The authors of this paper have previously defined an isotropy test based on the socalled BiPolar power spectrum, with the goal of analysing the WMAP data with respect to possible deviations of isotropy. In this paper they apply this machinery to the WMAP ILC map and to Tegmark's cleaned map, both of which are full sky "foreground cleaned" maps, without finding significant deviations of isotropy.
I always found this result very puzzling for the following reason. I made an endtoend Monte Carlo analysis of the ILC map (astroph/0403098) some time ago, in order to assess how much residual foregrounds are left in that map. And the results were not comforting: About half of the dust present in the Wband map is still present in the ILC map (in the highlatitude Kp2 region), and the galactic plane is strongly biased toward low values, by as much as 50 muK (figure 1 of mentioned paper).
To me it is therefore somewhat surprising that a test that is designed to measure deviations from isotropy returns a nullresult when applied to these strongly foregroundcontaminated maps. Does anybody have any useful ideas or comments on this issue?
[astroph/0406354] Statistical Isotropy of the Wilkinson Microwave Anisotropy Probe Data: A Bipolar Power Spectrum Analysis
Authors:  Amir Hajian, Tarun Souradeep, Neil Cornish 
Abstract:  The statistical expectation values of the temperature fluctuations of the Cosmic Microwave Background (CMB) are assumed to be preserved under rotations of the sky. We investigate the Statistical Isotropy (SI) of the CMB anisotropy maps recently measured by the Wilkinson Microwave Anisotropy Probe (WMAP) using Bipolar Power Spectrum (BiPS) proposed in [Hajian & Souradeep 2003]. The method can probe specific regions in multipole space using appropriate window functions. The BiPS is estimated for full sky CMB anisotropy maps based on the first year WMAP data using a range of window functions. The BiPS spectra computed for both full sky maps for all our window functions are consistent with zero, roughly within $2 \sigma$. The null BiPS results may be interpreted as an absence of strong violation of statistical isotropy in the firstyear WMAP data on angular scales larger than that corresponding to $l\sim60$. However, pending a careful direct comparison, our results do not necessarily conflict with the specific SI related anomalies reported using other statistical tests. 
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[astroph/0406354] Statistical Isotropy of the Wilkinson Mic
This is a very nice paper, but I think the authors dilute the power of their test by reducing the information to a "anisotropy power spectrum" which is only a function of [tex]\ell[/tex]. This makes the test less sensitive to statistical anisotropy in any particular direction in the sky.

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[astroph/0406354] Statistical Isotropy of the Wilkinson Mic
Sorry for the long post, but we thought it will help to give a bit of background (we have only put out the letters which are perhaps terse). Also we wanted air our thoughts in details and get useful feedback and suggestions.
Statistical isotropy (SI) can be violated in many different ways. Different measures and tests are by design more sensitive to different aspects/type of SI violation. Our method is looking for nonSI structure/patterns in the underlying total Covariance ( in any component of [tex]C = C_T+ C_N + C_{res}[/tex]).
The bipolar power spectrum (BiPS) is particularly sensitive to real space correlation patterns (prefered directions, etc.) on characteristic angular scales. In harmonic space too, the BiPS at multipole [tex]\ell[/tex] sums power in offdiagonal [tex]<a_{lm}a_{l'm'}>[/tex] elements such the `angular momentum' addition of states [tex]l m[/tex], [tex]l' m'[/tex] have nonzero overlap with a state with angular momentum [tex]\ell[/tex]. Signature, like [tex]a_{lm}[/tex] and [tex]a_{l+n m}[/tex] being correlated over a significant range [tex]l[/tex] are ideal targets for BiPS. These are typical of SI violation due to cosmic topology and the predicted BiPS in these models have a strong spectral signature in the bipolar multipole [tex]\ell[/tex] space (eg. astroph/0301590). Here, the orientation independence of BiPS is an advantage since one can obtain constraints on cosmic topology that do not depend on the unknown specific orientation of the pattern (eg., prefered directions). The average over the bipolar [tex]M[/tex] index also beats down the cosmic variance by an additional [tex]2\ell+1[/tex] factor.
The extent to which BiPS probes foreground residuals is yet to be fully studied and explored. It is true we do not see any significant effect of the residual foregrounds in ILC and the Tegmark et al. foreground cleaned maps that Hans mentioned in his post. It is a puzzle but not a concern yet. First, we need to know what should have been seen in the BiPS. Is the signal strong enough? Within our angular [tex]l[/tex]space window, is the effect too smeared out in bipolar multipole space, or (as Ben says) we are blinded because of directional indendence of BiPS? On the other hand, BiPS does show a strong signal for the Weiner filtered map (Tegmark et al). The signal similar (not identical) to what one gets for a sky map with a galactic mask, presumably because the Weiner filter tampers with (supresses) the power in foreground infested galactic plane. So at some level, BiPS is sensitive to galactic residuals. We have not had time to quantify/study this. It would be useful to have maps varying levels of foreground contanimination. Suggestions are most welcome :)
Imortantly, as Ben points out, the directional independece of BiPS does `dilute' the sensitivity to galactic foregrounds where the orientation is known. Measuring individual BipoSH coeffcient [tex]A_{ll'}^{\ell M}[/tex] will have a huge cosmic variance (It is same as measuring individual elements of [tex]<a_{lm} a_{l'm'}>[/tex] matrix). We are now sifting through the analysis of WMAP maps to spot sets of anamalously large BipoSH coefficients, say, to check for bipolar index [tex]M[/tex] dependent excesses. Of course, one would not like to create a combination of BipoSH coefficients that are peculiar to WMAP. On the other hand, the reduction of the set of BipoSH to BiPS was motivated by orientation independence and there are other reductions which retain different amounts of orientation information.
It will also be interesting to express the WMAP `anomalies' detected in terms of BiPS or BipoSH measures. For eg., if the variance of the CMB fluctuation is varies on the sky, one could model that as real (angular) space Covariance matrix with nonuniform diagonal. The BiPS then is simply related to modsquare of the harmonic decomposition of the varying variance (in the simple approx of diagonal Covariance ).
Cheers
Amir & Tarun
Statistical isotropy (SI) can be violated in many different ways. Different measures and tests are by design more sensitive to different aspects/type of SI violation. Our method is looking for nonSI structure/patterns in the underlying total Covariance ( in any component of [tex]C = C_T+ C_N + C_{res}[/tex]).
The bipolar power spectrum (BiPS) is particularly sensitive to real space correlation patterns (prefered directions, etc.) on characteristic angular scales. In harmonic space too, the BiPS at multipole [tex]\ell[/tex] sums power in offdiagonal [tex]<a_{lm}a_{l'm'}>[/tex] elements such the `angular momentum' addition of states [tex]l m[/tex], [tex]l' m'[/tex] have nonzero overlap with a state with angular momentum [tex]\ell[/tex]. Signature, like [tex]a_{lm}[/tex] and [tex]a_{l+n m}[/tex] being correlated over a significant range [tex]l[/tex] are ideal targets for BiPS. These are typical of SI violation due to cosmic topology and the predicted BiPS in these models have a strong spectral signature in the bipolar multipole [tex]\ell[/tex] space (eg. astroph/0301590). Here, the orientation independence of BiPS is an advantage since one can obtain constraints on cosmic topology that do not depend on the unknown specific orientation of the pattern (eg., prefered directions). The average over the bipolar [tex]M[/tex] index also beats down the cosmic variance by an additional [tex]2\ell+1[/tex] factor.
The extent to which BiPS probes foreground residuals is yet to be fully studied and explored. It is true we do not see any significant effect of the residual foregrounds in ILC and the Tegmark et al. foreground cleaned maps that Hans mentioned in his post. It is a puzzle but not a concern yet. First, we need to know what should have been seen in the BiPS. Is the signal strong enough? Within our angular [tex]l[/tex]space window, is the effect too smeared out in bipolar multipole space, or (as Ben says) we are blinded because of directional indendence of BiPS? On the other hand, BiPS does show a strong signal for the Weiner filtered map (Tegmark et al). The signal similar (not identical) to what one gets for a sky map with a galactic mask, presumably because the Weiner filter tampers with (supresses) the power in foreground infested galactic plane. So at some level, BiPS is sensitive to galactic residuals. We have not had time to quantify/study this. It would be useful to have maps varying levels of foreground contanimination. Suggestions are most welcome :)
Imortantly, as Ben points out, the directional independece of BiPS does `dilute' the sensitivity to galactic foregrounds where the orientation is known. Measuring individual BipoSH coeffcient [tex]A_{ll'}^{\ell M}[/tex] will have a huge cosmic variance (It is same as measuring individual elements of [tex]<a_{lm} a_{l'm'}>[/tex] matrix). We are now sifting through the analysis of WMAP maps to spot sets of anamalously large BipoSH coefficients, say, to check for bipolar index [tex]M[/tex] dependent excesses. Of course, one would not like to create a combination of BipoSH coefficients that are peculiar to WMAP. On the other hand, the reduction of the set of BipoSH to BiPS was motivated by orientation independence and there are other reductions which retain different amounts of orientation information.
It will also be interesting to express the WMAP `anomalies' detected in terms of BiPS or BipoSH measures. For eg., if the variance of the CMB fluctuation is varies on the sky, one could model that as real (angular) space Covariance matrix with nonuniform diagonal. The BiPS then is simply related to modsquare of the harmonic decomposition of the varying variance (in the simple approx of diagonal Covariance ).
Cheers
Amir & Tarun