Alignment tests for low CMB multipoles

We investigate the large scale anomalies in the angular distribution of the cosmic microwave background radiation as measured by WMAP using several tests. These tests, based on the multipole vector expansion, measure correlations between the phases of the multipoles as expressed by the directions of the multipole vectors and their associated normal planes. We have computed the probability distribution functions for 46 such tests, for the multipoles $\ell =2-5$. We confirm earlier findings that point to a high level of alignment between $\ell =2$ (quadrupole) and $\ell =3$ (octopole), but with our tests we do not find significant planarity in the octopole. In addition, we have found other possible anomalies in the alignment between the octopole and the $\ell =4$ (hexadecupole) components, as well as in the planarity of $\ell =4$ and $\ell =5$. We introduce the notion of a global anomaly statistic to estimate the relevance of the low-multipoles tests of non-Gaussianity. We show that, as a result of these tests, the CMB maps which are most widely used for cosmological analysis lie within the $\sim 10\%$ of randomly generated maps with lowest global anomaly statistics.

Figure 1

Normalized PDF’s for the $S$ statistic found by simulating $3\times {10}^5$ mock maps, binned in intervals of 0.01. From left to right, top to bottom: $S_{22}$, $S_{23}$, $S_{24}$, $S_{25}$, $S_{33}$, $S_{34}$, $S_{35}$, $S_{44}$, $S_{45}$ and $S_{55}$. In all panels, the horizontal axis correspond to the value of each individual test, and the vertical axis to its normalized PDF.

Figure 2

Normalized PDF’s for the $D$ statistic. From left to right, top to bottom: $D_{23}$, $D_{24}$, $D_{25}$, $D_{33}$, $D_{34}$, $D_{35}$, $D_{44}$, $D_{45}$ and $D_{55}$.

Figure 3

Normalized PDF’s for the $R$ statistic. From left to right, top to bottom: $R_{22}$, $R_{23}$, $R_{24}$, $R_{25}$, $R_{33}$, $R_{34}$, $R_{35}$, $R_{44}$, $R_{45}$ and $R_{55}$.

Figure 4

Normalized PDF’s for the $N$ and $W$ statistics—alignments of the multipole ($N$) and normal ($W$) vectors with a particular direction in the sky. Top line, from left to right: $N_2$, $N_3$, $N_4$, $N_5$. Bottom line, from left to right: $W_2$, $W_3$, $W_4$, $W_5$.

Figure 5

Normalized histograms of ${log}_{10}{L}_{38}(S,D,R,B)$ (left panel), ${log}_{10}{L}_{29}(S,D,R)$ (center) and ${log}_{10}{L}_{29}(S,R,B)$ (right), obtained by simulating 25 000 mock maps.