Homogeneous Cosmology

Assume the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); the local calculations below use no further choice principle.

Let $\mathrm{\Sigma }$ be a smooth manifold, let $\gamma$ be a Riemannian metric on $\mathrm{\Sigma }$ (Riemannian metric and riemannian manifold) and let $\lambda >0$ be a constant. Write ${\gamma }_{\lambda }:=\lambda \gamma$ for the rescaled metric. Then:

(a) Connection. The Levi-Civita connection of ${\gamma }_{\lambda }$ equals the Levi-Civita connection of $\gamma$ (Christoffel formula for the levi civita connection).

(b) Curvature operator. If $R_{\lambda }$ and $R$ are the curvature operators of the two Levi-Civita connections in the convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ (Coordinate formula for the curvature tensor, Curvature is a type (1,3) tensor), then $R_{\lambda }=R$: the two operators are equal as $(1,3)$-tensor fields.

(c) Ricci and scalar curvature. With the Ricci curvature of Ricci curvature and the scalar curvature of Scalar curvature,

$\mathrm{Ric}({\gamma }_{\lambda })=\mathrm{Ric}(\gamma ),S({\gamma }_{\lambda })={\lambda }^{-1}S(\gamma )\mathrm{.}$

(d) Lowered curvature and sectional curvature. With the Riemann four-tensor of Riemann curvature four-tensor,

${\mathrm{Rm}}_{{\gamma }_{\lambda }}(X,Y,Z,W)=\lambda {\mathrm{Rm}}_{\gamma }(X,Y,Z,W),$

and for every tangent two-plane $\sigma$ on which $\gamma$ is defined the sectional curvature scales inversely, $K_{{\gamma }_{\lambda }}(\sigma )={\lambda }^{-1}{K}_{\gamma }(\sigma )$ (Sectional curvature).

• Scope. The lemma is stated for smooth manifolds with a Riemannian metric; it is pointwise and local, and it holds on every such manifold. The constant $\lambda$ must be strictly positive so that $\lambda \gamma$ is again positive definite. The raised-index Ricci tensor satisfies $\mathrm{Ric}({\gamma }_{\lambda }{)}^{ab}={\lambda }^{-2}\mathrm{Ric}(\gamma {)}^{ab}$, which is the statement used if an index-raised form is wanted.

• Use on this page. Constant rescaling preserves the covariant Ricci tensor and the sign of sectional curvature, while its magnitude changes inversely with the metric factor. Thus a normalized spatial metric with curvature $k$ has curvature $k\mathrm{/}a(t{)}^2$ when multiplied by $a(t{)}^2$; clause (c) of The constant-curvature spatial model of a homogeneous cosmology distinguishes this scale from the normalized parameter.

This item assumes ${\mathrm{A}\mathrm{C}}_{\omega }$ as inherited through the Riemannian items cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no further choice principle is used.

(a) The datum. Let $\mathrm{\Sigma }$ be a connected smooth three-manifold without boundary (Smooth manifolds and their smooth charts) and let $\gamma$ be a Riemannian metric on $\mathrm{\Sigma }$ (Riemannian metric and riemannian manifold). Let $k\in \{-1,0,1\}$. The pair $(\mathrm{\Sigma },\gamma )$ is a spatial model of constant normalized curvature $k$, with normalized curvature parameter $k$, when

$\mathrm{Ric}(\gamma )=2k\gamma \text{at every point of }\mathrm{\Sigma },$

where $\mathrm{Ric}(\gamma )$ is the Ricci curvature of $\gamma$ (Ricci curvature). We then call $(\mathrm{\Sigma },\gamma ,k)$ a spatial model datum. The signs are named for orientation only: $k=+1$ is called the closed or positively curved case, $k=0$ the flat case and $k=-1$ the open or negatively curved case; no topological content is attached to these names by this definition.

(b) Status of the condition. The condition in (a) is pointwise and local. It neither asserts nor implies that $\mathrm{\Sigma }$ is complete, compact, simply connected or geodesically complete, and it does not determine the global model: this definition proves no classification theorem and this page assumes none. In dimension three the condition is equivalent to the statement that $\gamma$ has constant sectional curvature $k$; the companion lemma on constant-curvature three-manifolds, recorded in the Remarks, proves that equivalence. What is used by the cosmological deductions of this page is exactly the Ricci form displayed in (a).

(c) Normalization and scale. The condition (a) fixes the numerical curvature to $k\in \{-1,0,1\}$. For a constant $\lambda >0$, put ${\gamma }^{\mathrm{\prime }}={\lambda }^2\gamma$. Then

$\mathrm{Ric}({\gamma }^{\mathrm{\prime }})=\mathrm{Ric}(\gamma )=2k\gamma =2(k\mathrm{/}{\lambda }^2){\gamma }^{\mathrm{\prime }},$

and $K_{{\gamma }^{\mathrm{\prime }}}=k\mathrm{/}{\lambda }^2$ (Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature). The sign is unchanged, but for $k\mathrm{\ne }0$ the normalized condition (a) is preserved only when $\lambda =1$; the flat condition is preserved for every $\lambda >0$. Conversely, if $\mathrm{Ric}(h)=2\kappa h$ for a real constant $\kappa \mathrm{\ne }0$, the metric $\gamma =\mathrm{\mid }\kappa \mathrm{\mid }h$ satisfies $\mathrm{Ric}(\gamma )=2\mathrm{sgn}(\kappa )\gamma$ by the same Ricci invariance, so it restores the normalization. A physical model assigns its spatial length scale through the further factor in the model postulate.

(d) Standard models. For each $k\in \{-1,0,1\}$ at least one complete, connected, boundaryless, simply connected model satisfying (a) is available:

• For $k=+1$, take the round unit sphere $S^3\subset {\mathbb{R}}^4$ with its induced metric. Its constant curvature $+1$, completeness and connected boundaryless Riemannian structure are supplied by Round sphere model geometry with $n=3$, $R=1$ (the curvature calculation is also The round sphere has positive constant sectional curvature); its simple connectedness is $S^n$ is simply connected for every $n\ge 2$.
• For $k=0$, take ${\mathbb{R}}^3$ with the standard metric. Its Riemannian structure and vanishing curvature are Euclidean space has zero curvature. It is nonempty and convex, hence simply connected by Every nonempty convex subset of ${\mathbb{R}}^n$ is simply connected, and thus connected by Every path-connected space is connected, and every path component lies inside a component. For completeness, the constant metric coefficients give zero Christoffel symbols (Christoffel formula for the levi civita connection); for every initial position $p$ and velocity $v$, the all-time curve $t\mapsto p+tv$ solves the coordinate geodesic equation (Coordinate geodesic equation). Uniqueness of the maximal geodesic (Existence uniqueness and smooth dependence of geodesics) identifies it with that curve, so every maximal geodesic has domain $\mathbb{R}$. Hopf–Rinow (Hopf–Rinow theorem) then gives metric completeness; its hypotheses hold because ${\mathbb{R}}^3$ is nonempty, connected and boundaryless.
• For $k=-1$, take the upper half-space $\{x\in {\mathbb{R}}^3:x^3>0\}$ with $y=x^3$ and metric $y^{-2}{\sum }_{a=1}^{3}d{x}^a\otimes d{x}^a$. Its complete, connected, boundaryless, simply connected Riemannian structure and constant curvature $-1$ are exactly the $n=3$, $a=1$ instance of Upper half-space model geometry.

By the companion lemma recorded in the Remarks, constant sectional curvature $k$ in dimension three gives $\mathrm{Ric}(\gamma )=2k\gamma$, so each model satisfies (a) and each sign is realized. A model datum restricts to one of these global standard models only when its postulate says so.

• Why a Ricci condition and not a metric formula. The displayed form $\mathrm{Ric}(\gamma )=2k\gamma$ is the exact input of the Einstein reduction performed later on this page: the warped-product computation of the four-dimensional Ricci tensor uses only $\mathrm{Ric}(\gamma )$, never the explicit coordinate form of a constant-curvature metric. The equivalence with constant sectional curvature in clause (b) is what allows the standard models of clause (d), and the physical reading of the parameter, to be described in the usual geometric language.

• Global model data. Clause (a) imposes the displayed Ricci condition. The global spatial manifold is a separate part of the datum, and no classification of all manifolds satisfying this condition is asserted. The physical postulate specifies which global model is adopted.

• Companion items on this page. The equivalence between the Ricci condition of clause (a) and constant sectional curvature in dimension three, used in clauses (b) and (d), is Constant-curvature three-manifolds: equivalence of constant sectional curvature, the tensor identity and the Ricci condition ↗. The invariance of the Ricci tensor under a constant rescaling of the metric, used in clause (c), is Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature.

• Choice convention. The cited Riemannian suppliers assume ${\mathrm{A}\mathrm{C}}_{\omega }$ through the construction of sectional curvature, geodesic domains and orthonormal frames; this definition inherits that assumption and uses no further choice principle.

This item assumes ${\mathrm{A}\mathrm{C}}_{\omega }$ as inherited through Constant sectional curvature and space form; the pointwise tensor arguments below use no further choice principle (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Let $\mathrm{\Sigma }$ be a smooth three-manifold and let $\gamma$ be a Riemannian metric on $\mathrm{\Sigma }$ (Riemannian metric and riemannian manifold), with curvature operator $R$, Riemann four-tensor $\mathrm{Rm}$ (Riemann curvature four-tensor), Ricci curvature $\mathrm{Ric}$ (Ricci curvature) and scalar curvature $S$ (Scalar curvature). Let $k\in \mathbb{R}$. The following three statements are equivalent:

(a) The metric $\gamma$ has constant sectional curvature $k$: every tangent two-plane has sectional curvature $k$ (Constant sectional curvature and space form, Sectional curvature).

(b) $\mathrm{Ric}(\gamma )=2k\gamma$ at every point of $\mathrm{\Sigma }$; that is the curvature condition for the real parameter $k$. When $\mathrm{\Sigma }$ is connected and boundaryless and $k\in \{-1,0,1\}$, it is precisely the normalized spatial-model condition of The constant-curvature spatial model of a homogeneous cosmology.

(c) $R(X,Y)Z=k(\gamma (Y,Z)X-\gamma (X,Z)Y)$ for all smooth vector fields $X,Y,Z$ on $\mathrm{\Sigma }$ (Curvature tensor of constant sectional curvature).

Whenever these hold, the scalar curvature is $S=6k$, and the trace-free Ricci tensor vanishes, ${\mathrm{Ric}}_0=0$ (Kulkarni–Nomizu product, trace-free Ricci tensor, and Weyl curvature).

• Why the converse is available here. In dimension three the Weyl part of the curvature vanishes identically, so the full curvature tensor is determined by the Ricci tensor and the scalar curvature; the converse (b)$\Rightarrow$(a) of the lemma is a genuine dimension-three statement and fails in higher dimensions, where trace-free Ricci and Weyl data are independent. The lemma is used on this page only for three-dimensional spatial metrics.

• Normalization. The condition $\mathrm{Ric}(\gamma )=2k\gamma$ with $k\in \{-1,0,1\}$ is the normalization adopted in The constant-curvature spatial model of a homogeneous cosmology; by the lemma it is the same as saying that the spatial model has constant sectional curvature $k$ with the three standard signs, which is the form in which the constant-curvature models are usually presented.

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.

Let $(I,\mathrm{\Sigma },\gamma ,k,a)$ be an FLRW model datum with conversion constant $c>0$, with spacetime $M=I\times \mathrm{\Sigma }$, metric $g=-c^{2}d{t}^2+a(t{)}^2\gamma$ and comoving congruence $n=c^{-1}{\mathrm{\partial }}_t$ (FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter, The constant-curvature spatial model of a homogeneous cosmology). Then:

(a) The field $g$ is a smooth Lorentzian metric of signature $(-,+,+,+)$ on $M$, and $N:={\mathrm{\partial }}_t$ satisfies $g(N,N)=-c^2<0$ at every point; hence $(M,g,N)$ is a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

(b) The comoving congruence $n=c^{-1}{\mathrm{\partial }}_t$ is a smooth future-directed unit timelike vector field on $M$: $g(n,n)=-1$ and $g(n,N)=-c<0$ at every point (Timelike, null, spacelike and causal vectors of a Lorentz vector space). Its maximal integral curve through $(t_0,x)$ is ${\eta }_{t_0,x}(s)=(t_0+s\mathrm{/}c,x)$ on $c(I-t_0)=\{c(t-t_0):t\in I\}$. Reparametrizing by cosmic time gives the comoving curve ${\gamma }_x(t):=(t,x)$ on $I$; each is a future timelike worldline whose parameter $t$ is its proper time up to an additive constant: for $t_0,t_1\in I$ with $t_0<t_1$, the proper-time increment between ${\gamma }_x(t_0)$ and ${\gamma }_x(t_1)$ is $t_1-t_0$ (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent). The physical four-velocity of a comoving observer is $U={\mathrm{\partial }}_t=cn$, with $g(U,U)=-c^2$.

(c) The congruence $n$ is $g$-orthogonal to the spatial slices: for every $t_0\in I$, the slice inclusion ${\iota }_{t_0}:\mathrm{\Sigma }\to M$, $x\mapsto (t_0,x)$, is a smooth embedding, the pullback ${\iota }_{t_0}^{\ast }g=a(t_0{)}^2\gamma$ is a Riemannian metric on $\mathrm{\Sigma }$, and $g(n,{\iota }_{t_0\ast }v)=0$ for every $v\in T_x\mathrm{\Sigma }$. Moreover the slice metric ${\gamma }_{t_0}:=a(t_0{)}^2\gamma$ has Ricci curvature $\mathrm{Ric}({\gamma }_{t_0})=2k\gamma$ and constant sectional curvature $k\mathrm{/}a(t_0{)}^2$ (Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature, Constant-curvature three-manifolds: equivalence of constant sectional curvature, the tensor identity and the Ricci condition).

• What is not asserted. The lemma asserts neither completeness of the comoving observers beyond their computed domains (the maximal $n$-integral parameter domain is $c(I-t_0)$; their cosmic/proper-time domain is $I$), nor any global product structure of the spacetime beyond the product structure that defines the datum, nor any relation between the time parameter and a physical clock; the physical clock reading is supplied by the framework's clock postulate, not by this lemma.

• Sign of the curvature reading. The slice curvature $k\mathrm{/}a(t_0{)}^2$ is a real number whose sign is the sign of $k$. The Ricci form displayed in the final proof step uses the equivalence of Constant-curvature three-manifolds: equivalence of constant sectional curvature, the tensor identity and the Ricci condition for the real parameter $k\mathrm{/}a(t_0{)}^2$, which is not restricted to the three signs $\{-1,0,1\}$.

Let $(I,\mathrm{\Sigma },\gamma ,k,a)$ be an FLRW model datum with conversion constant $c>0$, with spacetime $M=I\times \mathrm{\Sigma }$, metric $g$ and comoving congruence $n=c^{-1}{\mathrm{\partial }}_t$ (FLRW model data: warped product metric, cosmic time, comoving congruence, scale factor and Hubble parameter), and let $\mathrm{\nabla }$ be the metric-selected connection of $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). Let $(x^0,x^1,x^2,x^3)$ be a product chart with $x^0=t$, where $(x^1,x^2,x^3)$ is a smooth chart of $\mathrm{\Sigma }$, and let ${}^{(3)}{\mathrm{\Gamma }}^i{}_{jk}$ be the Christoffel symbols of the Riemannian metric $\gamma$ in that three-dimensional chart (Christoffel formula for the levi civita connection). Write $\dot{a}$ for the derivative of the scale factor and $H=\dot{a}\mathrm{/}a$ for the Hubble parameter. Then in the product chart the Christoffel symbols ${\mathrm{\Gamma }}^a{}_{bc}$ of $\mathrm{\nabla }$ (Christoffel symbols of an affine connection) are:

(a) ${\mathrm{\Gamma }}^0{}_{00}={\mathrm{\Gamma }}^0{}_{0i}={\mathrm{\Gamma }}^0{}_{i0}=0$ and ${\mathrm{\Gamma }}^i{}_{00}=0$ for $i=1,2,3$;

(b) ${\mathrm{\Gamma }}^0{}_{ij}={\mathrm{\Gamma }}^0{}_{ji}={\displaystyle \frac{a\dot{a}}{c^2}}{\gamma }_{ij}$ for $i,j\in \{1,2,3\}$;

(c) ${\mathrm{\Gamma }}^i{}_{0j}={\mathrm{\Gamma }}^i{}_{j0}=H{\delta }^i{}_j$ for $i,j\in \{1,2,3\}$;