Covariance of CMB anomalies

Several unexpected features are observed at large angular scales in the cosmic microwave background (CMB) temperature anisotropy measurements by both WMAP and Planck. These include the lack of both variance and correlation, alignment of the lowest multipole moments with one another, hemispherical power asymmetry, and an odd-to-even parity excess. In this work, we study the statistics of eight representative large-angle CMB features in order to evaluate their covariance in the standard $\mathrm{\Lambda }\mathrm{CDM}$ model. We do so using two sets of simulated CMB temperature maps; an ensemble of 100 000 simple Gaussian simulations, and 1000 Full Focal Plane (FFP) simulations provided by the Planck collaboration. In measuring feature probabilities, we pay particular attention to analysis choices, making sure that we can reproduce previous results in the literature, and explain differences where appropriate. The covariance structure we find is consistent with expectations given that many of the features studied are functions of the angular power spectrum. Notably, we find significant differences in the covariance entries associated with the quadrupole-octopole alignments derived from the Gaussian and FFP simulations. We additionally perform a principal component analysis to quantitatively gauge what combinations of features capture the most information about how simulation measurements vary, and to provide an alternative assessment of the ways in which the real sky is anomalous. The first four principal components explain about 90% of the simulations’ variance, with the first two roughly quantifying the lack of large-angle correlations, and the next two quantifying the phase-dependent anomalies (multipole alignments and power asymmetry). Though the results of this analysis are fairly unsurprising, its comprehensive approach serves to tie together a number of previous results, and will therefore provide context for future studies of large-angle anomalies.

Figure 1

Comparison of the different angular power spectrum measurements described in Sec. 2c (left) and their corresponding angular correlation functions (right). The collection of grey lines behind them are from the full-sky $C_{\ell }$ measurements of the first 100 synfast simulations. The black dotted line shows theoretical expectation, and the gray dotted lines show the 68% confidence level cosmic-variance errors.

Figure 2

Plot of the lower-tail probability $p$ for the parity asymmetry as a function of the largest multipole ${\ell }_{\max }$ considered, for various SMICA power spectrum measurement and simulation ensemble combinations. Solid lines show the probabilities for the SMICA map assessed relative to the synfast simulations, and dashed lines show them relative to the FFP simulations. For the QML and full-sky $C_{\ell }$ SMICA measurements, the simulations are measured using full-sky $C_{\ell }$’s. For the cut-sky pseudo-$C_{\ell }$ SMICA measurements, cut-sky pseudo-$C_{\ell }$ simulation measurements are used. The vertical line denotes the ${\ell }_{\max }$ value at which where Planck XVI (I&S) found the most anomalous parity statistic; see text for details.

Figure 3

Summary plot where each row shows the results for one of the features discussed in Sec. 3. Grey histograms show statistics for the features of the CMB temperature map measured from our 100 000 synfast simulations (left column), and from 1000 FFP 8.1 simulations (right column). The grey text in the lower right of each panel denotes whether simulations measurements are based on the full-sky or the cut-sky, where cut-sky measurements use the UT78 mask. Vertical lines show values for the Planck SMICA map, as well as theory predictions from Planck’s best-fit cosmology. The $p$-values displayed are single-tail probabilities showing the percentage of simulations that are more extreme that the SMICA measurements. Arrows by the $p$-values indicate which measurement we think is most relevant for each feature.

Figure 4

Relationships between large-angle CMB features (see Table 1 for descriptions of these quantities). Gray contours show the 1, 2, and $3\sigma$ confidence regions based on measurements from our ensemble of 100 000 synfast simulations, using the same measurement choices that produced the gray histograms in the left column of Fig. 3. The 1D histograms on the diagonal are the same as those in the left column of Fig. 3. The marked data points for SMICA measurements and theoretical expectations are equivalent to the vertical lines in Fig. 3. Note that the statistics based on the phases of the $a_{\ell m}$, $S_{\mathrm{QO}}$, and $A_{\mathrm{LV}}$, do not have the corresponding theoretical expectations because they cannot be computed analytically from $C_{\ell }$.

Figure 5

Left: The feature covariance matrix $S_{\mathrm{syn}}$ measured from the synfast simulation ensemble. Right: The difference $\mathrm{\Delta }S$ [given in Eq. (24)], between the covariances measured from the FFP and synfast simulations in units of its sampling error ${\sigma }_{\mathrm{syn}}^{(1000)}$ estimated from sets 1000 synfast realizations [defined in Eq. (25)].

Figure 6

Left: Eigenvectors of the anomaly covariance for our fiducial set of $10^5$ synfast simulations. Each column is one PC; the one with index 1 points in the direction of the data’s maximum variance. The rows indicate the features’ contributions to each PC. Right: Fractional residual variance for these PCs and the eigenvalues associated with each PC as a fraction of the total sum of all eigenvalues.

Figure 7

Summary of feature statistics projected into our PCA basis. The row labels indicate the basis vectors of the PC basis, which are equivalent to the unit-eigenvectors of the synfast (left) and FFP (right) simulations’ feature covariance matrix. The gray histograms show the distribution of the components of simulation realizations in the direction of each PC. The red lines show the same projection of our fiducial SMICA map measurements (using the measurement methods whose $p$-values in Fig. 3 are denoted by an arrow). The red numbers in the top right corner of each panel show the percentage of simulations that are more extreme than the corresponding SMICA measurement.

Figure 8

Left: Absolute difference between the covariance matrix measured from 1000 FFP simulations and from 100 000 synfast simulations. Right: Sample variance error bars for covariance matrix entries measured from 100 subsamples of the 100 000 synfast realizations, each containing 1000 realizations. The right panel of Fig. 5 shows the ratio between the two panels of this figure.