Bianchi Cosmologies and Anisotropy

Bianchi cosmologies keep spatial homogeneity while relaxing spatial isotropy. They replace the single FLRW scale factor by a time-dependent metric on a three-dimensional Lie group. The Lie algebra determines which spatial curvature terms are allowed; the trace-free part of the expansion measures directional distortion. This makes the models a controlled test of which cosmological conclusions follow from homogeneity alone and which require the stronger assumption of isotropy.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian metrics and causal cones; Affine and Variational Geodesics supplies the geodesic description of the preferred observer congruence. Its Jacobi fields describe geodesic deviation and are distinct from the Lie-bracket Jacobi identity used below.

Helpful background. The Einstein Equation supplies the constraint and evolution equations, while Relativistic Matter and Compact Objects supplies perfect-fluid and anisotropic stress tensors.

Homogeneity without isotropy

Let a three-dimensional group of isometries act simply transitively on each spacelike slice $\Sigma_t$. “Transitively” means that the group can move any point of the slice to any other; “simply” means that the motion is unique. Choose invariant spatial vector fields $e_a$ and their dual one-forms $\omega^a$. Their commutators and exterior derivatives are

$$[e_b,e_c]=C^a{}_{bc}e_a, \qquad d\omega^a=-\frac12 C^a{}_{bc}\, \omega^b\wedge\omega^c .$$

The constants $C^a{}_{bc}$ encode the local homogeneous geometry. A metric orthogonal to the group orbits can then be written

$$ds^2=-dt^2+h_{ab}(t)\,\omega^a\omega^b .$$

All spatial dependence sits in the invariant one-forms, while Einstein’s equations reduce to ordinary differential equations for $h_{ab}(t)$ and the homogeneous matter variables. A diagonal $h_{ab}$ is consistent in many important non-tilted sectors, but homogeneity by itself does not require diagonality, an orthogonal fluid velocity, or vanishing anisotropic stress. The orthonormal-frame variables and the resulting constraint structure are set out in Wainwright and Lim 2004, §§2.3–2.4, pp. 4–8, eqs. (1)–(17) (PDF).

Structure constants and the Bianchi classes

After choosing an orientation and a positive reference inner product on the Lie algebra, decompose the structure constants as

$$C^a{}_{bc} =\varepsilon_{bcd}n^{da} +a_b\delta^a_c-a_c\delta^a_b, \qquad n^{ab}=n^{ba}.$$

The Jacobi identity $[e_{[a},[e_b,e_{c]}]]=0$ becomes the algebraic condition

$$n^{ab}a_b=0.$$

Class A has $a_b=0$ and is unimodular; class B has $a_b\ne0$. The table gives three representatives in a canonical basis. Nonzero entries may be rescaled, and signs depend on the invariant-frame convention. The explicit comparison axes are the pair $(n^{ab},a_b)$, unimodularity class, spatial-curvature content, and the FLRW specialization admitted by the algebra. The decomposition, Jacobi reduction, and class-A/B distinction follow Ellis and van Elst 1999, §6, pp. 44–45, eqs. (169)–(172) (PDF).

Type	$n^{ab}$	$a_b$	Class	Cosmological role
I	$\operatorname{diag}(0,0,0)$	$(0,0,0)$	A	Spatially flat; contains flat FLRW
IX	$\operatorname{diag}(1,1,1)$	$(0,0,0)$	A	Admits a positively curved isotropic sector
V	$0$	$(a,0,0)$, $a\ne0$	B	Admits a negatively curved isotropic sector

Each row satisfies $n^{ab}a_b=0$: trivially for the two class-A examples, and because $n^{ab}=0$ for type V. Type IX also illustrates why the Bianchi label does not mean “anisotropic”: equal metric coefficients give closed FLRW. Conversely, type I allows three unequal directional scale factors even though its spatial curvature vanishes.

This page uses only the classification data needed for cosmology. The full Lie-algebra taxonomy, convention changes, and global-quotient caveats belong to Bianchi Classification and Homogeneous Geometry; representative exact metrics belong to Exact Bianchi Solution Families.

Expansion and shear

The cleanest dynamical check is diagonal Bianchi I,

$$ds^2=-dt^2+a_1^2(t)\,dx^2+a_2^2(t)\,dy^2+a_3^2(t)\,dz^2 .$$

Take $x,y,z$ to be dimensionless, so the directional scale factors carry length. Constant rescalings of these coordinates change the normalization of the $a_i$ but not the directional Hubble rates or shear.

Define the volume scale factor, directional Hubble rates, and mean Hubble rate by

$$a=(a_1a_2a_3)^{1/3}, \qquad H_i=\frac{\dot a_i}{a_i}, \qquad H=\frac13\sum_{i=1}^{3}H_i=\frac{\dot a}{a}.$$

For observers orthogonal to the homogeneous slices, the shear is the trace-free part of the expansion:

$$\sigma^i{}_j =\operatorname{diag}(H_1-H,H_2-H,H_3-H), \qquad \sigma^2\equiv\frac12\sigma^i{}_j\sigma^j{}_i =\frac12\sum_{i=1}^{3}(H_i-H)^2 .$$

Thus $\sum_i(H_i-H)=0$, leaving two independent anisotropy degrees of freedom. The isotropic limit is not merely $a_1=a_2=a_3$ at one instant: it requires $\sigma^i{}_j=0$ throughout the evolution.

For an irrotational congruence orthogonal to the slices, the Hamiltonian constraint is

$$3H^2 =8\pi G\rho+\Lambda+\sigma^2-\frac12\,{}^{(3)}R .$$

This normalization follows from $\sigma^2=\tfrac12\sigma_{ab}\sigma^{ab}$. Setting $\sigma^2=0$ and $^{(3)}R=6\kappa/a^2$ in the normalized-curvature scale-factor convention recovers the FLRW relation $H^2=8\pi G\rho/3+\Lambda/3-\kappa/a^2$. Setting instead $^{(3)}R=0$ gives the Bianchi-I constraint: shear contributes positively to the mean expansion even though the spatial slices are flat. This generalized Friedmann check follows directly from the Gauss equation; see Ellis and van Elst 1999, §2.4, eq. (55), p. 13 (PDF).

The source-free Bianchi-I scaling

Continue with the diagonal Bianchi-I metric above. Let the matter flow be orthogonal and non-tilted, with vanishing energy flux and anisotropic stress, $q_i=0=\pi^i{}_j$, and use a nonrotating, Fermi-propagated spatial frame. Arbitrary isotropic pressure and $\Lambda$ are allowed. The trace-free spatial Einstein equations then reduce to

$$\dot{\sigma}^i{}_j+3H\sigma^i{}_j=0.$$

Multiplying by $a^3$ and using $\dot a/a=H$ gives

$$\frac{d}{dt}\left(a^3\sigma^i{}_j\right)=0, \qquad \sigma^i{}_j=\frac{\Sigma^i{}_j}{a^3}, \qquad \sigma^2=\frac{\Sigma^2}{a^6},$$

where $\Sigma^i{}_j$ is constant and $\Sigma^2=\tfrac12\Sigma^i{}_j\Sigma^j{}_i$. The $a^{-6}$ law is exact under these assumptions, not a universal law for all Bianchi models. Trace-free spatial curvature, anisotropic stress, fluid tilt, or non-orthogonal kinematics can source or redirect the shear. Ellis and van Elst derive the Bianchi-I result explicitly in Ellis and van Elst 1999, §5.3, p. 38, eqs. (156)–(157) (PDF).

The scaling makes shear act like an effective stiff contribution in the Hamiltonian constraint, but it is geometric, not an additional material fluid. Whether a model isotropizes must be judged using a dimensionless quantity such as $\sigma/H$ and a specified time limit. Small dimensional shear alone is not enough, and some models pass near an FLRW state only for a finite interval. The distinction between asymptotic and intermediate isotropization is discussed in Wainwright and Lim 2004, §3.2, pp. 15–18 (PDF).

Common pitfalls

Homogeneous does not mean isotropic. A three-dimensional transitive group removes spatial dependence from invariant scalars, but it need not supply rotations about every point. FLRW has a larger, six-dimensional spatial isometry group.

Flat slices do not imply flat spacetime. Bianchi I has $^{(3)}R=0$, yet unequal directional expansion can produce nonzero shear and Weyl curvature. Vacuum Bianchi I includes the non-flat Kasner solutions.

A class label is not a complete model. Matter content, tilt, the time-dependent metric, lapse, spatial topology, and initial constraints remain to be specified. Class-A Hamiltonian formulas cannot automatically be reused for class B.

Exercises

For the three representatives in the table, verify the Jacobi condition and identify the FLRW curvature sign available in each type.

Solution

For I and IX, $a_b=0$, so $n^{ab}a_b=0$. Type I has zero spatial structure constants and contains $\kappa=0$ FLRW. Equal scale factors in IX give $\kappa\gt0$ FLRW. For V, $n^{ab}=0$, so the Jacobi condition holds for nonzero $a_b$; its isotropic specialization has $\kappa\lt0$. These are local statements—global quotients require additional analysis.

Assume diagonal Bianchi I with isotropic stress. Starting from the shear propagation equation, derive the $a^{-6}$ scaling and show that the FLRW limit is preserved by the evolution.

Solution

Since $\dot{\sigma}^i{}_j+3H\sigma^i{}_j=0$ and $3H=d(\ln a^3)/dt$, integration gives $a^3\sigma^i{}_j=\Sigma^i{}_j$. Contracting the result yields $\sigma^2=\tfrac12\Sigma^i{}_j\Sigma^j{}_i/a^6=\Sigma^2/a^6$. If the initial shear vanishes, then $\Sigma^i{}_j=0$, so it remains zero. Together with equal directional scale factors up to constant coordinate rescalings, this is the flat-FLRW subfamily.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models (Cargèse Lectures 1998).” In Marc Lachièze-Rey, ed., Theoretical and Observational Cosmology, pp. 1–116. Dordrecht: Springer, 1999. DOI. Open PDF, arXiv:gr-qc/9812046v5.
• Wainwright, John, and Woei Chet Lim. “Cosmological Models from a Dynamical Systems Perspective.” arXiv:gr-qc/0409082v1, 21 September 2004. Open PDF.