We introduce new symmetry-based methods to test for isotropy in cosmic microwave background (CMB) radiation. Each angular multipole is factored into unique products of power eigenvectors, related multipoles and singular values that provide two new rotationally invariant measures mode by mode. The power entropy and directional entropy are new tests of randomness that are independent of the usual CMB power. Simulated Galactic plane contamination is readily identified. The ILC-WMAP data maps show seven axes well aligned with one another and the direction Virgo. Parameter free statistics find 12 independent cases of extraordinary axial alignment, low power entropy, or both having 5 per cent probability or lower in an isotropic distribution. Isotropy of the ILC maps is ruled out to confidence levels of better than 99.9 per cent, whether or not coincidences with other puzzles coming from the Virgo axis are included. Our work shows that anisotropy is not confined to the low l region, but extends over a much larger l range.

Recently, a symmetry-based method to test for statistical isotropy of the cosmic microwave background was developed. We apply the method to template-cleaned 3-and 5-years Wilkinson Microwave Anisotropy Probe-Differencing Assembly maps. We examine a wide range of angular multipoles from 2 < l < 300. The analysis detects statistically significant signals of anisotropy inconsistent with an isotropic cosmic microwave background in some of the foreground-cleaned maps. We are unable to resolve whether the anomalies have a cosmological, local astrophysical or instrumental origin. Assuming the anisotropy arises due to residual foreground contamination, we estimate the residual foreground power in the maps. For the W-band maps, we also find a highly improbable degree of isotropy we cannot explain. We speculate that excess isotropy may be caused by faulty modelling of detector noise.

We estimate CMB polarization and temperature power spectra using WMAP 5-year foreground contaminated maps. The power spectrum is estimated by using a model independent method, which does not utilize directly the diffuse foreground templates nor the detector noise model. The method essentially consists of two steps, (i) removal of diffuse foregrounds contamination by making linear combination of individual maps in harmonic space and (ii) cross-correlation of foreground cleaned maps to minimize detector noise bias. For temperature power spectrum we also estimate and subtract residual unresolved point source contamination in the cross-power spectrum using the point source model provided by the WMAP science team. Our T T , T E and EE power spectra are in good agreement with the published results of the WMAP science team. The error bars on the polarization power spectra, however, turn out to be smaller in comparison to what is obtained by the WMAP science team. We perform detailed numerical simulations to test for bias in our procedure. We find that the bias is small in all cases. A negative bias at low l in T T power spectrum has been pointed in an earlier publication. We find that the bias corrected quadrupole power (l(l + 1)C l /2π) is 532 µK 2 , approximately 2.5 times the estimate (213.4 µK 2 ) made by the WMAP team.

We analyse the effect of foregrounds on the observed alignment of the cosmic microwave background radiation (CMBR) quadrupole and octopole. The alignment between these multipoles is studied by using a symmetry‐based approach which assigns a principal eigenvector (PEV) or an axis with each multipole. We determine the alignment between these multipoles and its significance using the internal linear combination (ILC) 5‐ and 7‐yr maps. We also use the maps obtained by applying the internal power spectrum estimation (IPSE) procedure on the corresponding Wilkinson Microwave Anisotropy Probe data sets to assess its significance. The effect of foreground cleaning is studied in detail within the framework of the IPSE method both analytically and numerically. By using simulated CMBR data, including foregrounds and detector noise, we study how the PEVs of the simulated pure CMB maps differ from those of the corresponding clean maps. We find that, in general, the shift in the PEVs is relatively small and in random directions. Because of the random nature of the shift we conclude that the residual foregrounds can only lead to misalignment rather than cause alignment of multipoles. We also directly estimate the significance of the observed alignment by using simulated clean maps. We find that the results in this case are identical to those obtained by simple analytic estimates or by using simulated pure CMB maps.

In recent years, there have been a large number of studies which support violation of statistical isotropy. Meanwhile there are some studies which also found inconsistency. We use the power tensor method defined earlier in the literature to study the new CMBR data. The orientation of these three orthogonal vectors, as well as the power associated with each vector, contains information about possible violation of statistical isotropy. This information is encoded in two entropy measures, the power-entropy and alignment-entropy. We apply this method to WMAP 9-year and PLANCK data. Here, we also revisit the statistics to test high-l anomaly reported in our earlier paper and find that the high degree of isotropy seen in earlier WMAP 5-year data is absent in the revised WMAP-9 year data.