Orthogonal bipolar spherical harmonics measures: Scrutinizing sources of isotropy violation

The two-point correlation function of the cosmic microwave background temperature anisotropies is generally assumed to be statistically isotropic (SI). Deviations from this assumption could be traced to physical or observational artifacts and systematic effects. Measurement of nonvanishing power in the bipolar spherical harmonic spectra is a standard statistical technique to search for isotropy violations. Although this is a neat tool allowing a blind search for SI violations in the cosmic microwave background sky, it is not easy to discern the cause of isotropy violation by using this measure. In this article, we propose a novel technique of constructing orthogonal bipolar spherical harmonic estimators, which can be used to discern between models of isotropy violation.

Figure 1

Top panel: The red and the blue bars denote the significance of detection of the BipoSH spectra evaluated from simulated modulated maps constructed by using a modulation field given by $P(\widehat{n})=0.1Y_{20}(\widehat{n})$. Bottom panel: The green, light blue, and pink bars representing the sources, lensing ($L$), anisotropic power spectrum ($A$), and noncircular beam ($B$), respectively, show highly discrepant model parameters. Note that the dark blue bars which represent the modulation ($M$) model are seen to be consistent with zero within $3\sigma$.

Figure 2

As in Fig. 1, but for simulated maps convolved with a noncircular beam. Note that the dark blue bars which represent the anisotropic beam ($B$) model are seen to be consistent with zero within $3\sigma$.

Figure 3

This plot depicts the significance of BipoSH spectra from WMAP 9 foreground reduced temperature maps (LAMBDA site). We evaluate the BipoSH spectra in the ecliptic coordinate system and use a KQ75 temperature mask to negotiate with foregrounds. Note that there are highly significant detections in ${\alpha }_{ll}^{20}$ and ${\alpha }_{ll+2}^{20}$ modes in both the $V$ band and the $W$ band.

Figure 4

The green bars denote the significance of the parameter difference defined via Eq. (9) for the lensing-independent estimator, while the pale-blue and pink bars denote the same for the modulation and anisotropic power spectrum-independent estimators, respectively. To retain spectral information, the orthogonal BipoSH measures were evaluated by computing the minimizing variance sum over multipoles of bin width $\mathrm{\Delta }l=50$.