Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 9D / b / Solution

For an expanding universe with $0\le w\le 1$, the coefficient $(1+3w)\mathcal{H}$ is positive. Thus $\Omega =1$ is unstable toward the future: an initial $0<\Omega <1$ decreases away from one toward zero, whereas an initial $\Omega >1$ increases and eventually corresponds to recollapse. In radiation domination, $w=1/3$ andso a phase-line sketch has arrows away from the fixed point $1$ on both sides. Consequently the observed near-flatness today requires extremely fine tuning of the early value of $\Omega$.