$\rho(t)\gt or \lt \rho_{critical}(t)$ depends upon $k$ for expansion or contraction in cosmology?

Question

From friedmann equation $$1=\frac{\rho(t)}{\rho_c(t)}-\frac{k}{a^2H^2},$$$$\dot a(t)=+-\sqrt\frac{k}{\frac{\rho(t)}{\rho_c(t)}-1}$$ for $k\gt0$,$$\rho(t)\gt\rho_c(t)$$ and for $k\lt0$,$$\rho(t)\lt\rho_c(t)$$ therefore is it that for expansion or contraction $\rho(t)$ attains a greater or lesser value than the critical density according to curvature of space? And at any point expansion or contraction is equally possible and nature choose to expand?