K>0. How to proceed further and prove that lamda is positive?
When including a cosmological constant, the acceleration equation reads $$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3c^2}(\epsilon+3P)+\frac{\Lambda}{3}$$ In a pressureless universe, $P=0$, and in a static universe $\frac{\ddot{a}}{a}=0$. The equation becomes: $$\frac{4\pi G}{3c^2}\epsilon=\frac{\Lambda}{3}$$Energy density is always positive and non-zero, so $\Lambda>0$.
The solution is unstable since after a slight perturbation, if $a$ grows, $\epsilon$ will go down ($\epsilon_m\propto a^{-3}$ for matter for example), which will result in an increase of the acceleration, making $a$ go up even more.
