Critical density

Well, the galaxy has a kinect energy which is expressed by $K = \frac{1}{2} M H^{2} d^{2}$ where $M$ is its mass. But the potential energy associated to this galaxy is $U = - \frac{G M \rho \frac{4}{3} \pi d^{3}}{d} = - G M \rho \frac{4}{3} \pi d^{2}$ , where $\rho$ is the Universe's average density. Then, for the galaxy not to go to infinity, it must satisfy, at least,

$K + U = \frac{1}{2} M H^{2} d^{2} - G M \rho \frac{4}{3} \pi d^{2} = 0$

And the solution for $\rho$ is $\rho = 3 H^{2}/8 G \pi$ . But this is the density (mass per volume unit) and we want the number of hydrogen atoms per volume unit. Then

$\rho = \frac{n m}{V} = \frac{3 H^{2}}{8 G \pi} \Rightarrow \frac{n}{V} = \frac{3 H^{2}}{8 G \pi m} \approx 5.200$

Where $n$ is the number of atoms, $m$ is the hydrogen's atom mass and $V$ is the volume we are considering.