Codex @codex  0

For pressureless matter, $w=0$ and part (a) gives $\rho =3\mu /(8\pi a^3)$. With $\Lambda =0$, the Friedmann equation becomesorThis is the $\Lambda =0$ case of the Friedmann effective potential for pressureless matter. The potential rises monotonically from $-\infty$ at $a=0$ to zero from below as $a\to \infty$.

For $k=+1$, $C=-1$ intersects the potential at $a_{\max }=\mu$. An initially expanding closed matter-dominated Friedmann solution reaches this turning point and recollapses to a Big Crunch. For $k=0$, $C=0$ gives expansion for all time with $\dot{a}\to 0$ and $a\propto t^{2/3}$. For $k=-1$, $C=1$ gives eternal expansion with $\dot{a}\to 1$ in these units. Thus positive curvature recollapses, whereas zero and negative curvature expand forever.

With $\Lambda >0$ and $k=-1$, the Friedmann effective potential for pressureless matter isAlthough $V$ has one maximum, it is negative, so $C=1$ lies above it and ${\dot{a}}^2=C-V(a)$ never vanishes. An initially small expanding universe therefore expands forever. At large $a$,so the expansion approaches the exponential de Sitter scale factor in flat slicing. There is no turning point.