Statistical diagnostics to identify Galactic foregrounds in B-mode maps
aa r X i v : . [ a s t r o - ph . C O ] O c t Statistical diagnostics to identify Galactic foregrounds in B-mode maps
Marc Kamionkowski and Ely D. Kovetz Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA and Theory Group, Department of Physics and Texas Cosmology Center,The University of Texas at Austin, TX 78712, USA (Dated: October 14, 2018)Recent developments in the search for inflationary gravitational waves in the cosmic microwavebackground (CMB) polarization motivate the search for new diagnostics to distinguish the Galacticforeground contribution to B modes from the cosmic signal. We show that B modes from theseforegrounds should exhibit a local hexadecapolar departure in power from statistical isotropy (SI).We present a simple algorithm to search for a uniform SI violation of this sort, as may arise in asufficiently small patch of sky. We then show how to search for these effects if the orientation ofthe SI violation varies across the survey region, as is more likely to occur in surveys with moresky coverage. If detected, these departures from Gaussianity would indicate some level of Galacticforeground contamination in the B-mode maps. Given uncertainties about foreground properties,though, caution should be exercised in attributing a null detection to an absence of foregrounds.
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The BICEP2 collaboration recently reported [1] evi-dence for the signature [2] of inflationary gravitationalwaves [3] in the B-mode component [4, 5] of the po-larization of the cosmic microwave background (CMB).The extraodinary stream of papers [6] that have followedthis announcement provides some indication of the sig-nificance of a B-mode detection. However, the remark-able implications of this measurement—the detection ofa new relic from inflation—demand that the results re-ceive the deepest possible scrutiny. Discussions that havetaken place since the March 2014 announcement indicatethat more work must be done to establish, with the typeof confidence such an extraordinary result warrants, thatthe B-mode signal cannot be attributed fully to polarizedemission from interstellar dust (see, e.g., Refs. [7, 8]).The gold standard to distinguish CMB from fore-grounds (primarily synchrotron and dust emission fromthe Milky Way) has typically been to obtain high–signal-to-noise maps at multiple frequencies. Important stepsin this direction should soon be taken for the BICEP2 B-mode signal with new data from the 100-GHz Keck Array[9] and from polarization measurements from Planck [10]at higher frequencies, and soon indepenently with otherexperiments (e.g., Ref. [11–14]). However, these measure-ments may, like any others, ultimately have limits. Forexample, extrapolation of measurements of the B-modepower from dust obtained with Planck’s 353-GHz chan-nel to BICEP2’s 150 GHz channel may suffer from the-oretical uncertainties in the frequency dependence of thedust polarization . The use of spatial cross-correlationsbetween different frequency channels may be imperfect Indeed, the frequency-dependent models 1 and 3 of Ref. [15] (seeFig. (8)) predict an opposite trend with frequency than observed(see Fig. (13) in Ref. [16]), indicating our theoretical uncertainty. if the depths in the interstellar medium probed by thosetwo frequencies differ. Even if the dust contribution turnsout to be small enough that such subtleties do not pre-vent the confident establishment of a gravitational-wavesignal, every detail about the early Universe that we canextract from detailed characterization of the B-mode sig-nal will be priceless. It is thus imperative that we remainever vigilant in our quest to find new ways to root outcontaminants to the cosmic B-mode signal.Here we propose two statistical tests that can be per-formed on an observed B-mode map .—either a single-frequency map or one that has been cleaned with multi-frequency information—to help identify foreground con-tamination. The idea is simple: The departures in theinflationary gravitational-wave signal from Gaussianityand statistical isotropy (SI) are expected to be extremelysmall [17]. Any statistically significant departure fromGaussianity or SI would thus indicate some non-cosmiccontamination.The question, though, is what type of non-Gaussianityor SI violation should we be seeking? Here we arguethat the polarization due to foregrounds over a suffi-ciently small region of the sky induces a hexadecapolaranisotropy in the B-mode power, something that shouldbe relatively simple to seek. We then show how to lookfor a spatially-varying SI violation of this sort, something In principle, one might just look in the data for a preferred ori-entation in the polarization map. Most CMB experiments, how-ever, measure only differences in polarization and thus are notequipped to measure the average orientation. Moreover, much ofwhat we discuss below for B modes also applies to E modes, butthe additional information in E modes is likely to be swampedby cosmic variance from the dominant density-perturbation con-tribution to E modes. Still, higher-frequency E-mode maps maybe useful for constructing dust orientation templates for cross-correlation with B-mode maps. that is more likely to describe the foreground polarizationpattern on larger patches of sky.Let us begin by understanding how this SI violationarises, in particular for the case of dust. Polarized emis-sion from dust stems from the alignment of spinning dustgrains with the Galactic magnetic field [15] (which alsodetermines the synchrotron polarization). Galactic mag-netic fields are known to have long-range correlations,implying an orientation angle that is fairly coherent onlarge regions of the sky [18], and perhaps larger than thepatch covered by BICEP2. There may, of course, be sig-nificant changes in that orientation angle in small skypatches if there are regions of high-density plasma in theISM in that patch. The BICEP2 patch, however, whichlies in the “Southern Hole,” was chosen for the expecta-tion that it was relatively clean [19] and thus likely freefrom rapid variation in the orientation angle (as shown inFig. (13) of Ref. [18], the typical angle dispersion of dustpolarization is lowest in the highest polarization-fractionregions of the sky, those cleanest and most suitable forB-mode measurements). Furthermore, measurements ofpolarized absorption of starlight (which is correlated withpolarized dust emission [20]) in the BICEP2 region mayprovide some empirical indication that the orientation ofthe dust polarization in the BICEP2 patch is roughlyuniform, as noted by Ref. [7]. However, as this data liesnear the edges of the field, it cannot provide a robustconstraint on the entire patch.Let us therefore consider a B-mode signal from a mapin which the orientation angle of the polarization is con-stant. The Stokes parameters Q ( ~θ ) and U ( ~θ ), measuredas a function of position ~θ = ( θ x , θ y ) on a flat region ofsky, are components of a polarization tensor, P ab = 1 √ Q ( ~θ ) U ( ~θ ) U ( ~θ ) − Q ( ~θ ) ! . (1)The polarization map is then decomposed into scalar andpseudoscalar components E ( ~θ ) and B ( ~θ ) by ∇ E = ∂ a ∂ b P ab ; ∇ B = ǫ ab ∂ a ∂ c P cb , (2)where ǫ ab is the antisymmetric tensor. The Fourier com-ponents of E ( ~θ ) and B ( ~θ ) are˜ E ( ~l ) = 2 − / h cos 2 ϕ ~l ˜ Q ( ~l ) + sin 2 ϕ ~l ˜ U ( ~l ) i , (3)˜ B ( ~l ) = 2 − / h − sin 2 ϕ ~l ˜ Q ( ~l ) + cos 2 ϕ ~l ˜ U ( ~l ) i , (4)in terms of the Fourier transforms ˜ Q ( ~l ) and ˜ U ( ~l ) of theStokes parameters and the angle ϕ ~l that ~l makes withthe θ x axis.If the polarization is constant across the map with ori-entation α = (1 /
2) arctan(
U/Q ) with respect to the θ x axis, then the Fourier modes for E and B will be,˜ E ( ~l ) = ˜ P ( ~l ) √ (cid:2) α − ϕ ~l ) (cid:3) , (5)˜ B ( ~l ) = ˜ P ( ~l ) √ (cid:2) α − ϕ ~l ) (cid:3) , (6)where ˜ P ( ~l ) is the Fourier transform of the polarizationamplitude P ( ~θ ) ≡ ( Q + U ) / ( ~θ ). We thus see that if the orientation angle of polarization is constant, theB modes that result are not statistically isotropic . Theyare, rather, modulated by sin (cid:2) (cid:0) α − ϕ ~l (cid:1)(cid:3) .An estimator for this departure from statisticalisotropy in the B-mode map can be obtained through astraightforward augmentation of the usual algorithm todetermine the amplitude of the B-mode power. Eq. (6)—which is what we expect if the observed B modes are dueentirely to dust and if the dust polarization has uniformorientation—implies that the mean-square amplitude ofeach B-mode coefficient is, (cid:28)(cid:12)(cid:12)(cid:12) ˜ B ( ~l ) (cid:12)(cid:12)(cid:12) (cid:29) = AC fl (cid:2) − cos 4 α cos 4 ϕ ~l − sin 4 α sin 4 ϕ ~l (cid:3) , (7)where C fl parametrizes an assumed fiducial l dependence(e.g., C fl ∝ l − . , as current measurements suggest [18])and A an amplitude of the signal. Note that although themodulation of the Fourier amplitudes is quadrupolar ( ∝ e iα ), the departure from statistical isotropy in the powerspectrum is a hexadecapole; it has an e iα dependence.More generally, if the orientation of the dust polariza-tion is not perfectly uniform, but is rather spread oversome small range δα , then the modulation in Eq. (7) willbe reduced by a factor ∼ ( δα ) /α . Thus, to test for dust,we should aim to measure the parameters in the angle-dependent power spectrum, (cid:28)(cid:12)(cid:12)(cid:12) ˜ B ( ~l ) (cid:12)(cid:12)(cid:12) (cid:29) = AC fl (cid:2) − f c cos 4 ϕ ~l − f s sin 4 ϕ ~l (cid:3) , (8)where f s , f c < α =(1 /
4) arctan( f s /f c ).The minimum-variance estimator for the isotropic am-plitude A is the usual one, b A = P ~l (cid:12)(cid:12)(cid:12) ˜ B ~l (cid:12)(cid:12)(cid:12) C fl /σ l P ~l (cid:16) C fl (cid:17) /σ l , (9)where the sum is over all Fourier modes ~l with amplitudes˜ B ( ~l ) each measured with variance σ l (which may re-ceive contributions from detector noise and from lensing-induced B modes [21, 22]). The minimum-variance esti-mators for the amplitudes of the SI-violating terms arelikewise, d Af c = P ~l (cid:12)(cid:12)(cid:12) ˜ B ~l (cid:12)(cid:12)(cid:12) C fl cos 4 ϕ ~l /σ l P ~l (cid:16) C fl cos 4 ϕ ~l (cid:17) /σ l , (10)and similarly for the f s term with the replacement cos → sin. If there is no prior information about the orien-tation of the dust polarization, then the parameters f s and f c are both obtained simultaneously and indepen-dently from the data. If, however, there is some priorinformation about the expected orientation—e.g., fromstarlight polarization—then the ratio f s /f c can be fixedand the sensitivity to dust-induced SI violation thus ac-cordingly improved. Either way, any statistically signifi-cant detection of nonzero f s and/or f c indicates at leastsome contamination of the cosmic signal. If, moreover,either of the inferred values f c or f s differs significantlyfrom zero, then there is good evidence that the signalis predominantly non-cosmic. If there is strong reasonto believe that the foreground-polarization orientation isindeed uniform across the survey, then a strong null re-sult may imply that the observed signal is not foregrounddominated. If, though, that orientation is uncertain, thena null result in this SI test cannot be used to rule outforeground contamination.The variances and covariances with which the param-eters A , f s , and f c can be measured are easily derived.However, they will depend considerably on the detailsof any given experiment and perhaps a bit on the factthat the lensing-induced B-mode map is not preciselyGaussian. We thus leave these covariances to simula-tions of the complete analysis pipelines. Heuristically,though, the estimator measures the difference in the B-mode power for modes oriented perpendicular/parallel tosome axis versus those oriented at 45 ◦ . If there is a & σ detection of power, and if that power is due entirely touniformly oriented dust, then the violation of statisticalisotropy should appear with high statistical significance.Indeed, a crude estimate for the minimum amplitude A that can be measured at 1 σ is given by σ − b A = X ~l (cid:16) C fl (cid:17) /σ l ∼ Ω Z d l (2 π ) (cid:16) C fl (cid:17) /σ l = 4 πf sky Z d l (2 π ) (cid:16) C fl (cid:17) /σ l = 2 f sky Z ldl (cid:16) C fl (cid:17) /σ l (11)and σ d Af c = 2 σ b A (as R π dϕ = 2 R π cos (4 ϕ ) dϕ = 2 π ).The signal-to-noise in a particular experiment is governedby the sensitivity per Fourier mode σ l = s f sky (2 l + 1) (cid:16) C lens l + f sky w − ( T ) e l σ b (cid:17) , (12) where the pixel noise σ pix = s/ √ t pix is determined bythe detector sensitivity s and the observation time t pix = T /N pix dedicated to each pixel, and where we used thedefinition w − ( T ) ≡ πs /T .It should be noted that some of the dust-polarizationtemplates used by BICEP2 and investigated in subse-quent work were constructed assuming a uniform dust-polarization orientation. The departures from SI con-sidered above are then effectively incorporated into thedata-template cross-correlation analyses done already.Those cross-correlations, though, may still vanish if ei-ther (1) the assumed orientation angle is incorrect, or(2) the spatial variation of the polarization amplitude isnot correctly represented, as can be seen in Ref. [7]). TheSI-violation analysis suggested above, though, does notrely on prior knowledge of the spatial variation of theamplitude nor the assumed orientation angle.So far we have supposed that the sky patch is smallenough that a uniform dust-polarization orientation maybe reasonably hypothesized. However, future experi-ments will cover larger regions of the sky (e.g., Ref. [11–13]), and it is increasingly likely that the foreground-polarization orientation will meander across the surveyregion as the size of that region increases. The fore-ground polarization may thus be modeled in terms of anamplitude that has rapid small-scale variation with anorientation that has longer-range correlations. This canbe sought in a straightforward fashion by simply mea-suring the correlations in the polarization amplitude andin the orientation angle. If the signal is cosmic, the cor-relations in both should be similar. Evidence that thosetwo correlation lengths differ could indicate a non-cosmicsource of contamination. Such an analysis, though, willlikely be limited by cosmic variance from the dominantdensity-perturbation–induced polarization.Instead, we now spell out a diagnostic for spatial vari-ations of the type of SI-violation above that parallels al-gorithms developed to search for spatially-varying cos-mic birefringence [23], optical depth (“patchy screening”)[24], and cosmological parameters [25], and before those,weak lensing [26] (which has now been detected [22, 27]).For clarity, we work here in the flat-sky limit; the gen-eralization to the full sky is straightforward and followsthis other previous analogous work.We suppose that there are variations of the orientationangle that vary slowly across the sky with small-scalefluctuations in the polarization amplitude. We thus as-sume the polarization can be written, P ab ( ~θ ) = P oab ( ~θ ) φ ( ~θ ) (13)in terms of a smooth “orientation field” P oab ( θ ) withStokes parameters Q o ( ~θ ) and U o ( ~θ ) and a more rapidly-varying polarization-amplitude field φ ( ~θ ) (which for dustshould be correlated with the dust-intensity field, al-though we do not use any such information here). Theorientation field can be decomposed in the usual man-ner into E and B modes E o ( ~θ ) and B o ( ~θ ). There is anambiguity in the definitions of P oab ( ~θ ) and φ ( ~θ )—one canbe increased while the other is reduced without changing P ab —that can be removed by demanding, e.g., that thepolarization amplitude field have unit variance or somespecific maximum value.Consider a spatial variation of the orientation that con-sists of a single Fourier mode of wavevector ~L of eitherthe E type or the B type. The orientation pattern inthe first case always has only nonzero Q (measured withrespect to axes aligned with ~L ) and in the latter caseonly nonzero U . Thus, in the first case (E-mode orien-tation), the polarization is always aligned/perpendicularto ~L , and in the second (B-mode orientation), the po-larization is always aligned at axes rotated by 45 ◦ from ~L . Therefore, in either case—a pure-E orientation or apure-B orientation—the orientation of the SI violation inthe polarization B modes are everywhere the same, eventhough the orientation angle is changing. Thus, in eitherof these two cases, there will be SI violation in the ob-served B modes that is uniform across the sky, and thesimple SI-violation test above will capture the effect inits entirety and have a positive result.To make things a bit more interesting, consider an ori-entation that rotates clockwise as we move in the θ x direction, completing a full revolution after a distance θ x = 2 π/L . I.e., (cid:18) Q o U o (cid:19) ( ~θ ) = R ~L (cid:18) cos Lθ x sin Lθ x (cid:19) . (14)This is a linear combination of an E mode and a B mode,both of the same ~L , added out of phase—i.e., E + iB —and R ~L is the amplitude of this Fourier mode. Moreprecisely, (cid:18) Q o U o (cid:19) ( ~θ ) = (cid:20) R ~L √ (cid:18) i (cid:19) e i~L · ~θ + cc (cid:21) , (15)where now we have allowed R ~L to be complex to allow aphase different from that in Eq. (14). We then supposethat the observed polarization is obtained by multiply-ing this slowly-varying orientation field with a rapidly-varying amplitude φ ( ~θ ); i.e., (cid:18) QU (cid:19) ( ~θ ) = (cid:20) R ~L √ (cid:18) i (cid:19) e i~L · ~θ + cc (cid:21) φ ( ~θ ) . (16)Since the orientation varies over all possible values, theobserved B modes will be statistically isotropic when av-eraged over the whole field, and the SI-violation test sug-gested above will give a null result. Still, the observed Bmodes will exhibit local departures from SI.We now explain how to detect this position-dependentlocal SI violation. The polarization pattern in Eq. (16)yields B modes,˜ B ( ~l ) = i h R ~L ˜ φ ( ~l − ~L ) e iϕ ~l − R ∗ ~L ˜ φ ( ~l + ~L ) e − iϕ ~l i . (17) Before proceeding, recall that the B modes due toinflationary gravitational waves are expected to beGaussian and statistically isotropic which implies that D ˜ B ( ~l ) ˜ B ∗ ( ~l ′ ) E = 0 for ~l = ~l ′ . However, we now find thatthe polarization pattern in Eq. (16) has expectation val-ues, D ˜ B ( ~l ) ˜ B ∗ ( ~l ′ ) E = 14 h | R ~L | ( C φ | ~l − ~L | + C φ | ~l + ~L | ) δ ~l,~l ′ − ( R ∗ ~L ) C φ | ~l + ~L | e − i ( ϕ ~l + ϕ ~l ′ ) δ ~l ′ ,~l +2 ~L − ( R ~L ) C φ | ~l − ~L | e i ( ϕ ~l + ϕ ~l ′ ) δ ~l ′ ,~l − ~L i , (18)where C φl is the power spectrum of the modulation field φ ( ~θ ), and δ ~l,~l ′ is shorthand for (2 π ) δ D ( ~l − ~l ′ ), the Diracdelta function. The first term in Eq. (18), the onlyone that is nonvanishing for ~l = ~l ′ , provides the (angle-averaged) B-mode power spectrum for the map. Roughlyspeaking, it is the amplitude power spectrum C φl smearedin l space by L . As argued above, this first term indicatesthat there is no departure from statistical isotropy whenpower is averaged over the entire map.The second two terms in Eq. (18), though, describethe local SI violation of a polarization field due to thesmall-scale modulation of a longer-range orientation field.They indicate a cross-correlation of a Fourier mode ofwavevector ~l with those of wavevectors ~l ′ = ~l ± ~L . Theappearance of 2 ~L (rather than just ~L ) is related to thehexadecapolar nature of the power asymmetry.Eq. (18) implies that each pair of Fourier amplitudes˜ B ( ~l ) and ˜ B ( ~l ′ ) with ~l − ~l ′ = 2 ~L provides an estimator, \ ( R ∗ ~L ) = − B ( ~l ) ˜ B ∗ ( ~l ′ ) e i ( ϕ ~l + ϕ ~l ′ ) C φ | ~l + ~L | , (19)for the Fourier amplitude R ∗ ~L (or actually, its square) ofthe orientation amplitude. One then adds the estimatorsfrom each such ~l,~l ′ pair with inverse-variance weightingto obtain the optimal estimator for ( R ∗ ~L ) . The procedureis directly analogous to that for weak-lensing, cosmic-birefringence, and patchy-screening reconstruction, andwe leave the details to be presented elsewhere. If any R ~L (for any wavevector ~L that can be accessedwith the map) is found to be nonzero with statistical sig-nificance, it indicates a likely contamination from fore-ground. Naturally, when searching for deviation from SIin multiple independent L modes, the “look elsewhereeffect” must be properly taken into account. It shouldbe possible, however, in a map that covers a sufficientlylarge region of sky with sufficient signal to noise, to mea-sure a large number of amplitudes for E + iB and E − iB modes and thus to reconstruct the orientation-angle map P oab ( ~θ ) as a function of position on the sky.Reassuringly, in the limit L →
0, where the orientationangle becomes uniform (and taking R ~L to be real, so thatthe orientation is aligned with θ x ), Eq. (18) simplifies to, D ˜ B ( ~l ) ˜ B ∗ ( ~l ′ ) E = R ~L C φl (1 − cos 4 ϕ ~l ) δ ~l,~l ′ , (20)recovering the expected hexadecapolar power anisotropy.To conclude, we have argued that polarization fromdust is likely to give rise to non-Gaussianity in the Bmodes they induce, that appears as a local hexadecap-olar departure from statistical isotropy. A simple testthat will seek this SI violation in the event that the ori-entation of the dust-induced polarization is roughly con-stant was presented. We also showed how an orientationthat varies across the survey region can be sought. Herewe have only sketched out how these tests can be done.Much more work will be needed before they are imple-mented in real data. This will include the full develop-ment of the optimal estimators, full-sky formalisms, toolsto deal with imperfect sky coverage, etc. Still, these de-velopments should parallel the analogous developmentsfor, e.g., weak lensing. The estimators for the effectswe deal with here differ in detail from those, e.g., forweak lensing (here we seek a local hexadecapolar SI vi-olation, while lensing induces a quadrupolar effect), butsome thought should be given to possible confusion in alow–signal-to-noise scenario.We do not advocate that the foreground diagnosticswe discuss here replace multifrequency component sepa-ration. Rather, they can be implemented in the event oflimited multifrequency information or, in the event thatmultifrequency maps uncover a cosmic signal, as a wayto check for consistency or identify residual foregroundcontamination in the maps.We thank Sam Gralla and Hirosi Ooguri for useful dis-cussions and the Aspen Center for Physics for hospital-ity. 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