Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 3

Separate conservation of a fluid with fixed $w=P/\rho$ gives $\dot{\rho }+3H(1+w)\rho =0$, hence $\rho \propto a^{-3(1+w)}$. The $w=-1/3$ component therefore has ${\rho }_Q\propto a^{-2}$. Flatness today fixes its normalization to ${\rho }_{Q0}=3H_0^2(1-{\Omega }_{M0})/(8\pi G)$, soThis is a coasting fluid: its contribution to $\rho +3P$ is zero, so it does not itself drive accelerated expansion. Its background Friedmann contribution resembles curvature, although the actual spatial geometry remains flat.

Write $\Omega ={\Omega }_{M0}$, ${\beta }^2=H_0^2(1-\Omega )$ and $\mathcal{H}=a^{\prime }/a=\dot{a}$. The Friedmann equation isThe Raychaudhuri equation has only the matter component on its active-gravity side, soCombining these expressions provesThis is the conformal Riccati equation for matter and a coasting fluid. The prime denotes conformal time, whereas $H_0$ is the physical present Hubble parameter.

Choose the Big Bang at $\tau =t=0$. For $0<\Omega <1$, the expanding singular solution of the Riccati equation is $\mathcal{H}=\beta \coth (\beta \tau /2)$. Integrating $a^{\prime }/a=\mathcal{H}$ gives $a=C{\sinh }^2(\beta \tau /2)$. The Friedmann constraint fixes $C=\Omega /(1-\Omega )$, rather than leaving an independent amplitude. ThusNow $dt=ad\tau$. Integrating from the singular origin givesThis derives the flat matter-coasting-fluid Friedmann solution. For small $\tau$, $a\propto {\tau }^2$ and $t\propto {\tau }^3$, hence $a\propto t^{2/3}$. At late time $a\propto t$, as expected when the coasting fluid dominates.

Set $a=1$ in the parametric solution. With $b=1-\Omega$,ConsequentlyThe endpoint values are obtained by limits. A particularly direct proof of the age bound avoids subtracting nearly equal terms: from the Friedmann equation,For nonnegative component densities, $0\le \Omega \le 1$ and $a\le \Omega +(1-\Omega )a\le 1$. The integrand lies between $\sqrt{a}$ and $1$. Integrating therefore givesEquality holds for pure matter and pure coasting fluid, respectively. These are the age of a flat matter-coasting-fluid universe and the coasting-fluid age bound; the positivity range is necessary for the stated universal inequality.