= Solution
For <pressureless matter>, $w=0$ and part (a) gives $\rho=3\mu/(8\pi a^3)$. With $\Lambda=0$, the <Friedmann equation> becomes
$$
\dot a^2+k=\frac{\mu}{a},
$$
or
$$
\boxed{\dot a^2+V(a)=C,\qquad
V(a)=-\frac{\mu}{a},\qquad C=-k.}
$$
This 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 $\dot a\to0$ and $a\propto t^{2/3}$. For $k=-1$, $C=1$ gives eternal expansion with $\dot a\to1$ in these units. Thus \b[positive curvature recollapses, whereas zero and negative curvature expand forever].
Back to article page
