Too few spots in the cosmic microwave background
aa r X i v : . [ a s t r o - ph . C O ] F e b Too few spots in the cosmic microwave background
Youness Ayaita, Maik Weber, Christof Wetterich
Institut f¨ur Theoretische Physik, Universit¨at HeidelbergPhilosophenweg 16, D-69120 Heidelberg, Germany
We investigate the abundance of large-scale hot and cold spots in the WMAP-5 temperaturemaps and find considerable discrepancies compared to Gaussian simulations based on the ΛCDMbest-fit model. Too few spots are present in the reliably observed cosmic microwave background(CMB) region, i.e., outside the foreground-contaminated parts excluded by the KQ75 mask. Evensimulated maps created from the original WMAP-5 estimated multipoles contain more spots thanvisible in the measured CMB maps. A strong suppression of the lowest multipoles would leadto better agreement. The lack of spots is reflected in a low mean temperature fluctuation onscales of several degrees (4 ◦ –8 ◦ ), which is only shared by less than 1% (0 . . . . I. INTRODUCTION
The precise measurement of anisotropies in the cosmicmicrowave background (CMB) has played a key rolein amplifying our knowledge about the structure andevolution of the Universe. The best data available todayis provided by the Wilkinson Microwave Anisotropy Probe(WMAP) satellite mission from five years of observation.Its results are powerful enough to put various cosmologicalmodels to stringent tests. They helped establishingtoday’s standard model of a spatially flat universe withGaussian initial perturbations, possibly generated duringan early inflationary epoch. According to the standardΛCDM model, the present Universe is essentially made upfrom dark energy in the form of a cosmological constantΛ and cold dark matter (CDM). Under the assumptionsof Gaussianity and statistical isotropy, all the informationabout the temperature fluctuations in the CMB are en-coded in the angular power spectrum C ℓ from a harmonicdecomposition of the temperature field. A crucial resultof the WMAP analysis therefore is an estimate of themultipoles C ℓ which is in good agreement with the ΛCDMbest fit (Nolta et al., 2009) except for the well-known dis-crepancies of the low multipoles, especially the quadrupole C (Hinshaw et al., 2007). Nonetheless, many issues arestill under intense discussion. Repeatedly, authors haveclaimed to detect non-Gaussian signals (McEwen et al.,2008; Yadav and Wandelt, 2008) or statisticalanisotropy (Bernui et al., 2006; de Oliveira-Costa et al.,2004; Eriksen et al., 2004; Hansen et al., 2009;Hoftuft et al., 2009; Land and Magueijo, 2005). Sincethe power spectrum is insensitive to these anomalies,it is necessary to perform additional investigations ofthe temperature sky map. These are done in harmonic,wavelet, and pixel space (Cabella et al., 2004). Even ifGaussianity holds, it may still give new insights to switchto another representation of the statistical properties ofthe temperature maps since a phenomenon can be moreeasily detected in one representation than in another.The goal of this work is to provide a clear and intuitiveanalysis in pixel space regarding abundances of large-scalehot and cold spots identified as regions whose mean tem- perature contrasts exceed some (variable) threshold. Weanalyze both observed CMB maps and Gaussian simu-lations based on ΛCDM. The comparison reveals severedeviations. Other authors who worked with statistics oflocal extrema in the temperature field also observed sig-nificant anomalies (Hou et al., 2009; Larson and Wandelt,2004, 2005).We start by recalling some basic results that connectpixel-space analyses with the angular power spectrum inSec. II. A comprehensive description of our method followsin Sec. III including the preparation of adequate Gaussiansimulations, the working principle of our spot searching al-gorithm, and an error estimation. Our results are presentedin Sec. IV. We consider both cut-sky maps (with unreliablepixels excluded by the KQ75 temperature analysis mask)and the Internal Linear Combination (ILC) full-sky map,and quantify deviations from Gaussian simulations. Wesum up and conclude in Sec. V. II. PRELIMINARY CONSIDERATIONS
The most robust comparison between predicted and ob-served spot abundances of CMB sky maps relies on simu-lated maps since analytic methods can hardly care for com-plications due to masking and beam properties. Creatinga number of simulated maps and treating them in exactlythe same way as the original map therefore is the clear-est method. Nonetheless, it is instructive to recall somewell-known analytic results that connect the pixel-spaceanalysis to familiar harmonic space.The spot abundances in a CMB sky map are dictated bythe angular correlations of temperature fluctuations. Themost popular theories stick to Gaussianity and statisticalisotropy. Then, the ensemble average of the angular cor-relation between two directions ( θ, ϕ ) and ( θ ′ , ϕ ′ ) only de-pends on the angle Θ between them. This leads to thedefinition of the angular correlation function C (Θ) = (cid:28) ∆ T ¯ T ( θ, ϕ ) × ∆ T ¯ T ( θ ′ , ϕ ′ ) (cid:29) . (1)We can switch to harmonic space by decomposing the tem-perature field into spherical harmonics:∆ T ¯ T ( θ, ϕ ) = X ℓ,m a ℓm Y ℓm ( θ, ϕ ) , (2)where the crucial assumption of statistical isotropy implies h a ℓm a ∗ ℓ ′ m ′ i = δ ℓℓ ′ δ mm ′ C ℓ . (3)So, in this case, all the statistical information is in the co-efficients C ℓ , the angular power spectrum. More generally,we may define C ℓ = 12 ℓ + 1 X m (cid:10) | a ℓm | (cid:11) . (4)When searching for spots of a given size, we will aver-age the temperature fluctuations in regions of that size.These regions are defined by window functions W ( θ, ϕ ).The mean temperature contrast in such a region is∆ T = Z dΩ ∆ T ( θ, ϕ ) W ( θ, ϕ ) . (5)In our sense, a spot is characterized as follows. When athreshold ∆ T is fixed, a hot spot is found if ∆ T ≥ ∆ T ,whereas a cold spot is found if ∆ T ≤ − ∆ T . The charac-teristic scale for ∆ T is the mean temperature contrast forthese regions ∆ T rms = p h ∆ T i . Clearly, if ∆ T ≪ ∆ T rms ,most regions will be spots, if ∆ T ≫ ∆ T rms , only a few ornone.The transformation to harmonic space can be done bydecomposing the window function W ( θ, ϕ ) into sphericalharmonics with coefficients W ℓm and defining W ℓ = X m | W ℓm | . (6)Together with Eqs. (2) and (3), it is straightforward tocalculate ∆ T rms = X ℓ ℓ + 14 π C ℓ W ℓ ¯ T . (7)This result shows that the mean temperature fluctuation∆ T rms is given by the multipoles C ℓ weighted by W ℓ . The W ℓ strongly depend on the angular scale of the regions.Their magnitude will suppress large ℓ values correspond-ing to scales smaller than the window. By virtue of theaddition theorem for spherical harmonics, we can write W ℓ = Z dΩ Z dΩ ′ W ( θ, ϕ ) W ( θ ′ , ϕ ′ ) P ℓ (cos Θ) . (8)This allows us to calculate the W ℓ for a chosen window.An example is shown in Fig. 1.In our case, it is adequate to approximate the sphere bythe tangent plane at a region, replacing the direction ( θ, ϕ )by points x on the plane. For our purposes, it is most con-venient to work with top hat windows because they haveclear boundaries. This is the easiest way to avoid ambi-guities arising from overlapping spots. Exemplary choicesmay be the top hat circle with window function W ( x ) = 1 πR Θ( R − | x | ) (9) PSfrag replacements ℓ W ℓ FIG. 1 Coefficients W ℓ for the top hat circle window functionat scales a = 1 ◦ (right plot), 6 ◦ (left plot). The plots showwhich multipoles predominantly determine ∆ T rms . For smallerangular scale a , higher ℓ values enter the analysis. or a square with window function W ( x ) = 1 a Θ( a − x ) Θ( a − x ) . (10)Following Durrer (2008, p. 218), we can approximate the W ℓ by an angular average over the Fourier transform of W ( x ) which considerably reduces the computational effort: W ℓ ≈ π Z π d α | ˜ W ( l ) | . (11)For the aforementioned window functions, we can use thisequation to easily calculate ∆ T rms by Eq. (7). The resultsare plotted for the ΛCDM best-fit power spectrum in Fig. 2.For the sake of comparability, we use the parameter a whichequals the square root of the windows’ area; in the case ofsquares, it simply is the side length. We also show therelative deviation due to the different window functions.We conclude that the result is not sensitive to the exactgeometry if the covered surface area is the same. III. METHOD
Our strategy consists of performing an identical analysisof spot abundances both for observational maps and mapsgenerated from simulations of Gaussian fluctuations. Forthe simulated maps, we use the best-fit ΛCDM model and aGaussian fluctuation model based on the C ℓ quoted by theWMAP collaboration. The comparison with maps fromobservation tests Gaussianity.Because of the excellent data products of the WMAPteam available at the legacy archive and the comprehen-sive HEALPix package (Gorski et al., 2005), it is possi-ble to obtain reliable CMB sky maps and to create mapsfrom Gaussian simulations. We summarize the steps in http://cmbdata.gsfc.nasa.gov http://healpix.jpl.nasa.gov
40 45 50 55 60 65 70 75 80 85 1 2 3 4 5 6 7 8 9 10
PSfrag replacements a [ ◦ ] ∆ T r m s [ µ K ] -1-0.5 0 0.5 1 1 2 3 4 5 6 7 8 9 10 PSfrag replacements a [ ◦ ]∆ T rms [ µ K] a [ ◦ ] R e l a t i v e d e v i a t i o n i n % FIG. 2 Mean temperature fluctuation for various spot sizes andthe ΛCDM power spectrum. The plots for circles and squaresare visually indistinguishable. The difference between the resultfor circles and the result for squares is shown in the secondfigure.
Sec. III.A. We developed an algorithm searching for hotand cold spots (in the sense of Sec. II) within these temper-ature sky maps. Its working principle and properties arepresented in Sec. III.B. The treatment of statistical errorsis described in Sec. III.C.
A. Maps and data preparation
Whenever the original signal is to be extracted fromCMB data, it is crucial to minimize the influence offoreground contamination. The frequency dependence ofthe foreground components (e.g., synchrotron emission,free-free emission, and thermal dust) allows to reducethe contamination with the help of various foregroundmodels (Gold et al., 2009). The WMAP team providesforeground-reduced maps for the Q (35–46 GHz), V (53–69GHz), and W (82–106 GHz) bands. Since the V band hasa better signal-to-noise ratio than the W band and is lessforeground contaminated than the Q band (Hinshaw et al.,2007), it is the natural choice to use the foreground-reduced V map. Further noise minimization by constructing lin-ear combinations of the maps is possible but does not af-fect our analysis which focuses on large scales. But still,large parts of the temperature map are unreliable and mustbe excluded from the analysis. We therefore apply theKQ75 mask, cutting out the contaminated galaxy region and point sources (Gold et al., 2009). Finally, the resid-ual monopole and dipole are removed with the HEALPixroutine remove dipole . Figure 3 shows the foreground-reduced V map and the KQ75 mask. FIG. 3 The foreground-reduced V map (temperature contrastin mK) and the KQ75 mask cutting out the contaminatedgalaxy region and point sources. Gaussian simulations based on some input C ℓ spectrumand a beam window function are achieved with the helpof the synfast HEALPix facility. These input data canbe obtained from the legacy archive. The power spectrawe used are the ΛCDM best fit and the original WMAP-5estimate both shown in Fig. 1 of Nolta et al. (2009). Subse-quently, we will refer to them by “ΛCDM” and “WMAP-5”power spectrum for short. We take care of treating simu-lated and original maps as equally as possible. This neces-sitates the additional simulation of the instruments’ noise,masking, and removal of monopole and dipole. Since theWMAP design minimizes noise correlation between neigh-boring pixels in a map (Page et al., 2003), it is legitimateto add white noise with the properties described by theWMAP team at the legacy archive.When studying possible anisotropy of the CMB, we needa full-sky (unmasked) map. Since the foreground contam-inations usually force us to mask parts of the sky, it is nota trivial task to reconstruct the full-sky CMB signal. How-ever, the WMAP team tries to tackle this job by combiningthe measurements of all bands and merge them into a single(ILC) map of the full sky (Gold et al., 2009). The appliedprocedure is independent of foreground models but has thedisadvantage of being doubtful on scales below approxi-mately 10 ◦ according to the WMAP product descriptionat the legacy archive. But since we are lacking any bet-ter alternative, we employ the 5-year WMAP ILC map forfull-sky analyses. B. Spot searching algorithm
The primary goal of the algorithm is to count hot andcold spots in CMB sky maps on various scales and tem-perature contrasts. A typical application will be to plotspot abundances against the threshold on the temperaturecontrast ∆ T for a specific angular scale. This applicationdirectly imposes several features the algorithm should have:(i) It must define sectors on the sphere of equal surfacearea (for some desired scale). Their mean tempera-ture contrasts will decide whether they are countedas spots.(ii) The areas must be chosen such that one can smoothlyscan through the map. Between two distinct areas,there must exist many others allowing for a smoothtransition.(iii) Double counting of spots has to be excluded. Theeasiest way to achieve this is working with top hatwindows which have clear boundaries. Overlappingspots will be counted as a single.(iv) For a statistically satisfactory comparison betweenobserved and simulated CMB maps, the algorithmwill have to analyze many sky maps. Given the hugeamount of data, one has to implement the algorithmcarefully in order to make this numerically tractable .The algorithm is designed such that it allows for an ap-proximate pixelization of the sphere into distinct areas ofa given scale. Calculating their temperature contrasts de-termines the mean temperature fluctuation ∆ T rms on thatscale. By virtue of the ergodic theorem, this is a goodestimate for the ensemble average introduced in Sec. II.
1. Working principle
The first task is to define the sectors S of equal surfacearea on the sphere satisfying the requirements explainedabove. We choose them to be intersections of latitude andlongitude rings. A latitude ring R lat consists of all pointsbetween two latitude angles θ and θ , a longitude ring R lon of all points between two longitude angles ϕ and ϕ . The rings have two nice properties. First, as neededfor spot searching, one can smoothly go from one ring toany other ring by smoothly changing its boundary angles;second, as needed for calculating ∆ T rms , it is an easy taskto discretize a sphere into distinct rings. Since sectors areintersections S = R lat ∩ R lon , they share these properties.We thereby satisfy the requirement of smooth scanning toall directions.We impose [meeting the requirement (i) above] equalarea A for all sectors: A = Z S dΩ = Z ϕ ϕ d ϕ Z θ θ d θ sin θ. (12)Once we have decided to define sectors like this, we stillhave some freedom to choose the boundaries θ , θ , ϕ , ϕ . In order to avoid the influence of small scales, wemust reasonably choose the sectors such that they are by no means degenerated. We therefore fix this freedom byadding another constraint. For any sector S , the boundarylines in the north-south direction and the longer boundaryin the east-west direction are chosen to be of equal length:( ϕ − ϕ ) sin θ ∗ = ( θ − θ ) . (13)On the northern hemisphere θ ∗ = θ , on the southern hemi-sphere θ ∗ = θ . Note that these sectors behave well. In thelimiting case near the equator, they correspond to squaresin flat space. At the poles ( θ = 0 or θ = π ), they becomeequilateral triangles.In practice though, the temperature field is not givenas a smooth function of θ and ϕ . The WMAP tempera-ture sky maps are lists assigning a temperature contrast∆ T i to each HEALPix pixel p i . The mapping p i ( θ, ϕ )is given in the form of a table. But since our approachdefines sectors by means of angles, we need the reverse.Given the list p i ( θ, ϕ ), finding the appropriate pixel p i for given angles ( θ, ϕ ) corresponds to searching throughthe list. Whereas searching in an unsorted list is very ex-pensive, an adequate sorting may considerably reduce theeffort. The algorithm performs the following steps startingat the north pole θ = 0:1. For given θ and area A , calculate θ and ∆ ϕ bysolving Eqs. (12) and (13).2. Collect the pixels { p i } belonging to the latitude ring R lat between θ and θ . This can be done efficientlyif the map was prepared by transforming to sortedlatitude angles (HEALPix ring ordering).3. Using a fast routine, sort the list { p i } with respectto longitude angles. This new sorting allows one todirectly identify the pixels out of { p i } belonging toa longitude ring R lon with boundaries ϕ and ϕ —these pixels form the sector S = R lat ∩ R lon . Startat ϕ = 0 and ϕ = ∆ ϕ and smoothly scan (byincreasing ϕ , ϕ by a small step size h ) throughall longitude rings. For every sector, calculate thesector’s mean temperature contrast ∆ T by averagingover the pixel values ∆ T i and compare it with thethreshold ∆ T . If it exceeds the threshold, count aspot if the sector does not overlap with a previouslyfound spot.4. Choose the next ring by slightly increasing θ θ + h . It is profitable to exploit the fact that thesorting for longitude angles (point 3) need not be re-peated completely. The algorithm saves the previoussorting and uses it for a presort such that as muchinformation is transferred as possible.Having increased the threshold ∆ T , again searching forspot abundances in a map can be optimized by noticingthat spots at a higher threshold cannot be found wherethere were not spots at a lower threshold. Our algorithmcan focus on areas around previously found spots once thisbecomes advantageous.If we slightly adapt the algorithm, we can use it to mea-sure ∆ T rms . Now, the algorithm jumps between distinctsectors instead of smoothly transforming them. The dis-tinct sectors are visualized in Fig. 4. In every distinct sec- FIG. 4 Exemplary decomposition of the sky into N sec distinctsectors S j for measuring ∆ T rms . For searching spots, the algo-rithm analyzes many more sectors S (those in between, sharingpixels with the illustrated sectors S j ). Nonetheless, N sec limitsthe maximum number of spots since overlapping spots are notmultiply counted. tor, the mean temperature fluctuation is calculated. Thesquares are averaged to give ∆ T rms . Although the shapesof the sectors vary, the results of Sec. II ensure that ∆ T rms is only marginally affected.
2. Treatment of masked maps
The sectors defined by our algorithm may include none,some, or many masked pixels. We must define selectionrules determining which sectors are to be included in thestatistics. We used the following two rules. The most re-strictive choice is to only consider sectors with no maskoverlap ( strict selection for short). These sectors will onlycontain reliable pixels. But since especially on large scales,only a minority of sectors will belong to this group, badstatistics are the price to pay. The alternative choice is toalso consider sectors with a slight mask overlap ( tolerantselection ). This is a compromise between good statisticson the one hand and reliable results on the other. Wetypically allow for 5% masked area within a sector whichguarantees that usually the majority of sectors fall into thisgroup. In any case, we emphasize that masked pixels, evenif included in the statistics, are assigned zero temperaturefluctuation. This will avoid misinterpreting foregroundsas a CMB signal. Note however, that the pixels of zerotemperature fluctuation reduce ∆ T rms . For comparisonsbetween observed maps and Gaussian simulations, we em-ploy the tolerant selection for the sake of better statistics;the comparison is still trustworthy.
3. Alternative shapes
The algorithm works with the shapes defined inSec. III.B.1 and illustrated in Fig. 4. But we can easilytreat other shapes by embedding them into the previoussectors. This corresponds to a multiplication of the pre-vious window function W with the window function W of the desired shape where W must be large enough toensure W ≡ W is non-zero. The condition (12) of equal area now concerns the new shape and reads Z ϕ ϕ d ϕ Z θ θ d θ sin θ W ( θ, ϕ ) = A. (14)As an example, we compare the standard shape with tophat circles of equal area [cf. Eq. (9)] and plot the resultin Fig. 5. For low thresholds, the abundances are system- PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f s p o t s FIG. 5 Mean spot abundances in 100 simulated ΛCDM full-skymaps showing the results for different window functions of scale a = √ A = 6 ◦ . atically higher for the standard window function. This isdue to the fact that circles do not exhaust the area with-out space in between. The effect becomes important wheremany spots are found and overlapping is frequent but dis-appears for large thresholds where the results agree.
4. Step size dependence
In the ideal case, the boundary angles of the sectorswould vary in a perfectly smooth manner when searchingfor spots in a map. But numerically, we have to choosea finite step size h (introduced in Sec. III.B.1). A goodchoice of h balances sensitivity and numerical effort. Fig-ure 6 shows detected spot abundances against h in simu-lated maps. We chose h = 0 . ◦ for which we conclude thatour sensitivity to detect spots is satisfactory. C. Errors and cosmic variance
There are statistical uncertainties simply due to the fi-nite number of Gaussian simulations. Moreover, the CMBsignal itself can be regarded as the outcome of a statisti-cal process. It is therefore subject to statistical variation,quantified by the concept of cosmic variance.Let us assume that N Gaussian maps are analyzed forspots (area and threshold fixed). If n ( k ) spots are detectedin map k , the mean spot abundance is¯ n = N X k =1 n ( k ) N . (15)
45 50 55 60 65 70 0.1 1
PSfrag replacements
Step size h [ ◦ ] A bund a n c e s o f s p o t s FIG. 6 Mean spot abundances for a fixed threshold (80 µ K)against a varying step size. 100 masked ΛCDM simulated mapswere scanned for a = 6 ◦ , the error bars quantify the statisticalerror. The statistical uncertainty of the mean value ¯ n and thestatistical deviation of the single values ¯ n ( k ) are σ n = P Nk =1 ( n ( k ) − ¯ n ) N ( N − , σ n ( k ) = N σ n . (16)The same procedure applies if we measure the mean tem-perature fluctuations ∆ T ( k ) rms in the maps and calculate amean value ∆ ¯ T ( k ) rms .We now consider cosmic variance. When we observe aspot abundance n , we must expect a certain deviation fromthe theoretically predicted ensemble average h n i . The ex-pectation value of this deviation, σ n = (cid:10) ( n − h n i ) (cid:11) , quan-tifies cosmic variance. For a very large number N of simu-lated maps, we may replace the ensemble average by anaveraging over the set of simulations. We then obtain σ n ≈ σ n ( k ) with the latter calculated according to Eq. (16).This can be done equally for the mean temperature con-trast ∆ T rms . Whenever we specify cosmic variance (e.g.,in the form of error bars), we estimated it by this method. IV. RESULTS
The application of the spot-searching algorithm de-scribed in Sec. III shows that the standard model ΛCDMtogether with Gaussianity predicts more large-scale hotand cold spots than are actually present in cut-skyWMAP-5 data (see Sec. IV.A). Removing the quadrupoleor using the original WMAP-5 C ℓ (instead of the ΛCDMfit) considerably reduces the discrepancies. While only0 . .
62% of Gaussian ΛCDM simulations fall below theobserved mean temperature fluctuations on angular scalesof 4 ◦ –8 ◦ , this increases to 2 . A. Cut-sky maps
The spots’ size is characterized by their area A . Weuse the parameter a = √ A to specify the angular scale ofthis size. Since on the one hand, we aim at large scales,and on the other hand, we want reasonable statistics, weare forced to find a compromise. We chose an angularscale a = 6 ◦ . The spot abundances of the WMAP-5 V map and 500 Gaussian ΛCDM simulations (created as de-scribed in Sec. III.A) are found for varying threshold ∆ T .The HEALPix resolution parameter of the maps is 8, corre-sponding to N pix = 12 × = 786 ,
432 pixels. Statisticaluncertainties and cosmic variance are displayed as errorbars even though the spot abundances for different thresh-olds are of course correlated. The results for hot and coldspots are plotted in Fig. 7. The striking feature of the plots
PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f h o t s p o t s PSfrag replacementsThreshold ∆ T [ µ K]Abundances of hot spots
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s FIG. 7 Spot abundances in the CMB sky (with cosmic vari-ance) as compared to 500 ΛCDM simulations (with statisticalerrors) on an angular scale of a = 6 ◦ . The fractions of Gaussiansimulations with smaller values of s [Eq. (17)] are p hot s = 0 . p cold s = 1 . is the discrepancy between theory and observation. Theyonly agree in the limit of very small thresholds ∆ T whereit is obvious that almost every area is counted as a spotanyway. The discrepancy is seemingly more drastic for hotspots. In the plot for cold spots, it is seen that there is oneconsiderable cold spot nearly reaching 150 µ K. But eventhis spot does not surpass the ΛCDM prediction. We notethat this spot is localized in the region of the famous Vielvacold spot (Vielva et al., 2004). It is insightful to look at thespot distributions of single Gaussian simulations in order toget an impression of their typical behavior. Five examplesare plotted in Fig. 8.
PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f h o t s p o t s FIG. 8 Spot abundances in five randomly chosen Gaussian sim-ulations based on the ΛCDM best-fit power spectrum and themean curve from Fig. 7 (hot spots).
Because of the strong correlation between the spot abun-dances n i for different thresholds ∆ T i , it is difficult to judgethe significance of the discrepancies by eye. A possiblequantity that can be used for a comparison of the observedCMB map with Gaussian simulations is obtained by sum-ming up the spot abundances at different thresholds, s = X i n i , (17)where the lowest threshold included is chosen to be thecharacteristic scale ∆ ¯ T rms . We denote the fraction ofGaussian simulations k with s ( k ) smaller than found in the V map by p s . For the spot abundances shown in Fig. 7,we find p hot s = 0 .
2% for hot spots and p cold s = 1 .
8% for coldspots.The discrepancies are reflected in the mean temperaturefluctuation ∆ T rms which on large scales is higher in ΛCDMsimulations than in the observed CMB sky. We have simu-lated 5000 ΛCDM maps and compared their mean temper-ature fluctuations to the value of the V map. We employedthe tolerant selection of our algorithm (see Sec. III.B.2).For a = 6 ◦ , we find the mean value ∆ T rms = 39 . µ Kfor the V map, as compared to the mean value ∆ ¯ T rms =47 . ± . µ K for ΛCDM, where the error is only statisticalwhile cosmic variance amounts to 4 . µ K. Only a fraction p = 0 .
6% of the simulations had a smaller ∆ T rms than the V map. This does not improve at other large angular scaleswhich can be seen in Table I. It is interesting to see howthis behavior changes when going to smaller scales. How-ever, the results on smaller scales (approaching 1 ◦ ) becomesensitive to noise and beam properties. Since the WMAPteam offers the latter for the differencing assemblies V V V map,it is the easiest to switch to the V T rms against the scale a for the V TABLE I The fraction p of Gaussian ΛCDM simulations witha ∆ T rms smaller than found in the V map on the angular scale a . Scale a Fraction p ◦ . ◦ . ◦ . ◦ . ◦ . ΛCDM simulations (again with tolerant selection). We see
30 35 40 45 50 55 60 65 70 75 80 1 2 3 4 5 6 7 8V1 mapcosmic variancesimulations (LCDM Cl)statistical error
PSfrag replacements
Angular scale a [ ◦ ] ∆ T r m s [ µ K ] FIG. 9 The mean temperature fluctuation for different angularscales a in 50 Gaussian ΛCDM simulations and the V that the deviations decrease when going to smaller scales.This is also suggested by the C ℓ spectrum which is in goodagreement with the ΛCDM fit for large ℓ which dominateon small scales. But still, Monteserin et al. (2008) find atoo low CMB variance which approximately correspondsto ∆ T rms on scales even smaller than 1 ◦ .For the results in Fig. 9, we used the highest availableHEALPix resolution 9 corresponding to N pix = 12 × =3 , ,
728 pixels in a map. As stated above, the plots arehighly influenced by the beam function and noise. Thebeam function acts as an extra window function whichsuppresses the growth of ∆ T rms for small scales. Thewhite noise instead leads to a diverging 1 /a behavior onthe smallest scales (with an effective pixel noise ampli-tude σ pix and the number of pixels N a = N pix × a / π within a sector of scale a , the noise contribution will be∆ T noise rms = σ pix / √ N a ∝ /a ).On large scales, the first multipoles of the C ℓ spectrumplay an important role (see, e.g., Fig. 1). It is there-fore a natural idea to suspect the well-known quadrupoleanomaly (Hinshaw et al., 2007) to be responsible for theobserved discrepancies. We check this by repeating theanalysis after removing the quadrupole from the ΛCDMsimulations as well as the observed CMB map. The re-sults, summarized in Table II, confirm the influence of thequadrupole anomaly. Now, the fractions p of GaussianΛCDM simulations reach p = 7 .
3% for a = 6 ◦ . These TABLE II The fraction p of 1000 Gaussian ΛCDM simulationswith a ∆ T rms smaller than found in the V map on the angularscale a , after removing the quadrupole from the maps.Scale a Fraction p ◦ . ◦ . ◦ . ◦ . ◦ . numbers still do not show good agreement, but they arenot statistically significant anymore.We now investigate whether there are still discrepanciesif we compare the observed V map with Gaussian simula-tions based on the original WMAP-5 C ℓ spectrum ratherthan the ΛCDM best fit. This tests whether the observedmap is a typical Gaussian realization of the WMAP-5power spectrum. Figure 10 shows the spot abundances.The effect arising from changing the power spectrum is PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f h o t s p o t s PSfrag replacementsThreshold ∆ T [ µ K]Abundances of hot spots
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s FIG. 10 Spot abundances in the CMB sky (with cosmic vari-ance) as compared to 500 simulations (with statistical errors)based on the WMAP-5 C ℓ spectrum on an angular scale of a = 6 ◦ . The fractions of Gaussian simulations with smallervalues of s [Eq. (17)] are p hot s = 3 .
4% and p cold s = 13 . clearly visible and reduces the discrepancies to some ex-tent. But although closer to the spot abundances in the TABLE III The fraction p of 1000 Gaussian simulations(WMAP-5 C ℓ ) with a ∆ T rms smaller than found in the V mapon the angular scale a .Scale a Fraction p ◦ . ◦ . ◦ . ◦ . ◦ . observed cut-sky CMB map, the numbers of hot and coldspots are still too high. Again, this is reflected in the factthat most simulated maps have a larger ∆ T rms than the V map. Even though the values, listed in Table III, are lessdrastic, we emphasize that the WMAP-5 estimation of the C ℓ relies on similar data, i.e., cut-sky CMB maps. If theobserved CMB map was a typical Gaussian realization ofthe extracted C ℓ spectrum, we would expect agreement.Bearing in mind, however, that power spectra refer tothe full sky whereas we only look at regions outside themask, an explanation could be that the missing spots werelocated in the hidden part of the sky. In the next section,we investigate whether the WMAP-5 ILC map indicatesthis violation of isotropy. B. ILC full-sky map
The five-year ILC map is the best approximate full-skyCMB map available. We therefore analyze it even thoughthe quality of the reconstruction is not high enough to guar-antee robustness of the results (see, also, Sec. III.A). Weanalyze the ILC full-sky map and 100 Gaussian full-skysimulations based on the WMAP-5 power spectrum andseparately consider the results in three sky regions. First,we analyze the full sky. Second, we collect the spots ofthose regions that have also been studied in the V map, i.e.,regions with no or little overlap with the KQ75 mask (tol-erant selection). Finally, we consider the remaining spotsthat consequently lie in sectors completely inside the maskor with considerable mask overlap (rejected by tolerant se-lection). We loosely refer to the three regions as full sky , outside , and inside mask . The results are plotted in Fig. 11.In the previously analyzed region (outside the mask), wesee too few spots, as before. But there are by far too manyspots in the complementary region. The variances provid-ing the error bars are, as always, obtained from Eq. (16).Although there are less statistics inside the mask than inthe full sky, the error bars in the corresponding figure aresmaller. This can be intuitively understood as follows. If,for simplicity, we assumed that the spot abundances out-side and inside the mask were statistically independent, thevariances σ , σ would add to σ in the full sky, whence σ in < σ full . The loss of statistics when counting spots in-side the masked region only causes the relative fluctuationsbetween two Gaussian simulations to increase. The errorbars in the central figure (outside the mask) visually ap- PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s FIG. 11 Spot abundances in the ILC map (with cosmic vari-ance) compared to simulations (with statistical errors) based onthe measured WMAP-5 C ℓ on an angular scale of a = 6 ◦ forthree different parts of the sky. The corresponding values of p s are p full s = 58%, p out s = 7%, and p in s = 96%. pear larger due to the logarithmic plotting but are in factsmaller than for the full sky.The values of p s confirm the uneven distribution ofspots in the ILC map. For the full sky, we have p full s =58% in good agreement with the simulations. Outsidethe mask, there are too few spots, p out s = 7%, whereasinside the mask, we find p in s = 96%. The ILC mapis clearly anisotropic. Other authors draw the same conclusion (Bernui and Reboucas, 2009; Copi et al., 2009;Hajian, 2007).Anisotropy of the CMB is a possible explanation of thediscrepancies revealed in Sec. IV.A and quantified in Ta-ble I, and indeed, the ILC map contains this anisotropy.But since there is not enough reliable information aboutthe CMB signal in the galactic plane, we cannot finallyjudge whether this is the true solution to the problem. Wehave also studied if, additional to the galactic plane, theorientation of the galactic halo defines a preferred direc-tion. Therefore, we divided the ILC map into two halves,one around the galactic center and one covering the oppo-site direction. We have seen no signal of anisotropy in thisdirection. C. Modified power spectra
We have pointed out that anisotropy is a potential expla-nation. It is however unsatisfying to assume that so manyadditional spots lie in the contaminated regions hidden bythe KQ75 mask. This would be a surprising coincidence ofCMB signals and the orientation of the galactic plane. Al-ternatively, we may stick to statistical isotropy; then, ourresults may be due to some non-Gaussian signal.In this section, we investigate whether our results implynon-Gaussianity or statistical anisotropy by themselves.We do this by analyzing the effect of modifications to the C ℓ spectrum.So far, ∆ T rms has proved to be a good parameter toquantify the visible effects. We can perform a quick checkwhether our data supports the hypothesis that ∆ T rms isthe decisive parameter. Out of the 500 simulations withthe WMAP-5 power spectrum used in Sec. IV.A, we collectthose with a ∆ T rms smaller or equal than those found inthe V map. Figure 12 shows their spot abundances whichagree well with the V map.If there is a C ℓ spectrum that produces ∆ T rms valuessimilar to the ones found in the V map, our results alonedo not imply non-Gaussianity or statistical anisotropy. Inorder to keep the analysis as generic as possible, we donot use any specific cosmological model but only modifythe C ℓ of the original WMAP-5 spectrum. Figure 9 sug-gests that only large scales are affected which is why weconcentrate on a few low multipoles ℓ . Copi et al. (2009)found that the correlation function is essentially zero onangular scales above ≈ ◦ . Since this scale is roughlylinked to the multipole range ℓ ≤
3, our first modifica-tion simply consists in setting C ℓ ≡ ℓ ≤ C ℓ for ℓ ≤ T rms . The plots show the discrepancies betweenthe ΛCDM prediction, the WMAP-5 spectrum, and obser-vation. We also show the results for a combined powerspectrum, replacing the first 32 multipoles by the valuesquoted by WMAP-1 (Hinshaw et al., 2003). For this rangeof multipoles the WMAP analysis changed after the 1-yearrelease, following the suggestion of Efstathiou (2004). Thedifference between WMAP-5 and WMAP-1 may serve as0 PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f h o t s p o t s PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s FIG. 12 Spot abundances of Gaussian simulations k (errorsstatistical) with ∆ T ( k ) rms ≤ ∆ T V map rms in comparison to the V map(with cosmic variance). For these plots, we have p hot s = 68%and p cold s = 86%, showing agreement. Since we only considerGaussian simulations with ∆ T ( k ) rms smaller than in the V map,it is no surprise that the p s values lie above 50%. an illustration that the extraction of reliable C ℓ for low ℓ isa nontrivial task. Modifications I and II of the power spec-trum succeeded in reconciling Gaussian simulations andobserved CMB sky. This is confirmed by measuring thespot abundances in simulated maps based on the modifiedspectra, seen in Fig. 14. We conclude that our results arenot incompatible with Gaussianity. However, if we stick toGaussianity, they favor (although statistically not signifi-cant, cf. Table III) even lower values of the first multipolesthan currently estimated. V. CONCLUSIONS
The study of spot abundances has revealed discrepan-cies between the cut-sky CMB temperature maps and thestandard best-fit ΛCDM model or, but less significant, aGaussian spectrum for the C ℓ estimated by WMAP-5. Wehave shown in Sec. IV.C that a good parameter to quantifythem is the mean temperature fluctuation ∆ T rms which weinvestigated on large scales. On scales a between 4 ◦ and
30 40 50 60 70 80 2 3 4 5 6 7 8V1 mapLCDM ClWMAP-5 ClWMAP-1 Cl (l<33)modified Cl I
PSfrag replacements
Angular scale a [ ◦ ] ∆ T r m s [ µ K ]
30 40 50 60 70 80 2 3 4 5 6 7 8V1 mapLCDM ClWMAP-5 ClWMAP-1 Cl (l<33)modified Cl II
PSfrag replacements
Angular scale a [ ◦ ] ∆ T r m s [ µ K ] FIG. 13 The mean temperature fluctuation for large angularscales a . We compare the V T rms ), simulations based on the WMAP-5 C ℓ (a bitlower), the WMAP-1 C ℓ for ℓ <
33 (still lower), and on twomodified spectra. The first modification is created by setting C ℓ = 0 for ℓ ≤
3, the second by halving the C ℓ for ℓ ≤
5. Themodified spectra agree well with the V ◦ , only 0 .
16% to 0 .
62% of Gaussian simulations based onthe ΛCDM best-fit power spectrum fall below the ∆ T rms value of the observed V map. If this merely was an imprintof the anomalously low quadrupole, we would expect thediscrepancies to disappear when removing the quadrupolefrom the Gaussian simulations and the V map. The differ-ence in fact reduces, the aforementioned fractions changeto 2 .
5% to 8 . C ℓ , yielding 2 .
4% to5 . C ℓ themselves are estimated from the cut-sky CMBmaps (Nolta et al., 2009).Non-Gaussianity and also statistical anisotropy are pos-sible explanations. In our case, anisotropy means thatmany spots have to be hidden behind the masked re-gion. Unfortunately, this hypothesis can hardly be testedas there is currently no method to reliably extract theCMB signal in the highly foreground-contaminated regions.Nonetheless, we have employed the WMAP-5 ILC full-skyCMB map and found evidence for anisotropy in this map.This agrees with results obtained by Hajian (2007) and1 PSfrag replacements
Threshold ∆ T [ µ K] A bund a n c e s o f h o t s p o t s PSfrag replacementsThreshold ∆ T [ µ K]Abundances of hot spots
Threshold ∆ T [ µ K] A bund a n c e s o f h o t s p o t s PSfrag replacementsThreshold ∆ T [ µ K]Abundances of hot spotsThreshold ∆ T [ µ K]Abundances of hot spots
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s PSfrag replacementsThreshold ∆ T [ µ K]Abundances of hot spotsThreshold ∆ T [ µ K]Abundances of hot spotsThreshold ∆ T [ µ K]Abundances of cold spots
Threshold ∆ T [ µ K] A bund a n c e s o f c o l d s p o t s FIG. 14 Spot abundances in the CMB sky (with cosmic variance) as compared to 100 simulations based on the two modifiedspectra, respectively, (errors statistical) on an angular scale of a = 6 ◦ . The first modification yields p hot s = 30% and p cold s = 55%,the second modification p hot s = 37% and p cold s = 67%. Copi et al. (2009) who found that most of the power onthe largest scales comes from the (masked) galaxy region.Though possible, this unnatural alignment of the CMB sig-nal with the galactic plane would be intriguing and lacksso far any explanation.Our analysis of Sec. IV.C shows that our results forcut-sky maps do not suggest non-Gaussianity or statisticalanisotropy by themselves. They agree well with Gaussianfluctuations if one performs a modification of the lowestmultipoles. In doing so, no fine-tuning of the C ℓ is neces-sary in order to reconcile the spot abundances from Gaus-sian simulations and the observed CMB. It is sufficient tolower the first multipoles by a substantial amount. Whenstudying local extrema in the temperature field, Hou et al.(2009) similarly found discrepancies that disappeared whenexcluding the first multipoles. We recall, however, that the C ℓ and the assumption of Gaussianity completely fix theexpected spot abundances. If both the extraction of the C ℓ by WMAP-5 and our analysis of spot abundances are cor-rect, our results may indicate non-Gaussianity or statisticalanisotropy.If the discrepancies are not caused by mere statisti-cal coincidence or unknown secondary effects, we have toleave open the question whether we see the consequenceof non-Gaussianity or anisotropy, or whether our results strengthen the evidence for a severe lack of large-scalepower. The first case would challenge fundamental assump-tions, the second would make it difficult to understand theCMB maps on large scales within standard ΛCDM cosmol-ogy. If the discrepancies between the C ℓ , as determined byWMAP-5, and the spot abundances persist, this can beinterpreted as a signal for non-Gaussian fluctuations. Acknowledgements
We would like to thank Christian T. Byrnes for usefuldiscussions. We also thank the WMAP team for produc-ing great data products and publishing them on LAMBDA(the Legacy Archive for Microwave Background Data Anal-ysis). Support for LAMBDA is provided by the NASAOffice of Space Science. We acknowledge the use of theHEALPix package (Gorski et al., 2005) that we employedfor many tasks, most notably the creation and preparationof Gaussian simulations.
References
Bernui, A. and Reboucas, M. J. (2009). Searching for non-Gaussianity in the WMAP data.
Phys. Rev. , D79:063528, arXiv:0806.3758.Bernui, A., Villela, T., Wuensche, C. A., Leonardi, R., andFerreira, I. (2006). On the CMB large-scales angular cor-relations. Astron. Astrophys. , 454:409–414, arXiv:astro-ph/0601593.Cabella, P., Hansen, F., Marinucci, D., Pagano, D., and Vitto-rio, N. (2004). Search for non-Gaussianity in pixel, harmonicand wavelet space: compared and combined.
Phys. Rev. ,D69:063007, arXiv:astro-ph/0401307.Copi, C. J., Huterer, D., Schwarz, D. J., and Starkman, G. D.(2009). No large-angle correlations on the non-Galactic mi-crowave sky.
Mon. Not. Roy. Astron. Soc. , 399:295–303,arXiv:0808.3767.de Oliveira-Costa, A., Tegmark, M., Zaldarriaga, M., andHamilton, A. (2004). The significance of the largest scaleCMB fluctuations in WMAP.
Phys. Rev. , D69:063516,arXiv:astro-ph/0307282.Durrer, R. (2008).
The Cosmic Microwave Background . Cam-bridge University Press.Efstathiou, G. (2004). A Maximum Likelihood Analysis of theLow CMB Multipoles from WMAP.
Mon. Not. Roy. Astron.Soc. , 348:885, arXiv:astro-ph/0310207.Eriksen, H. K., Hansen, F. K., Banday, A. J., Gorski, K. M.,and Lilje, P. B. (2004). Asymmetries in the CMB anisotropyfield.
Astrophys. J. , 605:14–20, arXiv:astro-ph/0307507.Gold, B. et al. (2009). Five-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: Galactic Fore-ground Emission.
Astrophys. J. Suppl. , 180:265–282,arXiv:0803.0715.Gorski, K. M. et al. (2005). HEALPix – a Framework forHigh Resolution Discretization, and Fast Analysis of DataDistributed on the Sphere.
Astrophys. J. , 622:759–771,arXiv:astro-ph/0409513.Hajian, A. (2007). Analysis of the apparent lack of power inthe cosmic microwave background anisotropy at large angularscales. arXiv:astro-ph/0702723.Hansen, F. K., Banday, A. J., Gorski, K. M., Eriksen, H. K., andLilje, P. B. (2009). Power Asymmetry in Cosmic MicrowaveBackground Fluctuations from Full Sky to Sub-degree Scales:Is the Universe Isotropic?
Astrophys. J. , 704:1448–1458,arXiv:0812.3795.Hill, R. S. et al. (2009). Five-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: Beam Maps andWindow Functions.
Astrophys. J. Suppl. , 180:246–264, arXiv:0803.0570.Hinshaw, G. et al. (2003). First Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: Angular PowerSpectrum.
Astrophys. J. Suppl. , 148:135, arXiv:astro-ph/0302217.Hinshaw, G. et al. (2007). Three-year Wilkinson MicrowaveAnisotropy Probe (WMAP) observations: Temperature anal-ysis.
Astrophys. J. Suppl. , 170:288, arXiv:astro-ph/0603451.Hoftuft, J. et al. (2009). Increasing evidence for hemisphericalpower asymmetry in the five-year WMAP data.
Astrophys.J. , 699:985–989, arXiv:0903.1229.Hou, Z., Banday, A. J., and Gorski, K. M. (2009). The Hot andCold Spots in Five-Year WMAP Data. arXiv:0903.4446.Land, K. and Magueijo, J. (2005). The axis of evil.
Phys. Rev.Lett. , 95:071301, arXiv:astro-ph/0502237.Larson, D. L. and Wandelt, B. D. (2004). The Hot and ColdSpots in the WMAP Data are Not Hot and Cold Enough.
Astrophys. J. , 613:L85–L88, arXiv:astro-ph/0404037.Larson, D. L. and Wandelt, B. D. (2005). A Statistically Robust3-Sigma Detection of Non- Gaussianity in the WMAP DataUsing Hot and Cold Spots. arXiv:astro-ph/0505046.McEwen, J. D., Hobson, M. P., Lasenby, A. N., and Mort-lock, D. J. (2008). A high-significance detection of non-Gaussianity in the WMAP 5-year data using directionalspherical wavelets. arXiv:0803.2157.Monteserin, C. et al. (2008). A low CMB variance in theWMAP data.
Mon. Not. Roy. Astron. Soc. , 387:209–219,arXiv:0706.4289.Nolta, M. R. et al. (2009). Five-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: Angular PowerSpectra.
Astrophys. J. Suppl. , 180:296–305, arXiv:0803.0593.Page, L. et al. (2003). The Optical Design and Characterizationof the Microwave Anisotropy Probe.
Astrophys. J. , 585:566–586, arXiv:astro-ph/0301160.Vielva, P., Martinez-Gonzalez, E., Barreiro, R. B., Sanz, J. L.,and Cayon, L. (2004). Detection of non-Gaussianity in theWMAP 1-year data using spherical wavelets.