Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 1

Use cosmic time $t$ here; derivatives with respect to conformal time will be used from Question 2 onward. Conservation of pressureless matter and radiation in cosmology makes ${\rho }_m{a}^3$ and ${\rho }_r{a}^4$ constant. DefineThe Friedmann equation is a mechanical energy equation,For nonnegative dust and radiation, at least one nonzero, $V$ tends to $-\infty$ as $a\to 0$ and to $+\infty$ as $a\to \infty$, whileThus every energy line meets the potential once. An expanding solution reaches this turning point and then contracts: $a_{tt}=-V^{\prime }(a)/2<0$. The Friedmann effective potential for matter, radiation and vacuum is sketched below, alongside the positive-vacuum case.

To bound the full cosmic time from the initial zero of $a$ to its final zero, use the Friedmann acceleration equation:Put maximum expansion at $t=0$, with $a(0)=a_{\max }$ and $a_t(0)=0$. The forced-oscillator identity isUntil the first zero and for $0\le t\le \pi /(2\omega )$, the kernel is nonnegative and the bracket is nonpositive. Hence $a(t)\le a_{\max }\cos (\omega t)$, so contraction cannot remain at positive $a$ beyond $\pi /(2\omega )$. Applying the same argument backward from maximum expansion bounds the expanding half. Therefore the negative-vacuum Friedmann lifetime bound isAt either endpoint, nonzero dust or radiation has divergent density. If both are absent, the same estimate instead bounds the open Anti-de Sitter spacetime coordinate patch; its coordinate zeros are not a finite lifetime of the maximally extended vacuum spacetime.

Set $L=8\pi G{\rho }_{\Lambda }/3>0$. The Friedmann effective potential for matter, radiation and vacuum now readsWith nonzero nonnegative dust or radiation, $V$ tends to $-\infty$ at both ends. Its stationary point satisfiesEquivalently $V^{\prime }(a)=0$ when $2L=M/a^3+2Q/a^4$; the right-hand side decreases strictly, so there is exactly one stationary point. It is a maximum becauseThe energy line in the preceding sketch therefore gives five broad types of history:

At the critical energy there are also marginal branches within the monotonic classes: expansion from zero can approach $a_E$ asymptotically, or expansion can start asymptotically at $a_E$ and proceed to infinity; contraction gives their reversals. They must not be mistaken for reaching the static solution in finite time. Indeed near the maximum,so $dt\simeq da/[\sqrt{-V^{\prime \prime }(a_E)/2}|a-a_E|]$ diverges logarithmically. The static equilibrium is unstable. Since $V(a_E)<0$, the recollapsing, bouncing and static cases require positive spatial curvature $k>0$.

For an Einstein static universe, both $a_t$ and $a_{tt}$ vanish. The Friedmann acceleration equation givesand $a_E>0$ therefore impliesThe Friedmann equation additionally fixes the spatial curvature:Thus the acceleration balance is necessary but must be accompanied by the static energy constraint.

Without radiation, the Friedmann acceleration equation changes sign when ${\rho }_m=2{\rho }_{\Lambda }$. The cosmological redshift satisfies $1+z=a_0/a$, and conservation of pressureless matter gives ${\rho }_m(z)={\rho }_{m,0}(1+z{)}^3$. HenceThe expanding spatially flat model is decelerating above this redshift and accelerating below it. Positive ${\rho }_{\Lambda }$ is required for such a transition; a negative value gives no accelerated expanding epoch. If the displayed redshift is negative, the transition lies in our future.