FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter

This item assumes ${\mathrm{A}\mathrm{C}}_{\omega }$ through the tangent-bundle and manifold-product suppliers cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no further choice principle is used.

(a) Data. Let $c>0$ be a real constant. An FLRW model datum with conversion constant $c$ is a tuple

$(I,\mathrm{\Sigma },\gamma ,k,a)$

consisting of:

1. a nonempty open interval $I\subseteq \mathbb{R}$, the time interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length);
2. a spatial model datum $(\mathrm{\Sigma },\gamma ,k)$: a connected smooth three-manifold $\mathrm{\Sigma }$ without boundary, a Riemannian metric $\gamma$ on $\mathrm{\Sigma }$, and a normalized curvature parameter $k\in \{-1,0,1\}$ with $\mathrm{Ric}(\gamma )=2k\gamma$ (The constant-curvature spatial model of a homogeneous cosmology, Riemannian metric and riemannian manifold, Smooth manifolds and their smooth charts);
3. a scale factor: a smooth map $a:I\to (0,\mathrm{\infty })$, with values in the open interval of strictly positive reals ($C^r$ and smooth maps between smooth manifolds).

Its spacetime is the product manifold $M:=I\times \mathrm{\Sigma }$ with the product smooth structure (Products of smooth manifolds have a canonical product smooth structure), a smooth four-manifold without boundary. We write $t:M\to I$ for the first coordinate function, the cosmic time, and $\dot{a}:=\frac{d}{dt}a$ for the derivative of the scale factor (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set).

(b) The FLRW metric. In product charts $(x^0,x^1,x^2,x^3)$ with $x^0=t$, where $(x^1,x^2,x^3)$ is a smooth chart of $\mathrm{\Sigma }$, define the component functions

$g_{00}:=-c^2,g_{0i}:=g_{i0}:=0,g_{ij}:=a(t{)}^2{\gamma }_{ij},i,j\in \{1,2,3\}\mathrm{.}$

These functions are smooth on the chart domain, and they define a smooth symmetric covariant two-tensor field $g$ on $M$ (Smooth covariant 2-tensor fields on a smooth manifold): the component transformation law is that of a symmetric $(0,2)$-tensor because the definitions agree on chart overlaps, the time component is constant and the spatial components are multiplied by the common smooth factor $a(t{)}^2$. The field $g$ is the FLRW metric of the datum; we write the standard informal display $g=-c^{2}d{t}^2+a(t{)}^2\gamma$, meaning the symmetric tensor $-c^{2}dt\otimes dt+a(t{)}^2\gamma$. That $g$ is a Lorentzian metric of signature $(-,+,+,+)$ on $M$ and that the coordinate field ${\mathrm{\partial }}_t$ is a time orientation for it are not asserted by this definition but proved in 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 ↗ on this page; the same holds for the clauses below that use the metric's causal classification.

(c) Comoving congruence and cosmic time direction. The comoving congruence of the datum is the smooth vector field $n:=c^{-1}{\mathrm{\partial }}_t$ on $M$ (A smooth vector field is a smooth section of the tangent bundle). By 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 field $n$ is a smooth future-directed unit timelike field, $g(n,n)=-1$, and the comoving observers of the datum are its maximal integral curves; in product coordinates the integral curve through $(t_0,x)$ is $s\mapsto (t_0+s\mathrm{/}c,x)$ on $c(I-t_0)$. Its proper-time parametrization is the comoving curve $t\mapsto (t,x)$ on $I$, with constant spatial coordinate $x\in \mathrm{\Sigma }$ and velocity $U=cn={\mathrm{\partial }}_t$.

(d) Hubble parameter and expansion. The Hubble parameter of the datum is the smooth function

$H:=\frac{\dot{a}}{a}:I\to \mathbb{R}\mathrm{.}$

No field equation, no matter content, no equation of state, no measured value and no claim about nature is part of this definition: an FLRW model datum is a geometric object, and the physical reading of $c$, of $t$ as a clock reading, of $n$ as the matter rest congruence and of $H$ as an expansion rate is recorded in the physical postulates of the framework, not here.

• Slices. For $t_0\in I$ the subset ${\mathrm{\Sigma }}_{t_0}:=\{t_0\}\times \mathrm{\Sigma }$ is a smoothly embedded copy of $\mathrm{\Sigma }$ in $M$, the spatial slice at cosmic time $t_0$; its induced metric, the pullback of $g$ by $x\mapsto (t_0,x)$, is $a(t_0{)}^2\gamma$. The induced metric is Riemannian because $\gamma$ is, and by Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature it has constant sectional curvature $k\mathrm{/}a(t_0{)}^2$; the reading of the sign of $k$ as the sign of the spatial curvature is therefore independent of $t_0$, while the curvature scale depends on the scale factor. These readings are made precise, with the submersion and pullback operations, in 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 ↗.

• Coordinate conventions and units. In the physical framework the cosmic time coordinate $t$ is measured in seconds, the scale factor $a$ in metres, the parameter $c$ in metres per second and the Hubble parameter in inverse seconds; the spatial coordinates and the metric coefficients ${\gamma }_{ij}$ are dimensionless in these coordinate conventions, so that $a(t{)}^2{\gamma }_{ij}$ carries the dimension of $a^2$. The units and the physical readings are fixed by the framework's unit conventions and postulates, not by this mathematical definition.

• Scale redundancy and normalization. For a constant $\lambda >0$, the substitution $a^{\mathrm{\prime }}=\lambda a$, ${\gamma }^{\mathrm{\prime }}={\lambda }^{-2}\gamma$ leaves $g=-c^{2}d{t}^2+a^2\gamma$ unchanged. Ricci invariance (Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature) gives $\mathrm{Ric}({\gamma }^{\mathrm{\prime }})=2({\lambda }^{2}k){\gamma }^{\mathrm{\prime }}$, so the curvature parameter of this presentation is $k^{\mathrm{\prime }}={\lambda }^{2}k$, and $k^{\mathrm{\prime }}\mathrm{/}{a}^{\mathrm{\prime }2}=k\mathrm{/}{a}^2$. For $k=0$ this is again a normalized FLRW datum for every $\lambda >0$; for $k=\pm 1$ it is a normalized datum only when $\lambda =1$. Otherwise it is an unnormalized presentation of the same spacetime metric, and restoring the three-sign normalization returns the normalized spatial metric and scale factor. The definition keeps $k\in \{-1,0,1\}$ throughout.