Bianchi universes

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Andrew Pontzen (2016), Scholarpedia, 11(4):32340. doi:10.4249/scholarpedia.32340 revision #182483 [link to/cite this article]
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Curator: Andrew Pontzen

Bianchi universes are the class of cosmological models that are homogeneous but not necessarily isotropic on spatial slices, named after Luigi Bianchi who classified the relevant 3-dimensional spaces. They contain, as a subclass, the standard isotropic models known as Friedmann-Lemaître-Robertson-Walker (FLRW) universes. Calculations of nucleosynthesis and microwave background anisotropies in Bianchi models have been compared against data from the real Universe, typically giving null results which can be translated into upper limits on anisotropy. Tentative detections of non-zero anisotropic shear by Jaffe et al (2005) are currently believed to be inconsistent with other known cosmological parameters (Planck Collaboration et al 2015) and with polarisation of the microwave background (Pontzen & Challinor 2007). However the models remain widely-studied for their pedagogical value: homogeneity in space implies that the Einstein equations reduce from partial to ordinary differential equations in time, making them tractable exact solutions of Einstein's field equation.



The simplest examples of Bianchi universes are known as ‘Type I’, which are an immediate generalisation of the FLRW flat metric to the case with separate scale factors in each cartesian direction:\[\mathrm{d}s^2 = - \mathrm{d}t^2 + a_x(t)^2 \mathrm{d}x^2 + a_y(t)^2 \mathrm{d}y^2 + a_z(t)^2 \mathrm{d}z^2\textrm{,}\]

where we have adopted the convention that the speed of light \(c=1\). At any given time \(t\), the metric is homogeneous (does not depend on \(x\), \(y\) or \(z\)) but not generally isotropic (e.g. the coordinates \(x\) and \(y\) cannot be interchanged unless \(a_x=a_y\)). Comoving test particles in this metric follow geodesics of constant \(x\), \(y\) and \(z\); a set of such particles will change its volume and, unlike the FLRW case, also its shape. The rate of shape deformation is described by the shear \(\sigma\) which is constructed from the difference between the expansion rate in \(a_x\), \(a_y\) and \(a_z\) scale factors. While this geometry permits significantly more freedom than the typical FLRW case, Type I models are still a very special case. It is not immediately apparent how to generalise from the metric above to the case with spatial curvature, for example.

To capture the full range of Bianchi cosmologies we must start from a more general but consequently slightly abstract framework. One still has in mind a stack of spatial slices which are individually homogeneous, now defining this to mean that on each slice there are at least three Killing vector fields \(\vec{\xi}_A\) (where \(A=1,2,3\) labels the fields) that are linearly independent at every point. The Killing property states that \(\mathcal{L}_{\vec{\xi}_A} h_{ab}=0\), where \(h_{ab}\) is the 3-space metric tensor and \(\mathcal{L}\) is the Lie derivative,

\(0=\mathcal{L}_{\vec{\xi}_A} h_{ab} \equiv \xi_A^{c} \partial_c h_{ab} + h_{cb} \partial_a \xi_A^c + h_{ac} \partial_b \xi_A^c\textrm{,}\tag{1}\)

with \(\xi_A^c\) denoting the \(c\)th spatial component of the \(A\)th Killing vector and \(\partial_c\) indicating partial differentiation along the \(x^c\) coordinate direction. The covariance of the Lie derivative arises from a more formal definition in differential geometry (see e.g. Szekeres 2004), or can be confirmed by hand from the expression above. Colloquially, the Killing property implies that the spatial metric does not change if the entire space is moved a fixed distance along the integral curves of \(\vec{\xi}_A\).

The linearity of expression (1) implies that any linear combination of Killing vector fields \(\vec{\xi}_A\)s is also Killing. Furthermore, since the Bianchi definition demands three linearly-independent \(\vec{\xi}_A\) at each point, any two infinitesimally separated points can be connected by following some linear combination of the \(\vec{\xi}_A\), and so the space is homogeneous (provided it is connected). Equation (1) holds in the Type I special case above (where \(h_{ab} = \mathop{\mathrm{diag}}(a_x,a_y,a_z)\)) for any vector \(\vec{\xi}_A\) which is constant.

The three Killing fields \(\vec{\xi}_A\) specify how the metric is transported from one point to another in the space. The relationship between the three fields therefore specifies much of the nature of the space, and is the subject of the Bianchi classification. One needs to add the components of the metric at a single point in space to gain a complete description, so that within any one Bianchi type there are many different specific spaces.

The classification is based on the commutator \([\vec{\xi}_A, \vec{\xi}_B]\) of two Killing vector fields \(\vec{\xi}_A\) and \(\vec{\xi}_B\). This is the gap left when attempting to construct an infinitesimal closed loop; if we push the space along the integral fields of \(\vec{\xi}_A\) and then \(\vec{\xi}_B\), the resulting displacement is not typically identical to that achieved by following \(\vec{\xi}_B\) and then \(\vec{\xi}_A\). Because it can be constructed from operations leaving the space invariant, the difference between these paths must itself form a Killing vector field; its \(c\)th component can be written as

\([\vec{\xi}_A, \vec{\xi}_B]^c \equiv \xi_A^a \partial_a \xi_B^c - \xi_B^b \partial_b \xi_A^c\textrm{,}\)

which is again a covariant expression. By substituting this definition in equation (1), the Killing property of this ‘gap’ field can be explicitly confirmed. In a model where homogeneity is the only symmetry, the set of three \(\xi_A\)s is already complete, so each commutator must be expressable as a linear combination of the existing fields,

\([\vec{\xi}_A,\vec{\xi}_B] = C_{AB}^D \vec{\xi}_D,\)

where \(C_{AB}^D\) is a set of numbers known as the structure constants. The Bianchi classification amounts to defining the inequivalent choices for these \(C_{AB}^D\). Any construction must take into account the requirements of antisymmetry and that the Jacobi identities be satisfied, while factoring out the freedom to make linear transformations to the initial choice of fields \(\vec{\xi}_A\) (Estabrook, Wahlquist & Behr 1968). The result is that there are eight independent types (known as types I, II, IV, V, VI\(_0\), VII\(_0\), VIII and IX) plus two continuous families with free parameter \(h\) (type VI\(_h\) and VII\(_h\)). This nomenclature follows Bianchi’s original classification; Type III is missing because it turns out to be a special case of type VI. Types I, VII\(_0\), V, VII\(_h\) and IX have received particular attention as they contain the isotropic FLRW flat, flat, open, open and closed universes respectively. Other types can be arbitrarily close to, but not exactly, isotropic.

The spacetime 4-geometry can be completed by specifying the components of the spatial metric \(h_{\mu\nu}\) at a single point and a unit timelike normal vector to the hypersurface, \(\vec{n}\) such that the 4-metric is specified by \(g_{\mu\nu} = h_{\mu\nu} - n_{\mu} n_{\nu}\). Here, Greek indices denote spacetime components of vectors. By definition the spatial metric at any other point is found by integrating equation (1); additionally, to qualify as a Bianchi cosmological model, the \(\vec{n}\) vector field must be group-invariant, \([\vec{\xi}_A, \vec{n}]=0\). It follows from the Jacobi identities applied to \([\vec{n}, [\vec{\xi}_A, \vec{\xi}_B]]\) that the structure constants \(C_{AB}^D\) are then preserved over time. Since the evolution equations for \(\vec{n}\) and \(h_{\mu\nu}\) are derived from Einstein's equations, the energy-momentum tensor must also be invariant under the action of the \(\vec{\xi}_A\)s to maintain the invariance throughout the spacetime. This is important because it makes the spatial classification invariant throughout the spacetime, and so makes it meaningful to talk of the Bianchi type of a particular model universe.

Physical behaviour

Early studies of the Bianchi models in the context of Einstein’s equations were presented by Taub (1951), Heckmann & Schücking (1962) and Ellis and MacCallum (1969). The general phenomenology is highly complex and the subject of a significant body of work in mathematical physics (see e.g. Wainwright & Ellis 1997).

From an observational standpoint, one of the most important results is that of Wald’s (1983) theorem, which (alongside later works that expand the applicability) shows that universes with accelerating expansion tend towards isotropy. At face value this implies that, if the universe underwent an early period of inflation our present-day universe will seem highly isotropic; and, furthermore, since the universe is now starting to accelerate again, any anisotropy will remain small into the far future. We will sketch an explanation below, but note that there are limitations to the generality of the result (e.g. Goliath & Ellis 1999) that we will not discuss here.

All the above works start by separating the overall volume expansion of space from the anisotropic effects. This is complicated by the freedom in choosing \(\vec{n}\); here we consider only a geodesic, irrotational congruence for which

\(\nabla_{\mu}n_{\nu} = \sigma_{\mu\nu} + H h_{\mu\nu}\textrm{,}\)

where \(\nabla_{\mu}\) is the covariant derivative, \(H\) denotes the Hubble parameter (representing isotropic expansion) and \(\sigma_{\mu\nu}\) is the symmetric trace-free shear tensor (representing volume deformation). Two further terms representing rotation and acceleration appear when the irrotational geodesic assumption is relaxed.

Dynamics follow from the Einstein equations coupled to suitable matter content for the universe. Much of the behaviour is familiar from the FLRW models; for example, projecting the equations onto the \(\vec{n}\) direction gives an expression resembling the Friedmann equation,

\(\begin{aligned} 3H^2 & = 8 \pi G \rho + \sigma^2 - \hspace{0.1em}^3\hspace{-0.1em} R/2 + \Lambda\tag{2}\end{aligned}\)

where \(G\) is the gravitational constant, \(\rho\) is the source energy density, \(\sigma^2 \equiv \sigma_{\mu \nu} \sigma^{\mu\nu}/2\) is known as the shear scalar, \(\hspace{0.1em}^3\hspace{-0.1em} R\) is the scalar Ricci 3-curvature and \(\Lambda\) is the cosmological constant. The only new term here compared to the FLRW case is \(\sigma^2\) which modifies the expansion rate when shear is sufficiently large. Some entirely new relations also fall out from the Einstein equations; for example, the trace-free spatial projection of the equations (which vanishes in the FLRW case) yields the evolution of the shear,

\(h^{\sigma}_{\mu} h^{\tau}_{\nu} \dot{\sigma}_{\sigma\tau} = -3H \sigma_{\mu \nu} -\hspace{0.1em}^3\hspace{-0.1em} S_{\mu\nu} + \pi_{\mu\nu}\textrm{,}\tag{3}\)

where an overdot denotes the covariant derivative along the timelike \(n^{\mu}\) direction, \(\hspace{0.1em}^3\hspace{-0.1em} S_{\mu\nu}\) is the trace-free part of the spatial Ricci tensor and \(\pi_{\mu\nu}\) is the trace-free, spatial part of the energy-momentum tensor. This equation is exact, but its simple form masks significant complexity; in particular \(\hspace{0.1em}^3\hspace{-0.1em} S_{\mu\nu}\) is a function of the geometry (i.e. the structure constants and the metric) and has its own evolution equation to consider. Furthermore the projection on the left-hand-side can introduce complications such as Fermi-relative rotation of any coordinate system that one chooses to work in.

Yet it is possible to gain some qualitative insights from just these two equations, especially with a perfect fluid that does not move relative to the \(\vec{n}\) frame (so that \(\pi_{\mu\nu}=0\)). In the simplest models such as Type I where the anisotropic curvature vanishes (\(\hspace{0.1em}^3\hspace{-0.1em} S_{\mu\nu}=0\)), the shear will generically decay as \(\sigma \propto a^{-3}\). More generally, if an accelerating phase lasts long enough, the curvature is stretched out until it is negligible – so, colloquially, this is the Wald result: provided the spatial curvature scale is sufficiently larger than the Hubble scale, the shear rapidly dies out and the universe limits towards flatness and isotropy.

A considerable amount of effort has also been devoted to understanding how Bianchi models behave near an initial singularity in the absence of accelerated expansion. In contrast to the inflationary case, in the classical big bang the comoving horizon shrinks as one goes backwards in time. This naively suggests that at sufficiently early times, curvature can be ignored and \(H \propto \sigma \propto a^{-3}\), in turn implying an averaged scale factor which evolves as \(a \equiv (a_x a_y a_z)^{1/3} \propto t^{1/3}\) (in contrast to the radiation-dominated hot big bang for which \(a \propto t^{1/2}\)). Such behaviour can indeed be found in some models and is described as a Kasner phase.

On the other hand, the analysis is incomplete. In the high-redshift limit, the \(\sigma^2\) term in equation (2) dominates over any reasonable source of energy density. Consequently the shear is comparable to the Hubble expansion, meaning that the universe can contract along some axes while expanding along others, bringing curvature back inside the horizon. The most famous example of this phenomenon occurs in Misner’s (1969) Mixmaster universe, which alternates between Kasner phases and sharp ‘bounces’ where the curvature in equation (2) suddenly overwhelms the shear and reverses the direction of expansion. By adding a period of chaotic behaviour along these lines to the start of our own universe, and so making it older than the classical hot big bang extrapolation, Misner hoped to solve the horizon problem. This approach has, however, been all but abandoned in favour of the inflationary picture.

There are also a class of models that approach isotropy as they near the initial singularity, discussed by Collins and Hawking (1973) and Lukash (1976). This is physically accomplished by choosing initial conditions with zero shear but non-zero anisotropic curvature. The existence of these "growing mode" solutions means that it is possible to live in a universe which apparently has a regular FLRW-type big bang but becomes anisotropic at late times, even if there is no anisotropic pressure in the energy-momentum tensor.

Observational constraints

The level of anisotropy in the cosmos today is of prime interest, as any detection would point to relics from the very early (pre-inflationary) universe.

Figure 1: The temperature patterns in the cosmic microwave background expected in a Bianchi VII universe which was close to isotropy at early times. Inflation could occur in such a universe, albeit with some fine tuning; it would, at first order, superimpose stochastic, statistically-isotropic fluctuations on top of this pattern. From Pontzen (2009), based on the dynamical analysis later published by Pontzen and Challinor (2011).

Assuming the generic \(a^{-3}\) zero-curvature behaviour for the shear, which can occur in any Type under appropriate conditions, the strongest constraints come from considering early times. For example, differential expansion would lead to anisotropy in the temperature of the cosmic microwave background radiation. Schematically, the anisotropy \(\Delta T/T\) scales proportionally to \(\int \sigma \mathrm{d}t\), which immediately allows us to translate the observed CMB fluctuations with \(\Delta T/T \sim 10^{-5}\) into an upper limit of \((\sigma / H)_0 < 10^{-9}\). Stronger constraints are available if the polarisation of the CMB is considered as well as its temperature fluctuations.

Nucleosynthesis (at scalefactor \(a\sim 10^{-9}\)) is also sensitive to shear, primarily through the modified expansion rate (see equation 2) which leads to a higher neutron-to-proton ratio. From the calculations of Rothman & Matzner 1984 one can read off that \((\sigma/H)_{\mathrm{BBN}} < 0.3\) to achieve the accepted observed helium abundance and baryon densities (\(Y<0.26\), \(\Omega_b h^2 = 0.022\)), translating into a constraint of \((\sigma / H)_0 < 10^{-9}\) at the present day. The similarity between BBN and CMB upper limits is a coincidence. In both cases, the limits can be vastly weakened by assuming the anisotropy has a more complex dynamical behaviour, either by assuming an anisotropic matter source (non-vanishing \(\pi_{\mu\nu}\) in equation 3) or a non-trivial anisotropic curvature (non-vanishing \(\hspace{0.1em}^3\hspace{-0.1em} S_{\mu\nu}\)).

Jaffe et al. (2005) invoked a weak residual Bianchi Type VIIh anisotropy with amplitude \((\sigma/H)_0 \sim 3 \times 10^{-10}\) to explain purported microwave background anomalies: the low quadrupole amplitude, alignment of low-\(\ell\) modes, large-scale power asymmetry and cold spot. A weak statistical preference for including anisotropy persists in more recent Planck CMB data; however, the preference disappears when the model is required to be self-consistent in terms of the background FLRW parameters (Planck collaboration 2015). Moreover the physical approach is based on the work of Barrow et al (1985) and so exhibits the rapid shear decay \(\sigma \propto a^{-3}\) making it inconsistent with exiting inflation in an isotropic state and with the level of polarisation in the observed CMB (Pontzen & Challinor 2007). More general approaches (taking \(\hspace{0.1em}^3\hspace{-0.1em} S\ne 0\)) evade all these limitations by employing the Collins and Hawking "growing" mode (see above). Such cases completely evade nucleosynthesis constraints and give rise to smaller polarisation strengths for a fixed temperature anisotropy amplitude (Pontzen 2009); however the morphology of the anisotropy (Figure 1) is different from that of Jaffe et al (2005) and no systematic attempt to search for these different patterns in data has yet been made.


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