Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 4

The cosmological principle assumes statistical spatial homogeneity and cosmological isotropy on sufficiently large scales. Spatial homogeneity means that coarse-grained statistical properties do not depend on location; cosmological isotropy means that they do not depend on direction. The claim concerns a large-scale description, allowing individual galaxies, clusters and voids. It motivates the FRW metric and makes cosmic expansion describable by a single scale factor and a small number of matter variables.

Its adoption combines empirical large-scale regularity with the Copernican principle, the assumption that our location is typical. The resulting models provide a coherent account of cosmic expansion, the cosmic microwave background and primordial abundances. Neither the Einstein field equations alone nor cosmological isotropy around just one observer proves global spatial homogeneity: a spherically symmetric universe specially centred on that observer would also look isotropic. Observations of a finite past light cone support the principle over sampled scales without establishing it beyond the observed region.

Three supporting observations probe complementary aspects. First, after the observer-motion dipole and local foregrounds are removed, the cosmic microwave background is extremely isotropic, with intrinsic temperature fluctuations of order $10^{-5}$ and a nearly identical thermal spectrum across the sky. Second, suitably corrected all-sky counts of distant radio sources and extragalactic background intensities show no large preferred direction; Galactic obscuration and survey selection must be accounted for. Third, three-dimensional galaxy redshift surveys show that clustering and voids average toward statistically similar distributions in sufficiently large regions: large-volume number counts approach volume scaling and correlations weaken on large separation scales. The first two primarily test cosmological isotropy, while the third adds evidence for large-scale spatial homogeneity. The principle is a well-supported large-scale working hypothesis whose observational and typicality assumptions remain explicit.

Interpret the quoted values as the paper's 2006-era rounded cosmological benchmark. The Hubble constant is the present slope of the Hubble law, $v\simeq H_{0}d$ at sufficiently small redshift. A cosmic distance ladder begins with geometrical distance anchors, calibrates the Cepheid period-luminosity relation of Cepheid variables, and uses those distances to calibrate secondary indicators such as Type Ia supernovae. Distances to objects far enough away that peculiar velocities are a small fraction of recession velocity then determine the slope. Extinction, metallicity effects, distance-anchor calibration, sample selection and peculiar velocities contribute to its uncertainty.

The Hubble Space Telescope Key Project final result was $H_0=72\pm 8\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, combining random and systematic uncertainty, supporting the rounded value $70$. Independent approaches include distances inferred from gravitational-lens time delays and from cluster X-ray and Sunyaev-Zeldovich effect measurements; each brings different model uncertainties. Joint fits to the cosmic microwave background and large-scale structure also constrain $H_0$ within an assumed cosmological model. Thus the evidence for a value near $70$ combines calibrated distances with independent cross-checks; the approximate precision in the question should not be read as an identical uncertainty for every method.

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 numerical fractions describe physically very different components. Use $c=1$ so that pressure and energy density have the same units. The stress-energy conservation equation for a homogeneous component, together with its equation of state $p_i=w_i{\rho }_i$, givesNonrelativistic baryons and cold dark matter have $w\simeq 0$ and dilute as $a^{-3}$; radiation has $w=1/3$ and dilutes as $a^{-4}$; vacuum energy has $w=-1$ and remains constant. The curvature term scales as $a^{-2}$ in the Friedmann equation but is a geometric contribution, not an additional material fluid.

There is consequently a coincidence in the present mixture. Normalize $a_0=1$. ThenAt early times vacuum is negligible, while at late times it dominates overwhelmingly. Observing order-unity matter and vacuum fractions during their comparatively recent crossover motivates the cosmic coincidence problem: why should this crossover occur near our observing epoch?

The absolute scale gives a second puzzle. Restoring SI units, $H_0\simeq 2.27\times 10^{-18}{\mathrm{s}}^{-1}$ yieldsIts energy density is ${\rho }_{\Lambda ,0}{c}^2\simeq 5.8\times 10^{-10}\mathrm{J}{\mathrm{m}}^{-3}$. Quantum vacuum contributions associated with ordinary particle-physics scales can be far larger than this observed value. Explaining the tiny net value, including cancellations among contributions, is the cosmological constant problem. The required negative pressure also differs qualitatively from ordinary dilute matter.

The matter budget supplies another explanatory gap: only $0.045$ is baryonic, while $0.255$ must be nonbaryonic matter in this benchmark. Together with $0.7$ in vacuum energy, most of the cosmic density is in components whose microscopic explanation is not supplied by fitting their fractions. Finally, near-zero curvature raises the flatness problem. For nonzero $k$,During radiation domination $H\propto a^{-2}$, so $|{\Omega }_k|\propto a^2$; during matter domination $H\propto a^{-3/2}$, so $|{\Omega }_k|\propto a$. Near-flatness today therefore requires extremely small early curvature in an unmodified decelerating history. Cosmic inflation offers a mechanism that reduces this curvature fraction through accelerated expansion.

The apparent contrivances are the recent matter-vacuum coincidence, the tiny vacuum-energy scale, the near-flat initial geometry and the unexplained microscopic dark components. Their observationally inferred values impose concrete requirements on a deeper physical account.