The error budget of the Dark Flow measurement
F. Atrio-Barandela, A. Kashlinsky, H. Ebeling, D. Kocevski, A. Edge
aa r X i v : . [ a s t r o - ph . C O ] J un The error budget of the Dark Flow measurement.
F. Atrio-Barandela , A. Kashlinsky , H. Ebeling , D. Kocevski A. Edge ABSTRACT
We analyze the uncertainties and possible systematics associated with the“Dark Flow” measurements using the cumulative Sunyaev-Zeldovich effect com-bined with all-sky catalogs of clusters of galaxies. Filtering of all-sky cosmicmicrowave background (CMB) maps is required to remove the intrinsic cosmo-logical signal down to the limit imposed by cosmic variance. Contributions tothe errors come from the remaining cosmological signal, that integrates downwith the number of clusters, and the instrumental noise, that scales with thenumber of pixels; the latter decreases with integration time and is subdomi-nant for the Wilkinson Microwave Anisotropy Probe 5-year data. It is provenboth analytically and numerically that the errors for 5-year WMAP data are ≃ p /N clusters µ K per dipole component. The relevant components of the bulkflow velocity are measured with a high statistical significance of up to & − . σ for the brighter cluster samples. We discuss different methods to compute errorbars and demonstrate that they have biases that would over predict the errors,as is the case in a recent reanalysis of our earlier results. If the signal is causedby systematic effects present in the data, such systematics must have a dipolepattern, correlate with cluster X-ray luminosity and be present only at clusterpositions. Only contributions from the Sunyaev-Zeldovich effect could providesuch contaminants via several potential effects. We discuss such candidates apartfrom the bulk-motion of the cluster samples and demonstrate that their contri-butions to our measurements are negligible. Application of our methods anddatabase to the upcoming PLANCK maps, with their large frequency coverage,and in particular, the 217GHz channel, will eliminate any such contributions anddetermine better the amplitude, coherence and scale of the flow. F´ısica Te´orica, Universidad de Salamanca, 37008 Salamanca, Spain; [email protected] SSAI and Observational Cosmology Laboratory, Code 665, Goddard Space Flight Center, Greenbelt MD20771 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822 Department of Physics, University of California at Davis, 1 Shields Avenue, Davis, CA 95616 Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK
1. Introduction.
Peculiar velocities are deviations from the uniform expansion of the Universe. In thegravitational instability model, they are generated by the inhomogeneities in the matterdistribution. Most determinations of the peculiar velocities are based on surveys of individualgalaxies. Early measurements by Rubin et al (1976) found peculiar flows of amplitudes ∼ ∼ h − Mpc were streaming at ∼ ∼ h − Mpc, a result that was in agreement witha later analysis by Willick (1999). Using the brightest galaxy as a distance indicator for asample of 119 rich clusters, Lauer & Postman (1994 - LP) measured a bulk flow of ∼ ∼ h − Mpc, but a re-analysis of these data by Hudson & Ebeling(1997) taking into account the correlation between the luminosity of brightest galaxy andthat of its host cluster found a reduced bulk flow pointing in a different direction. Using theFP relation for early-type galaxies in 56 clusters Hudson et al (1999) found a similar bulkflow as LP and on a comparable scale, but in a different direction. A sample of 24 SNIashowed no evidence of significant bulk flows out to ∼ h − Mpc (Riess et al 1997), anda similar conclusion was reached with a TF-based study of spiral galaxies by Courteau etal (2000). Kocevski & Ebeling (2006) analyzed the contribution to the peculiar velocity ofthe Local Group due to structures beyond the Great Attractor and found that the dipoleanisotropy of the all-sky, X-ray-selected cluster sample compiled there suggested that most ofthe flow was due to overdensities at & h − Mpc. Watkins et al (2009) developed a methodto suppress the sampling noise in the various galaxy surveys and showed that all the data(except for the LP sample) agreed with substantial motion on a scale of ≃ − h − Mpc.In a follow-up study Feldman et al (2009) estimated the source of the flow to be at an effectivedistance larger than 200 h − Mpc; they suggested that the absence of shear is consistent withthe attractor being at infinity, as proposed in Kashlinsky et al (2008, hereafter KABKE1).Cosmic Microwave Background (CMB) temperature fluctuations in the direction of clus-ters of galaxies provide an alternative method to measure peculiar velocities. The scatteringof the microwave photons by the hot X-ray emitting gas inside clusters induces secondaryanisotropies (Sunyaev & Zeldovich 1970, 1972) that are redshift independent and, if thenoise is isolated, can be used to probe the velocity field to much higher redshifts than withgalaxies. The Sunyaev-Zeldovich (SZ) effect has two components: thermal (tSZ), due tothermal motions of electrons in the potential wells of clusters and kinematic (kSZ) due tomotion of the cluster as a whole with respect to the isotropic CMB rest frame (see reviewby Birkinshaw 1999). However, such measurements for individual clusters are dominated 3 –by large errors. On all-sky CMB maps, the bulk flow motion of clusters of galaxies can beobtained by using large all-sky cluster samples and evaluating the CMB dipole at clusterlocations (Kashlinsky & Atrio-Barandela, 2000 - hereafter KAB). We have applied the KABmethod using the largest - at that time - sample of galaxy clusters in conjunction with the3-yr WMAP data and have uncovered a large scale flow of amplitude 600 − ≃ h − Mpc (Kashlinsky et al 2008, 2009a - KABKE1,2). That analysishas now been extended to a still larger and deeper sample of over 1,000 clusters and 5-yrWMAP data (Kashlinsky et al 2009b - hereafter KAEEK). The KAEEK analysis confirmsthe KABKE1,2 results and shows that the flow remains coherent and extends to at leasttwice the distance probed in KABKE1,2. A larger cluster sample enabled KAEEK to binthe signal by cluster X-ray luminosity ( L X ). The dipoles evaluated for binned subsamplesincrease systematically with increasing L X -threshold, as expected if the signal is producedfrom the kSZ effect by all clusters participating in the same motion, a correlation that wouldnot exist if the signal was produced by a rare excursion from noise or primary CMB.Upcoming data from both the long integration WMAP data and the PLANCK missionwill bring more accurate CMB maps. PLANCK data will be particularly important becauseof its wider frequency coverage, finer angular resolution and lower instrument noise. It isimperative to identify the prospects and limitations of the applications of the current KABmethodology to these future datasets. Also, alternative methods can test the existence of alarge scale flow (see Itoh, Yahada & Takata, 2009; Zhang, 2010).In this paper we present a detailed analysis of the uncertainties affecting the measuredbulk flow providing the necessary details to support the KAEEK results. In Sec. 2 we brieflysummarize our method followed by a summary of data and dipole analysis. In Sec. 3 wepresent a theoretical derivation of the error bars, showing when they become dominated bycosmic variance of the cosmological CMB residual that remains in the maps after filtering.It is demonstrated that for the CMB sky of our Universe and an isotropic all-sky clustercatalog the errors in the KABKE/KAEEK dipole measurements are ≃ p /N cl µ K perdipole component.
Given the filtering scheme adopted in our studies , these errors cannotbe reduced much in CMB data with lower instrument noise. Rather the strategy of furtherincreasing the signal-to-noise in the measured dipole value must be through increasing thenumber of clusters. Particularly important would be to increase the number of observedclusters at the bright end of the cluster luminosity function, where the much larger clusteroptical depth, τ , compensates for the decrease in the abundance of such clusters (KAEEK). InSec 4 we discuss different methods to estimate error bars describing their various biases. Sec.5 addresses the overall statistical significance of the measurement. Given that the measureddipole increases with the X-ray luminosity threshold, the signal found in KABKE1,2 andKAEEK cannot arise in primary CMB, but tSZ dipole contributions can potentially provide 4 –confusion to the measurement. Section 6 discusses such possible systematic effects due totSZ and we show there that all are of negligible amplitude and none could have generatedthe measured signal. Finally, in Sec. 7 we present our conclussions.
2. Methodology and Analysis.2.1. KA-B method
If a cluster at angular position ~y has the line-of-sight velocity v with respect to theCMB, the SZ CMB fluctuation at frequency ν at this position will be δ ν ( ~y ) = δ tSZ ( ~y ) G ( ν ) + δ kSZ ( ~y ) H ( ν ), with δ tSZ = τ T X /T e , ann and δ kSZ = − τ ( v p /c ) cos θ , being θ the angle between thecluster peculiar velocity ~v p and the line of sight. Here G ( ν ) ≃ − .
85 to − .
35 and H ( ν ) = 1if the thermodynamic CMB temperature is measured over the range of frequencies probedby the WMAP data, τ is the projected optical depth due to Compton scattering, T X is thecluster electron temperature and k B T e , ann =511 KeV. When averaged over many isotropicallydistributed clusters moving with a significant bulk flow with respect to the CMB, the kine-matic term generates a dipole contribution that could dominate, enabling a measurementthe bulk motion V bulk of the cluster sample. Thus, KAB suggested measuring the dipolecomponent of δ ν ( ~y ) at N cl cluster locations on the CMB sky.We denote by ( a , a x , a y , a z ) the monopole and three dipole components evaluated oversome locations in the sky and follow the same conventions as in KABKE1,2 and KAEEK: a = h ∆ T i and a i = h ∆ T n i i with i = ( x, y, z ) and n x,y,z = (sin θ cos φ, sin θ sin φ, cos θ )are the direction cosines of a vector with angular coordinates ( θ, φ ). Brackets representaverages taken over the cluster population of our catalog. The definition of monopole anddipole above follows the convention used in the Healpix remove dipole routine. The dipolepower is defined as C = P m =1 m = − | a m | , where a m are the three dipole components. Withour normalization, C , kin is such that a coherent motion at velocity V bulk would lead to C , kin = T h τ i V /c , where T CMB = 2 . p C , kin ≃ h τ i / − )( V bulk / / sec) µ K.When the dipole is computed at the position of N cl clusters, it will have contributionsfrom 1) the instrument noise, 2) the tSZ component, 3) the primary cosmological CMBfluctuation component from the last-scattering surface, and 4) the various foreground con-tributions at WMAP frequencies. The latter can be significant at the K and Ka WMAPchannels, so we restricted our analyses to the WMAP Channels Q, V and W which havenegligible foreground contributions. 5 –For N cl ≫ δ ν becomes a m ≃ a kSZ1 m + a tSZ1 m + a CMB1 m + σ noise √ N cl (1)Prior to any analysis, the CMB dipole due to our motion with respect to the isotropicCMB frame is removed from the data. The KSZ effect measures velocities with respect tothe CMB frame which also is taken to be the frame of the universal expansion. This doesnot change when all-sky dipole or any other ℓ -pole moments are subtracted in the all-skymaps. This dipole subtraction removes our peculiar velocity, v local , contributions down to O [( v local /c ) ] contributions to the quadrupole. To check that the latter does not contributeto the measurement, we also ran the pipeline subtracting the all-sky quadrupole from theoriginal maps and detected only negligible differences in the final results. As shown in KAB,in this way the kSZ term can be isolated in eq. 1. The process that enabled us to isolate the kSZ term is described in detail in KABKE1,2.Briefly:(A) An all-sky catalog of X-ray selected galaxy clusters was constructed using availableX-ray data extending to z ≃ . and quadrupole from the original maps and demonstrated that the quadrupole did not contributeto the results. This removes v local down to O [( v local /c ) ] contribution to the octupole.(C) The cosmological CMB component was removed from the WMAP data using aWiener-type filter, constructed using the ΛCDM model that best fit the data. It was con-structed in order to minimize the difference h ( δT − noise)) i . Next, filtered maps wereconstructed using all multipoles with ℓ ≥ ℓ ≤ remove dipole routine ascribing to each cluster a given circularaperture. Due to the variations of the Galactic absorbing column density and ROSATobserving strategy, cluster selection function and X-ray properties may vary across the skyintroducing possible systematics. In KABKE1,2 we used the measured X-ray extent ofeach cluster, θ X and computed the dipole for different apertures, in multiples of θ X and,to avoid being dominated by a few very extended nearby clusters like Coma, we introduceda cut so the final extent of any cluster was always smaller than 30’. There we computedcore radii directly from the data and from an L X − r c relation. Analyses using both setsgave consistent results, consistent with the X-ray systematic effects not affecting our resultssignificantly. More important, variations in the final aperture were already tiny in theKABKE1,2 analysis and KAEEK used altogether a fixed aperture were the mean monopolevanishes. The KAEEK results are consistent with the previous (KABKE1,2) measurements.Fixing the same aperture for all clusters simplifies the statistical analysis and this is theapproach taken in this article.(F) We compute the monopole and dipole for different angular apertures. At smallapertures ( ∼ ′ ), clusters show a clear tSZ decrement, but the amplitude of the signal fallsoff with increasing angular aperture. The final dipole is computed at the aperture wherethe mean monopole of the clusters vanishes. This ensures that the TSZ contribution to themeasured dipole is negligible and does not confuse the KSZ component.(G) Our final result is a dipole measured in units of thermodynamic CMB temperature.To translate the three measured dipoles into three velocity components, we need to determinethe average cluster optical depth to the CMB photons, h τ i , on the filtered maps. Sincefiltering reduces the intrinsic CMB contribution, it also modifies its optical depth, τ . InKABKE1,2 we introduced a calibration factor C , that gave the kSZ dipole in µ K of abulk motion of amplitude V bulk = 100 km/s . The calibration factor depends both on the filterand on the cluster profile. In KABKE1,2 and KAEEK it was estimated using a β model andthe angular X-ray extent of the cluster.We defer to Sec. 5 a discussion on the statistical significance of our measurements. Weemphasize that in the filtered maps we measure monopole and dipole simultaneously. Themonopole is dominated by the tSZ component and its amplitude sets an upper limit on a tSZ m (see eq. 1), the tSZ dipole due to an inhomogeneous cluster distribution on the sky. Wefound a dipole at cluster positions with a high confidence level and we obtained this dipoleat the (fixed) cluster aperture when the tSZ monopole component was zero . Since the tSZ 7 –component from the clusters vanishes, only a contribution from the kSZ component, due tolarge-scale bulk motion of the cluster sample, remains.The main current uncertainty in our method is the calibration, currently parameterizedwith the C , quantity, which generally is a matrix. At present, we do not have enoughinformation on the tSZ profile of the clusters in our catalog to increase the accuracy ofour calibration. Sec. 8 of KABKE2 discusses the issues and points out that we may be overestimating the velocity amplitude in the current cluster catalog by ∼ −
3. Noise and Intrinsic CMB Residual Contributions.
The KAB method to measure bulk flows using clusters of galaxies as tracers of thevelocity field requires the intrinsic CMB component to be removed from the data. To thisend we have designed a filter which minimizes h ( δT − noise) i . As we show below thisfilter removes the primary CMB anisotropies down to the fundamental limit imposed by thecosmic variance. In Fourier space this filter is expressed as F ℓ = | d ℓ | − C thℓ B ℓ | d ℓ | (2)where | d ℓ | = (2 ℓ + 1) − P m | a ℓm | is the power measured in each Differencing Assembly(DA) corrected for the mask sky area, and C thℓ B ℓ is the power spectrum of the theoreticalmodel that best fits the data, convolved with the antenna beam B ℓ of each DA. Although thisfilter removes much of the intrinsic primary CMB contributions, it leaves a residual CMBcomponent since the theoretical model does not reproduce perfectly the data measured atour location. This residual will be common to all frequencies and, since it is correlatedbetween the various DA’s, it limits the accuracy down to which the primary CMB can beremoved in the KAB method.Because of the cosmic variance, the power of the CMB sky at our location C LOCℓ differsfrom the theoretical model C thℓ and so a residual CMB signal from primary anisotropies isleft in the filtered maps. To estimate the contribution of noise and the CMB residual to thetotal power on these maps, let δT (ˆ n ) = P F ℓ a ℓm Y ℓm (ˆ n ) be the temperature anisotropy of 8 –the filtered maps expanded in spherical harmonics Y ℓm . The variance of any filtered map is: σ fil = 14 π X (2 ℓ + 1) F ℓ | d ℓ | = 14 π X (2 ℓ + 1) ( | d ℓ | − C thℓ B ℓ ) | d ℓ | . (3)As indicated, δT (ˆ n ) contains the cosmological CMB signal and noise, | d ℓ | = C LOCℓ B ℓ + N ℓ .The power spectrum at our location differs from the underlying power spectrum by a randomvariable of zero mean and (cosmic) variance ∆ ℓ = ( ℓ + ) C thℓ /f sky , where f sky is the fractionof the sky covered by the data (Abbot & Wise 1984). Then, due to cosmic variance, C LOCℓ = C thℓ ± ∆ / ℓ . The above limits on C ℓ bound the range of σ fil , eq. (3), to: σ fil = 14 π X (2 ℓ + 1) (cid:20) ∆ ℓ C thℓ + ∆ ℓ + N ℓ + N ℓ C thℓ + ∆ ℓ + N ℓ (cid:21) = σ CV,fil + σ N,fil ( t obs ) (4)In this last expression, the variance of the filtered map depends on two components: theresidual CMB left due to cosmic variance σ CV,fil and the noise σ N,fil , that is not removed bythe filter. The latter component integrates down with increasing observing time t obs as t − / and becomes progressively less important in WMAP data with longer integration time.We denote by σ q ≡ π (2 q + 1)(∆ q + N q )( C thq + ∆ q + N q ) − and let σ ( ℓ ) = P ℓq =4 σ q bethe cumulative variance of the residual map. With these definitions, the total variance of thefiltered map is σ fil = σ fil ( ℓ max ). For Healpix maps with N side = 512 the maximal multipoleis ℓ max = 1024 (Gorski et al 2005). In Figure 1 we plot this cumulative contribution of eachmultipole ℓ , σ fil ( ℓ ), to the total rms of the map. The solid lines represent the mean andrms σ fil ( ℓ ) of filtered maps of 4,000 realizations of the Q1 DA; the shaded area representsthe dispersion of those realizations, the dot-dashed line is the same quantity but for thefiltered Q1 WMAP 5-year data. The lower dashed lines represent σ CV,fil , the residual CMBcomponent, and upper dashed line, the total variance of the map [eq (4)]. The dot-dashedline also contains any contributions from foreground emissions; the fact that it lies so closeto the to the region expected from simulating CMB sky implies that foreground emissioncontributions to σ fil are small. Figure 1 clearly shows that for multipoles below ℓ ∼
200 thecumulative variance of the 5-year WMAP maps σ ( ℓ ) is dominated by the residual primaryCMB signal from the cosmic variance, even though the total variance of the filtered maps isdominated by noise. For the Q1 WMAP channel, the mean variance of our simulations was σ fil ∼ µ K) out of which ∼ µ K) come from the residual primary CMB signal.Finally, Figure 1 indicates that our filter removes the intrinsic CMB down to the fun-damental limit imposed by cosmic variance. In this sense the filter is close to optimal, sinceit minimizes the errors contributed to our measurements by primary CMB. In principle,one can define a more aggressive filter that, together with the intrinsic CMB, also removesthe noise leaving only the SZ signal. But filtering is not a unitary operation and does not 9 –preserve power. Such a filter would then remove an important fraction of the SZ componentand would probably reduce the overall S/N. In general, a different filter would give differentdipole (measured in units of temperature) and would require a different calibration. Discus-sion of filtering schemes that maximize the S/N ratio and minimize the systematic error onthe calibration will be given elsewhere.
4. Monopole and dipole uncertainties.
Here we consider how the two components present in the filtered maps, i.e. 1) residualprimary CMB and 2) instrument noise, contribute to the uncertainty in the measurementof bulk flows. In KABKE1,2 we adopted two methods to estimate the uncertainties: (I)evaluating monopole and dipole on the filtered maps outside cluster locations and (II) usingthe same cluster template on simulated maps. Both methods are different but complemen-tary. Errors estimated using method I include any contribution originated by foregroundresiduals and CMB masking while in Method II we account for the inhomogeneity of thecluster distribution on the sky.It is important to emphasize that the filtered maps have no intrinsic monopole or dipole by construction. Since we measure these two moments from a small fraction of the sky, ourlimited sampling generates an error due to (random) distribution of these quantities aroundtheir mean zero value. The sampling variances of h a i and h a i i are V ar ( h a i ) = h a i /N , V ar ( h σ i i ) = h a i i /N , where N is the number of independent data points. Direct computationshows that: σ ≡ h a i = h (∆ T ) i σ i ≡ h a i i = h (∆ T ) ih n i i , i = ( x, y, z ) (5)In this expression, n i are the direction cosines of clusters. If clusters were homogeneouslydistributed on the sky then h n i i = 1 / σ i = √ σ .Thus the error on the monopole serves as a consistency check in any such computation.Sec. 3 discussed the two components of the variance of the filtered map. As before, σ CV,fil and σ N,fil represent the contribution to the total variance due to the residual CMBcomponent and the noise, respectively. When we estimate error bars by placing randomclusters on the real filtered maps outside clusters (Method I) or the real clusters on simulatedfiltered data (Method II), N cl clusters occupy N pix in N DA Differencing Assemblies. As theresidual CMB signal is correlated from map to map, it will decrease only as the number ofclusters increases, but the noise term will decrease much faster since it is uncorrelated frommap to map and pixel to pixel and integrates down with N DA and the integration time. 10 –Then, the resulting sampling variance will be σ = σ CV,fil N cl + σ N,fil ( t obs ) N DA N pix , σ i = σ h n i i = 3 σ (6)As expected from eq. 5, for an homogeneous cluster catalog the variance in each dipolecomponent is three times larger than on the monopole since three quantities are derivedfrom the same data set. From Figure 1 we obtain that σ CV,fil ≃ µ K and σ N,fil ≃ µ K.When clusters are not homogeneously distributed in the sky, the basis of direction cosines isno longer orthogonal and error bars need to be estimated numerically.The results presented in Figure 1 together with eq. (6) indicate that Method II willgive slightly larger error bars. If monopole and dipole are evaluated at cluster positions on simulated maps then σ CV,fil and σ N,fil in eq. (6) will be close to the average CMB residualand noise of the simulated maps. As Figure 1 indicates, they are larger than the filtereddata (shown by a dot-dashed line) corresponding to the CMB realization representing ourUniverse. The latter, however, is the only CMB sky relevant for the true error analysis inthis measurement. This fact was already noticed in KABKE2 where such comparison wasmade and the errors were found to be 10-15% larger if using Method II.To avoid this bias, we introduce Method IIa: error bars are computed from randomrealizations of the power spectrum of the filtered maps . In Figure 2 we plot the histogramsof the monopole and dipole components of 4,000 simulations of 1000 clusters with constantangular size of 30 ′ with both Method I (random clusters located outside the mask on the realdata) and Method IIa (the cluster template is fixed and the sky is simulated; the spectrumare gaussian realizations of the measured power of the filtered maps). From left to rightwe display the histogram of the monopole and ( x, y, z ) components of the dipole. The rmsdeviations, given to the left and right of each plot, correspond to Method I and Method IIa,respectively. Solid lines represent the histograms in Method I and dashed lines in Method IIa.We find that, to good accuracy, the distribution of the monopole and dipoles is Gaussian withzero mean. More importantly, we see no systematic differences between both methods. Then,neither foreground residuals nor cluster inhomogeneities have a significant contribution tothe estimated error bars. Instead the errors are dominated by the sampling/cosmic variancewhen measuring the monopole and dipole from a limited fraction of the sky.To test the validity of eq. (6) we carried out another 4,000 simulations with differentnumber of clusters: N cl =100, 180, 320, 570 and 1000 in accordance with Table 1 of KAEEK.In Method I, we placed N cl clusters at random on the sky. To be fully consistent with howcluster samples are selected from the data, the smaller samples are subsets chosen randomlyfrom the full sample. In Figure 3 we plot the rms deviation of the monopole (open triangles)and the three dipole components. Filled circles, diamonds and solid triangles correspond to 11 –the ( x, y, z ) dipole components. Solid lines connect the results when clusters are assignedradii of 30 ′ , while dashed lines correspond to results with 20 ′ clusters and follow the sameordering as the solid lines. The figure shows that σ (0 ,x,y,z ) ∝ N / cl with great accuracy.As expected, the errors are larger when the cluster size is smaller because of the differentnumbers of pixels entering the instrument noise contribution in eq. (4). In Figure 3, thedifferences between the dipole components come from differences in sky coverage. The x and y components, that are in the plane of the Galaxy, are determined with progressively lessaccuracy since the CMB data in the Galactic plane is dominated by foreground emission.Still the difference from the uncertainty of the z -component is small ( . y -component of the dipole.We can use eq. (6) to estimate how accurately we measure any dipole component com-pared to the monopole. In Figure 4a we plot σ ( x,y,z ) /σ , the ratio of the rms deviation ofdipole components to the rms of the monopole of the 4,000 simulations generated usingMethod I, as described above. Filled circles, diamonds and triangles correspond to the ratioof the ( x, y, z ) rms deviation of the dipole to that of the monopole, respectively. In Figure 4bwe plot the same magnitudes for Method IIa. The dotted line represents σ ( x,y,z ) /σ = √ N / cl , as seen also in Figure 3. In Figure 4b the behavior is very similar: the erroron the x-component is largest. In this case, and since in Method IIa the cluster templateis fixed, the scaling is not as exactly ∝ N / cl , reflecting the inhomogeneities present in thecluster distribution. However, these deviations are not very significant.To study the effect of cluster inhomogeneities potentially present in studies based onother catalogs, we carry a different analysis. In Fig 4c we excise clusters from the KAEEKcatalog as a function of galactic latitude. We plot h n i i − / evaluated over the cluster dis-tribution when all clusters with | b | ≤ b cut are removed. Thick and thin solid and dashedlines correspond to the x, y, z components, respectively. The dotted line is h n i i − / = √ b cut . ◦ there is little deviation from theKAEEK errors. For much larger values of b cut , the error on the x and y components increaseswhile the error on the z component approaches that of the monopole. Then, eq (6) permitsto write the error bars as σ ( x,y,z ) = (1 . , . , . × p /N cl µ K, i.e., the expected accu-racy for each of the components would be only 12 and 5% worse that for an all sky survey,compared with that of the monopole, while the z component would be 12% better since theGalaxy removes the region of the sky where there is no contribution to it. 12 –Comparing the different panels in Figure 4, we see that σ y /σ may be smaller thanthe value estimated from the geometry of the catalog as is evident when comparing thisratio for N cl = 1000 in Figure 4b with Figure 4c. However, while in (c) the ratio of theerrors is computed from the cluster geometry, in (b) they are estimated from simulationswhereby in Method IIa we use simulations of the power spectrum of the filtered CMB data.Since monopole and dipole are sensitive to different parity multipoles (even vs odd), theslightly lower value of the dipole components with respect to the monopole is reflecting apower asymmetry between odd and even multipoles in the filtered map. So, on average themonopole is larger than in a random sky and the dipole is smaller. This effect introduces anextra variance and enhances the differences between Fig 4a and b.When this paper was being completed, Keisler (2009) replicated the analysis of KABKE1,2compiling his own X-ray cluster catalog using publicly available data. Analogously toKAEEK and this study, he noticed that the errors on WMAP 3-year data were alreadydominated by the residual CMB and not by the noise. He confirmed the measured centraldipole values of KABKE2, but claimed significantly larger errors than KAEEK, particularlyfor the y -component. (We note again that if the KABKE1,2 dipole originated from primaryCMB and/or noise, its magnitude should display no correlation with the cluster luminositythreshold that was demonstrated to exist in KAEEK). Specifically, in the final configurationhis catalog contained ∼
700 clusters and his claimed errors were σ Keisler x,y,z ≃ (1 . , . , . µ K.Those errors are larger than those quoted in KAEEK. A small increment (of order of 10-15%) can be accounted for by his treatment of the errors using simulations of the CMB skyaround the theoretical ΛCDM model and thereby pumping up the cosmic variance compo-nent (see Fig. 1), as well as anisotropies in his catalog. Keisler (2009) uses a catalog withoutrecomputing cluster properties from X-ray data, a procedure done in Kocevski & Ebeling(2006). That dataset is then less complete, especially at low latitudes, but that in itself canaccount only for a small increase in the errors. However, Keisler (2009) claims an increasein errors by a factor of > √
20 compared to KABKE2. Clearly, the effect of residual CMBcorrelations between the N DA = 8 WMAP channels can at most increase the KABKE1,2errors by a factor of √ N DA < √
8. (In reality, because the instrument noise is also present,the errors on individual dipole components in Table 2 of KABKE would be increased for3-year WMAP data by a factor of ≃ √ . µ K at the largest redshift bin.) Alarger increase, as we have demonstrated above, cannot happen. In our computations we donot reproduce Keisler (2009) errors with proper analytical and numerical procedures, evenusing his methodology.Interestingly, we recover the magnitude of his claimed errors if one important aspectof the KABKE processing is omitted. When working on simulated data such as in MethodII, care must be taken to replicate all the details of the data analysis done in KABKE1,2. 13 –The filter must be constructed using the theoretical model and the simulated data. Sinceonly modes with ℓ ≥ the fraction of the sky outside the mask does not . To test this effect we carry out twosets of 4,000 simulations S1 and S2, starting with the same initial seed and using MethodII. In S1 simulations we removed monopole and dipole outside the Galactic mask; in S2 wedid not. In Figure 5 we show the histograms with the distribution (from left to right) of themonopole and the (x,y,z) components of the dipole. The solid and dashed line shows theresults for the S1 and S2 simulations, respectively. The labels on the left (right) give the rmsdeviation for S1 (S2). The differences can be easily explained: if the monopole and dipoleare not subtracted the measured monopole and dipole at cluster locations are not differentfrom zero simply because we are sampling the signal over a very small fraction of the sky.Rather they are not zero because we are measuring the monopole and dipole present on thefraction of the sky outside the mask. We checked that when in the S1 simulations we addthe variance of the monopole/dipole subtracted outside the mask and the variance of themonopole/dipole computed at cluster positions, we obtain exactly the variance measuredin the S2 simulations. For instance, the variances on ( a , a x , a y , a z ) outside the mask are(0 . , . , . , . µ K) . The variances on the monopole/dipoles measured at the location of1000 clusters of our catalog are (0 . , . , . , . µ K) ; if added with the previous variances,the monopole/dipoles error bar increase by (46,36,41,5)%, respectively. As expected wesee that, since the z axis is perpendicular to the galactic plane, the error bars are boostedpreferentially in the x and y directions. This explains that in S2 simulations the error inthe monopole σ - as well as σ x,y - is larger than σ z . Only in the case of the faulty S2processing do we recover the magnitude of errors found by Keisler. We cannot claim thatthis step was necessarily overlooked by him but we do find this coincidence puzzling especiallywhen considering the deviation of his ratios of σ y /σ z and σ x /σ y from the (analytically)explained ratios (Fig. 4) and Fig. 1.
5. The Statistical Significance of the “Dark Flow”.
Because of the correlations in the final filtered maps between the eight WMAP DA’s,the S/N of the Dark Flow measurement is smaller than suggested in Table 2 of KABKE2for individual dipole components, although not fatally so. This was corrected in KAEEK,where it was also demonstrated that the dipole correlates strongly with the cluster X-rayluminosity L X , as it should if the dipole signal originated from the kSZ effect and not from 14 –the primary CMB. The minimal S/N of the dark flow measurement is, of course, given bythe single DA map processing. Fig. 8 of KABKE2, which plots the mean CMB temperaturedecrement over cluster pixels versus the cosine of the angle between the cluster and the apexof the motion, shows that KABKE1,2 already detect the dipole at cluster positions at the ≃ (2 . − σ level in each of the eight DAs. The overall S/N cannot then be lower than thisfloor level. KAEEK further increase this significance and measure the motion to a muchlarger scale. The systematic uncertainties in our calibration procedure do not yet allow us toquantify the properties of the flow better, but we hope to accomplish this task in the comingyears.In the KAEEK catalog, the error on the y component is only 5% larger than whatit would be for a homogeneous cluster catalog. Future versions of the catalog will includeclusters at higher redshifts that will help to probe the velocity field on even larger scales.A great effort is devoted to produce a spatially homogeneous and flux limited sample. Ifthe “Dark Flow” is but a large scale flow that affects all the scales out to the horizon,one could argue that the signal is uniform on the entire sky and would be unaffected byanisotropies on the cluster distribution in alternative catalogs, but this is not so, as Fig. 4indicates: incompleteness and asymmetries increase the error bars and could make somecluster catalogs insensitive to the flow.The original evidence in favor of the measurement being real were three (KABKE1,2): (a) the motion was found at cluster positions, (b) it was persistent when the number ofclusters increased from . to & , (c) the dipole kSZ signal was measured when thetSZ monopole vanished. Since the thermal and kinematic components are both generatedby the X-ray gas, it was thought that a measurement of the kSZ effect could be obtainedonly when enough frequency coverage allowed to remove the thermal contribution, becauseof their different frequency dependence. However, in Atrio-Barandela et al (2008 - hereafterAKKE) we showed - for the first time - that cluster gas distribution follows an NFW profile(Navarro, Frenk & White 1996). Then, cluster temperature falls with radius and, by addingthe contribution from the cluster outskirts, the kinematic component dominates over thethermal in the KAB method (KABKE2). If clusters were isothermal, the thermal SZ signaldipole due to the inhomogeneous distribution of the sky could be large enough to make thekSZ effect undetectable at WMAP frequencies.In KAEEK we provided further evidence in support of the cluster bulk flow being a realeffect. The cluster catalog used there was large enough to allow the analysis to be carriedout in luminosity bins. The kSZ signal is ∆ T ksz ∼ τ v B . Since τ is proportional to the clusterelectron density, it correlates with X-ray luminosity. If the velocity does not correlate withcluster luminosity, for example if the flow is homogeneous across the cluster sample, we 15 –would expect the dipole evaluated at different cluster subsamples to be larger for the moreluminous clusters. In KAEEK we were able to carry out such test by decomposing thesample in luminosity bins and the analysis conclusively showed that (d) the measured dipolecorrelates with X-ray luminosity , strengthening the evidence against a possible undiagnosedsystematic effect.In KAEEK it was shown that clusters with the highest luminosity dominate the S/Nof the measured flow. To quantify the level of statistical significance there, we generate10,000 dipole components drawn from a gaussian distribution with zero mean and rms themeasured error of each component as shown in Table 1 of KAEEK. The significance is thenthe percentage of simulated values that deviate from zero less than the measurement. Forinstance, when we consider the measured dipoles for L X ≥ × erg/sec clusters with z ≤ .
25 we measured ( a x , a y , a z ) = (3 . ± . , − . ± . , . ± . µ K. If the dipoles a i are Gaussian-distributed random variables, the amplitude of the flow for these clustersis detected at the 99.95% level consistent with our simulations (in Method I we find just 2realizations out of 4,000 with such parameters). For some other configurations in Table 1of KAEEK the confidence level would be even higher. Foreground contributions, by theirnon-Gaussian nature, can in principle alter the above percentiles, but the fact that ourUniverse lies so close to the lines in Fig. 1 generated from pure primary CMB, impliesthat foreground emissions contributions are small in our calculations. We do not necessarilyadvocate the above levels to be highly precise, but this discussion clearly shows that werecover a very statistically significant dipole. While the dipole components are less significantin lower L X -bins, presumably because of the lower τ ’s for these clusters, the a y component isalways negative and a z almost always positive in all three L X -bins, while the a x componentoscillates and is the least accurately measured component. In this case, the possibility that a x is zero can be rejected at more than 95% and a z , a z at the 99% confidence level. Dueto the changing sign, the measurements of the lower L X bins reduces the significance of thedetection of a x and we can not claim any measurement but the other two components arestill significant at more than 95%. Finally, these probabilities would become even higher ifone folds in the directional coincidence of the recovered dipole to that measured by Watkinset al (2009) from galaxy surveys data on smaller scales, .
100 Mpc.
6. Possible, but negligible, L X -dependent (SZ) systematics The only possible systematic effect that could mimic our measurements would have tobe present exclusively at cluster positions, produce zero monopole and also give a dipolewhich increases with increasing L X . Such systematics cannot come from primary CMB, and 16 –would have to originate from contributions by the SZ components, which depend on L X in the appropriate manner. Since in KAEEK we showed that the measured dipole corre-lates with the X-ray luminosity threshold, it is important to discuss possible L X -dependentcontributions even if only to rule them out because of their negligible magnitudes. Giventhat we evaluate the dipole at the aperture where the monopole vanishes, there are threeways that could potentially confuse the measurement: 1) Systematic effects that could foldthe Doppler-shifting due to the local motion into the tSZ contributions, 2) cross-talk effectsbetween the tSZ monopole and dipole terms in sparse/small samples (Watkins & Feldman1995); and 3) inner motions of the intracluster medium (ICM) as opposed to the coherentflow of the entire cluster sample.We discuss all three of these contributions below and demonstrate that they are negligi-ble. Before we go into the rest of the section, we emphasize again that the dipole at clusterpositions is measured at zero monopole. That monopole vanishes within the noise with 1- σ uncertainty of ≃ / √ N cl µ K or amplitudes significantly below 1 µ K for N cl & . The intrinsic CMB dipole due to the motion of the Sun is over two orders of magnitudelarger than the measured cluster dipole. This motion is known to be u ⊙ ≃ ± l, b ) = (264 , ), close to the direction (276 , ) of the Local Group withrespect to the same reference frame (Kogut et al, 1993) and is not far within the errorsfrom the direction measured in KAEEK: (290 ± , ± . An undiagnosed systematiceffect, present in the time ordered data or in our pipeline that affect preferentially the tSZsignal, could fold the motion of the Sun into our measurement. For example, a residualof the CMB all-sky dipole (∆ T ) res coupled to the thermal SZ effect would correlate with 17 –X-ray luminosity and would satisfy the same properties (a-d) as the kSZ effect, except itsfrequency dependence. The amplitude of such undiagnosed systematic dipole will be boundby (∆ T ) res < (∆ T ) tSZ ( u ⊙ /c ). In AKKE we showed that the tSZ amplitude of clusters inunfiltered maps is of the order of ∼ − µ K and this amplitude is reduced a factor of ∼ T ) res < − µ K, more than 2 orders of magnitude smaller than themeasured effect.
Since clusters are not randomly distributed on the sky, the tSZ signal will give riseto a non-trivial dipole signature that, in principle, may confuse the kSZ dipole. The tSZdipole for a random cluster distribution is given by a tSZ1 m ∼ h (∆ T ) tSZ i (3 /N cl ) − / decreasingwith increasing N cl . This decrease could be altered if clusters are not distributed randomlyand there may be some cross-talk between the monopole and dipole terms especially forsmall/sparse samples (Watkins & Feldman 1995). As discussed in KABKE2, the dipolefrom the tSZ component varies with the cluster sub-sample, contrary to measurements, andalso has negligible amplitude because it is bound from above by the remaining monopoleamplitude of h (∆ T ) tSZ i ≪ µ K measured at the final aperture (see Table 1 of KAEEK).In order to assess that there is no cross-talk between the remaining monopole and dipolewhich may confuse the measured kSZ dipole, we proceed in the same manner as in KABKE2(see Fig 6 there) repeating the following experiment: 1) The tSZ and kSZ components fromthe catalog clusters were modeled using cluster parameters derived for our current catalog.To exaggerate the effect of the cross-talk from the tSZ component, the latter was normalizedto h (∆ T ) tSZ i = − µ K, a value significantly larger than the monopoles in Table 1 of KAEEKat which the final dipole was measured; the results for even larger monopoles were alsocomputed and can be scaled as described below. For the kSZ component each cluster wasgiven a bulk velocity, V bulk , in the direction specified in Table 1 of KAEEK, whose amplitudevaried from 0 to 2,000 km/sec in 21 increments of 100 km/sec. The resultant CMB mapwas then filtered and the CMB dipole, a m (cat), over the cluster pixels computed for eachvalue of V bulk . 2) At the second stage we randomized cluster positions with ( l, b ) uniformlydistributed on celestial sphere over the full sky for a net of 500 realizations for each value of V bulk . This random catalog keeps the same cluster parameters, but the cluster distributionnow occupies the full sky (there is now no mask) and on average does not have the samelevels of anisotropy as the original catalog. We then assigned each cluster the same bulk flowand computed the resultant CMB dipole, a m (sim), for each realization. The final a m (sim) 18 –were averaged and their standard deviation evaluated.Fig. 6 shows the comparison between the two dipoles for each value of V bulk for themost sparse sub-samples from Table 1 of KAEEK. We also made the computations at tSZmonopole values still larger than above (see upper left panel for one such example). Theoverall contribution from the tSZ component to the dipole is ∝ h ∆ T tSZ i , so in the absenceof cross-talk effects the amplitude of the scatter in the simulated dipoles is made of twocomponents: 1) remaining tSZ ∝ h ∆ T tSZ i and 2) genuine kSZ dipole with amplitude ∝ V bulk to within the calibration. One can see that there is no significant offset in the CMB dipoleproduced by either the mask or the cluster true sky distribution. The two sets of dipolecoefficients are both linearly proportional to V bulk and to each other; in the absence of anybulk motion we recover to a good accuracy the small value of the tSZ dipole marked withfilled circles. As discussed in KABKE2, since the bulk flow motion is fixed in direction andthe cluster distribution is random, one expects the calibration parameterized by C , to bedifferent from one realization to the next, e.g. in some realizations certain clusters may bemore heavily concentrated in a plane perpendicular to the bulk flow motion and the measured C , would be smaller. In our case, the mean C , differs by .
10% suggesting that ourcatalog cluster distribution is close to the mean cluster distribution in the simulations. Thisdifference in the overall normalization would only affect our translation of the dipole in µ Kinto V bulk in km/sec, but we note again the systematic bias in the calibration resulting fromour current catalog modeling clusters as isothermal β -model systems rather than the NFWprofiles required by our observations (AKKE, KABKE2). We have no progress to reporton this issue beyond discussion in sec. 8 of KABKE2 and this paper does not address themeasured velocity amplitude stemming from calibration; this work is in progress and will beaddressed after the recalibration of our catalog has been successfully completed. The intra-cluster medium (ICM) may not be at rest in the cluster potential wells asa result of mergers during cluster formation process. In principle, our measurement andinterpretation then may be affected by turbulent motions that give rise to a kSZ effect thatwould be larger for the more massive clusters. However, since the motions are randomlyoriented with respect to the line of sight, they will not produce a significant effect. In order,to reach the value comparable to V bulk ∼ ,
000 km/sec, a typical cluster in our sub-sampleof N cl would need to have thermal motions of ∼ V bulk N / , over an order of magnitude largerthan the velocity dispersion of Coma-type clusters. Rather these motions will enter theoverall dispersion budget (noise, gravitational instability and this component) around the 19 –coherent bulk flow component shown in Fig. 2 of KAEEK.
7. Conclusions.
We have analyzed the statistical significance of the results presented in KAEEK. Wehave identified the main contributions to the error budget: noise and the residual CMBcontribution. While the instrument noise was important in WMAP 1-year data, it wasmuch less so in the 3- and 5-year data. With our filtering scheme, there remains a residualcontribution due to cosmic variance, which correlates at different frequencies and decreasesonly as the number of clusters increases. We have discussed methods to compute the errorsand presented analytical discussion to estimate the various contributions to the final errorbudget. Measuring dipoles with a fixed template over simulated skies increases the error barin two respects: clusters do not sample the sky homogeneously and different maps will havedifferent CMB residuals. Since the measured CMB sky in our Universe has less power thanthe average ΛCDM realization, this also can boost the errors, but by only ∼ y component the increment is about 5% compared to the ideal case.We have argued that a proper method to compute error bars would be to perform randomsimulations of the measured power of the filtered maps corresponding to the CMB sky ofour Universe. However, we found that the difference with taking random clusters outsidethe mask, but using real data, was insignificant.We have discussed the evidence supporting the existence of the Dark Flow. Indepen-dently, different groups using galaxies as tracers of the density or velocity field are showingthe amplitude and direction of the local flow that are consistent, albeit at a much smallerscale, with the Dark Flow motion (Kocevski & Ebeling 2006, Watkins et al 2009, Feldmanet al and 2009, Lavaux et al 2009). This analysis with the forthcoming PLANCK data willprovide an important consistency check. With a scanning strategy different from WMAPand with better frequency coverage, it will permit us to characterize still better any possibleundiagnosed systematic. The 217 GHz band with ∼ ′ resolution will be specially usefulsince it will allow to measure the kSZ signal from central parts of the clusters in our cataloguncontaminated by the thermal component.This work was supported by NASA ADP grants NNG04G089G and 09-ADP09-0050.FAB acknowledges financial support from the Spanish Ministerio de Educaci´on y Ciencia(grant FIS2009-07238) and the Junta de Castilla y Le´on (grant GR-234). 20 – REFERENCES
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Cumulative rms deviation as a function of multipole. Solid line and shaded area showmean and rms of 4,000 simulated Q1 filtered maps. Dashed lines represent the residual CMBcomponent of the filtered maps due to cosmic variance, computed using eq. (4) and the residualCMB plus the noise components. The dot-dashed line corresponds to the actual Q1 band of WMAP5-yr data.
Histograms of the distribution of monopoles and the three dipole components computedusing the filtered Q1 WMAP 5-yr map data. Solid, dashed lines correspond to Method I andMethod IIa of 4,000 simulations (see text), respectively. Also indicated is the rms dispersion (inmicro Kelvin) for Method I (left) and Method IIa (right). (a) Rms deviation of the monopole and three dipole components computed scaled by thenumber of clusters. Open triangles, circles, diamonds (blue) and solid triangles (red) correspond tothe monopole and (x,y,z) components of the dipole. Solid lines joint the symbols of clusters with30’ radius, while dashed lines follow the same ordering than solid lines but correspond to clusterswith 20’ radius.
Dipole to monopole error bar ratio. (a) (Black) circles, (blue) diamonds and (red)circles correspond the the ratio of the (x,y,z) component of the dipole to the monopole, respectively.Monopole and dipole were computed using Method I. (b) Same as (a) but monopole and dipolesare computed using Method IIa. (c) Ratio of the dipoles to monopole error bars for our clustercatalog. The horizontal axis, b cut indicates that clusters with | b | ≤ b cut are excised from the catalog.In all three plots, the dotted line represents the ratio for a perfectly isotropic cluster catalog. Histograms of 4,000 realizations of the CMB sky using Q1 DA parameters. Solid, dashedlines correspond S1, S2 simulations; in S1 (S2) the monopole and dipole outside the mask are (arenot) removed. The left, right rms dispersion corresponds to S1, S2, respectively. The figures aregiven in micro Kelvin.