Correlating anomalies of the microwave sky

At the largest angular scales the presence of a number of unexpected features has been confirmed by the latest measurements of the cosmic microwave background (CMB). Among them are the anomalous alignment of the quadrupole and octopole with each other as well as the stubborn lack of angular correlation on scales $>60°$. We search for correlations between these two phenomena and demonstrate their absence. A Monte Carlo likelihood analysis confirms previous studies and shows that the joint likelihood of both anomalies is incompatible with the best-fit $\Lambda$ cold dark matter model at $>99.95\%$ C.L. Extending also to higher multipoles, a common special direction (the “Axis of Evil”) has been identified. In the search for an explanation of the anomalies, several studies invoke effects that exhibit an axial symmetry. We find that this interpretation of the “Axis of Evil” is inconsistent with three-year data from the Wilkinson Microwave Anisotropy Probe (WMAP). The data require a preferred plane, whereupon the axis is just the normal direction. Rotational symmetry within that plane is ruled out at high confidence.

Figure 1

Mollweide projection of the sky with quadrupole (upper row) and octopole (lower row) multipole vectors [Eq. (5)]. The mesh consists of steps in 30°. Displayed are ten pairs of quadrupole vectors (small dots) and their ten area vectors [Eq. (6) (big dots)] as well as ten triples of octopole vectors (small dots) and their area vectors (big dots); togetherness is indicated by color. The arbitrary sign of the vectors has been used to gauge them all to the northern hemisphere. The statistically isotropic and Gaussian case (left column) is broken by the imprint of a strong axial effect $a_{\ell 0}=1000\mu \mathrm{K}$ (right column) whereupon multipole vectors move to the pole and area vectors move to the equatorial plane. The onset of the shown separation of multipole vectors and cross products can already be observed at moderate axial contributions of $a_{\ell 0}\sim 100\mu \mathrm{K}$.

Figure 2

Evolution of the Monte Carlo likelihood of the alignment statistics $S_{\mathit{n}\mathit{n}}$ (7) and $S_{\mathit{w}\mathit{w}}$ (8). The effect of an axis in the CMB is modeled via increasing additional zonal harmonics with coefficients $a_{\ell 0}$. At $a_{\ell 0}=1000\mu \mathrm{K}$ the multipoles become purely zonal in good approximation. Regarding WMAP’s ILC(3 yr) map $S_{\mathit{n}\mathit{n}}$ is unexpected at 98.3% C.L. and $S_{\mathit{w}\mathit{w}}$ is odd at 99.5% C.L. with respect to the statistically isotropic and Gaussian sky (bold histograms). The best improvement is reached for both statistics at roughly $a_{\ell 0}=100\mu \mathrm{K}$.

Figure 3

Contour of the scatter of intrinsic alignment (7) versus full-sky power squared (4). The shape can be understood from the folding of the two respective distributions. The total number of Monte Carlo points is $N={10}^5$. The measured data point from WMAP three-year data is included. The maximum of likelihood requires $S_{\mathit{w}\mathit{w}}$ far smaller than obtained from ILC(3 yr). Consistency with the data can be excluded at 99.95% C.L. Contours correspond to lines of $1/2^n$ times the maximal likelihood, with $n=1,\dots ,5$.

Figure 4

The sign of additional axial contributions $a_{\ell 0}$ has no physical effect on the statistics $S_{\mathit{n}\mathit{n}}$ and $S_{\mathit{w}\mathit{w}}$. For the quadrupole this follows from the symmetry of the Legendre polynomial $P_2$ [see Eq. (12)]. The quadrupole contribution is kept fixed at $a_{20}=100\mu \mathrm{K}$ while the axial contribution to the octopole is varied both in magnitude and in sign. Respective pairs of $\pm a_{30}$ histograms lie virtually on each other and their statistics are thus indistinguishable. The reference histograms following from the axially unmodified $\Lambda \mathrm{CDM}$ model (bold histograms in Fig. 2) lie nearly on top of the displayed $a_{20}=100\mu \mathrm{K}$ and $a_{30}=\pm 10\mu \mathrm{K}$ cases, and are thus not shown.

Figure 5

Scatter contour of pairs of $S_{\mathit{w}\mathit{w}}$ and $S_{\mathrm{full}}^{\mathrm{trunc}}$ after an axial modification of $a_{\ell 0}=70\mu \mathrm{K}$ has been applied; this is the contribution involving maximal improvement in $S_{\mathit{w}\mathit{w}}$ (see Fig. 2). The total number of Monte Carlo pairs is $N={10}^5$. Note that the horizontal axis now runs from zero to $1.4\times {10}^{-6}\mathrm{m}{\mathrm{K}}^4$, whereas in Fig. 3 the maximal displayed value is $4\times {10}^{-7}\mathrm{m}{\mathrm{K}}^4$. The inclusion of a preferred axis leaves the all-sky multipole power and intrinsic alignment totally uncorrelated and inconsistent with the WMAP(3 yr) data. Contour lines are defined as in Fig. 3.

Figure 6

Contour plot of the scatter of pairs $(S_{\mathit{w}\mathit{w}},S_{\mathrm{full}}^{\mathrm{trunc}})$ after a strong axial contingent of $a_{\ell 0}=1000\mu \mathrm{K}$ is induced to the multipole vectors (see also Figs. 1, 2). The total number of Monte Carlo pairs is $N={10}^5$. The all-sky power statistic reacts heavily as the scale on the $S_{\mathrm{full}}^{\mathrm{trunc}}$-axis is shifted by 4 orders of magnitude with respect to the case of $a_{\ell 0}=70\mu \mathrm{K}$ (Fig. 5). The likelihood maximum departs very articulately from the WMAP(3 yr) data. The contour lines are defined like in Fig. 3.