It must be remembered that Einstein’s theory of general relativity is essentially a theory of geometry and does not predict the geometry, homogeneity, isotropy or any global topol- ogy13. The anomalies above have raised speculation that the universe may not have a
trivial topology and may not be isotropic. As mentioned above, the flat FRW model is one solution for the special case of an isotropic and homogeneous universe. It is the simplest and as such favoured in the absence of evidence to the contrary. (There is still debate about what would constitute the “simplest” or most “natural” topology, with arguments made for both compact and infinite possibilities.) But it is possible that some of the anomalies described above may be the first such evidence.
1.4 Non-standard Cosmological models 31
Below are introduced a few examples of homogeneous but non-standard models that are particularly interesting in light of WMAP data and will be discussed further in the following chapters.
1.4.1
Compact Topologies
Many, including Einstein, have argued that a compact universe is simpler and more “nat- ural” than an infinite universe (Levin et al. 1998 and references therein, or Misner et al. 1973, p. 704).
De Oliveira-Costa et al. (1995) among others considered “small universe” models as a possible explanation for the observed homogeneity of the universe. In a compact, multiply connected universe, if the size of the fundamental cell is much smaller than the current horizon, then we observe many copies of the same universe, which would then appear homogeneous at the largest scales. This idea was put forward as an alternative to the inflationary paradigm. It is interesting to note that these models were considered unin- teresting based on the COBE observation of too much power at large scales to allow a small enough cell size to explain the homogeneity. At the time, the quadrupole was not considered anomalous, in part due to the loose constraints on the CMB power spectrum.
But as the power spectrum was measured with increasing accuracy, the lack of power on large scales and the axis of evil revived speculation that the universe may not be infinite. If instead it is compact, then fluctuations would not be larger than the size of the fundamental cell. They would show a drop-off in power at scales approaching this size and non-Gaussian correlations between such modes. In the multi-connected case such as the torus, an asymmetry in the fundamental cell size in different directions could lead to the alignments of the largest modes. If the fundamental size is smaller than the horizon, an observer would see multiple copies of an individual object in different directions. For the CMB, this would correspond to identical temperature fluctuations along “circles in the sky” at the self-intersections of the surface of last scattering. (See, for example, Fig. 7 of Riazuelo et al. 2004b.)
De Oliveira-Costa et al. (2004) speculated that this axis of evil might be due to a compact universe model with one smaller dimension in the direction of Virgo. They apply two tests for compact models, the S-statistics (discussed in§6.2.4) and matched circles, but find no evidence that the data are inconsistent with an infinite universe. An independent analysis by Cornish et al. (2004) searching for back-to-back circles also turned up no evidence of a compact universe. Note that these searches must assume a certain topology and range of sizes to test, since computational resources do not allow all possible pairs of circles to be tested in a feasible amount of time. Kunz et al. (2006) apply a method using the full covariance matrix to test the sky for a given topology, and again find no evidence for a finite universe for the cases they examine.
Another interesting topology is the Poincar´e Dodecahedron, which can be imagined in its 2-dimensional analogue of a sphere tiled with 12 pentagons. Luminet et al. (2003) show how this model is consistent with the large-scale power measured by WMAP. Roukema et al. (2004) find a possible confirmation with a matched-circles test, but the statistical
32 1. Introduction
significance of the possible detection is marginal.
Yet these models remain of interest. Linde (2004) discusses the preference of certain inflation theories for such compact models. Barrow et al. (2001), by contrast, discusses how compact topologies may explain the observed universe without the need for inflation.
1.4.2
Bianchi Models
In the context of Riemannian geometry, before general relativity was invented, the Bianchi classification system defined the isometry classes of Riemannian 3-manifolds. In other words, the Bianchi types I through IX describe the possible geometries of the Universe. Those which admit the FRW model are types I, V, VII0, VIIh, and IX. (Note that in this
work, in the context of Bianchi models, the topology is assumed to be non-compact. As mentioned above, the topology can limit the geometry, and not all Bianchi models are allowed in compact topologies. See Barrow et al. 2001, Table 1.)
Types VII0, VIIh and IX are the most general solutions to the Einstein equations for
homogeneous, anisotropic universes in the flat, open, and closed cases respectively. (Types I and V are special cases of VII0 and VIIh.) Each contains the corresponding FRW model
as a special case. The generic anisotropic cases allow universal rotation or vorticity about a given axis and differential expansion or shear along that axis. Vorticity means that at every point in the universe, the rest of the universe appears to be rotating about an axis running through that point. Shear means that objects in the direction along that axis appear to be receding at different velocities from objects away from it.
This anisotropic geometry imprints an anisotropic pattern on the CMB due the rotation of the geodesics themselves. Photons propagating from the surface of last scattering to the observer will be red- or blueshifted differently depending on the path they take. Examples are shown in Fig. 2.1. Note that a universe with vorticity must then have a handedness. Another important aspect of the type VIIh models is the resulting asymmetry; in the case
of hyperbolic, negatively curved geometry, the geodesics are focused in one direction along the symmetry axis. Flat type VII0 models have no such asymmetric focusing, but retain the
spiral structure and are consistent with type VIIh in the limit as the density approaches
critical from below. By contrast, closed type IX models have no spiral structure, and therefore, as Barrow et al. point out, a detection of any such spiral would unambiguously rule out a closed universe, however large the uncertainty about the value of Ωtot.
Note that these models are not consistent with the inflationary paradigm. The early exponential expansion is expected to wipe out any anisotropy just as it would any curvature. The CMB has been used to place ever lower limits on the amount of vorticity and shear, starting with Collins & Hawking (1973), who used the fact that no anisotropies had yet been detected in the CMB to place fairly loose limits on vorticity. Kogut et al. (1997) and Bunn et al. (1996) tightened these limit significantly with COBE-DMR data. Their limits came from the lack of detection of any such spiral pattern. But as will be shown in Chapter 2, the Bianchi type VIIh are particularly interesting in light of the anomalies in