Measuring Bulk Flow of Galaxy Clusters using Kinematic Sunyaev-Zel'dovich effect: Prediction for Planck
aa r X i v : . [ a s t r o - ph . C O ] J a n Measuring Bulk Flow of Galaxy Clusters using KinematicSunyaev-Zel’dovich effect: Prediction for Planck
D.S.Y. Mak, E. Pierpaoli
University of Southern California, Los Angeles, CA, 90089–0484
S.J. Osborne
Stanford University, 382 Via Pueblo, Varian Building, Stanford, CA 94305
ABSTRACT
We predict the performance of the
Planck satellite in determining the bulk flow through kineticSunyaev-Zeldovich (kSZ) measurements. As velocity tracers, we use ROSAT All-Sky Survey(RASS) clusters as well as expected cluster catalogs from the upcoming missions
Planck andeRosita (All-Sky Survey: EASS). We implement a semi-analytical approach to simulate realistic
Planck maps as well as
Planck and eRosita cluster catalogs. We adopt an unbiased kinetic SZfilter (UF) and matched filter (MF) to maximize the cluster kSZ signal to noise ratio. We findthat the use of
Planck
CMB maps in conjunction with the currently existing ROSAT clustersample improves current upper limits on the bulk flow determination by a factor ∼ ∼ Planck and to 24 km/s for EASS; whilethe systematic bias decreases from 44% for RASS, 5% for
Planck , to 0% for EASS. The 95%upper limit for the recovered bulk flow direction ∆ α ranges between 4 ◦ and 60 ◦ depending oncluster sample and adopted filter. The kSZ dipole determination is mainly limited by the effectsof thermal SZ (tSZ) emission in all cases but the one of EASS clusters analyzed with the unbiasedfilter. This fact makes the UF preferable to the MF when analyzing Planck maps.
Subject headings:
Cosmology: cosmic microwave background, observations, diffuse radiation
1. Introduction
Peculiar velocities, along with inhomogeneities,can be used to constrain cosmology. A coher-ent, large scale peculiar velocity, also called bulkflow, may originate from spatial inhomogeneitiesin the mass distribution of large scale structuresaround us. The standard inflationary model pre-dicts that the rms bulk velocity within a sphereof radius R decreases linearly with comoving dis-tance in the ΛCDM universe with V rms ( r > −
100 Mpc h − ) ≈
100 Mpc h − r ) km/s (Kashlin-sky & Jones 1991). Galaxy cluster peculiar ve-locity surveys at scales R ≤
60 Mpc h − gener-ally agree with theoretical predictions of the clus-ter bulk velocities. However, recent measurementsat larger scales ( R ≥
100 Mpc h − ) indicate thatthe bulk flow velocity is significantly larger thanthe ΛCDM prediction with statistical significance up to 3 σ (Kashlinsky et al. 2008; Feldman et al.2010).At scales R ≤
60 Mpc h − , an enhancementof the bulk flow with respect to the predictedΛCDM value in the local universe is attributed toa large scale void or overdensity at these depth.This is thought to be the cause of the LocalGroup (LG) motion with respect to the CMBrest frame, with velocity v = 627 ±
22 km/s to-wards l = 276 ◦ , b = +30 ◦ , in alignment with theCMB dipole (Kogut et al. 1993). The measuredvalue is within the cosmic variance limit. Dressleret al. (1987) and Lynden-Bell et al. (1988) iden-tified the Great Attractor (GA), a mass concen-tration of ∼ M ⊙ , as the origin of the flowat <
60 Mpc h − . On large scales where ΛCDMpredicts bulk flows with negligible amplitude, ob-servations show the contrary; Lauer & Postman(1994) found a strong bulk flow signature with1 = 561 ±
284 km/s towards l = 220 ◦ , b = − ◦ ( ± ◦ ) at a depth of 110 Mpc h − . Using a sam-ple of 119 Abell clusters within 150 Mpc h − theyfound that the flow originate from a mass con-centration beyond 100 Mpc h − . Similarly, Feld-man et al. (2010) found a bulk flow on scales of ∼
100 Mpc h − with v = 416 ±
78 km/s towards l = 282 ◦ ± ◦ , b = +6 ◦ ± ◦ , in disagreement withthe WMAP5 cosmological parameters at 99.5%confidence. The direction and scale of these bulkflows are shown in Figure 1.Several theories have been suggested to ex-plain the high bulk flow velocities at large scales.One explanation is that pre-inflationary fluctua-tions in scalar fields on superhorizon scales givesa titled universe (Turner 1991, Kashlinsky et al.1994, Mersini-Houghton & Holman 2009). In thispicture, matter slides from one side of our Hubblevolume to the other, producing an intrinsic CMBdipole anisotropy as seen in the matter rest frame.This inhomogeneity generates a bulk flow with cor-relation length of order the horizon size. Alter-natively, Wyman & Khoury (2010) have showedthat a strengthened gravitational attraction at latetimes can speed up structure formation and in-crease peculiar velocities. Using N-body simula-tions, they found an enhancement in large scale, R >
100 Mpc h − , bulk flow velocities of up to ∼
40% relative to the ΛCDM cosmology. A sim-ilar approach using modified gravity is discussedin Afshordi et al. (2009) and Khoury & Wyman(2009).Kashlinsky & Atrio-Barandela (2000) have pro-posed a method aimed at determining the largestscale bulk flows from galaxy cluster peculiar veloc-ities measured using the kinetic Sunyaev-Zeldovich(kSZ) effect. If many galaxy clusters are mov-ing with a coherent motion with respect to theCMB rest frame, the kinematic part of the SZ sig-nal acquires a dipole moment. Since the kSZ sig-nal is proportional to line of sight velocity, sucha measurement directly probes the bulk flow, freeof distance measurement errors. Several authorsattempted to measure the bulk flow using themeasured kSZ effect in
WMAP data. Kashlinskyet al. (2008) (hereafter KAKE, and later Kashlin-sky et al. 2010) first utilized this method, claim-ing a large-scale flow with v >
600 km/s out to ∼
575 Mpc h − , without sign of convergence tothe ΛCDM predicted value. However, by repeat-ing the same method Keisler (2009) did not de-tect a statistically significant bulk flow. Osborneet al. (2010) (hereafter OMCP), used filters con-structed to enhance the signal to noise of the ki- netic signal and found no significant velocity dipolein the WMAP 7 year data. More specifically, theyfound a 95% bulk flow upper limits of the orderof 4600 km/s in the direction of the KAKE Theyalso showed that the matched filter outperformsthe unbiased one when WMAP data are used,and demonstrated that CMB and instrument noisedominate the uncertainties.In this work, we apply the scheme of OMCPto study the capability of
Planck data and futurecluster surveys to measure bulk flows. The use ofPlanck maps is expected to produce improved re-sults with respect to the WMAP case because ofreduced instrument noise, wider frequency cover-age (ensuring better foregrounds’ subtraction) andincreased spatial resolution of this mission. In ad-dition,
Planck will also produce the first all-sky SZsurvey with a median redshift of z = 0 . Planck
Blue Book) and will ensure a better performancein separating the tSZ from the kSZ signal for anycluster sample considered.We investigate to what extent the use of Planckmaps, in combination with data for existingROSAT clusters, improves on bulk flow determi-nation from WMAP. We also assess the expectedperformances of bulk flow measurements for up-coming all–sky cluster catalogs, such as the onesderived from
Planck and eRosita satellites. Suchsamples are more abundant and extend to higherredshifts than the one in hand.Our goals are: (i) to determine the sensitivityof the cluster velocity dipole measurement withthe
Planck specifications and assess the nature ofthe uncertainty; (ii) to study which cluster surveycan best constrain the bulk flow; (iii) to study theperformance of the filters used in OMCP with the
Planck setup.This paper is organized as follows. The bulkflow velocity expected from the ΛCDM model iscalculated in section 2. In section 3, we briefly de-scribe the procedure we use to measure the bulkflow velocity. In sections 4 and 5 we give detailsof the SZ and X-ray cluster catalogs we use anddescribe the procedure we adopt to generate sim-ulated SZ maps. In section 6, we present the twofilters we use to reconstruct the kSZ signal fromthe CMB maps. In section 7, we describe the anal-ysis pipeline we use to measure and calibrate thecluster dipole. In section 8, we describe the sys-tematic effects that may contaminate our results.The results are presented in section 9, followed byour conclusions in section 10. Throughout this pa-per, we assume a ΛCDM cosmological model with2ig. 1.— The dipole direction in Galactic coordinates of current bulk flow measurements. (a) CMB b =48 . ± . ◦ , l = 263 . ± . ◦ (Jarosik et al. 2010); (b) Local Group b = 30 ± ◦ , l = 276 ± ◦ (Kogutet al. 1993); (c) KAKE b = 34 ◦ , l = 267 ◦ ; (d) Watkins et al. (2009) b = 8 ± ◦ , l = 287 ± ◦ ; (e) Lauer &Postman (1994) b = − ± ◦ , l = 220 ± ◦ ; (f) Hudson et al. (2004) b = 0 ± ◦ , l = 263 ± ◦ ; (f) Feldmanet al. (2010) b = 6 ± ◦ , l = 282 ± ◦ . The size of the colored region is proportional to the amplitude of themeasured bulk flow. The color code represents the convergence depth of the bulk flow, in units of Mpc h − .3 m = 0 .
3, Ω Λ = 0 . h = 0 . w = − σ = 0 .
2. Theoretical Aspects2.1. The Kinetic SZ effect as a probe ofvelocity
The SZ effect (Sunyaev & Zeldovich 1970) is asecondary CMB anisotropy caused by the scatter-ing of CMB photons by high energy electrons, suchas those inside the intra-cluster medium. The frac-tional change in temperature of the CMB photonsis the sum of two componets: δT = ∆ T kSZ +∆ T tSZ ,where the first term and second terms are the ki-netic (kSZ) and termal (tSZ) Sunyaev-Zeldovicheffect respectively. The kSZ is caused by theDoppler shifting of CMB photons due to the pe-culiar motion of the galaxy cluster with respect tothe CMB rest frame. The fractional temperaturechange is: (cid:18) ∆ TT CMB (cid:19) kSZ = − v p c τ (1)where v p is the line-of-sight peculiar velocity ofthe cluster, τ = σ T R n e dl is the optical depth ofthe cluster, n e is the electron density, and σ T isthe Thompson scattering cross section. For typ-ical galaxy clusters with τ ∼ − and v ∼ T kSZ ∼ µK at the location of a galaxy clus-ter. To date, the kSZ effect has not been mea-sured in individual galaxy cluster. Nevertheless, ifmany galaxy clusters are moving in the same direc-tion, their velocity field creates a dipolar patternin the CMB radiation at large scale and leads toa net dipole moment C ,ksz = T h τ i V /c (KAKE), where h τ i is the mean optical depth ofthe cluster sample.The thermal Sunyaev-Zeldovich (tSZ) effect isthe boosting in energy of CMB photons scatteredby hot electrons in the intra cluster medium andresults in a fractional temperature change of: (cid:18) ∆ TT CMB (cid:19) tSZ = yf ( x ) (2)where y = σ T R dl ( k B T e ) / ( m e c ) n e is the Comp-tonization parameter, T e is the electron tempera-ture of the cluster, f ( x ) = x ( e x + 1) / ( e x − − x = hν/k B T CMB . The minimum and maximumtemperature changes occur at ν = 143 GHz and ν = 353 GHz with a null at ν = 217 GHz. For typ-ical clusters, T e ∼
10 keV gives ∆ T tSZ ∼ µK . In individual clusters, the tSZ effect is typicallylarger than the kSZ effect:∆ T kSZ ∆ T tSZ ∼ . f ( x ) (cid:18) v p
300 km / s (cid:19) (cid:18) T e (cid:19) − (3)Nevertheless, the dipole component of the kSZ ef-fect may dominate over the statistical dipole com-ponent of the tSZ effect if a bulk flow is present.Furthermore, the different frequency dependenceof the two effects allows us to extract the kSZ sig-nal. Λ CDM model
In the linear theory of structure formation thepeculiar velocity field of galaxy clusters is relatedto the matter overdensity through the continuityequation: ~v k = if ( z ) δ k k ˆ k (4)where f ( z ) ≡ aD dD da = H ( z ) (cid:12)(cid:12)(cid:12) dD ( z ) dz (cid:12)(cid:12)(cid:12) , and D isthe growth factor (equation B2).At scales much larger than the size of a cluster,the peculiar velocity field is correlated within agiven region leading to a coherent motion, or bulkflow, of the objects inside the region. Assumingthe distribution of clusters is isotropic, the bulkflow velocity within a region of size R is given by: σ v ( z ) = f ( z ) Z dk P m ( k )2 π | W ( kR ) | (5)where W ( kR ) is the Fourier transform of the top-hat window function, R = R z c/H ( z ′ ) dz ′ is thecomoving radius and P m ( k ) is the present-day lin-ear matter power spectrum. The cosmology de-pendence is embedded in the matter power spec-trum P m ( k ) and its time evolution. In particularthe rms velocity, σ v , is proportional to the am-plitude of the matter power spectrum σ whichis known with an uncertainty <
10% (e.g. Larsonet al. 2010).We incorporate the effect of the different red-shift distributions of the clusters in the catalogswe use by weighting the bulk flow velocity by a se-lection function φ ( z ). The selection function takesinto account the fact that the survey is not com-plete at all of the redshifts we use. Then,4 v ( z ) = Z dk P m ( k )2 π | W ( kR ) | (cid:18)Z z φ ( z ′ ) f ( z ′ ) dVdz ′ dz ′ (cid:19) (6)where V ( z ) is the comoving volume, φ ( z ) is the co-moving number density or selection function. Wecalculate φ ( z ) from the halo mass function (de-scribed in appendix B.1) and cluster mass limitsfor each survey (section 4), φ ( z ) = ¯ n ( z ) = Z ∞ M min ( z ) dM dn ( M, z ) dM (7)where M min ( z ) is the limiting mass of object inthe cluster survey at redshift z. φ ( z ) is normalizedsuch that Z z φ ( z ′ )( dV ( z ′ ) /dz ′ d Ω) dz ′ = 1Figure 2 shows the rms bulk flow velocity pre-dicted by equation 6 for the three cluster samplesconsidered in this work. The effect of the selec-tion function is small that the velocities amongthe three cluster samples are no different than 1%at all redshifts. For comparison, the rms bulk flowcomputed with no selection function is also plot-ted. The sample variance of the velocity distri-bution is the largest source of uncertainty. Fora Gaussian density field the amplitude of the bulkflow has a Maxwellian distribution with the proba-bility of having a velocity lying between V and V + dV given by P ( V ) dV ∝ V exp( − . V /σ v ) dV .The 95% confidence limits on the measured bulkflow of amplitude V are then V / < V < . V .This is the shaded region in the plot.In addition to large scale correlated cluster ve-locities, each cluster has a peculiar velocity causedby matter inhomogeniety on cluster scale ( R ∼ − ). We include this components in oursimulations.
3. Methodology3.1. Assumptions and Definitions
In this section we define the different physicalprocesses contributing to the sky signal, as well asthe
Planck instrumentation.The observed dipole signal at clusters’ locationshas, in principle, several contributions: a m = a CMB1 m + a noise1 m + a tSZ1 m + a kSZ1 m + a point1 m + a Gal1 m (8) where a i1 m are the dipole coefficients of the CMB,instrument noise, thermal SZ, kinetic SZ, extra-galactic radio and infrared point sources, andgalactic signal. An appropriate temperature mapmask can be used to suppress the galactic com-ponents and fit for the cluster velocity dipoleoutside the galactic region. We therefore do notconsider the effect of galactic foregrounds in thiswork and apply a cut at | b | ≤ ◦ to the simu-lated Planck maps. The contribution from in-frared point sources is not well characterized atthe present time and no model is available to ac-curately describe their abundances within galaxyclusters. We therefore do not consider the infraredpoint source signal in this work, and only simulatethe first 4 terms in equation 8 as well as the radiopoint source term.Our goal is to separate a kSZ1 m from the othercomponents and calculate its amplitude in Planck maps using different cluster samples. The detectornoise levels and beam sizes we use are summarizedin Table 1. Although CMB observations by
Planck cover 9 frequencies from 30 GHz to 857 GHz, weonly consider the 6 channels between 44 GHz and353 GHz because of large potential foreground con-tamination outside this range.
The steps we take to estimate the cluster kineticSZ dipole velocity are:1. We simulate cluster catalogs using halomodel and cluster self similar scaling rela-tions. We do so for three different clustersamples from the ROSAT All-Sky Survey,
Planck , and eRosita All-Sky Survey.2. Full-sky maps of the simulated kinetic andthermal SZ are created in the 6
Planck fre-quency channels together with realizations ofthe CMB. These maps are convolved withGaussian beams and detector noise is addedusing the properties given in Table 1.3. The maps are filtered to enhance the kineticSZ signal while suppressing the CMB and in-strument noise terms and removing the ther-mal SZ signal.4. The filtered maps are used to calculate thedipole moment of the reconstructed kSZ sig-nal. The dipole is calculated at the clusterpositions using different redshift shells.5. Temperature dipoles (in units of Kelvin) areconverted into velocity dipoles (in units of5ig. 2.— The theoretical bulk flow velocity in ΛCDM cosmology, using selection functions for the threecluster surveys, smoothed over a top hat window function W ( kR ) with the comoving sphere centered onthe observer with radius R. Also plotted is the amplitude of the expected bulk flow with uniform selectionfunction. The shaded region is the uncertainty of the expected bulk flow from sample variance. Table 1Characteristics of the
Planck channels
Planck
Channel 1 2 3 4 5 6Center frequency ν (GHz) 44 70 100 143 217 353Resolution ∆ θ (FWHM) 26 . ′ . ′ . ′ . ′ . ′ . ′ σ N ( ∆ TT CMB , 10 − ) 1.1 2.2 0.6 1.0 1.6 4.9 Note.—
Characteristics of the
Planck LFI -receivers (column 2-3)and
HFI -bolometers (column 4-7): center frequency ν , angular res-olution ∆ θ in FWHM, and instrument noise variance per pixel σ N (thermodynamic temperature units).6m/s) by a calibration matrix M , which iscalculated from kSZ realizations with pre-defined bulk flow amplitudes, i.e. a V = Ma T , where a T and a V are the monopoleand dipole coefficients in temperature andvelocity units respectively.We iterate the above pipeline to study the prop-erties of the recovered bulk flows by (a) using dif-ferent filters; (b) varying the amplitude of the in-put bulk flow velocity; (c) using different clustercatalogs; (d) inputing different systematic compo-nents into the maps to determine their importance. The calibration of conversion from dipole am-plitude to flow velocity, through the calibrationmatrix M (section 7.2), requires cluster positionand optical depth. In this work, we assume op-tical depth is measured in the observed sampleswith negligible measurement errors and considerthe scatter in the Y − M relation for Planck andEASS clusters and L − T relation for RASS clusterswhen producing our simulations (appendix B.2).When dealing with clusters from observations, wehave to derive the optical depth of the cluster sam-ple to reconstruct the kinetic SZ signal. We brieflyoutline below how we can recover optical depthfrom SZ and X-ray clusters respectively.For SZ observation, Planck , the optical depthof each cluster is directly obtained from their SZflux, or equivalently the integrated compton Y-parameter, by τ = Y ( m e c ) / ( k B T e ) if the elec-tron temperature T e is known. The tempera-ture can be obtained from X-ray measurementsthrough the L − T scaling relations. For X-rayobservations, RASS and EASS, we can calculatethe optical depth from the electron density by τ = σ T R n e dl . We follow OMCP to determinethe cluster electron density (see equation 8 inOMCP) using the Bremsstrahlung emission lumi-nosity (Gronenschild & Mewe 1978) and the re-lation T X = (2 . ± . L . ± . (White et al.1997) to calculate the electron temperature of thecluster. For the RASS clusters, propagating theerror in observed L X to optical depth gives uncer-tainty of ∼ L X for the EASSclusters is likely to be smaller since the backgroundnoise for eRosita is expected to be lower. The un-certainty in the optical depth for RASS is thereforean upper limit for EASS clusters.
4. Cluster Catalogs
We construct cluster catalogs for three represen-tative all sky surveys: the combined ROSAT All-Sky Survey (hereafter RASS), the future
Planck cluster survey and eRosita All-Sky Survey (here-after EASS) . Each survey has its own advantages:the RASS sample already exists and has been usedfor previous bulk flow measurements (e.g. KAKEand OMCP ); the Planck cluster sample will con-tain about three–five times as many clusters, willextend to greater distances and,being an SZ sur-vey, will be subject to different selection effectsthan RASS; the EASS cluster sample will be largerthan both the ROSAT and
Planck samples andwill extend to greater redshift z ∼
1. All threesurveys provide isotropic samples outside regionsat low galactic latitudes. All surveys are flux-limited, and so at any redshift z only objects ofmass
M > M lim ( F lim ) are included in the catalog.A summary of the catalog properties is listed inTable 2.All surveys are flux-limited, and so at any red-shift z only objects of mass M > M lim ( F lim ) areincluded in the catalog. A summary of the catalogproperties is listed in Table 2. The eRosita mission is expected to be launchedin 2012 and perform the first all-sky X-ray imag-ing survey in the X-ray energy range up to 5keV with a limiting flux of F lim , . − = 1 . × − erg s − cm − . The EASS is expected toyield a catalog of a few tens of thousands of clus-ters out to redshift z ≈ .
3. Multi-band opticalsurveys are planned to provide photometric andspectroscopic redshifts for the EASS clusters withmasses above 3 . × M ⊙ h − (Cappelluti et al.2010). The second phase of the eRosita survey(the wide survey) will detect more clusters by us-ing longer exposures than the first phase (the allsky survey). However, it will only cover about halfof the sky resulting in a non-isotropic cluster sam-ple and so we only use clusters expected from allsky survey. We apply a galactic cut at | b | ≤ ◦ giving a sky fraction 0.727. The limiting mass ofcluster to be included in the sample at redshift zis given by (Fedeli et al. 2009): Planck ,and EASS catalogs. The cluster mass that we use in the catalogs is max[10 M ⊙ , M lim ( z )]. For the RASSsample, we fit equation 9 to the archived catalog and obtain the M lim values shown in this plot.Fig. 4.— The differential cluster redshift distribution per square degree for clusters in the Planck , EASSand RASS, with cluster mass
M > max[10 M ⊙ , M lim ( z )]. The ΛCDM cosmology is assumed and σ = 0 . Planck and EASS clusters. The observed number count of the RASSclusters are overlaid with the solid lines for comparison with the fitted number counts (dash dot line).8 lim , ( z )10 M ⊙ h − = 1 E ( z ) (cid:20) πd L ( z ) F lim , . − /c b . × ergs − (cid:21) / . (9)where E ( z ) ≡ H ( z ) /H is the normalized Hub-ble parameter, d L ( z ) is the luminosity distanceand c b is the band correction factor which convertsthe bolometric flux to the eRosita energy 0 . − c b by assuming a Raymond-Smith (Raymond & Smith 1977) plasma modelwith metalliticity of 0 . Z ⊙ , a cluster temperatureof 4 keV, and a Galactic absorption column den-sity of n H = 10 cm − . For consistency, we usethe virial mass definition throughout this work andwe convert M to M v (and write M v as M here-after) using the conversion fitting formula by Hu& Kravtsov (2003).To create mock catalogs, we assign a mass andredshift to each cluster in the EASS sample andsimulate the properties of the clusters using X-rayscaling relations (described in appendix B.2). Planck
Catalog
Planck is imaging the whole sky with an un-precedented combination of sensitivity (∆
T /T ∼ × − at 143 GHz), angular resolution (5 ′ at217 GHz), and frequency coverage (30 −
857 GHz).The SZ signal is expected to be measured froma few thousand galaxy clusters.
Planck will pro-duce a cluster sample with median redshift ∼ . | b | ≤ ◦ tominimize the Galactic signal. The SZ observ-able is the integrated Comptonization parameter Y = R y d Ω cluster (appendix B.2) with expectedvalues of Y > − arcmin (Malte Sch¨afer &Bartelmann 2007), where Y is the integratedcomptonization parameter within r . Fedeli et al.(2009) provide a fitting formula for the limitingmass as a function of redshift based on simulationsby Malte Sch¨afer & Bartelmann (2007): M lim , ( z )10 M ⊙ h − = (cid:26) e − . . − [( z − . . ] if z ≥ . e − . . z if z ≤ . (cid:27) (10) The RASS sample, consisting of clusters fromthe REFLEX (B¨ohringer et al. 2004), BCS (Ebel-ing et al. 2000b), eBCS (Ebeling et al. 2000a),CIZA (Ebeling et al. 2002; Kocevski et al. 2007), and MACS (Ebeling et al. 2001) catalogs com-prises a total of 827 clusters. All of the clustershave spectroscopic redshifts with z < .
5. We sim-ulate the SZ signal of the clusters in this catalogfrom the observed X-ray properties via scaling re-lations, while the redshifts and positions on thesky are taken from the catalog.Figure 3 shows the limiting mass for clusters tobe included in the RASS catalog. We fit equa-tion 9 to the 827 RASS clusters and find an effec-tive flux limit of F lim = 1 . × − ergs − cm − .In Figure 4 we show the cluster number counts dN/dz/d Ω, for our cluster catalogs.
5. Sky Simulations
We create full-sky SZ maps using a semi-analytical approach. We follow the method pre-sented in in Delabrouille et al. (2002) and Waiz-mann & Bartelmann (2009) that used the halomodel and cluster gas properties. The simulationsinclude SZ emission, primary CMB anisotropyand instrument noise, while diffuse Galactic fore-grounds are excluded.Clusters are simulated with properties expectedfrom the surveys described in section 4 with num-ber densities given by the halo mass function N ( z, M ). Each simulated cluster is then assigneda mass and redshift and the gas properties are thenderived. We leave the details of the number countsand the cluster models to appendix B and focus onthe preparation of the SZ maps here.In the nonrelativistic limit the distortion to theCMB temperature of the kinetic SZ effect is givenby equation 1 but with the peculiar velocity re-placed by ~v = ~V bulk + ~v peculiar . We give an over-all bulk velocity ~V bulk to the whole cluster sam-ple which is a free parameter of the simulation.Besides a bulk motion, we also give each clustera random peculiar velocity ~v peculiar drawn from aGaussian distribution with variance given by equa-tion 5 with R = 8 Mpc h − .Maps of CMB anisotropies are generated fromthe angular power spectrum using the CMBfastcode (Seljak & Zaldarriaga 1996) for a flat ΛCDMcosmology. The CMB and SZ maps are com-bined and then smoothed with a Gaussian beamwith sizes given in Table 1. Finally, the instru-ment noise is added with the noise variance andthe beam size of the Planck channels are listedin Table 1. Synthesized maps are generated in theHEALPix pixelization scheme (G´orski et al. 2005),with resolution of Nside=1024.9 able 2Parameters of cluster catalogs
Catalog RASS
Planck
EASSunits f . − Y f . − ergs − cm − arcmin ergs − cm − Flux limit 1 . × − − . × − z ∗ median h N cl i
827 2700 33000 h τ i (7 . ± . × − (8 . ± . × − (6 . ± . × − Note.— ∗ The median redshift is computed from the simulated clustersample with a cut-off redshift z=0.5 for all three cluster samples.
6. Filtering: Reconstruction of the KineticSZ signal
The kinetic SZ signal from a cluster is embeddedin the CMB and instrument noise, and tSZ emis-sion is also present at the same location. We there-fore filter the maps to increase the signal-to-noiseof our cluster dipole measurement. We use twotypes of linear multifrequency filters to reconstructthe cluster kinetic SZ signal. The filters are con-structed with the aim of minimizing the CMB andnoise variance in the map. We leave the derivationof the filter shapes to the appendix A. The first fil-ter is a matched filter (hereafter MF; Haehnelt &Tegmark 1996; Herranz et al. 2002) that is opti-mized to detect the kSZ signal. The second filter(hereafter UF) is subject to the additional con-straint of removing the tSZ signal at the clusterlocation. The UF was first proposed by Herranzet al. (2005) for use on flat patches of the sky andlater Sch¨afer et al. (2006) extended the scheme foruse on full sky maps. Both filters use all of the fre-quency channels and take into account the statis-tical correlation of the CMB and instrument noisebetween different frequencies.The filter kernels are:matched filter : Φ MF l = C l − γ B l (11)unbiased filter : Φ UF l = C − ∆ ( α B l − β F l ) (12)where α = P l max l =0 F T l C − F l , β = P l max l =0 B T l C − F l , γ = P l max l =0 B T l C − B l , and ∆ = αγ − β .We assume the clusters are point sources inthe Planck maps and so B l are the spherical har- monic coefficients of the Planck beam function, F l = f ( ν ) B l where f ( ν ) gives the tSZ frequencydependence, C l = C CMB l + C noise l is a matrixat every multipole giving the sum of the crosspower spectra of the CMB and instrument noisebetween frequency channels. The filters are nor-malized such that the filtered field gives the am-plitude of the kinetic SZ signal at the central pixelof each cluster. Given the spatial resolution of Planck we compute the filter kernels out to multi-pole l max = 3000. Figure 5 shows the MF and UFfor the 6 Planck frequency channels we use, usingthe
Planck beam FWHM and pixel noise variance.
The MF is less complicated than the UF sincemost of the features are only seen at the clusterscales (1000 ≤ l ≤ ν = 100, ν = 143, ν = 217 GHz are giventhe largest weights. The matched filter is moreefficient than the UF at removing the CMB andinstrument noise components because it forces thescales at which the noise and CMB dominant todisappear. However, since the MF is not designedto remove the thermal SZ signal, the amplitudesof the filter kernels are all positive. The UF is constructed to give an unbiased esti-mate of the kinetic SZ signal at the clusters’ loca-tions. At large and intermediate scales ( l ≤ ν = 100 GHz are subtractedfrom those above. The CMB fluctuations whichdominate the signal at these scales are therefore10ig. 5.— (Upper) Multi-frequency matched filtersigned to detect the kSZ signal. (Lower) Unbi-ased multi-frequency matched filters designed toremove the tSZ components . Both filters areconstructed using the Planck beam profiles as thecluster radial profiles. suppressed. At cluster scales (1000 ≤ l ≤ σ N ≈ µK ) and so the 100 GHzchannel also has substantial weight. At smallerscales ( l ≥
7. Analysis Pipeline7.1. Weighted Least Square Fitting ofDipole
We fit the real spherical harmonic coefficients ofthe monopole, a , and dipole terms, a x , a y and a z , to the filtered maps using a weighted leastsquare fit which is based on the Healpix IDL pro-cedure remove dipole (G´orski et al. 2005): a T = ( X T WX ) − X T Wu (13)where a T is a vector of best fit monopole anddipole coefficients, u is the filtered map (e.g.equation A2), W is a matrix with diagonal elements equal to theweight given to each pixel of the map, and X is amatrix giving the contribution of the fitting func-tion to each pixel. We give the central cluster pixelof the map a weight W i = 1 /σ N,i , where σ N,i is thei–th pixel noise variance, all other pixels are givenzero weight. The noise variances are calculatedfrom 100 filtered CMB and noise realizations. Wealso tried weighing each pixel by an estimate ofthe signal to noise, i.e. W i = τ i /σ N,i , but we finda larger systematic bias due to tSZ contamina-tion in the recovered bulk flow than when usingthe former weighting scheme. We therefore usea weighting scheme that involves only the pixelnoise.We fit the dipole only at central cluster pixelsbecause our filters are optimized to reconstruct thesource amplitude if the source were centered atthat pixel. µK to km/s We construct a 4 × a V = Ma T to con-vert the dipole from temperature units ( µK ) to ve-locity units (km/s), in which the matrix elementshaving units of km/s/K. This involves the use of11imulated kinetic SZ maps that can account forthe attenuation of the optical depth by the beamconvolution and the filtering process. In principleone can perform this unit conversion in observa-tions, e.g. Planck temperature sky maps, usingcalibrated SZ simulations based on estimation ofthe optical depth of each clusters. This is the casein OMCP that they used the simulated kSZ mapsof the RASS clusters to calibrate
WMAP data.On the other hand, in this work we assume theoptical depth, hence the kSZ signal, of each clus-ter is known and calculate the matrix M from oursky simulations. Note that the matrix elements de-pend implicitly on the average optical depth of thecluster sample, therefore it is specific to individualcluster experiment and we construct it separatelyfor the three cluster surveys. We now describe howwe create this matrix.We generate four different sets of kinetic SZ sig-nal only realizations with a given bulk flow ampli-tude: one with a monopole velocity and the otherthree with a dipole velocities in the x, y, z direc-tions. We choose an amplitude of of 10,000 km/s,which is large enough such that the recovered sig-nal is not confused by uncertainty of optical depthand random peculiar velocity. Since the kinetic SZsignal of each cluster is proportional to the opticaldepth, the elements of the matrix M depend im-plicitly on the average optical depth of the clustersample. Each set of maps are passed through thepipeline to obtain the a m . The four a m from eachset of realizations are the elements of each row of M − . By repeating this procedure for the four ki-netic SZ signal only realizations, we construct the16 elements of the calibration matrix. We checkthat the off-diagonal elements of M are at most1% of the diagonal ones for all of the cluster sam-ples considered.We perform this exercise 20 times and assignthe average value from these 20 realizations to thematrix elements. For each of the RASS, Planck ,and EASS sample, we calculate one average M andapply it to the filtered temperature maps. Due tothe intrinsic scatter of the Y − M relation intro-duced in the simulations, the average optical depthof any simulated cluster sample might deviate fromthe expected value from the characteristics associ-ated with that catalog, this may lead to calibrationerror (section 8.3).
8. Error estimation
The error budget to the dipole coefficients con-tains several terms: σ a V = σ + σ + σ + σ (14)where σ CMB+noise is the residual from the CMBand instrument noise, σ is the error in the ki-netic SZ signal due to the intrinsic scatter of the Y − M relation and the random component of thegalaxy cluster peculiar velocity that is not part ofthe bulk flow, σ is the thermal SZ residual, and σ is the contamination from radio or infraredpoint sources associated with clusters. In princi-ple, the last three may be correlated, while we areassuming here that correlations are negligible.The errors on the three dipole coefficients a m are correlated and the uncertainty in the best fitvalue can be described by a covariance matrix N . We compute the covariance matrix of eachtype of error in equation 14 by passing simula-tions of the components through our pipeline andperforming the dipole fit on them. The scatter σ in dipole coefficients then provides an estimateof the noise correlations between the dipole direc-tions, i.e. N = (cid:10) a V a T V (cid:11) . Then, χ = ( a V , rec − a V , in ) T N − tot ( a V , rec − a V , in ) (15)where a V , rec is the best fit dipole coefficients of therecovered velocity and a V , in is the input velocitydipole coefficients. The confidence level of a given a V can be calculated from the χ probability dis-tribution with 3 degrees of freedom.Figure 6 shows the contribution from each sys-tematic effect to the dipole velocity. We discussthese results in the following subsections. The CMB signal is not completely removed byour filters because there is CMB emission on clus-ter scales with the same frequency dependence asthe kSZ signal. We evaluate the error from theCMB and instrument noise by measuring the bulkflow of 300 filtered CMB plus noise maps. As afurther check on this procedure we also calculatethe dipole on a single CMB plus noise realizationand repeat the procedure with 100 realizations ofthe cluster sample, each having a different spatialdistribution. The results from the two proceduresare similar with errors that are both gaussian withsimilar variance.As expected the UF errors are larger than theMF ones by about 30% for the three cluster sam-ples. We find that the error is largest for the RASS12ig. 6.— The systematic error contributions tothe bulk flow measurement using the RASS (top),
Planck (middle), and EASS (bottom) cluster sam-ples: instrument noise and CMB (cross), thermalSZ (asterisk), uncertainty in the kinetic SZ sig-nal due to scatter in both the optical depth andcluster peculiar velocity (diamond), and the con-tamination from radio point sources (triangle forour simulation and solid line for PSM simulation).The data points are the average values from 100realizations and the error bars are the standarddeviations of the one-sided distribution. sample and smallest for the EASS sample, sincethe EASS sample contains more clusters.
We expect the thermal SZ signal to be entirelyremoved when using the unbiased matched filter(UF). However, maps filtered with the MF sufferfrom a systematic bias by the unfiltered thermalSZ signal that contaminates the kinetic SZ signal.In addition to the effect of filtering, the intrinsicscatter of the cluster y-parameter would also in-troduce further incomplete removal of the thermalsignal and resulting in scatters around the meanrecovered velocities. To evaluate the level of ther-mal SZ bias and uncertainty, we generate simu-lated maps containing only tSZ signal.We find that in thermal SZ signal only mapsthe average systematic bias ( ± σ uncertainty asrepresented by the error bars in Figure 6) to thecluster dipole are v = 313 ±
48 km/s for RASS, v = 77 ±
33 km/s for
Planck , and v = 19 ± v = 885 ± v = 171 ±
77 km/s for
Planck ,and v = 47 ±
20 km/s for EASS when filtered bythe MF. The thermal bias is most serious for MFfiltered RASS clusters while the 1 σ uncertainty isthe largest for MF filtered Planck clusters. Thisis due to the fact that the average optical depthis the largest for
Planck clusters and hence thelargest intrinsic scatter (and uncertainty).We find a significant monopole: v a = 730(RASS), 1440 ( Planck ), and 870 (EASS) km/swhen filtered by the UF, and v a = 3500 (RASS),3890 ( Planck ), 2180 (EASS) km/s when filteredby the MF. The UF is more effective at removingthe thermal SZ signal than the MF, by a factorof ∼
5. Though the monopole velocities are largecompared to the thermal dipole, they have no ef-fect on the bulk flow measurement.
We find that the cluster random peculiar ve-locity not part of the bulk flow gives a negligibledipole with an average magnitude of v <
50 km/sfor all three cluster samples and is independent ofthe type of filter used. The error due to uncer-tainty in the optical depth depends on the ampli-tude of the peculiar velocity since σ kSZ ∝ vσ τ . Wefind that σ τ ≈
15% for the
Planck cluster sam-ple and σ τ <
15% (O10) for the RASS and EASSclusters. The uncertainty for
Planck clusters isrelatively larger because it contains more abun-13ant massive clusters and hence propagate largererrors to the optical depth for the same scatterin the cluster Y-parameter. As mentioned ear-lier, this uncertainty introduces a calibration er-ror when translating from µ K to km/s because ofthe difference of the average optical depth and thecalibration matrix M . However, since we choose alarge calibration velocity of 10,000 km/s, the cali-bration error is only about 5% and negligible whencompared to the error due to CMB and noise. At small scales extragalactic point sources aresignificant contaminants of CMB maps. They givea Poisson noise contribution to the measured angu-lar power spectrum and a non-Gaussian signaturein the maps (Colombo & Pierpaoli 2010). Radiopoint sources have a falling spectrum with α ∼ . S ∝ ν − α , whereas infrared point sourceshave a rising spectrum with typical spectral in-dices α ∼
3. Thus infrared sources tend to bethe more significant contaminants at higher fre-quencies where the SZ effect is large. Recent mea-surements such as the South Pole Telescope (SPT)detected 47 IR sources at S/N > . dn (log P ν ) d log P ν = Z dn (log P . ) d log P . f ( α ) dα (16)where P ν is the radio luminosity in units ofWHz − at frequency ν , α is the spectral index, and f ( α ) is the probability distribution of thespectral indices at 1.4 GHz, which is taken to bea Gaussian distribution with mean ¯ α = 0 .
51 andrms σ α = 0 . f ( α ) is taken to be f ( α + 0 . ν >
90 GHz to account for possible steepeningof α at high frequencies. The RLF at 1.4 GHz isgiven by the fitting formula:log (cid:18) dnd log P . (cid:19) = u − s b + (cid:18) log P . − xw (cid:19) − . P . (17)where u = 37 . b = 2 . x = 25 .
80, and w = 0 .
78 (Lin & Mohr 2007). Here n is the radiosource number density within r . We integrateequation 16 to obtain the expected number of ra-dio sources, N radio , in a cluster and assign fluxesto the N radio sources using the RLF. Since brightsources will be masked in the Planck maps we alsomask sources with
S > S lim ν , where S lim ν is theupper limit of the radio flux at frequency ν givenby L´opez-Caniego et al. (2006, Table 2) leaving aradio map with sources having S < mJy .We find no significant cluster dipole from radiopoint sources in the RASS,
Planck , or EASS clus-ter samples. In the redshift shell z = 0 − . v <
10 km/s for EASS and v ∼
20 km/sfor both
Planck and RASS clusters when filteredwith either the UF or MF. We can further verifythe result for the RASS sample by using more real-istic radio point source simulations that considersinformation from sources observed by NVSS, andsimulations from the Planck Sky Model (PSM).As before, we mask bright radio sources in thePSM simulation. The result from the PSM sim-ulation is shown in Figure 6. The radio dipolesfrom the PSM are consistently larger than our ra-dio simulations at all redshift shells, but are of asimilar order of magnitude with v ∼
50 km/s.
9. Results
We generate sets of simulations containingCMB, noise, thermal SZ signal, and kinetic SZsignal with 68 bulk flow velocities logarithmicallyspaced between 100 and 10,000 km/s. In this sec-tion we describe how well we recover the inputbulk flow. z ≤ . Top : RASS,
Middle : Planck , Bottom : EASS)from simulated maps with input bulk flow velocities logarithmically spaced between 100 and 10,000 km/s,filtered with the MF (black cross) and UF (blue diamond). The solid line represents perfect recovery. Thedash lines quote the 95% confidence interval of the input bulk flow values. All results are based on mapscontaining kinetic and thermal SZ signals, but simulated with different systematic components. Plots labelledNoise and CMB are from maps containing instrument noise and CMB; plots labelled thermal SZ are fromresults containing thermal SZ emission and hence estimate the level of thermal SZ bias.15ig. 8.— Recovered bulk flow velocity for three input bulk flows v =500 (black), 1000 (yellow), and 5000(cyan) km/s as a function of redshift for the Planck cluster sample. The recovered bulk flows are measuredfrom simulations that contain all systematic errors we consider.
Figure 7 shows the recovered bulk flow values V recovered versus the input values V input for all clus-ters with redshift z ≤ .
5. The dashed lines inthe figures are the 95% upper limits to the re-covered bulk flow velocities and are computed bygenerating 10,000 Gaussian distributed dipole co-efficients at each of the x, y, and z directionshaving mean equal to the input bulk flow veloc-ity and thermal bias, and variance equal to thenoise levels described in section 8, i.e. a sim1 m =( a in1 m + a bias1 m ) ± a noise1 m . The resulting magnitudeof the dipole velocity follows the χ distribution,the magnitude and angle error (section 9.2) at 95%are then computed.We determine the precision of the velocity mea-surement by computing the deviation of the val-ues V recovered from their corresponding V input witha parameter defined to quantify the deviations asfollows: σ log V = s P Ni =0 (log V recovered , i − log V input , i ) N − V input being considered and N = 68. This parameter al-lows us to compare the level of dispersion amongthe results from the various experiments and sur- veys. We provide the values for each cluster samplein Table 3.Taking a representative velocity of V input =500km/s we find that the recovered velocities are over-estimated by 120% with a scatter of 16% whenevaluated using the RASS clusters using the MF( V recovered = 1085 ±
82 km/s), but perfectly re-covered with only a scatter of 5% when evaluatedon the EASS clusters using the UF ( V recovered =500 ± Planck clusters, with ascatter of 25%, and is consistent with the erroranalysis that the uncertainty due to intrinsic scat-ter in Y-parameter is largest for
Planck clusters.With the sensitivities we find, measurements usingeither the
Planck or EASS cluster sample will beable to detect a bulk flow as large as that claimedby KAKE. If the bulk flow is consistent with theΛCDM prediction of v = 30 km/s for redshift shellextending to z = 0 .
5, our analysis pipeline wouldrecover a 95% upper limit to the velocity of, whenthe UF is used, v = 470 km/s for RASS cluster, v = 160 km/s for Planck cluster sample and v = 60km/s for EASS cluster sample.Figure 8 shows the recovered signal in differentredshift shells at three input bulk flow velocities: V input = 500, 1000, and 5000 km/s, estimated from1600 realizations. For illustration purpose, only theresult from the Planck cluster sample is shown. Atthese velocities, the recovered velocities are shownto be consistent with the input values within 1 σ at redshifts z = 0 .
1. At lower redshifts, the recov-ered velocities are noise dominated. At this red-shift limit, the
Planck cluster catalog will containapproximately 400 clusters. If a bulk flow of am-plitude ≈
500 km/s is present our results suggestthat a sample of about 400 clusters would allow
Planck observations to constrain the bulk flow.
The direction of any bulk flow is not predictedby the ΛCDM model. We can estimate the er-ror in the direction of a detected dipole for eachof the three surveys. For V input = 500 km/s to-wards l = 280 ◦ and b = 30 ◦ the average values ofthe dipole components, ( h V x i , h V y i , h V z i ) from 100simulations, are listed in Table 3. The directionis recovered with deviation <
20% for the
Planck and EASS cluster samples (except for h V x i of the Planck sample filtered with the MF), but is un-constrained for RASS clusters. We find that theerror is larger in the V x and V y directions due tothe galactic cut. We also calculate the angle errorsfor a range of input velocities for four different in-put directions: the three perpendicular directionsx ( l = 0 ◦ , b = 0 ◦ ), y ( l = 90 ◦ , b = 0 ◦ ) and z( l = 0 ◦ , b = 90 ◦ ), as well as the reference V input used above. The 95% upper limit to the angle er-rors, ∆ α , is shown in figure 9.For maps filtered with the MF (UF) the 95%confidence limit on the dipole direction for a 500km/s input bulk flow is ∆ α = 62 ◦ (34 ◦ ) forRASS clusters, 34 ◦ (14 ◦ ) for Planck clusters, and9 ◦ (4 ◦ ) for EASS clusters. The errors are smallerthan the discrepancies in the observed bulk flow di-rections. Therefore Planck should be able to bet-ter constrain the region where the bulk flow pointsto.
The dominant systematic error is tSZ emis-sion, and the dominant statistical error is CMBand instrument noise and uncertainty in the Y-parameter. They are velocity-independent anddominates for v < <
10% error in thevelocity. Using inverse variance weights in the dipole fit-ting (as opposed to weighting by optical depth) isfound to minimize the tSZ contribution. The re-covered bulk flows are generally overestimated by ≈
20% when W = τ /σ is used to weight pixelsin the dipole fit. The overall performance of both the MF andUF agree with expectations that maps filtered bythe MF have higher kSZ signal to noise ratios whilemaps filtered by the UF have smaller thermal SZbias. The UF is slightly a better filter in the
Planck data since the dominant uncertainty of the mea-surement comes from the thermal SZ bias, andthe recovered velocities are more accurately deter-mined. Nevertheless, the thermal SZ bias is notcompletely removed by the UF. This is becausewe assume a β model for the cluster profile, butassume a point source model to construct the fil-ters. We could assume a β model to construct thefilters but we do not consider this here.We now compare the filter performance in the WMAP channels which are studied in OMCP (seetheir Figure 11) with the
Planck channels in thiswork (Figure 7), with the use of the RASS clustersample which is used in both analysis. In OMCPthe MF was found to be more sensitive than theUF, but we find the opposite. The velocity de-tected at 95% CL with the MF (UF) is reducedfrom ∼ ∼ ,
000 km/s) in the
WMAP channels to ∼ ∼ Planck channels. Considering that the amplitudeof observed bulk flow velocities are a few hundredsof km/s at scales r <
300 Mpc h − (Figure 1),the filters in the WMAP channels do not havethe sensitivity to measure bulk flow velocities atthese scales. The differences between the
WMAP and the
Planck filters is expected: the error dueto instrument noise is largely reduced because thenoise levels of
Planck are much lower than
WMAP ;the wider frequency coverage in the SZ sensitiveregime also allows filters in the
Planck channels tosuppress the thermal SZ bias.
10. Conclusion and Discussion
We investigated
Planck performances in de-termining bulk flow velocities using the kineticSunyaev-Zeldovich effect. We characterize the sen-sitivity of the bulk flow velocity measurement us-ing simulated
Planck data combined with threerepresentative all sky galaxy cluster surveys: thearchived ROSAT All-Sky Survey, the
Planck clus-17ig. 9.— The 95% upper limit to the error in the angle measurements for all the three cluster surveys. The l = 30 ◦ , b = 280 ◦ data points correspond to the direction of the input bulk flow that the simulations inFigure 7 are based on. 18er catalog and the eRosita All-Sky Survey. Weemploy two different types of filters, a matched fil-ter (MF) and an unbiased kinetic SZ filter (UF),to maximize the cluster signal to noise ratio.The main results are (see also Table 3):1. The use of simulated Planck sky maps in-stead of
WMAP ones in combination withthe RASS catalog reduces the velocity thatcan be detected at 95% CL, from ∼ ∼ ,
000 km/s) to ∼ ≈ z ≤ . z = 0 . v = 30 km/s at z = 0 .
5, ouranalysis pipeline would obtain a 95% upperlimit to the recovered velocity (when the UFis used) of v = 470 km/s for RASS clus-ter, v = 160 km/s for Planck cluster sampleand v = 60 km/s for EASS cluster sample.This allows us to measure the departure fromΛCDM if the measured bulk flow is in excessof these estimates.4. Planck can also constrain a recovered bulkflow direction. For V input = 500 km/s, the95% upper limit to the angle errors (whenthe UF is used) are ∆ α ≈ ◦ for the RASSclusters, ∆ α ≈ ◦ for the Planck clus-ters, and ∆ α ≈ ◦ for the EASS clusters.These uncertainties are lower than the dis-crepancies observed in bulk flow directions atdifferent optical depth up to now (Figure 1).The errors in the directions contained withinthe galactic plane are larger than in the per-pendicular one due to the galactic cut at lowlatitudes.5. The error in the kinetic SZ dipole is dom-inated by the unfiltered CMB and instru-ment noise and intrinsic scatter of the Y-parameter, with a systematic thermal SZbias. The contamination from extragalacticradio sources is negligible and can be safelyignored. The noise level is lowest for the EASS sample since it contains the most clus-ters.6. The UF is more sensitive and effective thanthe MF in suppressing the bias induced bythe thermal SZ in the velocity reconstruc-tion. This is in contrast to the performanceof the two filters on the WMAP maps, wherethe MF was much more sensitive than theUF. Differences in performances are to beexpected, as the two experiments have dif-ferent characteristics. As Planck has muchlower noise, the tSZ contamination now playsa more major role in setting the bias anderrors.
Planck increased frequency coverageand resolution, however, enables the filters(UF in particular) to mitigate the effectsof the thermal SZ signal, especially when alarge cluster sample is considered.We have not considered here the effect of in-frared point sources, correlations in the signal be-tween CMB and clusters, large–scale (non bulkflow) peculiar velocities, and error determinationin extracting optical depth from the survey inhand. Some of these could present further im-provements to this initial study, which demon-strate the superior potentials
Planck has in de-termining the bulk flow.
11. Acknowledgements
DSY Mak acknowledges support from the USCProvosts Ph.D Fellowship Program. EP is an AD-VANCE fellow (NSF grant AST-0649899). DSYMak and EP acknowledge support from NASAgrant NNX07AH59G and JPL Planck subcon-tract 1290790 and the warm hospitality of As-pen Center for Physics. SJO acknowledges use-ful discussions with Neelima Sehgal and supportfrom the US Planck Project, which is funded bythe NASA Science Mission Directorate. Some ofthe results in this paper have been derived usingthe HEALPix (G´orski et al. 2005) package. Theauthors acknowledge the use of the Planck SkyModel, developed by the Component SeparationWorking Group (WG2) of the Planck Collabora-tion. The authors thank the JPL data analysisgroup, Stefano Borgani and Sarah Church for fos-tering initial conversations on this topic.19 able 3Summary of the results noise components RASS
Planck
EASSMF UF MF UF MF UF V noise + CMB + thermal SZ bias 1085 ±
82 720 ±
94 530 ±
126 516 ±
73 507 ±
33 500 ± h v x i (75 km/s) 129 ±
94 131 ±
138 108 ±
133 88 ±
68 80 ±
40 75 ± h v y i (-426 km/s) − ± − ± − ± − ± − ± − ± h v z i (250 km/s) 1065 ±
82 526 ±
66 240 ±
91 255 ±
49 256 ±
28 243 ± α ◦ ◦ ◦ ◦ ◦ ◦ σ log V, bin1 σ log V, bin2 σ log V, bin3 σ log V, bin4 σ log V, whole V ( h V rec i ) only thermal SZ bias 1100 ±
60 724 ±
82 530 ±
115 515 ±
63 511 ±
30 504 ± h v x i (75 km/s) 125 ±
73 142 ±
98 82 ±
109 82 ±
60 104 ±
30 101 ± h v y i (-426 km/s) − ± − ± − ± − ± − ± − ± h v z i (250 km/s) 1082 ±
63 534 ±
46 253 ±
86 250 ±
46 255 ±
29 244 ± α ◦ ◦ ◦ ◦ ◦ ◦ σ log V, bin1 σ log V, bin2 σ log V, bin3 σ log V, bin4 σ log V, whole Note.—
This table lists the main results of the bulk flow measurements for clusters within z ≤ . l = 280 ◦ and b = 30 ◦ . V is the average value of the recovered velocity, h v x i , h v x i , h v x i are the average of the individualdirections. For ideal recovery, they should have values (75, -426, 250) km/s. ∆ α is the 95% upper limit to the angle error. The last fivequantities σ log V, bini are the deviation parameters as defined in equation 18, where bin i refers to the binned velocity range considered in thecalculation: bin 1= 100–500 km/s, bin 2= 500–1000 km/s, bin 3= 1000–5000 km/s, bin 4= 5000–10000 km/s. The whole range is for velocities100–10000 km/s. REFERENCES
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This 2-column preprint was prepared with the AAS L A TEXmacros v5.2. . Derivation of Filter KernelA.1. Data Model In real space, the observation field can be described as s ν ( θ ) = y (0)( f ν − V ) B ν ( θ ) + n ν ( θ ) (A1)where y (0) is the y parameter at the cluster center (defined in equation B10), f ν = x ( e x + 1) / ( e x − − x = ( hν ) / ( k B T CMB ) is the frequency response of the tSZ signal, V = v r m e c/k B T e gives the signal inkSZ effect, B ν ( θ ) is the convolved cluster profile given by: B ν ( θ ) = Z d Ω ′ p ( θ ′ ) b ν ( θ − θ ′ )= ∞ X l =0 B l ,ν Y l (cos( θ ))where B l ,ν = p (4 π ) / (2 l + 1) b l ,ν p l ,ν , p ( θ ) and b ν ( θ ) are the cluster spatial profile and beam functionrespectively, and n ν is the noise consisting of instrumental noise and CMB . The noise map and the crosspower spectrum C of each of the two components satisfy h n lm,ν n l ′ m ′ ,ν i = C l,ν ,ν δ ll ′ δ mm ′ A.2. Filtered Field
Let Φ ν and u ν be the filter and filtered field respectively at frequency ν , then u ν ( θ ) = Z dθ s ν ( θ )Φ ν ( θ ) = X lm u lm,ν Y ml ( β ) (A2)with u lm,ν = q π l +1 s lm,ν Φ l ,ν The total signal in all frequency channels is u ( θ ) = P ν u ν ( θ )A useful quantity to consider is the variance of the filtered map: σ u = (cid:10) ( u ( θ ) − h u ( θ ) i ) (cid:11) We introduce vector notation for the set of frequency dependent quantities, e.g. ~B l = { B l,ν , B l,ν , · B l,ν N , } Since h u ( θ ) i = DP l h y c ( ~F l − V ~B l ) + ~n l i ~ Φ l E , where we have used the notation ~F l = { f ν B l ,ν } ν , then wehave σ u = *"X l ( y c ~F l − y c V ~B l ) ~ Φ l + ~n l ~ Φ l − ( y c ~F l − y c V ~B l ) ~ Φ l + = *"X l ~n l ~ Φ l + Therefore σ u = P l ~ Φ Tl C l ~ Φ l . 22 .3. Unbiased Matched Filter for kSZ signal In order to derive the optimal filter for the kSZ signal, we want to minimize σ u subject the the followingconstraints:1. The filter is an unbiased estimator of KSZ signal at the source location, such that X l ~B l ~ Φ Tl = 1 (A3)2. The filter should remove the TSZ signal at the source location, i.e. X l ~F l ~ Φ Tl = 0 (A4)The functional variation of σ u with respect to ~ Φ l subject to constraints A3 and A4 can be obtained usingLagrangian multipliers. The Lagrange function is defined as: L = X l ~ Φ Tl C l ~ Φ l + λ (1 − X l ~B l ~ Φ Tl ) + λ X l ~F l ~ Φ Tl Minimizing the Lagrange function with respect to the filter function ~ Φ l ,∆ ~ Φ Tl L = X l h C l ~ Φ l − λ ~B l + λ ~F l i = 0 ⇒ C l ~ Φ l = λ ~B l − λ ~F l ⇒ ~ Φ l = C − ( λ ~B l − λ ~F l ) (A5)The job now is to find the constants λ and λ . Using constraint A3 and A5, X l ~B l ~ Φ Tl ~ Φ l = X l h λ ~ Φ l ~B Tl C − ~B l − λ ~ Φ l ~B Tl C − ~F l i ⇒ λ X l ~B Tl C − ~B l − λ X l ~B Tl C − ~F l ⇒ λ γ − λ β (A6)Similarly, using constraint A4 and A5, X l ~F l ~ Φ Tl ~ Φ l = X l h λ ~ Φ l ~F Tl C − ~B l − λ ~ Φ l ~F Tl C − ~F l i ⇒ λ X l ~F Tl C − ~B l − λ X l ~F Tl C − ~F l ⇒ λ β − λ α (A7)where α = P l ~F Tl C − ~F l , β = P l ~F Tl C − ~B l , and γ = P l ~B Tl C − ~B l Solving A6 and A7, we obtain λ = βαγ − β , and λ = ααγ − β . Thus we get the filter kernal: ~ Φ UF l = C − αγ − β ( α ~B l − β ~F l ) (A8)It is easy to verify that this filter kernel satisfies the two constraints.The filter can be interpretated in this way: 23. ∆ ≡ αγ − β is the normalization such that P ~B Tl ~ Φ l = 12. The term α ∆ ~B l comes from the kSZ constraint, ensuring that the filter gives the kSZ signal at the sourcelocation.3. The term − β ∆ ~F l comes from the TSZ constraint. Its purpose is to suppress the tSZ signal such thattSZ signal vanishes at the source location.4. The filtered fields at each channel are weighted by the inverse of the covariance matrix C − and thencombined to form the final filtered signal. A.4. Matched Filter
The derivation of the matched filter is similar to that for the unbiased matched filter, except that con-straint A3 is used. Therefore we have the Lagrange function: L = X l ~ Φ Tl C l ~ Φ l + λ (1 − X l ~B l ~ Φ Tl ) ⇒ ∆ ~ Φ Tl L = X l h C l ~ Φ l − λ ~B l i = 0 ⇒ ~ Φ l = C − ( λ ~B l ) (A9)Using constraint A3 and solving for λ , we find λ = 1 /γ . Thus we get the filter kernel: ~ Φ MF l = C − γ ~B l (A10)24 . Simulations of full-sky SZ mapsB.1. Mass and Redshift Distribution of the cluster sample The number of galaxy clusters per mass and redshift bin can be estimated using the comoving massfunction: dNdM dz ( M, z ) = ∆Ω dndM ( M, z ) dVdzd Ω ( z )where ∆Ω is the solid angle in a given direction, and dndM ( M, z ) is the mass function which describes thenumber density of galaxy clusters per mass bin at a given redshift z. Here we adopt the Jenkins massfunction (Jenkins et al. 2001) with the fitting formula given by: dn ( M, z ) dM = ρ m M d ln σ − dM f ( σ − ) (B1) f ( σ − ) = 0 .
315 exp( − (cid:12)(cid:12) ln σ − + 0 . (cid:12)(cid:12) . )where ρ m is the present matter density, σ is the variance of the mass fluctuation σ M = π R ∞ k dkP m ( k ) | W R ( k ) | ,where P m ( k ) is the matter power spectrum P ( k, z ) = Ak n T ( k, z ) D ( z ),We use the transfer function T ( k, z ) from Eisenstein & Hu (1998) that accounts for all baryonic effects inthe matter transfer function on the large scale and an improved version of the BBKS transfer fitting formula.The growth function is defined as D + ( z ) = D ( z ) /D (0). We take the approximated form from Lahav et al.(1991) and Carroll et al. 1992: D ( z ) = (1 + z ) − m ( z )2 [Ω m z / − Ω Λ ( z ) + (1 + Ω m z/ Λ ( z ) / N ∆ M ∆ z = 4 π Z ∆ z dz Z ∆ M dM dNdM dz ( M, z ) (B3)We then assign N ′ clusters to a bin (∆ M, ∆ z ) from a random Poisson distribution with an averagevalue given by equation B3. The mass range is taken to be 10 − M ⊙ with a bin size of log(∆ M ) =0 .
1. The minimum mass of cluster at redshift z that can enter the simulated catalog is given by M min =max[10 M ⊙ , M lim ], where M lim is the limiting mass of the corresponding survey from equation 9 and 10.The redshift range z is 0 − . z = 0 .
02. The number of clusters from this halo model isa strong function of σ as shown in Figure 4.Having obtained the mass and redshift distributions of the clusters, we distribute the clusters over thewhole sky. For simplicity, we ignore the spatial correlation among clusters and assign a random position toeach cluster. B.2. Modeling SZ signal of individual clusters
The SZ signal of galaxy clusters is characterized by the Compton y parameter, which in turn depends onthe gas properties of individual clusters. The next step is to model the electron density n e ( r ), temperature T e ( r ) and the Integrated Compton Y parameter. B.2.1. Gas properties from scaling relations
Assuming that clusters are self-similar, virialized and isothermal, we obtain the mass-temperature relationfrom the virial theorem (Kaiser 1986): k B T e = β − (1 + z ) (cid:18) Ω m ∆ V ( z )Ω m ( z ) (cid:19) / (cid:18) M cl M ⊙ h − (cid:19) / keV (B4)25here β T = 0 .
75 is the normalization constant under the assumption of hydrostatic equilibrium and isother-mality, ∆ V is the mean overdensity of a virialised sphere which we have calculated using the fitting formulafrom Pierpaoli et al. (2001), M cl is the mass of the cluster within the virial radius r vir , M cl = 4 π/ V ρ crit r .Solving for r vir gives, r vir = 9 . z (cid:18) Ω m ∆ V ( z )Ω m ( z ) (cid:19) − / (cid:18) M cl M ⊙ h − (cid:19) / h − Mpc (B5)The assumption that clusters are isothermal is in good agreement with
XMM-Newton observations of outercluster regions which find that the cluster temperature profiles are isothermal within ±
10% up to ≈ r vir / β model ( Waizmann & Bartelmann 2009 andreference herein), n e ( r ) = n e0 (cid:18) r r c (cid:19) − / β (B6)where r c is the core radius. Geisb¨usch et al. (2005) obtain a relationship between the core radius and thevirial radius: r c ( z ) = 0 . z ) / r vir (B7)We choose β = 2 / n e0 is normalized by R r vir n e ( r ) dV = N e , in which N e is the total number ofelectrons within the cluster given by: N e = (cid:18) f H m p (cid:19) f gas M cl (B8)where f H is the hydrogen fraction, f gas = Ω b / Ω m is the baryonic gas mass fraction of the total cluster mass.Here we take f gas = 0 .
168 from WMAP 5 year results and f H = 0 .