Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 37E (General Relativity)

The cosmological perfect-fluid continuity equation and the constant barotropic equation of state $P=w\rho$ giveOn every interval where $a$ is monotone, separation of variables yieldsHence the constant-equation-of-state density scaling iswhere $\mu$ is a constant. The same expression extends through a regular turning point by continuity.

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.

Write $\Lambda =-|\Lambda |$. For $k=0$,The potential is strictly increasing from $-\infty$ to $+\infty$ and crosses the energy line atThe expanding universe reaches this turning point, reverses, and ends in a Big Crunch. Thus a negative cosmological constant forces recollapse in this spatially flat pressureless model.

For $\Lambda >0$ and $k=1$,The unique maximum occurs atThe critical closed matter-cosmological-constant universe is obtained from $V_{\max }=-1$:

If $\Lambda >{\Lambda }_{\mathrm{crit}}$, then $V_{\max }<-1$: the energy line lies above the barrier, and a universe expanding from $a=0$ passes over it and expands forever toward de Sitter behavior. If $\Lambda ={\Lambda }_{\mathrm{crit}}$, the exact solution at $a=a_E$ is the Einstein static universe; the Big-Bang branch approaches it only asymptotically, while a separate branch can emerge from it and expand forever. If $0<\Lambda <{\Lambda }_{\mathrm{crit}}$, the equation $V(a)=-1$ has two positive roots $a_{-}<a_E<a_{+}$. The Big-Bang branch expands to $a_{-}$ and recollapses, while the disconnected large-$a$ branch contracts to $a_{+}$ and undergoes a Friedmann bounce for pressureless matter and positive cosmological constant. These are all qualitatively different possibilities.