The official notes, PDF pp. 17–18, handwritten pp. 10–11 give the matter density, curvature and cosmological constant scalings and the requested logarithmic sketch. For the closed model the curvature contribution to the Friedmann equations is negative; the sketched curvature magnitude must be read with that sign restored.
The official notes, PDF p. 18, handwritten p. 11 explain the imminent maximum expansion, recollapse, remaining time of order the present age, and total lifetime of approximately twice that age.
The official notes, PDF p. 19, handwritten p. 12 integrate the radial null geodesic, express the radial coordinate in terms of comoving distance, and give its maximum value in that coordinate chart. The maximum radial coordinate is not the total path length around a closed spatial slice.
The official notes, PDF pp. 20–21, handwritten pp. 13–14 combine the Friedmann equations and Raychaudhuri equation to derive the density and scale factor of the Einstein static universe.
The official notes, PDF p. 21, handwritten p. 14 give the acceleration signs and the instability conclusion for the Einstein static universe. One wording correction is necessary: their word “oscillatory” for the overdense perturbation is a slip. Starting near rest, an overdense model contracts and its matter density rises, whereas an underdense model expands and its matter density falls; both move away from the static balance.
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