Test of cosmic isotropy in the Planck era
TTest of cosmic isotropy in the Planck era
Yabebal T. Fantaye
Departement of Mathematics, University of Tor Vergata Roma2, Rome, Italy Email:[email protected] of the talk is presented on behalf of the Planck Collaboration.
The two fundamental assumptions in cosmology are that the Universe is statistically homoge-neous and isotropic when averaged on large scales. Given the big implication of these assump-tions, there has been a lot of statistical tests carried out to verify their validity. Since the firsthigh-precision Cosmic Microwave Background (CMB) data release by the WMAP satellite,many anomalies that challenges the isotropy assumption, including dipolar power asymmetryon large angular scales, have been reported. In this talk I will present a brief summary of thetest of cosmic isotropy we carried out in the latest WMAP and Planck temperature data.
The European Space Agency (ESA) Planck satellite mission has recently produced the mostaccurate picture of the Universe by measuring the CMB with unprecedented precision. Thescientific findings of this mission is presented in a series of science papers, which are mostlyconsistent with that of WMAP, the previous CMB satellite experiment by NASA. These papers a present a simple but challenging picture of the Universe. Despite being consistent with thestandard picture, the Universe seen by Planck has some anomalies whose interpretation mightrequire a new physics.Amongst the Planck confirmed CMB anomalies the hemispherical power asymmetry , isone of the major one. This anomaly implies that the distribution of power in one side of theuniverse is different from that of the opposite one, leading to a breakdown of cosmic isotropy.In general, it is assumed that on large scales, scales that are not processed by non-linear gravity,the Universe is homogeneous and isotropic. The former implies that if we are able to zoom outand look at larger patches of the Universe, the statistical property of these patches should be thesame; while the latter means if we set ourselves at one point, like on earth, and see all around usthe statistical property of the universe in one direction should be similar to another one, hencerotation invariant.The significance of the hemispherical power asymmetry, however, has often been calledinto question, in particular, due to the alleged a-posteriori nature of the statistics used; thesignificance of the anomaly is calculated using a statistical method that is designed to detect theobserved feature. The challenge to this criticism was first given by who showed that the 5-yearWMAP data show a similar trend up to (cid:96) = 600. Moreover, using an alternative approach whichmodelled the power asymmetry in terms of a dipolar modulation field, as suggested by , founda 3 . σ detection using data smoothed to an angular resolution of 4 . ◦ FWHM, with an axis inexcellent agreement with previous results. a http://sci.esa.int/planck/51551-simple-but-challenging-the-universe-according-to-planck/. a r X i v : . [ a s t r o - ph . C O ] S e p n this talk I will present a brief summary of the test of cosmic isotropy we carried out inthe latest WMAP and Planck temperature data. There are in general two methods employed to test power asymmetry in the CMB map. Thefirst one is to compute local-power on a disc at different sky directions and compare theirconsistency with isotropic power distribution. The effect of mask, noise and other complicationsare incorporated in such a method by using a set of simulations which are processed in a similarway to the data. The excess mismatch between the distribution of local-power in the data andsimulations serve as a measure of significance of the power asymmetry. The second one is toassume a dipolar or higher order power asymmetry model and do a likelihood analysis. Thetwo common analyses in this category are the pure dipole modulation by and the generaliseddipole modulation model as implemented in Bipolar spherical harmonics (BiPoSHs) technique . In our analyses of the WMAP and Planck data in , , we quantified the degree of anisotropy bymeasuring the distribution of local-power, which is measured as the variance or power spectrumof the local patches, in the CMB maps. The variance of a map is related to its power spectrumas σ = 14 π (cid:96) max (cid:88) (cid:96) =0 (2 (cid:96) + 1) C (cid:96) , (1)and hence using both of these quantities offers a possibility to study any deviation fromisotropy in both real and harmonic space. Moreover, using power spectrum allows us easilystudy the scale dependence of a possible power variation in the CMB maps, while using variancewe avoid mask induced complications in harmonic decomposition.To incorporate a scale dependence study to the variance based method, we decomposed themap in to needlet components. β j ( n ) = B j +1 (cid:88) (cid:96) = B j − b ( (cid:96)B j ) T (cid:96) ( n ) , (2)where T (cid:96) ( n ) denotes the component at multipole (cid:96) of the CMB map T ( n ), e.g. T ( n ) = (cid:88) (cid:96) T (cid:96) ( n ) ,n ∈ S denotes the pointing direction, B is a fixed parameter (usually taken to be between 1and 2) and b ( . ) is a smooth function such that (cid:80) j b ( (cid:96)B j ) = 1 for all (cid:96). The advantage of using needlets is that the needlet filter has a very good localisation propertyboth in pixel and harmonic space, and the needlet component maps are minimally affected bymasked regions, especially at high-frequency. In particular, of course the multipole components T (cid:96) ( n ) cannot be reconstructed on masked data; nevertheless their linear combination (2) can beshown to be very robust to the presence of gaps, and more so on small scales/high frequencies.In what follows we will refer the local-power method with variance measure as local-variance and local-power with power spectra measures as local- C (cid:96) . A summary of the procedures wefollowed in both of these methods are as follows:1. create a binary patch mask which are centred on low resolution HEALPix pixels. Forboth local- C (cid:96) and local-variance methods we have considered 3072 ( N nside = 16) highlyoverlapping disc patches covering the entire sky. The radius of the discs varies from 1 to 90egrees. The final local- C (cid:96) results, however, are quoted from analysis performed using 12non-overlapping patches, the base HEALPix N nside = 1 pixels. We have shown that there isno significant difference between the results obtained using overlapping or non-overlappingpatches.2. looping over patch numbers, create an analysis patch mask by multiplying the binary discmask with that of a foreground confidence and point source masks. In the case of local- C (cid:96) analysis, we have apodised the foreground confidence and point source masks to reducecorrelations between modes.3. apply the analysis patch mask to the CMB + noise maps which are either the WMAP andPlanck data or realistic simulations.4. for each patch compute C (cid:96) s using the MASTER technique in bins of 16 multipoles; orvariance of the unmasked pixels. For the local- C (cid:96) case, the MASTER − (cid:96) blocks arefurther binned into 100 − (cid:96) blocks to reduce bin to bin correlations as well as to compareour results with previous similar analyses.5. for each patch estimate the mean and variance of local- C (cid:96) or local-variance using N sim realistic simulations. The mean is used to subtract the local mean-field from both dataand simulations, while the variance is used to weight the corresponding patch.6. estimate dipole amplitude and dipole directions of the local-power map using the HEALPix rou-tine remove dipole by applying an inverse variance weighting.
In this section I will present the summary of results presented in , , and some updates whichare not included in these published papers. In particular, I will describe the result we obtainedusing the local-variance analysis on needlet decomposed maps.As outlined in the previous section, our test statistics is based on dipole amplitudes anddipole directions of the data and simulation local-power maps. For local- C (cid:96) s, which are estimatedin 100- (cid:96) blocks, we obtained (cid:96) max / N nside = 1 local- C (cid:96) maps. For WMAP and Planck (cid:96) max = 600 and (cid:96) max = 1500 is used, which are corresponding to 6 and 15 dipole amplitudeand directions, respectively. On the other hand, for the local-variance analysis we used 20 discsizes with radius ranging from 1-90 degrees, and obtained 20 dipole amplitudes and directions.The later case is estimated for both the original real-space maps and the corresponding needletdecomposed maps.In all cases we have used p-values to quantify the significance of the observed power asym-metry. The p-values are computed by comparing the anisotropic signal measured by our test-statistics, magnitude of dipole amplitudes or clustering of dipole directions, with those obtainedfrom realistic simulations. For the WMAP case, we generated 1000 CMB-plus-noise Monte Carlo(MC) simulations based on the WMAP9 best-fit ΛCDM power spectrum . Noise realisationsare drawn as uncorrelated Gaussian realisations with a spatially varying RMS distribution givenby the number of observations per pixel. For Planck we adopt the 1000 “Full Focal Plane”(FFP6) end-to-end simulations produced by the Planck collaboration based on the instrumentperformance and noise properties. These simulations also incorporate lensing and componentseparation effects. The simulations are treated identically to the data in all steps discussedbelow.Using dipole modulated simulations, we found that clustering of dipole directions is moresensitive to measuring a small power asymmetry signal, small dipole modulation amplitudes,than comparing the magnitude of dipole amplitudes in data and simulations. The dipole di-rections clustering, however, can only be obtained when there are at least a few number ofndependent dipole direction estimates, which is only the case for local- C (cid:96) analysis. For thelocal-variance method, however, the dipole directions for different disc sizes are correlated, andhence we can not use the alignment of dipole directions as a measure of significance. We computep-values, therefore, based on excess of the data dipole amplitude from those of the simulations. C (cid:96) analysis In we performed a local- C (cid:96) analysis on WMAP 9-year data using 12 patches. We found that thepower asymmetry is statistically significant at the 3 . σ confidence level for (cid:96) = 2–600, where thedata is signal dominated. The preferred asymmetry direction is ( l, b ) = (227 , − H ); fractional densities of baryons (Ω b ), cold dark matter (Ω m ) and cosmic curvature (Ω k );the reionization optical depth ( τ ); the amplitude ( A s ) and spectral index ( n s ) of the initialscalar fluctuations. Since most of the information used to constrain these parameters comesfrom the CMB power spectrum, the observed power asymmetry may have been caused byasymmetry in one or more of these cosmological parameters. After doing local-MCMC analysis,we obtained a map of local best-fit values for each six parameters. Since we used all multipoleblocks of the MASTER power spectra for a given patch to estimate the parameters, we obtain onlya single dipole amplitude and direction per parameter per map. As a result of this we can notuse the dipole directions to measure significance of parameter asymmetry. The significance ofparameter asymmetry is, therefore, measured by comparing the data and simulation local-best-fit map dipole amplitudes. We found that none of the cosmological parameters show a significantasymmetry. This is probably because of the large error bars in the parameters. It is, however,interesting to note that the dipole direction for the parameters A s , n s and Ω b are in alignmentwith the power asymmetry direction, implying these parameters are the most sensitive to thepower asymmetry observed.The local- C (cid:96) analysis of the Planck collaboration confirms a similar power asymmetryobserved in the WMAP data, reassuring the power asymmetry is not due to systematic effects.This agreement across a wide range of scales as well as two different data sets clearly removesthe statistical a-posterior interpretation of the effect, and poses a new challenge for a detailinvestigation of the effect. In the Planck paper it was shown that the power asymmetry extendsup to (cid:96) ∼
600 with a significance of at least 3 σ . Beyond this scale, however, the doppler boosting,which is due to the motion of our solar system barycenter with respect to the CMB, becomesdominant. Boosting is a significant contamination for power asymmetry study, and it has to betaken into account when looking for power asymmetry at small scales, large multipoles. In we performed a local-variance analysis in both WMAP 9-year and Planck 1-year data. Leftpanel of Figure. (2) shows the results for Planck, which compares the dipole amplitudes of thedata local-variance map (green stars) with that of the 1000 FFP6 simulations (grey stars) - noneof the 1000 isotropic simulations have local-variance dipole amplitudes larger than the data overthe range 6 ◦ ≤ r disc ≤ ◦ . This implies that the variance in the Planck data exhibits a dipolar-like spatial variations that are statistically significant (at least) at the ∼ − . σ level or a p -value of less than 0.001. We showed that the distribution of the simulation dipole amplitudesare well fit by a Gaussian distribution for all discs sizes. The right panel of Figure. (2) visuallyillustrates mean-field subtracted local-variance map for the 6 ◦ radius disc. The dipole directionobtained from this map is ( l, b ) = (212 ◦ , − ◦ ).Similar analysis using the WMAP 9-year data yields a statistical significance of ∼ . σ , MAP1-PAWMAP5-DMWMAP9-PAPlanck-PAWMAP9-VA .. SEP . NEP -4545 dipole Planck-VAPlanck-DM
Figure 1 – Asymmetry directions found in different hemispherical power asymmetry analysis. The local varianceof the WMAP 9-year and Planck 2013 data [denoted by WMAP9-VA and Planck-VA], as well as the directionsfound previously from the latest likelihood analyses of the dipole modulation model [denoted by WMAP5-DMand Planck-DM], and the local-power spectrum analyses [denoted by WMAP1-PA, WMAP9-PA and Planck-PA]for the WMAP and Planck data. The background map is the CMB sky observed by Planck (SMICA). VA, DMand PA stand for variance asymmetry, dipole modulation and power asymmetry, respectively. This figure is takenfrom Akrami et. al. 2013 . and a direction fully consistent with those derived from Planck and other previous analyses.Our check shows that the difference in mask and input power spectra used to simulate theWMAP simulations seems to drive the difference in significance between Planck and WMAP,and a further investigation of this issue is necessary once the slight tension between WMAP andPlanck data is resolved.We have checked that the local-variance based results do not significantly change when weaccount for the doppler boosting effect either by including boosting in the simulations or bydeboosting the data, similar to what we did for the Planck 143GHz channel map with the local- C (cid:96) analysis. This is mostly because the local-variance method is mostly dominated by largescale modes where the effect of doppler boosting is small. As we will show below, however, thismethod actually is able to detect the boosting signal with > σ significance when considering ahigh-pass filtered map.The preferred directions obtained using local-variance and local- C (cid:96) methods from our papersas well as a number of similar results obtained in previous papers are summarised in Figure 1. To study the scale dependence of power asymmetry using the local-variance method, we de-composed the Planck SMICA data and FFP6 simulations into needlet components using (2),and performed the above analysis on each of the decomposed maps. The needlet parameters weused are such that a given map is decomposed into nine needlet component maps each havinga compact support over a multipole range defined by (cid:96) = [ B j − , B j +1 ] where j = 2 , , ..
10 and B = 2.As shown in Figure. (3), all but the j = 9 needlet component have dipole amplitudesconsistent with the FFP6 simulations - hence no significant power asymmetry. The j = 9 needletmap, which has a support to multipole ranges (cid:96) = [256 , . . .
87 24 1 0 0 0 0 3 4 11 15
Data Isotropic
Disk Radius [Degrees] V a r i an c e D i po l e A m p li t ude -0.11 0.13 ◦ ◦ ◦ ◦ ◦ ◦ -45 ◦ ◦ ∆ σ /σ Figure 2 – Left panel: local-variance dipole amplitude as a function of disc radius for Planck
SMICA data (ingreen) versus the 1000 isotropic FFP6 simulations (in grey). The labels above each scale indicate the number ofsimulations with amplitudes larger than the ones estimated from the data, and are located at the means of theamplitude values from the simulations. Right panel: mean-field subtracted, local-variance map computed with6 ◦ discs for Planck data. This figure is taken from Akrami et. al. 2013 . have detected in this component is not the power asymmetry we talked about in the previousparagraphs, but the well-know doppler boosting effect caused by our motion with respect tothe CMB frame. This effect has been detected for the first time by Planck using a completelydifferent algorithm , and is used to measure our solar system barycenter velocity independentof the CMB dipole component. In addition, as shown in Figure. (4), the dipole direction ofthe j = 10 component has a dipole direction right on the CMB dipole. The Planck data local-variance dipole amplitude for this particular needlet component, however, is compatible withthe FFP6 simulations. The reason for this may be due to the fact that the majority of themultipoles covered by the j = 10 component are sensitive to the boosting signal but with asmall signal to noise values, while all of multipoles in the j = 9 component are signal dominatedbut with less sensitivity to the boosting signal, or possibly contaminated by a non-vanishingpower asymmetry signal.Some interesting features to note from Figure. (3) & (4) are: the dipole directions for theneedlet component j = 2 − . j = 2 −
8, to the power asymmetry dipole direction may hint existence ofasymmetry at intermediate multipoles. Of course, a lot of verification work has to be done inthis regard.
In this talk I have discussed the unique nature of the CMB in testing the statistical propertyof the Universe. Thanks to the high precision experiments like Planck, the test of cosmologicalprinciples, the assumption of statistical isotropy and homogeneity, is now becoming a majorfield in the CMB and large scale data analysis.The current CMB data hints a lopsided Universe, more large scale structures in one side ofthe Universe than the other. The significance of this anomaly, however, is low, < σ , so onecan not rule out yet the effect being just a statistical fluke. Moreover, although the persistence
119 15 5 3 7 4 8 5 5 4 5 5 4 6 6 10
Data Isotropic
Disk Radius [Degrees] V a r i an c e D i po l e A m p li t ude Figure 3 – Dipole amplitudes of the local-variance map for the j = 9, 256 ≤ (cid:96) ≤ SMICA ) data (in green) versus the 1000 isotropic FFP6 simulations (in grey).The labels above each scale indicate the number of simulations with amplitudes larger than the ones estimatedfrom the data.
4 1024 .....
48 163264 128256 5121024 ◦ ◦ ◦ ◦ ◦ ◦ -45 ◦ ◦ ℓ central dipole low- ℓ WMAP -9 SEPNEP
Figure 4 – Dipole directions of the local-variance maps of the nine needlet components, j = 2 , , . . . ,
10 with adisc radius of 90 ◦ . The labels in this plot indicates the central multipole of a given needlet map. f the anomaly in both WMAP and Planck data seems to suggest the cause is unlikely to beforeground or systematic effects, we can not yet fully exclude the possibility of a local Universephenomena. For this and other cases more work needs to be done to confirm the signal we areobserving is truly cosmological. If that is proven, then this observation will represent anothermajor discovery about the nature of our Universe and might lead to a new insight about the innerworkings of inflation, which is responsible for laying out the initial conditions of the Universefrom a mere quantum fluctuations.In this endeavour there are different models in the literature trying to explain what isobserved in the data, but to date there is no any single theoretical model that explains all theobservations, power asymmetry both at large and intermediate scales.While theorists are working out a viable model, there are going to be results from differentexperiments. For example, the full Planck temperature and polarisation data will be releasedin near future and it will be interesting to see what the outcomes will be. The planned CMBexperiment PRISM promises to deliver a high-precision CMB polarisation maps. Since thestatistical nature of polarisation maps as well as its systematics and foregrounds are not neces-sarily similar to that of the temperature maps, PRISIM will be crucial in testing the isotropyhypothesis. On the other hand, EUCLID , a space-based survey mission from ESA, will mapthe large scale structure with an unprecedented precision. Such observations will be able to testthe isotropic assumption in the large scale structure, hence providing an independent verifica-tion. The future possibilities of the study and characterisation of the statistical nature of ourUniverse is therefore bright. These studies will ultimately contribute to our understanding ofthe processes that shaped the Universe from its birth to where it is now and where it is going. Acknowledgments
This conference proceeding is a summary of the work I did in collaboration with M. Axelsson,Y. Akrami, A. Shafieloo, H. K. Eriksen, F. K. Hansen, A. J. Banday and K. M. G´orski. Thiswork is supported by ERC Grant 277742 Pascal. I acknowledge the use of resources from theNorwegian national super-computing facilities, NOTUR. Maps and results have been derivedusing the
HEALPix (http://healpix.jpl.nasa.gov) software package developed by . eferences
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