How to find critical density?

Question

In Cosmology critical density is defined as the minimum density for a flat universe to keep expanding, by Friedmann Equation:

${\left({\frac {\dot {a}}{a}}\right)^{2}={\frac {8\pi G}{3}}\rho -{\frac {kc^{2}}{a^{2}}}\,}+\frac{{\Lambda}c^2}{3}$

Because the universe is flat, $k=0$ and define $\rho_\Lambda = \frac{\Lambda}{8\pi G c^2}$, this equation can be simplified into:

${\left({\frac {\dot {a}}{a}}\right)^{2} = {\frac {8\pi G}{3}}(\rho + \rho_\Lambda)}$

For the universe to expand, $\dot a \ge 0$,take $\dot a = 0$ for a stable universe, which gives $\rho_c = 0$, but in reality $\rho_c = \frac{3H^2}{8\pi G}$, so where did I messed up?

Answer 1

The Friedmann eqn is: $$H^2+\frac{k}{a^2}=\frac{\rho}{3M_p^2}$$ where $M_p^2=1/8\pi G$ is the Planck mass. In this eqn we take cosmology constant as dark energy, which becomes a part of $\rho$ above.

(For energy-momentum conservation eqn: $$\dot\rho+3H(\rho+p)=0$$ and for dark energy $\rho=-p$, such that $\rho$ is a constant.

And in your notation: $$H^2=\frac{8\pi G}{3}\rho_{m,r}-\frac{k}{a^2}+\frac{\Lambda}{3}$$ where $c=1$, $H=\dot{a}/a$ and $\rho_{m,r}$ is the energy density of matter and radiation, we take $M_p^2=1/8\pi G$ such that: $$H^2+\frac{k}{a^2}=\frac{1}{3M_p^2}\rho$$ where $\rho=\rho_{r,m}+\Lambda$, $\Lambda$ is the energy density of dark energy.)

$\rho_c$ is given by $\rho_c=3M_{p}^2H^2$. Such that if $\rho>\rho_c$, $k>0$, and vice versa. And when $\dot a=0$, $H=0$ and $\rho_c=3M_{p}^2H^2=0$. I don't think "the minimum density for a flat universe to keep expanding" is a good definition of critical density because it is not necessary to takes $\rho_c=3M_{p}^2H^2$ by this definition. Although, in the model of "matter + dark energy" flat universe, it's indeed a solution of Einstein static universe. (I'll show the detail of this modle if you need, you can tell me in comment)

The plot is like this: