Table Of Content

Cluster Mass Estimators from CMB Temperature and Polarization Lensing Wayne Hu,1 Simon De Deo,1 Chris Vale2 1Kavli Institute for Cosmological Physics, Department of Astronomy and Astrophysics, and Enrico Fermi Institute,University of Chicago, Chicago IL 60637 and 2Particle Astrophysics Center, Fermilab, P.O. Box 500, Batavia, IL 60510∗ Upcoming Sunyaev-Zel’dovich surveys are expected to return ∼ 104 intermediate mass clusters at high redshift. Their average masses must be known to same accuracy as desired for the dark energy properties. Internal to the surveys, the CMB potentially provides a source for lensing mass measurements whose distance is precisely known and behind all clusters. We develop statistical mass estimators from 6 quadratic combinations of CMB temperature and polarization fields that cansimultaneouslyrecoverlarge-scalestructureandclustermassprofiles. Theperformanceofthese estimators on idealized NFW clusters suggests that surveys with a ∼ 1′ beam and 10µK′ noise in uncontaminated temperature maps can make a ∼ 10σ detection, or equivalently a ∼ 10% mass 7 measurement for each 103 set of clusters. With internal or external acoustic scale E-polarization 0 measurements, the ET cross correlation estimator can provide a stringent test for contaminants 0 2 on a first detection at ∼ 1/3 the significance. For surveys that reach below 3µK′, the EB cross correlation estimator should provide the most precise measurements and potentially the strongest n control overcontaminants. a J PACSnumbers: 0 1 I. INTRODUCTION such as SPT and ACT should provide samples of 104 ∼ 1 intermediatemassclustersathighredshift,sothatapre- v ciseandunbiasedstatisticalmeasurementmaybecomea Upcoming surveys for clusters utilizing the Sunyaev- 6 realistic possibility in the near future. The CMB even 7 Zel’dovich (SZ) effect in the Cosmic Microwave Back- possesses some advantages over galaxy lensing, since the 2 ground(CMB)asadetectiontechniqueholdthepromise source statistical properties and redshift are extremely 1 tomeasurethepropertiesofthedarkenergytohighpre- 0 cision with 104 clusters at redshifts out to z 1. The well determined, and because lensing of the tempera- 7 ∼ ∼ ture and polarization provide strong consistency checks meanmassandvarianceofthesampleneedstobeknown 0 against possible contamination by cluster emission. to an accuracy comparable to that desired for the dark / h energyequationofstatetonotbealimitingfactor. Since Techniques which use CMB lensing to reconstruct the p the SZ effect is sensitive to the temperature weighted lensing convergence, κ, have been available for some o- baryon content of the cluster, it does not directly probe time [8, 9, 10], and although these were initially in- r the total cluster mass in a model independent way. tended as probes of large scale structure, they can in t principle recover the projected mass regardless of its s Fortunately,the same surveywhichidentifies the clus- a source. This certainly includes reconstructions of large ters in the SZ effect can potentially also constrain their : v masses through gravitational lensing of the CMB. Uti- galaxyclustersandassociatedstatisticalquantities,such Xi lizing the CMB as a source is also appealing in that its as the cluster-convergencecorrelationfunction. On cluster length scales, this is essentially an average density distance is both well determined and sufficiently far to r profile and mass measurement (e.g. [11, 12]). We will a probe even the highest redshift clusters in the sample. focus on estimators that are quadratic in the observed Studies of gravitational lensing of the CMB by clus- CMB fields [8, 9], since these can be implemented us- ters [1, 2, 3, 4, 5, 6, 7] typically focus on making de- ing fast algorithms, and can eliminate contamination by tailed measurements of individual high mass clusters or isolating pairs of modes. on statistical reconstructions of average cluster properties. TheCMBsuffersintheformercasewhencompared However,Amblardet al. [13]usedsimulationstoshow withgalaxyweaklensingbecausethebackgroundimage, thattheminimumvariancequadraticestimatorbuiltout inthiscasetheprimaryCMB,isaGaussianrandomfield of the temperature field is biased if the lensing field is whosepropertiesmustbe separatedfromthe clusterand non-Gaussian at the level expected for real structure. its emission. Subsequently Maturiet al. [6]helped to explainthis fact by deomonstrating that the reconstructed density field This disadvantage will be substantially reduced in the for this estimator is biased low in regions around large lattercasewherethegoalistomeasurestatisticalproper- clusters. tiesratherthanuniqueobjects,providedthatlargesam- ples of clusters are available for the analysis. Surveys In this paper, we analyze the origin of the bias and show that it can be nearly eliminated using the increas- inglywellmeasuredpropertiesoftheCMB,andthatthe modifications we introduce produce only a very modest ∗Electronicaddress: [email protected] degradation in the statistical reconstruction of cluster 2 properties. Since the temperature field estimators read- field. This approximation of course does not hold for all ilygeneralizetopolarization,thesecanbeusedasconsis- Fourier modes in the CMB fields andlens (see e.g. [18]). tencychecksoneachotherorindeedonanyothercluster It need only hold the modes that are correlated by the mass estimates. reconstruction. The lensed fieldT (nˆ) canbe prefiltered L Theoutline ofthepaperisasfollows. In IIwereview in Fourier space to isolate modes for which the gradient § the generalconstruction of quadratic estimators and the approximationis valid underlying approximations upon which they are based. We determine the origin of the biased estimates and LT(nˆ) TL(nˆ), ≡ showthattheyareassociatedwiththemiss-estimationof d2l sourcegradientsinthemoderatetostronglensingregime. TL(nˆ) = (2π)2eil·nˆWlTTl, (3) Thebiascanbeeliminatedatnegligiblecosttothesignal Z tonoisebyimposingastrongfilteragainstthisandother whereWT istheFourierfilter,thesubscript“L”denotes l contaminants. In III, we consider the idealized perfor- the lensing-filteredfieldand Tl is the Fourierrepresenta- § mance of the temperature and polarization estimators. tion of the full temperature field. For illustrative purposes, we use throughout a flat The lensed temperature field T can be approximated L ΛCDM cosmology with matter density Ωm = 0.24, by a Taylor expansion baryondensityΩ h2 =0.0223,Hubbleconstanth=0.73, b scalar tilt ns =0.958, amplitude σ8 =0.76. TL ≈T˜L+(∇T˜·∇φ)L, (4) which we will call the gradient approximation. For the case of cluster lensing, the typical deflections are < 1′ II. QUADRATIC ESTIMATORS compared with structure in the unlensed CMB with co- herence of 10′. Eq. (4) is therefore an excellent ap- Lensingisasurfacebrightnessconservingremappingof ∼ proximation for the full field even in the strong lensing the intrinsictemperatureandpolarizationfieldsfromre- regime. In this case the minimum variance filter can be combination. Given an unlensed temperature field T˜(nˆ) used [8] and Stokes Q˜(nˆ) and U˜(nˆ) fields, where nˆ denotes the angularpositiononthesky,thelensedfieldsaregivenby WT =(CTT +NTT)−1. (5) l l l T(nˆ) = T˜(nˆ+ φ), Here CTT is the lensed CMB power spectrum and NTT ∇ l l Q(nˆ) = Q˜(nˆ+ φ), is the noise power spectrum. More generally WlT can be ∇ chosen to suppress modes which violate the gradient ap- U(nˆ) = U˜(nˆ+ φ). (1) proximation or which are contaminated by foregrounds. ∇ Giventhatthegradientapproximationinducesacorre- Here φ is the deflection potential, φ is the deflection ∇ lationwiththeunlensedtemperaturegradient,oneforms angle, and they are related to the convergence as aquadraticestimatorbymultiplyingT byafilteredgra- L 2φ= 2κ. (2) dient of the lensed temperature field ∇ − GT(nˆ) T (nˆ), Allderivativesthroughoutareangularderivativesonthe ≡ ∇ G stikoyn. Winetwhiisllpfuarptehrerbmutoraellemexpplroeysstihoensflarteasdkyilyapgpernoexraimlizae- TG(nˆ) = (2dπ2)l2eil·nˆWlTTTl, (6) Z to the full sky with the replacement of ordinary derivatives with covariant derivatives and Fourier modes with where WlTT is another Fourier space filter. The filter [8] spherical harmonics [14, 15, 16]. WTT =C˜TT(CTT +NTT)−1 (7) The estimators of φ or equivalently κ introduced in l l l l [8, 9] rely on two related but independent linearization yieldsaminimumvariancereconstructionunderthefully approximations: the gradient approximation and lin- linearized approximation. Here C˜TT is the unlensed earization in the convergence. Only the latter, which is l CMBpowerspectrumandEq.(7)isessentiallyaWiener equivalentto consideringlensing as a small correctionto filter for the unlensed gradients (c.f. [6] for the more the source field, requires modification for use in cluster problematic Wiener filter for the total gradient). reconstruction. This filter has to be modified in the presence of rare Toseehowtheseapproximationsenter,letusfirstcon- non-Gaussian structure where lensing effects are moder- siderthecaseofthetemperaturefield. Theanalysisread- ate to strong. Expanding the product in the gradient ily generalizes to polarization as we shall see. approximation, we obtain Fundamentally the temperature estimator is built out ofalensinginducedcorrelationbetweenthetemperature GTLT ( T˜ )T˜ +( T˜ )( T˜ φ) G L G L field and its gradient [17]. This correlation arises from ≈ ∇ ∇ ∇ ·∇ +[( φ) ) T˜+( T˜ ) φ)] T˜ (8) the gradient approximation, valid when the deflections ∇ ·∇ ∇ ∇ ·∇ ∇ G L are small compared with the structure in the unlensed +[( φ) ) T˜+( T˜ ) φ)] ( T˜ φ) . G L ∇ ·∇ ∇ ∇ ·∇ ∇ ∇ ·∇ 3 The second approximation is that the quadratic term in φ is negligible. If so, averaging over realizations of the 10 unlensedtemperaturefieldyieldsaquantityproportional to φ with a proportionality coefficient related to the ∇ n) T well-determinedtwo-pointfunctionorpowerspectrumof mi theunlensedCMB.Thisquantityisthereforeaquadratic c r a estimator of the deflection angles φ or κ in the linear K/ 1 ∇ approximation. m( ms E The quadraticterminφ comesfromthe changeinthe Gr temperature gradients due to lensing. Qualitatively, its omission is equivalent to considering lensing as a small perturbationtotheunlensedimage. Quantitativelyitin- 0.1 volvestheassumptionthatthedeflectionanglesaresmall compared with structures in the lens. This approxima- 100 1000 tion is violated around a cluster and the Wiener filter, l G which is based on the variance of typical regions, does not sufficiently suppress this regionin the absence of de- FIG. 1: Gradient of the unlensed temperature T and polar- tector noise. ization E fieldsasafunctionofthemaximuml=lG ofatop To see the effect of the quadratic term, consider the hat low pass filter. For both fields, gradients are dominated lensedtemperaturefieldtobefilteredforsmallscalefluc- bymodes with l<2000. tuations as in Eq. (5) with NTT = 0 and likewise take l the unlensed temperature gradients T˜ const. In G that case, the estimator becomes ∇ ≈ low pass filter WlTT out to l =lG (GT )2 T˜ (nˆ) T˜ (nˆ) GTLT T˜+( T˜ ) φ) ( T˜ φ) . (9) rms ≡ h∇ G ·∇ G i ≈ ∇ ∇ ·∇ ∇ G ∇ ·∇ L = lG ldll2C˜TT . (10) h i 2π l Z1 Thequadratictermcarriesacoherentcontributionwhose strength depends on κ since 2φ = 2κ. As κ becomes Almost all of the gradient comes from l 2000 due to ≤ ∇ − diffusiondampingoftheacousticpeaks. Onthesescales, (1) in the interior of the cluster, the estimation will O the lensingcorrectiontothe gradientis tiny foratypical be biased low. In fact for κ > 1 the observed gradient cluster. This suggests that if we impose a sharpfilter on reverses and a second, flipped image of the intrinsic gra- thegradienttoexcludehighermultipoles,weloseverylit- dient appears at the center of an azimuthally symmetric tle signalandgaina cleanseparationbetweenthe lensed cluster. and unlensed structure. Such a filter is also desirable for Another way to see why the reconstruction is biased largescalereconstructiontoeliminatepairsofmodesthat low in the single image regime is that the cluster mag- violate the gradientapproximation. Furthermore,it also nifies the background image and decreases the observed reducescontaminationfromforegroundsandsystematics temperature gradient behind it. While this bias is miti- thatmightappearassignalwithoutacleanseparationof gatedby the Wienerfiltering ofEq.(7), the filter cannot scales between the two fields of the quadratic estimator. remove it entirely. Furthermore, a truly optimal filter Nowletusmaketheseconsiderationsconcreteandgen- would require knowledge of the cluster mass to be esti- eralize them to the full set of temperature and polariza- mated. tionfieldsX,Y T,E,B. E(nˆ)andB(nˆ)arerealvalued ∈ Thebiasarisesbecauseoftheoverlapinscalesbetween fields that are related to the Stokes parameters as the unlensed gradient field and the lensed temperature field. Maturi et al. [6] suggested that it can be fur- d2l ther mitigated by a combination of more strongly high Q(nˆ)+iU(nˆ)=− (2π)2[El+iBl]e2iϕleil·nˆ, (11) pass filtering the lensed temperature field in Eq. (5) and Z utilizing the direction of the large scale gradients. The where ϕl is the angle between l and the axis which de- efficacy of this technique depends on both the assumed fines Stokes Q. We shall assume that B is absent in the signal and the instrument noise. unlensed CMB. Insteadletusexploitthefactthatwehaverobustprior The angular space convergence estimators knowledge about the unlensed CMB spectrum. A hand- fsuhlapoef owuetllthdreoteurgmhitnheeddcaomsmpionlgogtiaciallwphaerraemtheteesrms fialxlesscaitles κˆXY(nˆ)= (2dπ2)l2eil·nˆκˆXl Y , (12) Z gradientspickupmostoftheircontribution. Fig.1shows theunlensedrmsgradientasafunctionofastepfunction are constructed from Fourier space estimators that are 4 quadratic in the observed fields Fig.1showsthatthisisalsoagoodchoicefortheE field. For l l , we retain the Wiener filter G κˆXY ≤ l = d2nê−inˆ·lRe [GXY(nˆ)LY∗(nˆ)] . AXl Y −Z ∇· WlXY =C˜lXY(ClXX +NlXX)−1, (l ≤lG), (21) (cid:8) (cid:9)(13) The gradientfieldGXY is built outofthe lensedX field for Y T,E and ∈ aseitheraspin-0orspin-2fielddepending onthe spinof WXB =C˜XE(CXX +NXX)−1, (l l ), (22) Y l l l l ≤ G 1, Y =T , for Y =B. GXl Y =ilWlXYXl× e2iϕl, Y =E,B. (14) Secondly, we distinguish between TE and ET estima- n tors. With the gradient filter employing only l < l , G LY represents the lensed fields the scale symmetry between the two fields is broken. It is then advantageous to separate the estimators. The 1, Y =T, LYl =WlYYl× e2iϕl, Y =E,B. (15) TE estimator uses T for the gradient field and E for the lensed field. If contamination from unpolarized clus- n ter emission is strong then this estimator can be used to Forthepolarizationfields,theyaretheStokesfieldQ+iU eliminate its effects. The ET estimator uses E for the filtered for the E and B components. The 2 gradi- gradient field and T for the lensed field. This estimator ent fields and 3 lensed fields produce 6 estimators from places less demanding requirements on the experiments the temperature and polarization fields for consistency in that only the relatively large E field of the acoustic checks. In addition, replacement of the divergence in peaks need be measured to high signal to noise. Indeed Eq. (13) with curl should leave estimators that are con- this measurement need not come from the same experi- sistent with noise. AXY is a normalization coefficient set to return unbi- ment that measures the cluster signal. Both estimators l strongly reject contaminants from foregrounds and sys- asedestimatorsunderthefullylinearizedapproximation. tematics. Only the background CMB will have T and For arbitrary filter functions, it is determined by noting E correlatedandanticorrelatedinthespecificoscillatory that [9] pattern of the acoustic peaks. Unfortunately, the sam- hXl1Yl2i=2fXY(l1,l2)κl, (16) ple variance of these estimators is also higher given the imperfect correlation between the two fields. where l=l1+l2 =0 and On the other hand, the sample variance of the EB es- 6 timatorisreducedbythefactthatB modesareassumed fTT(l ,l ) = C˜TT(l l )+C˜TT(l l ), 1 2 l1 · 1 l2 · 2 absentintheunlensedfields[9,10]. Inpracticethenoise fTE(l1,l2) = C˜lT1Ecos2ϕl1l2(l·l1)+C˜lT2E(l·l2), ofthe actualestimator willdepend onthe B-modes con- fTB(l1,l2) = C˜lT1Esin2ϕl1l2(l·l1), ttrhiebuEtTedebstyimfoarteogrr,ocurnodssscaonrdreloatthioerncoofntthaemlianragnetss.calLeikEe fET(l1,l2) = fTE(l2,l1), gradients with the small scale B-modes should help re- fEE(l1,l2) = [C˜lE1E(l·l1)+C˜lE2E(l·l2)]cos2ϕl1l2, dpuerciemaennytabliraesqauriirseimngenftrsomforcoEntBamairneansitms.ilaSrintcoe tthhaetexo-f fEB(l1,l2) = C˜lE1Esin2ϕl1l2(l·l1). (17) TE, TB, and EE but yield better prospects for con- straints, we focus on the EB, ET and TT estimators in Here ϕl l ϕl ϕl . The normalization is given in 12 ≡ 1 − 2 the next section. terms of these quantities and the filters as 1 2 d2l = 1 (l l )WXYWYcY(l ,l )fXY(l ,l ), AXY l2 (2π)2 · 1 l1 l2 1 2 1 2 III. IDEALIZED EXAMPLES l Z where To illustrate the performance of the estimators, let us take the idealization that all lenses are NFW [19] dark cT(l ,l ) = 1, 1 2 matter halos and the observed CMB fields have no con- cE(l1,l2) = cos2ϕl1l2, taminants aside from white detector noise. We will ad- cB(l1,l2) = sin2ϕl1l2. (18) dress more realistic cases in a future work [20]. The estimators should remain nearly unbiased since to pass For the lensed field filter, we retain the choice of [8, 9] through the reconstruction filters, a contaminant must be antisymmetric around the cluster center with an axis WY =(CYY +NYY)−1. (19) l l l whose direction is correlated with the large scale gradi- Asdescribedabove,thefirstmodificationisthatthegra- ents. In addition, contaminants will not appear in the 6 quadraticestimators,especiallythoseinvolvingcrosscor- dient weights are set to zero above l =2000 G relation, in the same way. They can however contribute WXY =0, (l >l ). (20) substantially to the noise and we will examine here the l G 5 performance of the estimators as a function of detector noise as a proxy for other contaminants. The NFW density profile is given by [19] 1 ρ(r) , (23) ∝ r/r (1+r/r )2 s s where the normalization coefficient can be expressed in terms of the mass of the halo. We define the mass to be that enclosed at r defined to be the radius which 180 enclosesthemassatanoverdensityof180timesthemean density 1/3 3M 1 r = , (24) 180 4π 180Ω ρ (cid:18) m c(cid:19) whereρ isthecriticaldensitytoday. Likewise,wedefine c the concentration of the halo in terms of this radius r 180 c = . (25) 180 r s The convergence profile for such a halo is [21] FIG. 2: Temperature (TT) reconstruction in the absence of detectornoiseandothercontaminants. Fromlefttoright,top 3 DLS MH0g(θ/θs) tobottomthereconstructionsarestackedwith1,10,102,103 κ(θ)= H D (1+z ) , (26) 4π 0 L DS L ρcrs2 f(c180) identical clusters of M14 = M/1014h−1M⊙ = 2 and z = 0.7. Shown here is a 35′×35′ region around thecluster. where the projected scale radius θ = r /D . D is the s s L comoving distance in a flat universe with subscripts “L” denoting the distance to the lens, “S” denoting distance Forthesimulatedreconstructionweconsiderafiducial to the source, and “LS” the distance between the lens experiment with a θFWHM = 1′ beam and varying levels and source. A primary advantage of the CMB is that of noise on the sky thesourceisbehindallclustersandatawell-determined NTT = ∆2B−2, distance DS =14.3Gpc. The concentration factor is l T l NEE = NBB =∆2B−2. (30) c l l P l f(c)=ln(1+c) , (27) − 1+c where the beam factor is and the functional form of the profile is [21] B =e−l2θF2WHM/16ln2 (31) l 1 2 x 1 with θFWHM in radians. It exponentiates the white de- g(x) = 1 atan − , (x>1) x2−1" − √x2−1 rx+1# taescstuomrenothisaeto∆n th=e b√ea2m∆d.ecoFnovroltvheisd fiskdyu.ciaWlebefuarmth,ear P T 1 2 1 x pixel scale of θ = 0.2′ suffices. The field is chosen to pix = x2−1"1− √1−x2atanhr1+−x#, (x<1) beW51e2le×ns51re2apliizxaetlison(1s0o0f′C×˜T1T0,0C′˜).EE accordingtothefull 1 l l = , (x=1). (28) non-linearprescriptionof Eq. (1) with 6th order polyno- 3 mial interpolation of the source image. This procedure For illustrative purposes, we take allows strong lensing effects to appear at the center of the cluster. To this lensed image we add a realization of M the noise power spectra above. M =2, (29) 14 ≡ 1014h−1M⊙ InFig.2,weshowthereconstructionwith∆T =0sim- ulations ofthe TT-estimatorfor N =1, 10,102, and103 c =3.2andz =0.7. Fortheseparametersθ =0.94′ stacked clusters. In this detector noise free limit, there 180 L s andκ(θ )=0.1. Note the lowvalue ofthe concentration is a clear detection with only one cluster. It appears as s reflects the low σ fiducial cosmology and is determined a concentrated spike on top of a noisy but slowly vary- 8 by the scaling of [22]. In a higher σ cosmology,clusters ingbackgroundfromchancecorrelationsoftheGaussian 8 at a fixed mass are more concentrated but correspond- random background field. The reconstructed single im- ingly larger clusters are more abundant. Other numbers agelackstheazimuthalsymmetryofthelensbutsymme- aretypicalofSZselectedclusterswithupcomingsurveys. try is recoveredby stacking images [6]. This asymmetry 6 comes from the fact that the deflections cannot be esti- 0.25 mated in the direction orthogonal to the gradient. The D =0m K’; N=103 T estimator however is unbiased once many realizations of 0.2 D =10m K’; N=103 T the gradient are stacked. mean (M =2) 14 D T=0m K’; N=103 0.15 true (M14=2) mean (M =2) kc trial (M14=2.6) 14 x true (M14=2) 0.1 0.1 0.05 kc x 0 0.01 5 10 15 q (arcmin) 2q FIG. 4: Beam filtered correlation function from 12000 clus- pix ters. The reconstruction is low pass filtered at l < lbeam(= 1 10 8095) to remove contributions below the beam scale. Shown q (arcmin) for comparison are the filtered input lens of M14 = 2 and a trial model of M14 = 2.6 for comparison. With a noise level FIG. 3: Recovered cluster-mass correlation function or of 10µK′ thetwo cases are clearly distinguished. stacked profile with 24000 detector noise free simulations. Bands represent the rms fluctuation per 1000 clusters in the sample. For M14 = 2, the reconstruction has no detectable Forcaseswithfinite noise,we explicitly removeeffects bias at the1−2% level within a few arcmin. below the beamscale fornumericalconvenience. We low pass filter both the lens and the reconstruction with a Stacking is equivalent to measuring the cluster- step function at l = √8ln2/θ . This also has beam FWHM convergence correlation function thedesirableeffectofmakingthereconstructionlesssen- sitive to potential contaminationnear the cluster center. dφ ξcκ(θ)= κˆ(θ,φ) , (32) ThefilteredcorrelationfunctionisshowninFig.4andre- 2πh i Z mainsunbiasedandwell-defined. Itisequivalenttomea- where the coordinatesare centeredatthe locationofthe suring the cross power spectrum at multipoles l <l . beam clustersuchthatθistheseparationfromthecenterandφ Fig. 4 also shows that with ∆ = 10µK′, M = 2 and T 14 istheazimuthalanglearoundthecluster. Theaveraging 2.6 are significantly distinguished. isovertheclustersinthesample. Bydefiningtheobserv- Toquantifythissensitivity,wecomputethecovariance able as the cluster-convergence correlation function, the matrix of the correlation function from the simulations estimator naturally generalizes to non-identical clusters andprojectedmassalongthelineofsightassociatedwith C(θ,θ′)= ξˆ (θ)ξˆ (θ′) ξ (θ)ξ (θ′), (33) cκ cκ cκ cκ h i− thecluster. Inpractice,weevaluatethecorrelationfunc- tion at discrete intervals in pixel units by interpolation where the discretization again is in bins of the pixel on the stacked image. Neighboring bins out to several width. Toestimatethesignificancewithwhichtwomod- arcminutesarethereforehighlycorrelatedascanbe seen els can be separated, we evaluate ∆χ2 between a test directly in Fig. 2. This correlation is in fact useful for model ξc′κ(θ) and the true model ξcκ(θ) distinguishing the signal from noise. In Fig. 3, we show the recovered correlation function ∆χ2 = [ξ′ (θ) ξ (θ)]C−1[ξ′ (θ′) ξ (θ′)], (34) cκ − cκ cκ − cκ from 24000 detector noise free realizations along with θ,θ′ X the scatter per 1000 clusters with M = 2. We have 14 smoothed both the lens and reconstruction with a 1.5 summed out to θ = 5′. Beyond a few scale radii of pixel FWHM beam after the fact for comparison. The theclusterthecorrelationfunctionisdominatedbylarge estimator has no detectable bias at the 1 2% level scale structure. ∼ − in the well-measured regime within a few scale radii. At In Tab. I, we show the detection significance, the dif- five times the fiducial mass or M14 = 10, a low bias ference between the fiducial M14 =2 lens and zero with of 8 9% develops for the fiducial gradient cut of the covariance matrix evaluated at zero signal. We have l ∼= 20−00. For these very rare clusters, the signal is scaled the significance per cluster by √N for N = 103. G large enough that a more aggressive l = 1000 yields At a noise level of 10µK′, the significance or signal to G a bias free reconstruction with sufficient signal to noise. noise is ∆χ2 0.4 per cluster. Above this noise level, ∼ Alternately, the bias can be calibrated out with simula- only TT and ET can provide significant detections. Al- p tions. though TT has roughly three times the signal-to-noise, 7 1 IV. DISCUSSION We have developed mass estimators from 6 quadratic 2 ET combinations of CMB temperature and polarization 1/ )3 fields that retain their minimum variance characteristic 0 EB N/1 for large scale structure and provide nearly unbiased re- M( 0.1 TT constructions for rare clusters where lensing effects are M/ moderate to strong. The key difference from previous work[8,9]isthattheWienerfilteringforunlensedCMB D gradients is augmented with a sharp filter that removes anygradientsatl >2000thatarenotpartofthesource G fields from recombination. q =1’; l<l 0.01 FWHM beam This sharp filter should also help prevent false signals 1 10 from cluster emission such as the thermal and kinetic D T=D P/21/2 (m K’) Sunyaev-Zel’dovich effect, point source emission as well asotherforegrounds. Toevenappearasexcessnoise,the FIG. 5: Idealized mass sensitivity for θFWHM = 1′, θ ≤ 5′. contaminantat 1′ musthaveacomponentthatisanti- ∼ The fractional errors on the idealized NFW mass with 1000 symmetricabouttheclustercenter. Tobiasthemeasure- clusters for theTT,ET and EB estimators. Unrelated mass ment,the directionoftheasymmetrymustbe correlated along the line of sight will add noise to the estimates as will with both the polarized and unpolarized contamination any emission from the cluster. No information from the re- at 10′inthesamewayastheacousticoscillationsinthe construction is used below the beam scale l≥lbeam. CM∼B. Note that projection effects from mass associated with the cluster should be considered part of the signal andcanbe calibratedbyN-body simulations. True con- ET canstillprovideausefulcheckagainstcontamination taminantshowevercansubstantiallyincreasethenoiseof andsystematicsforafirstdetection. Recallthatthefilter the estimators. An evaluation of the efficacy of the filter constructioninvolvestheoscillatoryacousticTE correla- in the presence ofreal worldcontaminants is beyond the tionandstronglyrejectsspurioussignals. Atnoiselevels scope of this paper but will be treated in future work between1 3µK′,theEB estimatorhasboththesignal- − [20]. to-noise and the ability to reject contaminants to make Here we have tested the estimators against idealized it competitive with TT. Below 1µK′, the EB estimator signalandnoise: identicalclusterslenseswithNFWdark dominates. matter profiles in the presence of beam filtered white In Fig. 5 we plot the sensitivity to the cluster mass. noise. At a noise level of 10µK′ and with a 1′ beam, Specifically we calculate the fractional change in mass the TT estimator can provide a 10σ detection per 103 that would generate a ∆χ2 = 1 with 103 clusters. The clustersof M 2 1014h−1M⊙∼orequivalently a 10% covariance is evaluated at the true model M =2 so as ∼ × ∼ 14 averagemassmeasurement. TheET estimator,basedon to include the samplevarianceofthe unlensedgradients. a separate measurement of E-polarization in the acous- The same features found for detection significance hold tic peaks and T measurements around the cluster, can formasssensitivity. Forexampleat10µK′,TT,ET,and provide an important cross check on a first detection. EB provide 13%, 43%, 53% mass estimates whereas at Sincetheestimatorfiltersforthe TE correlationandan- 1µK′ the numbers improve to 5%, 14%, 6%. ticorrelation of the acoustic peaks it provides a strong Finally, these sensitivities depend on the beam scale discriminant against contamination and systematics at only weakly near the fiducial θ = 1′ and ∆ = FWHM T the price ofa factorof 3 in signal-to-noise. Ultimately 1 10µK′ level. For example, at 3µK′, improving the ∼ with experiments that produce foreground free maps of be−am to 0.5′ has negligible effect on the signal to noise. the polarization to < 3µK′, the EB estimator will re- Enlargingthe beam to 2′, degradesthe TT andET esti- turn the best estimates, with the potential to ultimately mators by 30 40%in mass sensitivity. Since the EB provide 1% measurements. estimatoris∼nois−edominatedatthe1′ scaleevenwithout ∼ beam, it degrades negligibly. 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