Adopted homogeneous and isotropic cosmological model with a comoving perfect fluid

Adopt as the geometry and matter content of a homogeneous cosmological model of the classical gravitational framework, to be used together with Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions and the field law of Adopted Einstein field equation in the direct-equation presentation, the following assumptions.

(i) Spacetime geometry. The model's spacetime is an FLRW model datum $(I,\mathrm{\Sigma },\gamma ,k,a)$ with the framework's conversion constant $c>0$, its metric $g=-c^{2}d{t}^2+a(t{)}^2\gamma$ and its comoving congruence $n=c^{-1}{\mathrm{\partial }}_t$ (FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric). The spatial model $(\mathrm{\Sigma },\gamma ,k)$ is one of the standard complete simply connected models of The constant-curvature spatial model of a homogeneous cosmology clause (d), with normalized curvature parameter $k\in \{-1,0,1\}$: the round unit three-sphere for $k=+1$, Euclidean space for $k=0$, or hyperbolic three-space for $k=-1$; by Constant-curvature three-manifolds: equivalence of constant sectional curvature, the tensor identity and the Ricci condition each of these carries constant sectional curvature $k$ and $\mathrm{Ric}(\gamma )=2k\gamma$. The name homogeneous and isotropic cosmological model denotes here the explicit standard-spatial-model FLRW ansatz just specified. Its operative geometric assumptions are the chosen global standard model, the displayed metric and the curvature condition; the deductions do not use an additional undefined symmetry premise. The identification of the coordinate $t$ with the readings of ideal comoving clocks is not part of this clause: it is the separate clock rule of Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework, applied only where a physical clock reading is asserted. The mathematical content used by the deductions of this page is the FLRW form displayed above together with the curvature condition; no classification theorem, and no transitivity statement about isometry groups, is proved or used here.

(ii) Matter content. Exactly one of the following two sectors is adopted.

• Vacuum sector. No matter is present: the model's stress-energy tensor is the zero tensor, $\epsilon =p=0$.
• Comoving perfect-fluid sector. The model's matter is a perfect-fluid model $(\epsilon ,p,n_F)$ on the matter domain $U:=M$, the model's whole spacetime, in the sense of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy: that is, $U=M$ is the nonempty open domain, $\epsilon$ and $p$ are the smooth rest energy density and pressure on it, and the fluid's unit timelike four-velocity field $n_F$ is assumed to be the comoving congruence $n=c^{-1}{\mathrm{\partial }}_t$ of clause (i). The density and pressure are assumed homogeneous, that is, functions of cosmic time alone: $\epsilon =\epsilon (t)$ and $p=p(t)$ in the product charts, with values in the units of SI unit and coordinate conventions of the classical gravitational framework. The model additionally adopts an explicit equation of state, one of: a barotropic relation $p=P(\epsilon )$ with $P$ a smooth real function defined on an open interval containing the range of $\epsilon$; or the constant-parameter relation $p=w\epsilon$ with a real constant $w$ and the stated domain of positive $\epsilon$. The equation of state is an adopted constitutive restriction, not a consequence of the field law. The fluid stress-energy $T_{ab}=(\epsilon +p)n_a{n}_b+p{g}_{ab}$ is the gravitating source in $G_{ab}+\mathrm{\Lambda }{g}_{ab}=(8\pi G_N\mathrm{/}{c}^4)T_{ab}$ (Adopted Einstein field equation in the direct-equation presentation). The scale factor and fluid data must jointly satisfy that law when a solution is claimed; the fluid is not treated as test matter on a separately prescribed background.

(iii) Status, scope and exclusions. Clauses (i) and (ii) are adopted assumptions of the model. They are not derived from the field law, from mathematical symmetry, or from observation; the symmetry of the ansatz motivates but does not prove them, and no claim is made that any actual cosmological system is exactly homogeneous, isotropic, comoving or barotropic. The model contains no observed value: the scale factor, the densities, the parameters $k,\mathrm{\Lambda },w$ and the spacetime interval are model data, and every conclusion conditional on the model inherits the idealizations listed in the physical scope of this item. Anisotropic stresses, heat flow and dissipation, spatial inhomogeneities and perturbations, distributional or lower-dimensional sources, non-smooth fields and quantum effects lie outside the model and require additional content. Comparisons of the model's predictions with reported observations test the combined model — geometry, matter content, field law and the observation and analysis assumptions — and agreement is evidence about that combination, not a proof of this postulate.

• Why the standard spatial models are named. Clause (i) fixes the global spatial model as one of the three standard complete simply connected constant-curvature models rather than an arbitrary model satisfying the local curvature condition. This is the precise content of the homogeneity-and-isotropy idealization on this page: it selects a definite global model, avoids any unproved classification assertion, and matches the models for which the standard curvature computations are recorded in The constant-curvature spatial model of a homogeneous cosmology. A model built on a quotient or on a different global model is not excluded mathematically but is not the model adopted here.

• Clock readings are downstream. The postulate's geometric clause identifies a coordinate $t$; the assertion that ideal clocks carried by comoving observers read $t$ is the clock postulate Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework together with the proper-time computation of The FLRW metric is smooth Lorentzian; the comoving field is unit timelike and cosmic time parametrizes comoving curves by proper time; the slices carry the scaled spatial metric, and it is used only where a physical clock reading is claimed.

• Field law is separate. This postulate supplies the geometry and the matter content. The dynamics relating them is the field law of Adopted Einstein field equation in the direct-equation presentation; the conservation of the fluid stress-energy tensor and the continuity equation are consequences of that law for the model, proved on this page, not additional assumptions of this postulate.