Unveiling hidden patterns in CMB anisotropy maps

Bianchi ${\mathrm{VII}}_h$ models have been recently proposed to explain potential anomalies in the CMB anisotropy as observed by WMAP. We investigate the violation of statistical isotropy due to an embedded Bianchi ${\mathrm{VII}}_h$ template in the CMB anisotropy maps to determine whether the existence of a hidden Bianchi template in the WMAP data is consistent with the previous null detection of the bipolar power spectrum in the WMAP first-year maps. We argue that, although correcting the WMAP maps for the Bianchi template may explain some features in the WMAP data, it may cause other anomalies such as preferred directions leading to detectable levels of violation of statistical isotropy in the Bianchi corrected maps. We compute the bipolar power spectrum for the low density Bianchi ${\mathrm{VII}}_h$ models embedded in the background CMB anisotropy maps with the power spectrum that have been shown in recent literature to best fit the first-year WMAP data. By examining the statistical isotropy of these maps, we put a limit of $(\frac{\sigma }{H}{)}_0\le 2.77\times {10}^{-10}(99\%\mathrm{C}.\mathrm{L}.)$ on the shear parameter in Bianchi ${\mathrm{VII}}_h$ models.

Figure 1

Bianchi patterns in type ${\mathrm{VII}}_h$ models for $x=0.1$, 0.5, 1, and 5 (from top to bottom) and ${\Omega }_0=0.1$, 0.5, and 0.9 (from left to right). In low density models, the pattern is mostly focused in one hemisphere, but for nearly flat models the spiral patterns widen and extend to both hemispheres.

Figure 2

Adding a pattern template with a strength $\alpha$ to a random realization of the CMB anisotropy map. This introduces a preferred direction in the map and violates the statistical isotropy. Maps are rotated to place the galactic pole at the center for clear illustration.

Figure 3

Four Bianchi added CMB maps with different strength factors, rotated to place the galactic pole at the center for clear illustration. The Bianchi template in the above maps has been computed with $x=0.55$, ${\Omega }_0=0.5$, and strength factors $(\frac{\sigma }{H}{)}_0=1.83\times {10}^{-9}$ (top left), $(\frac{\sigma }{H}{)}_0=1.09\times {10}^{-9}$ (top right), $(\frac{\sigma }{H}{)}_0=7.3\times {10}^{-10}$ (bottom left), and $(\frac{\sigma }{H}{)}_0=3.66\times {10}^{-10}$ (bottom right). Although the pattern can hardly be seen in the fourth map, it has a considerable nonzero bipolar power spectrum (see Fig. 4).

Figure 4

Bipolar power spectrum of Bianchi added CMB maps with strength factors of $(\frac{\sigma }{H}{)}_0=4.76\times {10}^{-10}$, $3.66\times {10}^{-10}$, $2.56\times {10}^{-10}$, and $1.25\times {10}^{-10}$. We have used $W_l^S(4,13)$ to filter the maps in order to focus on the low-$l$ (large angular scale) properties of the maps.

Figure 5

Probability distribution of ${\chi }^2$ versus the shear for the ${\Omega }_0=0.5$ Bianchi template. Results of two different window functions, $W_l^G(25)$ (blue line) and $W_l^S(4,13)$ (red, solid line), are shown. By testing the statistical isotropy of Bianchi embedded CMB maps, we can put a limit of $(\frac{\sigma }{H}{)}_0\le 2.77\times {10}^{-10}$ (99% C.L.) on Bianchi ${\mathrm{VII}}_h$ models.