How does the critical density decide the fate of the Universe? Trying to teach myself cosmology. Given the Friedmann equation $$\left[\frac{1}{R}\frac{dR}{dt}\right]^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{R^{2}}$$
  and the critical density $$\rho_{c}\left(t\right)=\frac{3H^{2}\left(t\right)}{8\pi G}$$
I can see why, if $\rho>\rho_{c}$
  then $k=+1.$
  But I can't see why this leads to a collapsing universe? Conversely, if $\rho<\rho_{c}$
  then $k=-1.$
  But why does this lead to an expanding universe? I'm possibly confused because I've seen graphs of other FRW models with no correlation between the sign of $k$
  and whether the universe is contracting or expanding. Do I need to learn about the deceleration parameter (I've heard of it and that's all) in order to understand this?
Thank you
 A: This is something I, too, found confusing when first learning about cosmology. The correspondence that you mention between the sign of $k$ and the expansion fate of the universe only holds if there is no dark energy. It turns out that in our universe, which does indeed possess dark energy, $k$ might be zero but the universe may continue to experience accelerated expansion for all time (all available cosmological data is consistent with this scenario).
To solve for $a(t)$, you need not only the Friedmann equation but also a statement of energy conservation and an equation of state for each component of $\rho$. The energy conservation equation is \begin{equation} \dot{\rho} = -3 H/c^2 (\rho c^2 + p) \end{equation}
where $p$ is the pressure, related to $\rho$ through an equation of state
\begin{equation}
p = w \rho c^2
\end{equation}
where $w = 0$ for nonrelativistic matter, $w=1$ for radiation (anything traveling at or very near $c$), $w = -1$ for a cosmological constant, and $-1 < w < -1/3$ for any other type of dark energy. 
You can now solve the Friedmann equation for different initial densities and values of $k$. Generally you'll need to do this numerically. However, you can get some great insight by examining simple solutions that you can solve by hand. Barbara Ryden's textbook on cosmology is great for going through many of these partial solutions.
A: My answer do not correspond to any official position about this issue, but it's mine ;)
The above equation was written when everyone believed that matter had an infinite age.
Latter, it was found that it is not the case. The issue should be re-thinked.  I've some difficulty to accept such equation because the background premises, not written, had changed.   
Now, thinking without limitations, if the universe is infinite, if the gravitational interaction evolve at the speed of the light, i.e. is not instantaneous, then the universe has no conditions to collapse.  Every points in the universe are equivalent, then there is no central point where to collapse. The universe has no need to be infinite, it suffices that it is larger then the light cone in the perspective of the observer.   
If the universe is homogeneous, then the gravitational field is zero everywhere, irrespective of its density. Why should it collapse? What matters is not the absolute density but the gravitational potential, a derivative, a rate of change.
It will not collapse as well it is not expanding, imo. Atoms are shrinking giving us the illusion of space expansion, because we measure with atom properties.  
