Rogues’ gallery: The full freedom of the Bianchi CMB anomalies

Combining a recent derivation of the CMB evolution equations for homogeneous but anisotropic (Bianchi) cosmologies with an account of the full linearized dynamical freedoms available in such models, I calculate and discuss the various temperature and polarization anisotropy patterns that may be formed. Certain anisotropies can be hidden in superhorizon modes at early times, thus avoiding any constraints from nucleosynthesis, while nevertheless producing nontrivial redshift-zero temperature patterns in flat and open universes. The results are likely to be more of pedagogical than observational interest, but future work will assess whether such patterns can be matched to anomalies in WMAP results.

Figure 1

The $v$ (vector) modes of Type ${\mathrm{VII}}_h$ and its specializations. For a grid of parameters the temperature pattern is shown (upper panel) and $E$ and $B$ mode polarization exaggerated in scale by a factor 20 (small lower left and right panels, respectively). $x$ varies from top to bottom, while ${\Omega }_{\mathrm{M}}$ (and hence ${\Omega }_K$) varies from left to right. I assume ${\Omega }_{\Lambda }=0$ except in the rightmost panels for which the concordance values (${\Omega }_{\mathrm{M}}$, ${\Omega }_{\Lambda }=0.24$, 0.76) are used. The primary effect of nonzero $\Lambda$ is to increase the conformal time to last scattering, causing the final $z=0$ pattern to be more tightly wound. When ${\Omega }_{\mathrm{M}}+{\Omega }_{\Lambda }=1$ one obtains Type ${\mathrm{VII}}_0$; as $x\to \infty$ one obtains Type V (from ${\mathrm{VII}}_h$) or I (from ${\mathrm{VII}}_0$). The model with $({\Omega }_{\mathrm{M}},x)=(0.5,0.6)$ [shaded panel] is able to mimic known CMB temperature anomalies [11, 19].

Figure 2

The $s$ (scalar) modes of all open and flat types are insensitive to the spiralling of the basis vectors, controlled by $x$, because the quadrupole anisotropy is aligned along the symmetry axis ($m=0$ in the decomposition of PC07). Similarly, no $B$ modes can be generated. In the ${\mathrm{VII}}_0/\mathrm{I}$ limit, there is very little sensitivity to the value of ${\Omega }_{\Lambda }$ since the anisotropy is always a pure quadrupole; only the relative polarization strength is marginally affected (because of the increased conformal time to last scattering for larger ${\Omega }_{\Lambda }$).

Figure 3

The nucleosynthesis-compatible $t_1$ modes of Types ${\mathrm{VII}}_h$ and ${\mathrm{VII}}_0$. The polarization magnitude is exaggerated in scale by the factors specified in each subplot. The primary polarization varies according to Eq. (7) and, because the shear can grow after recombination, is generally much weaker relative to the temperature anisotropy than with $s$ and $v$ modes for which the shear always decays.