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Nonrelativistic matter dilutes as $a^{-3}$, while a cosmological constant has constant energy density. The ratio of vacuum energy density to matter density consequently scales as $a^3$, so comparable contributions occur during a limited epoch. The cosmic coincidence problem asks why observers find these very different components comparable near their own epoch. This timing puzzle is distinct from the cosmological constant problem, which concerns the tiny absolute vacuum-energy scale.

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. 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.

Separate the observables before combining their parameter constraints. The baryon fraction is included within total matter: for the benchmark, ${\Omega }_{\mathrm{CDM}}={\Omega }_m-{\Omega }_b=0.255$, rather than adding ${\Omega }_b$ to ${\Omega }_m$ a second time. With $h=H_0/(100\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1})=0.7$, the physical density combinations areFor baryons, Big Bang nucleosynthesis predicts primordial light-element abundances as a function of the baryon-to-photon ratio; deuterium is especially sensitive. Independently, baryon inertia changes the relative heights of compressional and rarefaction peaks in the cosmic microwave background through the baryon loading parameter. Agreement of these two physical-density estimates supports a baryon abundance near the quoted value. For example, the three-year WMAP cosmological analysis found ${\Omega }_b{h}^2=0.02229\pm 0.00073$ in its fitted model.

For total matter, matter-radiation equality affects the microwave-background peak pattern and fixes a characteristic turnover in the matter power spectrum. galaxy clustering therefore constrains matter density, with corrections for galaxy bias and peculiar-velocity distortions. Cluster baryon fractions combine observed gas and stars with total masses inferred dynamically or by gravitational lensing; comparison with the cosmic baryon fraction also constrains ${\Omega }_m$. Galactic rotation, cluster dynamics and gravitational lensing give independent evidence that visible baryons alone cannot supply the gravitating mass. The precise ${\Omega }_m$ estimate comes from quantitative joint fits, rather than from a rotation curve by itself. Different 2006 data combinations gave somewhat different best fits, so $0.3$ is a rounded benchmark rather than a universal exact fit.

For a cosmological constant, standardized high-redshift Type Ia supernovae measure the shape of the luminosity-distance relation. In a flat matter-vacuum model,The observed supernova dimming relative to simple decelerating matter models favours late accelerated expansion. The original evidence is given by Riess and collaborators' 1998 supernova analysis. Combining the distance curve with matter constraints selects a positive vacuum fraction close to the quoted benchmark; it gives $q_0={\Omega }_m/2-{\Omega }_{\Lambda }=-0.55$.

For curvature, the angular scale of the Cosmic microwave background acoustic peaks compares the sound horizon with the angular diameter distance to last scattering. This strongly constrains spatial geometry, but curvature and late-time expansion parameters can partly compensate one another. Independent distances, notably baryon acoustic oscillations and supernova distances, help break that degeneracy. The 2005 SDSS acoustic-peak detection supplies a low-redshift standard ruler complementary to the microwave-background ruler. In combination, these observations support near-flat geometry and the closure relation ${\Omega }_m+{\Omega }_{\Lambda }+{\Omega }_k\simeq 1$.

The benchmark is supported by mutually constraining abundance, clustering, angular-scale and distance measurements. Its uncertainties are correlated and depend on model assumptions. An exactly zero curvature parameter has no meaningful percentage error; observational evidence establishes consistency with zero and an absolute uncertainty, not a literal fractional accuracy better than ten per cent on zero.

The cosmological constant energy is constant, whereas the matter density scales as $a^{-3}$. HenceThe transition occurs when $\rho +3P={\rho }_{\mathrm{dust}}-2{\rho }_{\mathrm{cc}}=0$, so

A cosmological constant has fixed density and equation of state $w=p/(\rho c^2)=-1$. A separately conserved dynamical component obeysIf $w$ departs from $-1$, its history changes the Hubble parameter, luminosity distances and angular-diameter distances. Type Ia supernovae probe a relative distance-redshift relation, baryon acoustic oscillations provide standard-ruler distances and expansion information, and the cosmic microwave background fixes early-time scales and geometric combinations. Combining these breaks some degeneracies with matter density, curvature and calibration. Distance observables integrate the expansion history, so they smooth rapid variations in $w$ and cannot determine an arbitrary function exactly.

Structure-growth measurements add a dynamical test. Galaxy clustering, redshift-space distortions and weak gravitational lensing constrain the growth of matter perturbations and metric potentials. The late Integrated Sachs-Wolfe effect probes potential evolution through CMB-matter correlations. A scalar-field dark-energy model can have perturbations and an effective sound speed different from other models with a similar background expansion. Comparisons of geometry and growth can therefore distinguish some models and test gravity assumptions.

A statistically significant, robust departure from $w=-1$ or from constant dark-energy density would disfavor a pure cosmological constant within the stated framework. Consistency with $w=-1$ over a finite redshift range does not prove exact constancy: a slowly rolling field can mimic it. Supernova calibration/evolution, galaxy bias, photometric redshift errors, intrinsic galaxy alignments and foregrounds must be controlled; modified gravity can also change growth. Parameterizations such as $w(a)=w_0+w_a(1-a)$ are useful finite-dimensional tests rather than a complete classification of all dark-energy histories.