Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 1

For pressureless matter, the cosmological perfect-fluid continuity equation gives $\rho \propto a^{-3}$. The flat Friedmann equation then gives $a\propto t^{2/3}$, and conformal time $d\eta =dt/a$ gives $a\propto {\eta }^2$. Choose the initial conformal time to be zero. A galaxy seen at emission time ${\eta }_e$ has comoving radius $\chi ={\eta }_0-{\eta }_e$ and cosmological redshift $1+z=({\eta }_0/{\eta }_e{)}^2$.

Two opposite comoving galaxies are separated by $2\chi$. Direct one-way communication between their worldlines before emission requires the particle horizon ${\eta }_e$ to cover this separation: $2({\eta }_0-{\eta }_e)\le {\eta }_e$. Thus the threshold is ${\eta }_e=2{\eta }_0/3$, andThe distinction in direct cosmological communication and a shared causal past matters here. If causal contact means that their past light cones overlap, the condition instead is $2\chi \le 2{\eta }_e$, giving ${\eta }_e={\eta }_0/2$ and $\boxed{z_c=3\text{ for a shared causal past}}$. These are different operational meanings of the horizon criterion; the boundary corresponds to the limiting initial-time contact.

For radiation in cosmology, $\rho \propto a^{-4}$ by the cosmological perfect-fluid continuity equation. The flat Friedmann equation gives $a\propto t^{1/2}$, so $a\propto \eta$ in conformal time. Hence $1+z={\eta }_0/{\eta }_e$.

The opposite galaxies again have separation $2({\eta }_0-{\eta }_e)$. Direct communication requires this to be at most ${\eta }_e$, soFor overlap of their past light cones, the criterion is instead ${\eta }_e\ge {\eta }_0/2$, giving $\boxed{z_c=1\text{ for a shared causal past}}$. The two notions are precisely those distinguished in direct cosmological communication and a shared causal past.

A fixed comoving volume has physical volume proportional to $a^3$. The stretched domain walls within it have area proportional to $a^2$, and their energy per physical area is constant. The cosmological domain wall network therefore has $\rho \propto a^{-1}$, equivalently a barotropic equation of state with $w=-2/3$.

The flat Friedmann equation becomes ${\dot{a}}^2=Ca$ for a positive constant $C$. On the expanding branch, integration gives $a=(C/4)(t-t_B{)}^2$. Measuring age from the initial time $t_B=0$, $H=\dot{a}/a=2/t$. Thus

The scale factor of the cosmological domain wall network is $a\propto t^2$, so $\ddot{a}>0$: the expansion accelerates. Equivalently the Friedmann acceleration equation has $\rho +3P=\rho (1+3w)=-\rho <0$ for $w=-2/3$.

Set $a(t)=a_0(t/t_0{)}^2$. The remaining future conformal time isA light ray sent to a comoving reflector at separation $\chi$ and immediately returned consumes $2\chi$ of conformal time. The cosmological round-trip communication horizon therefore has supremum physical distanceFinite-time reception requires $d_0<d_{\max }$. Indeed the total travel equation is $2d_0/a_0=(t_0/a_0)(1-t_0/t_r)$, giving $t_r=t_0/(1-2d_0/t_0)$. At the limiting distance the return time tends to infinity. This round-trip radius is half the current physical cosmological event horizon radius.