Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 67 / 1

A comoving coordinate labels a position relative to the homogeneous expansion. For two nearby comoving positions separated by $\Delta \chi$, their proper distance in cosmology at time $t$ is $a(t)\Delta \chi$. Thus a fixed comoving separation expands physically with the scale factor, while a fixed proper separation corresponds to a shrinking comoving separation. Normalize $a_0=1$, so $a(z)=1/(1+z)$.

At a common cosmological redshift, a transverse proper size $l$ subtends $\theta \simeq l/D_A$, where $D_A$ is the angular diameter distance. At $z=3$, a $5$ comoving Mpc separation has proper size $5/(1+3)=1.25$ Mpc. HenceThe proper-size object has the larger angular size, by a factor of four. This comparison does not require a numerical angular diameter distance, since it cancels.

Pressureless matter obeys the cosmological perfect-fluid continuity equation, ${\dot{\rho }}_m+3H{\rho }_m=0$. Integrating gives ${\rho }_m={\rho }_{m,0}(a_0/a{)}^3$. Define the present critical density and cosmological density parameters byHere $c=1$ and the curvature contribution is an effective density parameter, not an additional matter fluid. Substitution into the Friedmann equation givesUsing the Hubble parameter $H=\dot{a}/a$ therefore yieldsAt $a=a_0$ the same equation implies ${\Omega }_{m,0}+{\Omega }_{k,0}+{\Omega }_{\Lambda ,0}=1$. The power in the matter term is $a_0^3$; the converted TeX's $a_0^2$ is an error not present in the original PDF.

On an expanding branch with $\dot{a}>0$, minimizing $\dot{a}$ is equivalent to minimizing its square. Differentiating the preceding Friedmann equation with respect to $a$ givesFor positive matter density and a positive cosmological constant, it vanishes uniquely atThe second derivative is $H_0^2[2{\Omega }_{m,0}{a}_0^3/a^3+2{\Omega }_{\Lambda ,0}]>0$. Thus this is the minimum expansion speed in a matter-Lambda universe. The matter and vacuum densities satisfy ${\rho }_m=2{\rho }_{\Lambda }$ there, precisely the transition from deceleration to acceleration in the Friedmann acceleration equation.

The conclusion concerns an expanding branch that actually reaches this value of $a$. A sufficiently closed model can recollapse before reaching it; positive $\Lambda$ alone does not rule that out. In flat or open models with the stated positive densities the minimum is reached. The degenerate zero-speed limiting case requires separate treatment and does not describe a finite-time passage through a positive-speed minimum.

The stationary-point relation is ${\Omega }_{m,0}{a}_0^3/a_{\min }=2{\Omega }_{\Lambda ,0}{a}_{\min }^2$. ConsequentlyTaking the positive root on the expanding branch and eliminating $a_0$ with the result of part (iii),The bracket must be positive for a genuine passage through this minimum expansion speed in a matter-Lambda universe. Negative values mean that this formal value of the scale factor is not accessible on the corresponding physical expanding solution.

For a spatially flat model, ${\Omega }_{k,0}=0$, so ${\dot{a}}_{\min }^2/a_{\min }^2=3H_0^2{\Omega }_{\Lambda ,0}$. With the squared definition of $E$ used in the paper,Equivalently $1+z_{\min }=a_0/a_{\min }=(2{\Omega }_{\Lambda ,0}/{\Omega }_{m,0}{)}^{1/3}$, and substitution into $H^2/H_0^2={\Omega }_{m,0}(1+z{)}^3+{\Omega }_{\Lambda ,0}$ gives the same answer. The minimum is in $\dot{a}$, not in the Hubble parameter $H$, which continues to decrease towards $H_0\sqrt{{\Omega }_{\Lambda ,0}}$.