Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 67 / 1

Take $c=1$ and normalize the Big Bang to $t=0$. The cosmological perfect-fluid continuity equation for pressureless matter makes $C=8\pi G{\rho }_m{a}^3/3$ a positive constant. The flat Friedmann equation consequently readsPutting $u=a^{3/2}$ gives ${\dot{u}}^2=\frac{9}{4}(C+\Lambda u^2/3)$, which is directly integrable. The expanding flat dust universe with a cosmological constant hasFor negative cosmological constant, the last expression describes expansion and contraction on $0<t<2\pi /\sqrt{3|\Lambda |}$. Its maximum is $a_m=(3C/|\Lambda |{)}^{1/3}$ at $t_m=\pi /\sqrt{3|\Lambda |}$; it reaches zero again at $2t_m$. Positive or zero cosmological constant gives unbounded expansion and no finite maximum. All three have infimum zero at their initial singularity; negative cosmological constant also gives zero at the final singularity, with no positive minimum.

For positive cosmological constant, expanding the hyperbolic sine shows $a\sim (9C/4{)}^{1/3}{t}^{2/3}$ at early times: pressureless matter dominates. At late times,Thus the expansion changes from matter-dominated deceleration to exponential acceleration. Indeed $\ddot{a}/a=-C/(2a^3)+\Lambda /3$ changes sign at $a^3=3C/(2\Lambda )$.

Normalize today's scale factor to $a_0=1$ and use $c=1$. A source of bolometric luminosity $L$ emits photons whose energies arrive smaller by $1+z$, and whose arrival intervals are longer by $1+z$. Their wavefront has physical area $4\pi a_0^2{r}_1^2$. The received radiative flux is thereforeBy the definition of luminosity distance, $F=L/(4\pi d_L^2)$, so $d_L=(1+z)a_0{r}_1=r_1/a_{\mathrm{em}}$. The final equality assumes the stated normalization of the scale factor; it is not a formula invariant under an arbitrary rescaling of $a_0$.

A radial null geodesic of the FRW metric obeys $dt/a=dr/\sqrt{1-K{r}^2}$. Using $1+z=1/a$ and $dz/dt=-(1+z)H(z)$ gives the radial comoving distanceFor $K>0$ this last integral is $\arcsin (\sqrt{K}{r}_1)/\sqrt{K}$, continued along the radial coordinate when necessary; for $K<0$ it is $\mathrm{arsinh}(\sqrt{|K|}{r}_1)/\sqrt{|K|}$. Inverting, with ${\Omega }_K=-K/H_0^2$, provesHere $S_K$ is sine for positive spatial curvature of an FLRW universe, hyperbolic sine for negative spatial curvature of an FLRW universe, and its argument for zero spatial curvature of an FLRW universe. The flat case is the continuous limit $d_L=(1+z)\chi$, not a literal division by zero. The original PDF has $H_0\sqrt{|{\Omega }_K|}$ in the denominator and $H=\dot{a}/a$; the converted TeX loses these details.

The cosmological perfect-fluid continuity equation gives $\rho \propto a^{-3(1+w)}$. In a flat universe the Friedmann equation then gives $H(z)=H_0(1+z{)}^{\alpha }$, where $\alpha =3(1+w)/2>0$. Writing $u=1+z$, the angular diameter distance obtained from the luminosity distance isIt is positive for $u>1$, starts at zero and tends to zero as $u\to \infty$ for every $\alpha >0$, including $\alpha <1$ where the radial comoving distance itself is unbounded. Differentiation gives the constant-equation-of-state angular diameter distance maximum:For $\alpha \ne 1$ the stationary equation is $u^{\alpha -1}=\alpha$, which has exactly one solution greater than one. For $\alpha =1$ the derivative has the sign of $1-\log u$. Hence this stationary point is always the unique maximum.

To determine the sketch's shape, differentiate $\log (1+z_m)=\log \alpha /(\alpha -1)$:with limiting value $-1/2$ at $\alpha =1$. The inequality follows because $\log \alpha -1+1/\alpha$ has its unique minimum zero at one. Thus $z_m$ decreases from infinity as $w\downarrow -1$ to $\sqrt{3}-1$ as $w\uparrow 1$. Useful points are $z_m=e-1$ at $w=-1/3$, $5/4$ at $w=0$, and $1$ at $w=1/3$. The endpoint $w=-1$ is excluded: there $d_A=z/[H_0(1+z)]$ increases to an asymptote and has no finite turning point.