Scope and empirical status of the homogeneous cosmological model: model predictions, reported observations and statistical inference are distinct

This note records what the items of this page do and do not establish, and how a comparison with reported observations would have to be made. It proves nothing and reports no measurement.

• Physical predictions on the page are conditional on the adopted model. The geometry, the matter sector and the equation of state are adopted in Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid; the field law is adopted in the framework. The reduction Reduction of the field equations of the homogeneous cosmological model to the Friedmann and acceleration equations, with derived continuity identifies the field equations of that model with the Friedmann and acceleration equations, and the solution theorems The flat barotropic power-law solutions of the homogeneous cosmological field equations and An expanding flat vacuum solution for positive cosmological parameter and non-existence for negative parameter verify particular data against those equations. The redshift relation Cosmological redshift: the observer-energy ratio of a null geodesic between comoving observers is the inverse scale-factor ratio is a consequence of the model metric and the light-ray rule for future affinely parametrized null rays, together with the comoving observer $n=c^{-1}{\mathrm{\partial }}_t$, the observer-energy definition $E=-cg(V,n)$ for the ray tangent $V$, and the redshift definition $1+z=E_E\mathrm{/}{E}_R$ stated in that theorem. No item on this page asserts that the universe is homogeneous, isotropic, flat, dust- or radiation-dominated, vacuum-dominated or described by any particular value of the parameters.

• Model predictions and observations are different objects. The items on this page produce model quantities: a scale factor $a(t)$, a Hubble parameter $H=\dot{a}\mathrm{/}a$, densities $\epsilon ,p$, a curvature parameter $k$, a cosmological parameter $\mathrm{\Lambda }$ and energy ratios along ideal null geodesics FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, SI unit and coordinate conventions of the classical gravitational framework. A reported observation, by contrast, is an apparatus- and calibration-dependent record: for example an identified spectral feature with a measured wavelength ratio, or a set of source fluxes, each with selection effects, resolution and uncertainty. No such record is cited, created or simulated on this page, and no numerical value of $H$, of a density, of a redshift, of a distance or of an age is asserted anywhere in the batch.

• Comparison is a separate statistical inference. To compare the model with data one must specify the preparation and selection of the sample, the measurement and calibration model of the apparatus, the relation between the recorded quantity and the model quantity (for instance a distance measure defined from fluxes and luminosities, or a temperature pattern), the statistical model of the sampling and the error budget, and the space of competing models. Each of these is an additional assumption, and the comparison tests the model together with them; agreement is evidence about the combination, and a discrepancy may equally indicate an incorrect auxiliary assumption, an apparatus effect or an analysis error (see the model's own exclusions in Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid clause (iii)). Statistical significance, a confidence interval and a posterior probability are properties of the inference, not of the theory, and a p-value is not the probability that a model is false.

• Idealizations the model does not capture. The adopted model has no spatial inhomogeneity or anisotropy, no anisotropic stress, no heat flow or dissipation, no structure formation and no quantum or non-smooth effects (Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid clause (iii), The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy). The comoving congruence is a specific idealization of the matter motion, and the equation of state is an adopted constitutive relation with a stated domain. The spatial model is fixed globally among the standard constant-curvature models (The constant-curvature spatial model of a homogeneous cosmology clause (d)); this is model data, not a consequence of the local curvature condition, and no classification theorem is used.

• What would be needed for an observation item. A future experiment item for this page would have to retrieve and read primary sources, record the apparatus, preparation, calibration and sampling model, give operational definitions of the measured quantities, report the observations with their uncertainties and limitations, and separate the reported statistical inference from the model prediction, exactly as required for experiment items of the framework. Until such items exist, the model predictions on this page remain conditional on the adopted model, and the cosmological interpretation of any observed redshift, distance or temperature pattern is a modelling and inference question, not a consequence of the mathematics proved here. This scope note itself is a methodological summary, not a model-derived result.