EDP Sciences
Planck 2015 results
Free Access
Volume 594, October 2016
Planck 2015 results
Article Number A16
Number of page(s) 62
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/201526681
Published online 20 September 2016

6681&amsec"> 3. Sim"aations

Thesres"aus pauth=tsdlin thss p"per arelaerived u&ing thesextens=ve full focal pssne (FFP8)ssim"aations described in , , assnck Cothaboration XII (.106).lOf mostlimportance arelthesaHern Caalo (MC)ssim"aations that provids thesrefeuthorssetlof Gau"&iunssky m"pslused fos thesnull tests tmployud h"ae. Theynalso form thesbasis of "aysdebiasing innthes5nalysis of thesreall1-15 "" required by ceauain stautsticalsmen=1ds.

Thessim"aations incluau both CMB signal "auninstrumentalsno="e real"zations that capture important chsracteristics of thesassnck sca=ning strat"gy, telescope, detector retponses, "aun1-15 reduct=on pipel"nelover thesfull mission period. In p"autcular, thessignal real"zations incluau FEBeCoP (, , M=tr8let al. .101) beam convolut=onl"tleach of thesassnck frequenctes, "aun5relpropagated th4ough thesvas=ous -25" data-sep"aas=onlpipel"nes u&ing thessame weights "s derived from thesassnck full mission data 5nalysis.

ThesFFP8sfiducial CMB power s"actrum h"s been adopted from our bestlesuimate of the cosmological p"aamsters from thesfirstlassnck 1-15 release (, , assnck Cothaboration I .104)ssThss co"autponds tola cosmology withnb8ryonldenr"ty given by 6681&amsimple-math">ωb69 = Ωb69h M. LcporA.,ncotdldark matuth (CDM)ldenr"ty 6681&amsimple-math">ωc69 = Ωc69h M. LcporA.,nneutrinosenergy utnr"ty 6681&amsimple-math">ων = Ωνh M. LcporA.,nutnr"ty p"aamster fos thescosmological constant 6681&amsimple-math">ΩΛ = 0.6823por">A.,nHubblelp"aamster 6681&amsimple-math">H069 = 100h km s M. L-liA. withn6681&amsimple-math">h = 0.6712A.,ns"actr5l ss="x of thespower s"actrum of thespaimordial curvature peauurbationn6681&amsimple-math">ns = 0.up>6<">A., "aun5mplttuau of thespaimordial power s"actrum (at 6681&amsimple-math">k = 0.05 Mpc M. L-liA.) 6681&amsimple-math">As = 2.09 ×l10A., "ad withnthesThom"on oputcal depthnth4ough reionizas=onldefined tolbe 6681&amsimple-math">τ = 0.065me<">A.ssEach real"zation of thesCMB sky i&lgenerat"d incluaing lanr"ng, Rayleigh scat"aring, "ad Doppler boosuing effects,ltheslatuth two of which 5relfrequency-depss="ntssUnfortunatthy, thesabssration contribut=on tolthesDoppler boosu was erroneouslynomiautd from thessim"aations, but, withnposs=blesexceptions described in Sect. , , 6, thss does notllead to s=y signiftca=tlimpact on thesres"aus innthss p"per. Assecond ordss thmpesature quadrupole (, , Kamionkowskis4-636 Knox .103) i&ladded to each sim"aation withnann5mplttuau co"autponding tolthesresidual unco"aucted contribut=on pauth=tsinnthesobserv"d a-15,nas described in , , assnck Cothaboration XII (.106).l

Howevtr, thesassnck m"pslwere effect=velynssnormal"zed by approximatthy 2%ltol3% in power s= thestimelbetwsen thelgeneration of thesFFP8ssim"aations "ad thesfinal maps. As discussed in , , assnck Cothaboration XII (.106), co"auct=on fos thislcalibration effect shou>C have nossigniftca=tlimpact on cosmological p"aamsters. As aucommss="d,lin thss p"per thesCMB -25" data of thessim"aations ss simply rescaled by a factorlof 6681&amsimple-math">1.0134por">A.lbefore 5nalysis.

Of somewhat moresimportance i&lanlobserv"d no="e mismatch betwsen thelsim"aations "ad thes1-15. Whilstlthss h"s essautially nolimpact on studies of tempesature "aisohropy, it i5" sesnimportant orm"tations on thesstautsticalsstudies of pol"rizas=onlsky m"pslthat cap>bssincluaud h"ae. Convhorely, "aalyses based onn1-pss="sstautstics, such "s thesvss=ance, "ad thes Msup>6681&amsimple-math">NA.-pss="sco"aul"t=on funct=on" have playeunimportant roles innestablishing thesnature of thss mismatch, which seemsltolbe scale-depss="nt withnann5mplttuau arouaun20% at lower resolut=on" butlfalling tola few per cent at higher resolut=onclAsla consequen"a, thss p"per onlynincluaussres"aus from a stacking 5nalysis of thespol"riz"d a-15, in which thesstacking of thes1-15 themrelvessneth"&"rily "cts tollower theseffect of thesno="e mismatch.lPol"rizas=onlstudies that do notlrelynon auto-stautstics cap>still yr">C s="aresuing new res"aus, as fouaunin , , assnck Cothaboration XIII (.106); , , assnck Cothaboration XVII (.106); , , assnck Cothaboration XVIII (.106).l

6681&amsec"> 4clTesus of non-Gau"&iun"ty

There i&lnolun"quelsignature of non-Gau"&iun"ty, butlthesappl=cation of a vss=ety of tests over a r"agolof "agular scaleslallows us tolprobenthes1-15 fos dep"auures from theoretically mos=v"ted Gau"&iunsstautstics. Onelof the moresimportant testslin thescontext of infaation8ry cosmology islrel"ted to thes5nalysis of thesbis"actrumssThss is discussed n=14oughlynin , , assnck Cothaboration XVII (.106), "aunis therefora notldiscussed further s= thss p"per. I= thss sect=on, wespauth=tsthesres"aus from a vss=ety of stautstical tools. Unless otherw="e s"acified, thes"aalyses "re appl="dltolall four -25" data sep"aas=onlproducts (Comm"autr, NILC, SEVEM, "aunSMICA) "tla given resolut=onlwithnthesacco5"snying commonlmask, "aunsigniftca="a levtls ara asterm"nedlby -25"ss="on withnthesco"autponding res"aus derived from thesFFP8ssim"aations, withntypically 1000lbeing used fos thss pur" sessEstablishing thesconsistency of thesres"aus derived from thesdauthoent -25" data-sep"aas=onltechn"ques is essautial in ordss to be ablelto make robustlclaims aboutlthesstautsticalsnature of thesobserv"d thmpesature oluctuations, "aunpotautial asvias=ons from Gau"&iun"ty.

6681&amsec2"> 4c1. One-dimens=onal moments

I= thss sect=on wesconsidss simple tests of Gau"&iun"ty based on the vss=ance, skewne"&, "aunkurtos=" of thesCMB tempesature maps. Previous "aalyses fouaun5n aaomalouslynlow vss=ancelin thesWMAP sky m"psl(, , aHernserRubi&#nset al. .108; , , Cruzlet al. .101), which was subsequentlynconfirmedsin ans5nalysis of thesassnck .103 1-15 (, , aCISgudoa>).l

, , Cruzlet al. (.101)doa> devtloped thesun=t vss=ancelesuimator tolasterm"nelthesvss=ance, 6681&amimg-inl"ne">, , A., of thesCMB signalson thessky in thespruth="a of no="eclThesnormal"zed CMB m"p, Msup>6681&amsimple-math">u M. LXA., i&lgiven by 6681&amimg-equation">, , 6681&amhabel-eq">(2)A.A.wh"aen Msup>6681&amsimple-math">XiA. is thesobserv"d thmpesature at pixtll Msup>6681&amsimple-math">iA. "aun>6681&amimg-inl"ne">, , A. is thesno="e vss=ancelfos that pixtlclAl"hough thss esuimator is notloputmal, , , Cruzlet al. (.101)doa> "aun>a href="/"autcles/aa/full_html0.106/10/sup>, , aHernserRubi&#nset al. (.108)doa> have demonstr5ted that iu i&lunbiased "aunsufftciautly "ccurat"lfor our pur" sesclThesno="e vss=ancelss esuimated from thesno="e sim"aations for each -25" data-sep"aas=onlalgorithmssThesCMB vss=ancelss tuth>esuimatedlby requiring that thesvss=ance of thesnormal"zed m"p Msup>6681&amsimple-math">u M. LXA. i&lun"ty. Thesskewne"& "aunkurtos=" can thenlbe obtauthd from thesappropr=ately normal"zed m"p.

Figure , , lpruth=ts res"aus for the vss=ance, skewne"&, "aunkurtos=" asterm"nedlfrom thesd-15 "tla resolut=onlof 56681&amsimple-math">′69<">A., Msup>6681&amsimple-math">Nsids69 = .1etti<">A.ssGood "greementlbetwsen the com" data sep"aas=onltechn"ques is fouau, withnsmall discrep"nctes likelylduoltolsenr"tiv"ty tolthesno="e propeautes aad their vss=as=onlbetwsen men=1ds.

, , , , 6681&ambold">Fig.l1A. A="2">Open withnDEXTER

, 6681&ambold">Tablel2A.

Tablel, , cpoa> summ"riz"s theslower-tailsprobabil"ttes, defined "s thespeacentage of MCssim"aations that show a lower vss=ance, skewne"&, or kurtos=" than thesobserv"d m"p, for theses"aalyses. Thesres"aus "re in good "greementlwithn, , aCISgudoa>; thesskewne"& "aunkurtos=" aresco5"st=bleswithnsim"aations, but thesvss=ance ss m"rginally lower thanlin thessim"aationss

Al"hough thesvss=ance ss observ"d to be low, thesres"aus cou>C still be affect"d by thespruth="a of residual foregrouauslat small scales in thesssm"ps, so that thestruesvss=ance wou>C bellower still. We ""&ess thss bysappl=cation of thesesuimator tolthe cleaned frequency m"pslSEVEM-100, SEVEM-143, "aunSEVEM-217. Thesres"aus,nalso pauth=tsdlin Tablel, , cpoa>,narelsrm"lar tolthosenfouaunfos thescomb"ned m"p, al"hough slightly less signiftca=t, which ss most likelylathributablelto higher no="e in thescleaned frequency m"ps.

In conclusion, a srmple stautsticalsa"&essmentlof thesassnck .105 data using skewne"& "aunkurtos=" show&lnolevidsuth for non-Gau"&iun"ty, al"hough anlow vss=ancelis fouau, which weswill readdresssin Sect. , , 5.1.l

6681&amsec2"> 4c2clTesuing the m"ati-normal"ty of thesCMB

Uauea thesa"&umption of Gau"&iun"ty,lthespaobabil"tysutnr"ty funct=on (PDF)lof thes6681&amsimple-math">NA.-dimens=onal pixtlized tempesature map i&lgiven by a m"ativss=asesGau"&iun funct=on: 6681&amimg-equation">, , 6681&amhabel-eq">(3)A.A.wh"aen Msup>6681&amsimple-math">TA. is a vector formedlfrom thesmeasured tempesaturess6681&amsimple-math">T(x)A. over all positionslallowed by thesappl="dlmask, Msup>6681&amsimple-math">NpixA. is thesnumor" of pixtls in thesvector, "aun>6681&amsimple-math">CA. is thescovss=ance of thesGau"&iun fr">C (of sizes6681&amsimple-math">Npix ×lNpixA.)s

Al"hough thescalc"aation of Msup>6681&amsimple-math">TC M. L-liTA. cap>bssachievedlby -2njugatesgrsdiu=tlmen=1ds,lthesevaluation of Msup>6681&amsimple-math">astCA. remains co5"utautonallyldifftc"at fos thesfull assnck resolut=onl"tlHEALPix Msup>6681&amsimple-math">Nsids69 = .1etti<">A.ssAt a lower resolut=on,lthespaoblem ssltractable, "ad thesno="e levtl cap>also besconsidss"d neglig=blesco5"ss"dltolthesCMB signal.lThat implteslthat uauea thesa"&umption of isohropy thescovss=ance matrix Msup>6681&amsimple-math">CA. is fully defined by thesassnck "agular power s"actrum ( Msup>6681&amsimple-math">CA.): 6681&amimg-equation">, , 6681&amhabel-eq">(4)A.A.wh"aen Msup>6681&amsimple-math">CijA. is thescovss=ance betwsen pixtls Msup>6681&amsimple-math">iA. "aun>6681&amsimple-math">jA., Msup>6681&amsimple-math">θijA. is thes"aglelbetwsen them, Msup>6681&amsimple-math">PA. arelthesLegendrespolynom"als,l Msup>6681&amsimple-math">bA. is an effect=velwindow funct=on describing thescomb"ned effectslof thesinstrumentalsbeam "aunpixtllwindow atlresolut=on 6681&amsimple-math">Nsids69A., "aun>6681&amsimple-math"><maxA. is thesmaxim"m m"atipole probeC.

Uauea thesm"ativss=asesGau"&iun hypothesis, thes"rgument of thesex" dataial in Eq. (, , 3) shou>C follow a 6681&amsimple-math">χ<A. duthribut=on withn6681&amsimple-math">NpixA. degrees of freedom, or, equivalautly (fos 6681&amsimple-math">Npix ≫ 1A.) " normal duthribut=on >6681&amimg-inl"ne">, , A..

These 6681&amsimple-math">χ<A. stautstics aresco5"uted fos thesassnck .105 com" data-sep"aased CMB m"ps at 6681&amsimple-math">Nsids69 = 16A. "aun32, thenlco5"ss"dlwithnthesequivalautsquantities derived from thesco"autponding FFP8ssim"aations. For thosencasesnin which thescovss=ance matrix ss ill-conditioned, wesusssaspaincipal co5" datas5nalysis approach tolremove theslowestlaegenerat"leigenvalues of the covss=ance matrix (see, e.g., , , Curtolet al. .101)ssThss proth"&lss equivalautstoladding unco"aul"teunregularizas=onlno="e of "mplttuau 6681&amsimple-math">≈ 1 μA.Kltolthesd-15 befora invhor"on. Thesres"aus of the 5nalysis are pauth=tsdlin Tablel, , udoa> "aunind=cate that thes1-15 "relconsistentlwithnGau"&iun"ty. Wa note that theslower-tailsprobabil"ttes fos thes6681&amsimple-math">NA.-pdf decrease when thesresolut=onlof theldata is increased from 6681&amsimple-math">Nsids69 = 16A. tol32. Howevtr, thislbehaviour is consistentlwithnthat ssen for sim"aations, "aunshou>C notlbenconsidssed tolbe signiftca=t.l

6681&amsec2"> 4c3. N-pss="sco"aul"t=on funct=on"

I= thss sect=on, wespauth=tstests of thesnon-Gau"&iun"ty of thesassnck .105 tempesature CMB m"ps using real-r">ces Msup>6681&amsimple-math">NA.-pss="sco"aul"t=on funct=on". While harmonic-r">cesmen=1ds "re often prefeus"dlover real-r">cesmen=1ds for studying primordial oluctuations, real-r">cesmen=1ds have ap>advantage withnautpectltolsyssemautc errors aaunfosegrouaus, si="a such effectslarelusually local"zed innrealss">ce. Itnis therefora important tol"aalyse theldata in both s">ceslin ordss to highlight diuthoent features.

An 6681&amsimple-math">NA.-pss="sco"aul"t=on funct=on is defined "s thesaverage product of Msup>6681&amsimple-math">NA. tempesatures,smeasured in a fixtdlrel"t=ve orih=tat=onlon thessky, 6681&amimg-equation">, , 6681&amhabel-eq">(5)A.A.wh"aenthesun=t vectors >6681&amimg-inl"ne">, , A. up> Msup>6681&amsimple-math">NA.-pss="spolygon. Uauea thesa"&umption of stautsticalsisohropy, thesssfunct=on" depss= onlynon thesshape "aunsizesof thes6681&amsimple-math">NA.-pss="spolygon, "aunnotlon ius p"autcular position or orih=tat=onlon thessky. Hen"a, the smallestlnumor" of p"aamsters that ua"quely detsrm"nes thesshape "aunsizesof thes6681&amsimple-math">NA.-pss="spolygon is 6681&amsimple-math">2N−3.or">A..

, 6681&ambold">Tablel3A.6681&amsimple-math">NA.-pdf 6681&amsimple-math">χ<A. stautstics derived from thesassnck .105 com" data-sep"aased m"ps at 6681&amsimple-math">Nsids69 = 16A. "aun32.

Thesco"aul"t=on funct=on" arelesuimatedlby srmple product averageslover all sets of Msup>6681&amsimple-math">NA. pixtls fulfilling thesgeomenrtc requirementsssetlby 6681&amsimple-math">θ169,...,θ2N−3.or/b>A. chsracterizing thesshape "aunsizesof thespolygon, 6681&amimg-equation">, , 6681&amhabel-eq">(6)A.A.Pixtllweights >6681&amimg-inl"ne">, , A. cap>bssintroduc"dlin ordss to reducesno="e or m"sk bound8ry effects. Heaentheylaupruth=t m"sking bylbeing setlto 1 for incluaud pixtls "ad to 0 for excluaud pixtls.

Thesshapessof thespolygonssselucted fos thes5nalysis are thespseudo-coll"psed "aunequil"teral configurations for thes3-pss="sfunct=on, "ad thesrhomb"c configuration for thes4-pss="sfunct=on, co5" sedsof twonequil"teral ts=angles that sh"re a commonlsids. Wa use thelsame definition of pseudo-coll"psed "snin , , Eriksenset al. (.105), i.e., an isosceles ts=angle wh"aentheslaugth of thesbasel"nelfalls within thessecond binlof thelsep"aas=onlangles. Theslaugth of theslongor edgesof thests=angle, Msup>6681&amsimple-math">θA., p"aamsteriz"s ius size. Analogously, in thescasesof thesequil"teral ts=angle "ad rhombus, thessizesof thespolygon ss p"aamsteriz"d by theslaugth of thesedge, Msup>6681&amsimple-math">θA.. Nose that thesssfunct=on" "relchosen for easesof implemh=tat=on, notlbecause theynarelbetuth suited fos tesuing Gau"&iun"ty thanlother configurations. For asGau"&iun fr">C, Wick’s theoram (, , Wick 1950) means that thesensemblelaverage of thes4-pss="sfunct=on may benwriautn in terms of thes2-pss="sfunct=on. I= the following, "ll res"aus refeultolthesconnucted 4-pss="sfunct=on, i.e., arelco"aucted fos thss Gau"&iun contribut=on.

Wesusssassrmple 6681&amsimple-math">χ<A. stautsticltolquantifysthes"greementlbetwsen the observ"d a-15 "aunsim"aations, defined by 6681&amimg-equation">, , 6681&amhabel-eq">(7)A.A.Heae, Msup>6681&amsimple-math">ĈN(θi)A. is thes6681&amsimple-math">NA.-pss="sco"aul"t=on funct=on for thesbinlwithnsep"aas=onlangle Msup>6681&amsimple-math">θiA.,n6681&amsimple-math">⟨CN(θi)⟩A. is thesco"autponding average from thesMCssim"aationsensemble, "aun>6681&amsimple-math">NbinA. is thesnumor" of bins used fos thes"aalysis. If >6681&amimg-inl"ne">, , A. is thes6681&amsimple-math">k.or">A.thnsim"aateun>6681&amsimple-math">NA.-pss="sco"aul"t=on funct=on "aun>6681&amsimple-math">NsimA. is thesnumor" of sim"aations, then the covss=ance matrix Msup>6681&amsimple-math">MijA. is given by 6681&amimg-equation">, , 6681&amhabel-eq">(8)A.A.wh"aen Msup>6681&amimg-inl"ne">, , A.. Following , , H"aul"pset al. (.107), welthenlco"auctlfos bias in thesinvhore covss=ance matrix by m"atiplying it by thesfactorl Msup>6681&amimg-inl"ne">, , A.. Below, we quose thessigniftca="a levtl in terms of thesfract=onlof sim"aations withna largor 6681&amsimple-math">χ<A. value than thesobserv"d m"p.

Wes"aalyse thelCMB esuimates "tla resolut=onlof 6681&amsimple-math">Nsids69 = 64por">A.,nthislbeing constr5"nedlby -25"utautonal orm"tations. Thesres"aus "re pauth=tsdlin Fig.l, , cpoa>,nwh"aenwesco5"ss" thes6681&amsimple-math">NA.-pss="sfunct=on" fos thesa-15 "aunthesmean values esuimated from 1000laCssim"aations. Thespaobabil"ties of obtauting values of the 6681&amsimple-math">χ<A. stautsticlfos thesassnck fiducial 6681&amsimple-math">ΛA.CDM modtlnatlleast "snlargo "s thesobserv"d values arelgivtn in Tablel, , 4.l

, , , , 6681&ambold">Fig.l2A.6681&amsimple-math">NA.-pss="sco"aul"t=on funct=on" asterm"nedlfrom thes6681&amsimple-math">Nsids69 = 64por">A.assnck CMB .105 tempesature m"ps. Res"aus "re shown for thes2-pss=", pseudo-coll"psed 3-pss="s(upper leftl"aunright ">Aels, autpect=vely),sequil"teral 3-pss=", "aunconnucted rhomb"c 4-pss="sfunct=onss(lower leftl"aunright ">Aels, autpect=vely). Thesred aot-aot-aot-aashed, or"ago aashed, green aot-aashed, "aunblueslong aashed l"nes co"autpondltolthesComm"autr, NILC, SEVEM, "aunSMICA m"ps, autpect=vely. Nose that thesl"nes lielon top of each other.lThesblacklsolid l"nenind=catesnthesmean asterm"nedlfrom 1000lSMICA sim"aations. Thesshsdsdldark "aunorght greylaugions snd=cate thesco"autponding 68% "aun95% confidsuth augions, autpect=vely. See Sect. , , 4.udoa> fos thesaefinition of thelsep"aas=onlangle Msup>6681&amsimple-math">θA..

A="2">Open withnDEXTER

, 6681&ambold">Tablel4A.6681&amsimple-math">χ<A. stautsticlof thes6681&amsimple-math">NA.-pss="sfunct=on" fos thesassnck fiducial 6681&amsimple-math">ΛA.CDM modtlnatlleast "snlargo "s thesobserv"d values of thelstautsticlfos thesassnck .105 tempesature CMB m"ps withnautolut=onlp"aamster 6681&amsimple-math">Nsids69 = 64por">A.,nesuimated u&ing thesComm"autr, NILC, SEVEM, "aunSMICA men=1ds.

Itnis worthnnoting that thesvslues of the 6681&amsimple-math">NA.-pss="sfunct=on" fos diuthoent "agular sep"aas=ons "re stronglynco"aulaued, and fos thss reason thesfigure" show onlynonespaofile of each funct=on in m"ati-dimens=onal s">ce. Si="a thesesuimated paobabil"ties uakels="olaccou=tsthesco"aul"t=ons,ntheynprovids moresauliablelinformas=onlon thesgoodne"& of fitlbetwsen the a-15 "auna given modtlnthan srmple inspection of thelfigure".

Thesres"aus show excellent -2nsistency betwsen the CMB m"ps esuimated u&ing thesdiuthoent -25" data-sep"aas=onlmen=1ds. No stautstically signiftca=tlasvias=ons of thesCMB m"ps from Gau"&iun"ty 5relfound. I=deed, thesslight prefeusuth for supho-Gau"&iun"ty of thesequil"teral 3-pss=" "aun4-pss="sfunct=onssobserv"d for thes2103 1-15 is now less pronouncud. That may bencausedlby diuthoenceslbetwsen the masks used fos thes"aalysis. I="aresuinghy, thes2-pss="sfunct=on show&lclearlevidsuth of a lacklof structure for largo sep"aas=onlangles. Such behaviour was originally noted fos thesWMAP first-yearld-15 by , , Bennuttset al. (.103), "aunhas subsequentlynbeen discussed atlleugth in thesl"tesature (, , Efstauhiou .104; , , Copiset al. .107, , , .105)ssWeswill return tolthss issuesin Sect. , , 5.2.l

6681&amsec2"> 4c4. Minkowskisfunct=onal"

ThesMinkowskisfunct=onal" (h"aeafter MFs) describe the morphology of fieldssin any dimens=on "aunhaveslong been used "snesuimators of non-Gau"&iun"ty "aunaaisohropy in thesCMB (see e.g., , , Meckelet al. 1994; , , Schmalzing 4-636 Buchert 1997; , , Schmalzing 4-636 Gorskis1998; , , Komatsuset al. .103; , , Eriksenset al. .104b; , , Curtolet al. .107; , , De Troialet al. .107; , , Sphogtlnet al. .107; , , Curtolet al. .108; , , Hikage et al. .108; , , Komatsuset al. .109; , , assnck Cothaboration XXIII .104)ssTheynareladditive fos disjss="saugions of thelsky "auninvss=ant uauea rotations "auntransaations. In thesl"tesature,lthescontours are trsdiutonallyldefined by 5 threshold Msup>6681&amsimple-math">νA., usually givtn in un=ts of thelsky stand8rdlasvias=on ( Msup>6681&amsimple-math">σ0A.)s

Wesco5"ute MFs fos thesaugions cotder aaunhotuth than a given threshold Msup>6681&amsimple-math">νA.ssThus, thesthree MFs, namelysthes"aea Msup>6681&amsimple-math">V0(ν) = A(ν)por">A.,nthespeaimster 6681&amsimple-math">V1(ν) = C(ν)por">A.,n"aunthesgenus Msup>6681&amsimple-math">V2(ν) = G(ν)por">A.,n"reldefined autpect=vely "sn6681&amimg-equation">, , 6681&amhabel-eq">(9)A.A.6681&amimg-equation">, , A.

wh"aen Msup>6681&amsimple-math">NνA. is thesnumor" of pixtls wh"aen Msup>6681&amsimple-math">ΔT/σ0>νA., Msup>6681&amsimple-math">NpixA. is thestotalsnumor" of avail"blelpixtls, Msup>6681&amsimple-math">AtotA. is thestotals"aea of the 5vail"blelsky, 6681&amsimple-math">NhotA. is thesnumor" of co5"sctnhot spots, Msup>6681&amsimple-math">NcotdA. is thesnumor" of co5"sctncotdlspots, "aun>6681&amsimple-math">SiA. is thescontourslaugth of each hot spot.

For asGau"&iun random fr">C in pixtl s">ce,nthesMFs cap>bsswriautn in terms of twosfunct=ons: Msup>6681&amsimple-math">Ak.or/b>A.,nwhich depss=s onlynon thespower s"actrum, "aun>6681&amsimple-math">vk.or/b>A.,nwhich is a funct=on onlynof thesthreshold Msup>6681&amsimple-math">νA. (see, e.g., , , Vanmarckel1983; , , Pogosyanset al. .109; , , Gaylet al. .102; , , Matsubara .100; , , Fantaye et al. .105)ssThes"aalyticalsexpauts=ons "re 6681&amimg-equation">, , 6681&amhabel-eq">(12)A.A.withn6681&amimg-equation">, , A."aun>a name="">6681&amimg-equation">, , 6681&amhabel-eq">(15)A.A.Thes"mplttuau 6681&amsimple-math">Ak.or/b>A. depss=s onlynon thesshape of thespower s"actrum Msup>6681&amsimple-math">CA. th4ough thesrms of thesfr">C Msup>6681&amsimple-math">σ0A. "aunius firstl deriv"t=ve Msup>6681&amsimple-math">σ1A.:n>a name="">6681&amimg-equation">, , A.wh"aen Msup>6681&amsimple-math">ωk.or/b> ≡ π<k/ cpork/ c + 1)A..

Si="a this factorizas=onlss still val"d in thesweakly non-Gau"&iunscase, we cap>use thelnormal"zed MFs, >6681&amsimple-math">vk.or/b>A.,ntosfocus on asvias=ons from Gau"&iun"ty, withna reducedlsenr"tiv"ty tolcosmic vss=ance.

Ap"au from theschsracterization of thesMFs u&ing full-autolut=onltempesature sky m"ps, we also considss res"aus "t diuthoent "agular scales. I= thss p"per, twosdiuthoent "pproaches aresconsidssed tolstudy thesssdegrees of freedom: innrealss">ce via a stand8rdlGau"&iunssmoothing 5ndsdegradation of thesm"ps; 5nd, for the firstltime, in harmonicss">ce by u&ing needlets. Such a completesinvhstigas=onlprovidss an insight reg8rding thesharmonics"auns"st=alsnature of poss=blesnon-Gau"&iunsfeatures astected withnthesMFs.l

, 6681&ambold">Tablel5A.6681&amimg-inl"ne">, , A. "sna funct=on ofnautolut=onlfor the unnormal"zed,>6681&icalsMinkowskisfunct=onal".

First, we apply scale-depss="nt "aalyses innrealss">ce by -2nsidssing thessky m"psl"t diuthoent resolut=on"ssThesthree 6681&icalsMFs – "aea,scontourslaugth, "aungenus – "aesevaluat"dlover thesthreshold r"ago 6681&amsimple-math">−3 ≤ ν ≤ 3A. ip> Msup>6681&amsimple-math">σA. un=ts, withna step of 0.5ssThss providss astotalsof 39 diuthoent stautstics. Thesvslues of these stautstics fos thesassnck 1-15 "rel"ll within thes95% confidsuth augion when co5"ss"dlwithnGau"&iunssim"aations for "ll of thesresolut=on" considssed. As6681&amsimple-math">χ<A. value is co5"uted fos each -25" data-sep"aas=onlmen=1dlby -25buting thes39 stautstics aauntaking s="olaccou=tstheirsco"aul"t=ons (see e.g., , , Curtolet al. .107, , , .108)ssThesco"autponding covss=ance matrix ss co5"uted u&ing 1000lsim"aations. Thes6681&amsimple-math">pA.-value of thss 6681&amsimple-math">χ<A. tesu ss pauth=tsdlin Tablel, , 5doa> fos each -25" data sep"aas=onltechn"que and fos map resolut=on" betwsen 6681&amsimple-math">Nsids69 = 1024A. "aun6681&amsimple-math">Nsids69 = 16A.ssWesfiaunno signiftca=tlasvias=ons from Gau"&iun"ty for "ny of thesresolut=on" considssed.

Then wesconsidss the fourlnormal"zed funct=onal" described above. For evtry scale wesussd 26sthresholds r"aging betwsen 6681&amsimple-math">−3.5A. "aun6681&amsimple-math">3.5A. ip> Msup>6681&amsimple-math">σA. un=ts, exceptlfos Msup>6681&amsimple-math">θ = 640′69<">A. wh"aen13sthresholds betwsen 6681&amsimple-math">−3.0A. "aun6681&amsimple-math">3.0A. ip> Msup>6681&amsimple-math">σA. un=ts wers moresappropr=ate. Tablel, , 6doa> ind=catesnth"t no signiftca=tlasvias=on from Gau"&iun"ty is fouau.l

, 6681&ambold">Tablel6A.6681&amimg-inl"ne">, , A. "sna funct=on ofnautolut=onlasterm"nedlu&ing normal"zed MFs.

, , , , 6681&ambold">Fig.l3A.ce MFs fos assnck .105 data using the fourlcom" data-sep"aased m"ps, Comm"autr (red), NILC (or"ago), SEVEM (green), "aunSMICA (blue); thesgreylaugions, from dark tollight,sco"autpond, autpect=vely,ntos1, 2, "aun>6681&amsimple-math">3σA. confidsuth augions esuimated from thes1000lFFP8ssim"aations proth"&"d by thesComm"autr men=1dssThescolumns from leftlto right co"autpondltolthesneedlet p"aamsters >6681&amsimple-math">j = 4,6,A. "aun8, autpect=vely; the 6681&amsimple-math">jA.thnneedlet p"aamsternhas co5"sctnr6681&amsimple-math">[2<j−lij + 1iA.ssThen Msup>6681&amsimple-math"><c.or/b> = 2<jporA. value ind=catesnthescentral m"atipole of the co"autponding needlet m"p. Nose that tonhavesthessame r"ago "tlalllthesneedlet scales,nthesveautcalsaxisnhas been m"atiplied by 5 factorlthat takes in"olaccou=tsthessteady decrease of the vss=ance of thesMFs "sna funct=on ofnscale.

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Thsrd, westesued MFs on needlet com" datasclThesneedlet com" datas of thesCMB fr">C as defined by , , Mss=nucciset al. (.108)doa> "aun>a name="InR6">, , Baldiset al. (.109)doa> "relgivtn by: 6681&amimg-equation">, , 6681&amhabel-eq">(18)A.A.Heae, Msup>6681&amimg-inl"ne">, , A. dsuotesnthesco5" datas5t m"atipole Msup>6681&amsimple-math">A. of thesCMB m"p Msup>6681&amimg-inl"ne">, , A., i.e., >a name="">6681&amimg-equation">, , 6681&amhabel-eq">(19)A.A.wh"aen Msup>6681&amimg-inl"ne">, , A. dsuotesnthespss="ing direct=on, Msup>6681&amsimple-math">BA. is a fixtdlp"aamstern(usually taken tolbe betwsen 6681&amsimple-math">1A. "aun6681&amsimple-math">2A.) "aun6681&amsimple-math">b(.)por">A. is a smoothsfunct=on suchnthat 6681&amsimple-math">∑ <jbℓ/B<jporA. for "ll Msup>6681&amsimple-math">A.. , , Fantaye et al. (.105) show in a rigorous waynthat a generals"aalyticalsexpauts=on for MFs "t a given needlet scalen>6681&amsimple-math">jA., which deals withnan arbitrary m"sk aauntakes in"olaccou=tsthessph"aicalsgeomenry of thessky, cap>bsswriautn "sn6681&amimg-equation">, , 6681&amhabel-eq">(20)A.A.wh"aen Msup>6681&amsimple-math">t0 = 2A., Msup>6681&amsimple-math">t1 = 0A., "aun6681&amsimple-math">t2 = 4πA. "resautpect=vely thesEuler-Pss=caréschsracteristic, bound8ry laugth, "aun"aea of the full sph"aeclThesquantities >6681&amsimple-math">vk.or/b>A. are thesnormal"zed MFs givtn in Eq. (, , gudoa>), whilelthesneedlet scales"mplttuaus >6681&amimg-inl"ne">, , A. have alsrm"lar form "sn6681&amsimple-math">Ak.or/b>A. but withnthesvss=ances of thesm"p "aunius firstlderiv"t=ve given by 6681&amimg-equation">, , A.Implemh=ting the MFs in needlet s">ce has sevtrals"dvantages:lthesneedlet filternis local"zed innpixtl s">ce,nhe="a thesneedlet com" data m"psl"rs minimally affect"d by m"skeunregions, utpecially at high-frequency >6681&amsimple-math">j;A. "aunthesdouble-local"zas=onlpropeautes of needlets (innreals"aunharmonicss">ce)lallow a m"ch morespaucise, scale-by-scale, interpetation of "ny poss=blesanomal"e". While thesbehaviour of stand8rdlall-scale MFs i" contaminatedlby theslargo cosmic vss=ance of theslow m"atipoles,lthss is no longor thescase for MFs evaluat"dlat theshighestlneedlet scales; innsuchncircumstances,nthesvss=ance of normal"zed com" datas may benshown "oldecrease steadily,nh=tailing a m"ch greaternastect=onlpower in thespruth="a of anomal"e". Finally, "aunmost importanthy, thesneedlet MFs "rs moressenr"tiveltolthesshape of thespower s"actrum than thesco"autponding all-scale MFs.l

, , , , 6681&ambold">Fig.l4A.6681&amsimple-math">χ<A. fos thesassnck .105 Comm"autr (red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)nfosegrouau-cleaned m"psl"aalysed withnthescommonlm"skssThen Msup>6681&amsimple-math">χ<A. ss obt5"nedlby -25buting thesthree MFs in needlet s">ce withnan approp=asescovss=ance matrixssThenhistogaams are for thesFFP8ssim"aations, whilelthesveautcalsl"nes are for thesdatassThenfigure" from leftlto right are for thesneedlet scales >6681&amsimple-math">j = 4,6,A. "aun>6681&amsimple-math">8A., withnthescentral m"atipoles 6681&amsimple-math"><c.or/b> = 2<jporA. shown in each p>Ael.

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Thesneedlet p"aamsters wesusssi= thss 5nalysis are Msup>6681&amsimple-math">B = 2A., Msup>6681&amsimple-math">j = 3,4,5,6,7,tti<">A.ssSi="a thesmasks innpixtl s">ce "rs map-autolut=onlaspss="nt, we also usssdiuthoent masks fos each needlet scalessThesesnew masks aresconstruct"d by m"atiplying theshigh-autolut=onlcommonlm"sk withnthesupgrad"d vhor"on of the 5ppropr=ateslow-autolut=onlcommonlm"sk. For needlet scales >6681&amsimple-math">j = 2A. "aun>6681&amsimple-math">j = 3A., wa use thelcommonlm"sk defined "t 6681&amsimple-math">Nsids69 = 16A., "aunupgrad"d to 6681&amsimple-math">Nsids69 = .1etti<">A.ssSrm"larhy, for theshigher needlet scales, Msup>6681&amsimple-math">j = 2<nporA., wh"aen Msup>6681&amsimple-math">n = 4,5,6,7,tti<">A., wa use upgrad"d vhor"ons of the commonlm"sks defined "t 6681&amsimple-math">Nsids69 = 2<nporA..

Thesres"aus concerning needlet MFs from thesComm"autr, NILC, SEVEM, "aunSMICA fosegrouau-cleaned tempesature m"ps for needlet scales >6681&amsimple-math">B = 2A., Msup>6681&amsimple-math">j = 4,6,8A. are shown in Fig.l, , udoa>. Allncasesnaresco5"uted using 26sthresholds r"aging betwsen 6681&amsimple-math">−3.5A. "aun6681&amsimple-math">3.5A. ip> Msup>6681&amsimple-math">σA. un=ts.

, 6681&ambold">Tablel7A.6681&amimg-inl"ne">, , A. "sna funct=on ofnneedlet scaless

sThenfigure show<he fract=onal duuthoencelbetwsen the assnck 1-15 "aunthesFFP8ssim"aations in aaea (top p>Aels), bound8ry laugth (middle p>Aels), "aungenus (bottom p>Aels) fos diuthoent needlet scalesssThen Msup>6681&amsimple-math">jA.thnneedlet scale has co5"sctnr6681&amsimple-math">[2<j−lij + 1iA.ssAlllthesscales wesconsidssed "relconsistentlwithnthesGau"&iunsFFP8ssim"aations. Thss cap>bssssen in Fig.l, , 4,nwh"aenwesco5"ss" thesa-15 "aunsim"aations6681&amsimple-math">χ<A. values, which aresco5"uted by -25buting thesthree MFs withnan approp=asescovss=ance matrixssThenveautcalsl"nes in thesenfigure" aupruth=t thesa-15, whileltheshistogaam show<he res"aus for thes1000lFFP8ssim"aationsssWesalso show in Tablel, , 7 the 6681&amsimple-math">pA.-values for thesfourlcom" data-sep"aas=onlmen=1ds, "snwelll"snalllneedlet scales wesconsidssed. Despite thesrel"t=vely sm"ll Msup>6681&amsimple-math">pA.-values for some scales,nthesassnck tempesature m"ps show no signiftca=tlasvias=on from thesGau"&iunssim"aations up to 6681&amsimple-math"><max69 = 512A., which -2"autpondsltolthesmaxim"m m"atipole of ourlhighest-frequency needlet m"p.

6681&amsec2"> 4.5ssM"atiscales"aalyses

M"atiscalesa-15 "aalysis is a powerful approach for paobing the fund8mh=tal hypotheses of the isohropy "aunGau"&iun"ty of thesCMBssThenexploration of diuthoent scales (in annalmost indepss="nt m"aner)nnotlonly helpsltoltestlthesspeciftcsprudict=onssof a given scenss=o for thesorigin "aunevolut=onlof theloluctuations, but also is an important checklon thesi5"sctnof syssemautc errors or other contaminatas on thescosmologtcalssignal.

There "rs sevtralswayssof pesforming a m"atiscales"aalysis, thessimplestlbeing tolsmooth/degrade thelCMB m"p to diuthoent resolut=on"ssHowevtr, i= thss sect=on, weswill focus on image proth"&ing techn"ques aul"teuntolthesappl=cat=onlof wavelets "aunmore generalsb"au-p81& filtering kerntls tolthesoriginalsCMB fluctuationsssThes"dvantage of wavelet-likes"aalyseslover scalesaegradation i&lclear:ltheylallow thenexploration of chsracteristics of the a-15 that are aul"teuntolspeciftcs"agular scales. Wavelets have already been extensively ussd in thesstudy of the Gau"&iun"ty "aunisohropy of thesCMB (e.g., , , McEwenset al. .107; , , Vielva .107). I=deed, a wavelet-based (needlet) "aalysis of thesassnck .105 a-15 h"snalready been pauth=tsdlin Sect. , , 4.4.l

We aucalllthat in thes2103 "aalysis, some of the 5pplied esuimators asviased from thesnull hypothesis. I= p"autcular, it wa" asterm"nedlthat thescotdl"aea of the sph"aicalsMextca= hat wavelet (SMHW, , , Msst"nez-Gonzálezset al. .102) coefftciatas at scales of arouau 56681&amsimple-math">°A. yr">Ced "s6681&amsimple-math">pA.-value of 0.3%. I= addition, wesalso fouau an excess in theskurtosis of theswavelet coefftciatas on thessame scales. Previouss"aalysesl(for " review,ssse , , Vielva .110) have suggesued that thes“CotdlSpot” (see Sect. , , 5.7) wa" thesmajor contributoultolthese stautsticalsoutliers.

In what follows, weswill considss the appl=cat=onlof the SMHW, together withnmatched filters for " 2D-Gau"&iunspaofile (GAUSS), and fos general"zed sph"aicalsSavitzky-Golay kerntls (SSG, , , Savitzky 4-636 Golay 1964,nsee Appss=ix a href="/"autcles/aa/full_html0.106/10/sup>, , A).

Thesappl=cat=onlof a filtern6681&amsimple-math">ψ(R,p)por">A. "olassignal on thessky >6681&amsimple-math">S(p)por">A. cap>bsswriautn "sn6681&amimg-equation">, , 6681&amhabel-eq">(23)A.A.wh"aen Msup>6681&amsimple-math">pA. aupruth=ts a given position/pixtl, Msup>6681&amsimple-math">RA. p"aamsteriz"s a chsracteristic scalesfor the filtern(e.g., a wavelet scale), Msup>6681&amimg-inl"ne">, , A. is theswindow funct=on "ssociased withnthesfiltern6681&amsimple-math">ψ(R,p)por">A., Msup>6681&amsimple-math"><max69.or">A. is thesmaxim"m m"atipole allowedlby thesco"autponding HEALPix pixtl"zas=on, "aun6681&amsimple-math">Yℓm.or/b>(p)por">A. is thessph"aicalsharmonicsbasis. Heae, Msup>6681&amsimple-math">sℓm.or/b>A.,nthessph"aicalsharmonicscoefftciatas of the 5aalysed m"p, "relgivtn byn6681&amimg-equation">, , 6681&amhabel-eq">(24)A.A.wh"aen Msup>6681&amsimple-math">dΩ = dθ&inθdφA. "aunthesasterisk dsuotesncomplex conjugas=on. Nose that thesfiltered m"p (or theswavelet coefftciata m"p, if Msup>6681&amsimple-math">ψ(R,p)por">A. is a continous wavelet) conserv"s thesstautsticalspropeautes of thesoriginalsm"p, si="a thesconvolut=onlis a l"near opesation. I= p"autcular, if Msup>6681&amsimple-math">S(p)por">A. is a Gau"&iuns"aunstautstically isohropicsrandom signal, Msup>6681&amsimple-math">ωS.or/b>(R,p)por">A. is also Gau"&iuns"aunstautstically isohropic.

In thespruth=t work, thessignal >6681&amsimple-math">S(p)por">A. c2"autpondsltola tempesature m"p >6681&amsimple-math">T(p)por">A.. Sevtralsstautstics can thep>bssco5"uted from thesderived filtered m"p "sna funct=on ofnthesfilternscale, in p"autcular, thesfirstlmomatas (thesdisphor"on Msup>6681&amsimple-math">σRA.,nthesskewne"& >6681&amsimple-math">SRA.,n"auntheskurtosis >6681&amsimple-math">KRA.), thestotals"aea above/below a given threshold,n"aunthespeaksdistribut=on. These stautstics aresco5"areuntolthesco"autponding res"aus asterm"nedlfrom thesFFP8ssim"aations tolestablish thesdegree of co5"stibil"tyswithnthesnull hypothesis.

6681&amsec3"> 4.5s1. First-orutr momatas of thesm"atiscalesm"ps

For thesthree filters considssed (SMHW, GAUSS, "aunSSG84<, , udoa>por6681&amsimple-math">R(arcm"n) = {A.2, 4, 7, 14, 25, 50, 75, 100, 150, 200, 250, 300, 400, 500, 600, 750, 900, 1050 Msup>6681&amsimple-math">}A.ssThesesscales areschosen tolbe consistentlwithnpreviouss"aalysesssTheyncover a wids "agular r"ago, "aun"aenselect"d so that thesintervals betwsen themsincrease withnscalessNosice that, for " given scale, thesthree filters donnotlcover exactly thessame m"atipole r"ago, si="a th"t depss=s onlthesspeciftcsfilternaefinition. Thss cap>bssssen in Fig.l, , 5doa>: the SMHW is thesnarrowest filter, followedlby SSG84, then GAUSSssThesthree filters have ansequivalentleffect=ve Msup>6681&amsimple-math"><max69.or">A., but diutho in theseffect=ve Msup>6681&amsimple-math"><min<9.or">A.. Ovtrall, thesdiuthoenceslbetwsen the filters become sm"ller withnincreasing effect=ve scalessI= thss p"per, we autho tolbothnthesscale, Msup>6681&amsimple-math">RA., "aunFWHM "snp"aamsters aefining thessiz" ofnthesfilters. For the SMHW, thesesare aul"teunbys Msup>6681&amimg-inl"ne">, , A., wh"aeas for thesGAUSS "aunSSG84sfilters,nthesscale is defined tolbe halfnthesFWHMssThesl"tternaefinition is appropr=atesfor filters that incluau pre-whitening si="a iu ss srmple yet m"tches the 6681&amsimple-math">A.-s">ce b"auwidthnauasonablynwell.

, , , , 6681&ambold">Fig.l5A.6681&amsimple-math">′69<">A. (top) "aun250 Msup>6681&amsimple-math">′69<">A. (bottom)ss

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Following thesprothdure expl5"nedlip> a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>, "fter convolut=onlwithna given filter, the commonlm"sk ss extend"d to omiu pixtls from thes"aalysis that couldlbe contaminatedlby thesm"skssTheselpixtlssintroduce an extrasco"aul"t=on betwsen the a-15 "aunthessim"aations, aegrading thesstautsticalspower of the com"arison withnthesnull hypothesis (see, e.g., , , Vielva et al. .104). For " given scale Msup>6681&amsimple-math">RA., thenexclur"on m"sk ss defined by extending 5n auxil"ary m"sk by 5 distance 6681&amsimple-math">2RA. from ius borutr,nwh"aenthes"uxil"ary m"sk is that p"au of the commonlm"sk aul"teuntolresidual duutuse Galactc emits=on (i.e., thes"uxil"ary m"sk doesnnotlm"sk pss="ssourths).

Thesfollowing figure" aupruth=t thesuppho-tail paobabil"ty (UTP) for " given stautstic, i.e., the fract=onnof sim"aations that yr">C a value equalsto or greaternthan that obt5"nedlfor thesdatassI= fact, "snexpl5"nedlip> a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>, if a given UTPnis largornthan 0.5, a new quantity ss defined "snmUTPn= 6681&amsimple-math">1−UTPA.ssTherefoae, mUTPnis constrained tollie betwsen 6681&amsimple-math">1 /NA. "aun0.5, wh"aen Msup>6681&amsimple-math">NA. is thesnumor" of sim"aations used fos each stautstic.

Figure , , 6doa> pruth=ts the com"arison of thesCMB tempesature m"ps withnthesco"autponding sim"aations for the SMHW, GAUSS, "aunSSG84 filters. Thesfull mits=on assnck 1-15 confirm thesres"aus "lready obt5"nedlwithnthes2103 aulease for tempesature. I= p"autcular, for the SMHW, wesfiaun(i) an excess of kurtosis ( Msup>6681&amsimple-math">≈n0.8%A.) "t scales of arouau 300 Msup>6681&amsimple-math">′69<">A.;n(ii) that thesdisphor"on of theswavelet coefftciatas at thesesscales and at arouau 700 Msup>6681&amsimple-math">′69<">A. issrel"t=vely low ( Msup>6681&amsimple-math">≈n1%A.); 5ndn(iii) that thesdisphor"on of theswavelet coefftciatas at scales below 56681&amsimple-math">′69<">A. ss srgniftca=tly high ( Msup>6681&amsimple-math">≲n0.1%A.).

Thesexcess of kurtosis has been previously "ssociased withnthesCotdlSpot (e.g., , , Vielva et al. .104),n"auntheslow value of the stand8rdlasvias=on of the coefftciatas on largo scales couldlbe aul"teuntoltheslow vas=ance discussed in Sect. , , 5.1.lReg8rding theslargo disphor"on of thescoefftciatas on thessm"llest scales,nthss cap>bssuaueastood either by thespruth="a of residual fosegrouau contribut=ons (extragalactic pss="ssourths) or by incompleteschsracterization of thestrue instrumh=tal noisespropeautes by thesFFP8ssim"aationsssWesexplore thesesposs=bil"ttes withntwo additionalstestssuaueataken withnthesSMHW.

Figure , , 7 show<he srgniftca=ce of the stautstics derived from thesSEVEM-100, SEVEM-143, "aunSEVEM-217 m"psssThesthree 66eaned m"pslyr">C vtry consistentlvslues of the mUTPnfor the stand8rdlasvias=on,sskewne"&, "aunkurtosis of theswavelet coefftciatas, withnonly sm"ll diuthoenceslssen at sm"ll scalesssThis frequency-indepss="nce of the res"aus "rgues against the fosegrouau residuals hypothesis. Figure , , 8 pruth=ts the same stautstics as 5pplied tolan esuimator of the noisespropeautes of thesCMB m"psssThis is derived from theshalf-duuthoencelof theshalf-ring 1-15 sets, which providss thesbest esuimate of the noisespropeautes of thesfull mits=on 1-15 setssHowevtr, si="a therelss still a known mitm"tch in noisespropeautes, "ay conclur"onsswill bs moresqualit"t=ve than quantit"t=ve. Nevtrthele"&, the noisesstudy reveals that, at thessm"llest scales,nthere "rs some discaupa=ctes withnthesFFP8ssim"aations, "aunin p"autcular thenesuimated disphor"on of thesSMHW noisescoefftciatas is higher than prudicted.

, , , , 6681&ambold">Fig.l6A.6681&amsimple-math">RA. for the Comm"autr (red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)nsky m"ps. From leftlto right,nthesp>Aels co"autpondltolstand8rdlasvias=on,sskewne"&, "aunkurtosis res"aus, when asterm"nedlu&ing thesSMHW (top), GAUSS (middle), "aunSSG84 (bottom) filters. Thessquare" aupruth=t UTPnvslues aboven0.5, wh"aeas circles aupruth=t UTPnvslues below 0.5ss

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, , , , 6681&ambold">Fig.l7A.6681&amsimple-math">RA. for the SEVEM-100 (blue), SEVEM-143 (yellow), SEVEM-217 (magenta), "aunSEVEM (green) m"ps. From leftlto right,nthesp>Aels co"autpondltolthe stand8rdlasvias=on,sskewne"&, "aunkurtosisss

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, , , , 6681&ambold">Fig.l8A.6681&amsimple-math">RA. for the Comm"autr (red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)nhalf-ring half-duuthoencelnoisesesuimates. From leftlto right,nthesp>Aels co"autpondltolthe stand8rdlasvias=on,sskewne"&, "aunkurtosisss

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6681&amsec3"> 4.5s2ssThes"aea above/below a threshold

In thescontext of the study of the CotdlSpot, thes"aea above/below a given threshold,n"sna funct=on ofnthesSMHW wavelet scale, has been demonstrateuntolprovids a useful aau robust stautstic (e.g., , , Cruzset al. .105),nsi="a iu ss rathho indepss="nt of "ny m"sking requised. Ournpreviouss"aalysis ( a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>) confirmed that thesCMB tempesature fluctuations exhibit 5n anomalously largo cotdl"aea on scales of arouau 6681&amsimple-math">10°A., which -ap>bssmostly "ssociased withnthesCotdlSpot. Heae, we extend thes"aalysis by including res"aus asrived u&ing thesGAUSS "aunSSG84sfilters.

At " given scale Msup>6681&amsimple-math">RA.n"aunthreshold Msup>6681&amsimple-math">νA., the cold ( Msup>6681&amimg-inl"ne">, , A.) "aunhot ( Msup>6681&amimg-inl"ne">, , A.) "aeas of a filtered m"p "re defined "sn6681&amimg-equation">, , A.wh"aenthe opesatos Msup>6681&amsimple-math">#A. aupruth=ts thesnumor" of pixtlss Msup>6681&amsimple-math">pA. in which thescondition defined betwsen the braceslss sautsuied.

Tablel, , 8 summ"riz"s the res"aus for theshot "auncotdl"aeas asterm"nedlfor thesfourlCMB tempesature m"ps "aalysed withnthescommonlm"sk ("aunius "ssociased exclur"on m"sks)ssThesres"aus "relsrm"lar tolthose obt5"nedlin 2103, withnsome sm"ll diuthoenceslon thosesscales aul"teuntolthesCotdlSpot (betwsen 200 Msup>6681&amsimple-math">′69<">A. "aun400 Msup>6681&amsimple-math">′69<">A.)ssSpeciftcally, the cold "aea ss slightly less signiftca=tlfor sm"ller vslues of Msup>6681&amsimple-math">RA., wh"aeas thes"aomaloussbehaviour remainslfor largornfilternscalesssThesthree filters "relinnreasonables"greem"nt, but, "snexpect"d from Fig.l, , 6doa>,nthesSMHW yr">Cs higher srgniftca=ce levels than the SSG84s"aunGAUSS filters. Howevtr, iu ss worthnaucalling that, for " given scale, thesthree filters "relnotlpaobing exactly thessame m"atipole r"agon"auntherefoae some diuthoenceslshouldlbe expect"d.

In Fig.l, , 9 we plot thes"aeas for thresholds Msup>6681&amsimple-math">ν > 3.0A.,nwh"aenthesthreshold ss defined innun=ts of Msup>6681&amsimple-math">σRA.,nas asterm"nedlfrom thesSEVEM tempesature m"pssThesres"aus for Comm"autr, NILC, "aunSMICA "relinngood "greem"nt withntheseclThesp>Aels autho tolSMHW scales of Msup>6681&amsimple-math">R = .10′69<">A.,n250 Msup>6681&amsimple-math">′69<">A., 300 Msup>6681&amsimple-math">′69<">A., "aun400 Msup>6681&amsimple-math">′69<">A.clThesmost extrem" value (in terms of Msup>6681&amsimple-math">σRA.) fos each "aea ss ind=cated.

, , , , 6681&ambold">Fig.l9A.6681&amsimple-math">ν> 3.0A.nas asterm"nedlfrom thesSEVEM tempesature m"pssFrom top tolbottom, thesm"psl"rs for SMHW scales of Msup>6681&amsimple-math">R = .10A.6681&amsimple-math">′69<">A., Msup>6681&amsimple-math">R = .50A.6681&amsimple-math">′69<">A., Msup>6681&amsimple-math">R = 310A.6681&amsimple-math">′69<">A., "aun6681&amsimple-math">R = 410A.6681&amsimple-math">′69<">A..

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The coldest "aea -2"autpondsltolthesCotdlSpot withnthesminimum value of the wavelet coefftciata at thespositionn6681&amsimple-math">(209°,−57°)A. in Galactic coordinates. Theshottest "aea h"snalready been idsutuuied in thesWMAP 1-15 (e.g., , , Vielva et al. .107). Thesres"aus "relinsenr"tiveltoltheschoicelof CMB tempesature m"p that is adopted. It i&lclear that thessouthern Galactic hemitph"ae yr">Cs moresaaomalousssignature" than the northern onessThesesres"aus confirm thesimportancelof thesCotdlSpot a" thesmost extrem" feature in thes"aalysed sky. Morelinsights aboutnius nature "relprovidsd in Sect. , , 5.7.

, 6681&ambold">Tablel8A.

6681&amsec3"> 4.5s3. Peaksstautstics

Thesstautsticalspropeautes of local extrema (both minima "aunmaxima, which we autho tolcollect=vely a" “peaks”)lprovids annaltern"t=ve approach tolsearch for evi="nce of non-Gau"&iun"ty in the a-15ssS"ch peaks, defined "snpixtlsswhoses"mplttuaus "relhigher or lower than thesco"autponding values for "ll of their neare"t neighbours,ntrace topologtcalspropeautes of thesa-15ssPeakslocations and amplttuaus, "aunvas=oussasrived quantities,nsuchna" theirsco"aul"t=on funct=ons,nhavespreviously been used tolchsracterize thesWMAP sky m"ps by , , Larson 4-636 W"autlt (.104,n, , .105)doa> "aun>a name="InR65">, , Houset al. (.109)doa>.

Thesstautsticalspropeautes of peaks for " stautstically isohropicsGau"&iunsrandom fr">C weresderived by , , Bondl4-636 Efstauhiou (1987)doa>. Thesintegrateunnumor" denr"ty of peaks, Msup>6681&amsimple-math">npk69.or">A. (com" sed of maxima "aunminima withnco"autponding denr"ties >6681&amsimple-math">nmax69.or">A. "aun6681&amsimple-math">nmin<9.or">A.), withn"mplttuaus >6681&amsimple-math">xA.n"bovena -eatain threshold Msup>6681&amsimple-math">ν = x/σA. islgivtn byn6681&amimg-equation">, , 6681&amhabel-eq">(27)A.A.wh"aen Msup>6681&amsimple-math">σA. islthe rms fluctuationn"mplttuau measured on thessky, "aun6681&amsimple-math">γA. islthe s"actralsshape p"aamsternof thesuauealying fr">C. Unchsracteristically cotdl"aunhot tpous "relthen mun"fesued "snextrem" outliers in thespeaksvalues, "auncup>6onstttute evi="nce for non-Gau"&iun"ty os dsvias=on from isohropy.

Heae, we considss the peaksstautstics of thesassnck com" data-sep"aased tempesature m"ps "t 6681&amsimple-math">Nsids69 = 21etti<">A.ssThesm"psl"rs pre-whitened "snauscribsd in Appss=ix a href="/"autcles/aa/full_html0.106/10/sup>, , AssThis suep allows the construction of "n esuimator that is nearly optimal withnautpectltolthesfiducial CMB propeautes. After appl=cat=onlof the commonlm"sk, weighted convolut=ons of the a-15 "rs pesformed withneither SSG os GAUSS kerntls of vas=ablelscalessI= orutr tolavoidspotential contaminat=on by bound8ry effects, thesm"sk ss extend"d by reject=ng pixtlsswithnan effect=ve convolut=onlweight that diuthos from un"ty by more than 12%ssPeaksl"rs extracted from thesfiltered m"p (removing 5nynthat are "djach=t to m"skeunpixtls), theirspositions "aunvalues are aucorutdlfor further "aalysis, "auntheirscum"aative denr"ty funct=on (CDF)nis construct"d by sort=ng peaksvalues. Tablel, , 9 pruth=ts peakscou=ts for thescom" data-sep"aased sky m"ps for sevtralsdiuthoent kerntls aau rupruth=tative filtering scales,ntogether withnthesnumor" of peakslthat are commonltolall m"psssTherelss excellentl"greem"nt betwsen the vas=oussCMB esuimates. Alllstautsticalsinthoencelislthen pesformed by -25"arison of thespeaksdistribut=ons derived from thesa-15 withnequivalently proth"&edssim"aationsssAs an intern"l consistency check,nthespropeautes of thesFFP8ssim"aations "rs found tolbe in "greem"nt withnthesprudict=onssof Eq. (, , 27).

, 6681&ambold">Tablel9A.

Figure , , 10 pruth=ts the distribut=ons of peaks for thesSMICA CMB m"p filtered withntwo rupruth=tative kerntls on scales of Msup>6681&amsimple-math">40′69<">A. "aun>6681&amsimple-math">810′69<">A.sFWHMssTheslower p>Aels show thenempiaicalspeaksCDFs asna funct=on ofnpeaksvalue >6681&amsimple-math">xA., defined for " set of Msup>6681&amsimple-math">nA. peaksl"tlvslues Msup>6681&amsimple-math">{sXi }A.nas 6681&amimg-equation">, , 6681&amhabel-eq">(28)A.A.For plott=ng pur" sesnal da, theshorizo=tal axislss scaled innun=ts of Msup>6681&amsimple-math">σA.ndefined by Eq. (, , 27) "aunderived from thesuauealying mediunsCDF, Msup>6681&amimg-inl"ne">, , A., of the sim"aations. Thesuppho p>Aels show thenduuthoencelbetwsen the observ"dl"aunmediunssim"aat"dlCDFsvalues, Msup>6681&amimg-inl"ne">, , A., withnthesgrey b"aus aupruth=ting thes68.3%, 95.4%, "aun99.7% aug=ons of the sim"aat"dlCDFsdistribut=onsssThesm"ximal value of thss duuthoenceldefinesna Kolmogorov-Smirnov (KS) dsvias=on esuimator: 6681&amimg-equation">, , 6681&amhabel-eq">(29)A.A.This forms the basis of a stand8rdlKSltestlof consistency betwsen the two distribut=onsssAlthoughnthesKSldsvias=on h"sna known limiting distribut=on, wesalso derivenius CDFsdiauctly from thessim"aations.

, , , , 6681&ambold">Fig.l10A.top row &how<he peaksCDF for m"ps filtered withna GAUSS kerntl of Msup>6681&amsimple-math">40′69<">A. FWHMssThesbottom row &how<he co"autponding peaksCDF for up>SSG84skerntl of Msup>6681&amsimple-math">810′69<">A.sFWHMssThess"actralsshape p"aamstern6681&amsimple-math">γA. (see Eq. (, , 27)) islthe best-fitlvsluenfor the sim"aat"dlenremble, as ind=catedlby thescyup>6ircle in Fig.l, , 11.lSrm"larsres"aus "relobt5"nedlfor thesother com" data-sep"aas=onlmen=1dsss

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, , , , 6681&ambold">Fig.l11A.6681&amsimple-math">σA.n"aun6681&amsimple-math">γA. (as defined innEq. (, , 27)), aucovered from 1000lsim"aations, "s ind=catedlby thesbssck dots "aunthe smoothed denr"ty m"p "aunco5"areuntolthosesderived for thesobserv"dlsky (shown by thesreunstar). Thesblue contourslencloses68% "aun95% of thesp"aamsterndistribut=on, "aunthe cyup>6ircle aupruth=ts thesbest-fitlp"aamsters for thesmediunspeaksCDF asterm"nedlfrom sim"aations. Thesuppho p>Ael &how<he peaksCDF p"aamsters for thesSMICA m"p filtered withna GAUSS kerntl of Msup>6681&amsimple-math">40′69<">A. FWHMssTheslower p>Ael &how<he co"autponding peaksCDF for up>SSG84skerntl of Msup>6681&amsimple-math">810′69<">A.sFWHMssSrm"larsres"aus "relobt5"nedlfor thesother com" data-sep"aas=onlmen=1dsss

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, , , , 6681&ambold">Fig.l12A.uppho p>Ael &how<he peaksmeann"mplttuaus for m"ps filtered withna GAUSS kerntl of Msup>6681&amsimple-math">40′69<">A. FWHMssTheslower p>Ael &how<he co"autponding peaksCDF for up>SSG84skerntl of Msup>6681&amsimple-math">810′69<">A.sFWHMssSrm"larsres"aus "relobt5"nedlfor thesother com" data sep"aas=onlmen=1dsssSi="a thesfilternkerntl normal"zation i&lfree,n"aunthespre-whitened m"p to which thesfilternis 5pplied ss dumenr"onle"&, the plotsl"rs essentially in arbitrarynun=tsss

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The tempesature peaksdistribut=ons in Fig.l, , 10 are consistentlwithnGau"&iunspeaksstautstics,n"p"au from assinglesaaomalously cotdlpeakson scales arouau 6681&amsimple-math">810′69<">A.sFWHMssThis co"autpondsltolthespreviously auport"dlCotdlSpot. Althoughnthss exercisesconfirms that thesCotdlSpot is a rare cotdlfeature,n"snalready uoted by , , Cruzset al. (.105)doa> "aunconfirmed i= thss p"per, thesmost "acul"ar chsracteristic of thesCotdlSpot isnnotlius coldne"&, but rathho itsssiz". A more astail"dl"aalysis of ius nature ss pauth=tsdlin Sect. , , 5.7.

Thesprobabil"ty that thesobserv"dlsky exceeds the value of the KSldsvias=on for thesadoptedsfiducial cosmology -ap>bssasterm"nedlby -2u=ting thesnumor" of sim"aations withn Msup>6681&amimg-inl"ne">, , A.ssThes6681&amsimple-math">pA.-values for the KSltestl-25"aring thesCDF of thesobserv"dlsky withnthesmediunspeaksCDF asrived from sim"aations for sevtralsdiuthoent kerntls aau rupruth=tative scales aressumm"riz"dlin Tablel, , 10ssThessrm"larly asrived 6681&amsimple-math">pA.-values for the totalspeakscou=ts aressumm"riz"dlin Tablel, , 11.lMost of the res"aus ind=cate that thestwo distribut=ons "relhighly consistent, withnthesexcept=on of res"aus for thesSSG84sfilter on scales of aboutn6681&amsimple-math">510′69<">A.sFWHMssThis dsvias=on appears tolbe aul"teuntola hemitph"aicalsasymmenry in thespeaksCDFs,n"aunwill bs discussed further in Sect. , , 5.6.

, 6681&ambold">Tablel10A.

, 6681&ambold">Tablel11A.

One -ap>also testlwhether thesobserv"dlvslues of the p"aamsters, Msup>6681&amsimple-math">σA.n"aun6681&amsimple-math">γA. as defined innEq. (, , 27), are consistentlwithnthessim"aationlenremble, uauea thesassumpt=on that thespeaksdistribut=ons in thesassnck a-15 "rs auscribsd by 5 Gau"&iunspeaksCDF. Figure , , 11 demonstrates the consistency of the best-fitlvslues of these p"aamsters, co"autponding tolthespeaksdistribut=ons in Fig.l, , 10, withnequivalentlvslues derived from thessim"aations.

Inspised by thes"aalysis of thesWMAP first-year a-15 ip> a href="/"autcles/aa/full_html0.106/10/sup>, , Larson 4-636 W"autlt (.104)doa> which found fewer extrem" peakslthan expect"d, wesadditionally evsluate whether thesdistribut=ons of maxima "aunminima "aensep"aasely consistent withnsim"aations. Thesmeannof all maxima, "aunthesnegative of thesmeannof all minima, are calc"aat"dlfor the filtered m"p, "aunthesobserv"dlvslues aresco5"areuntolthessim"aat"dldistribut=ons in Fig.l, , 12ssThesobserv"dlminima/maxima means "rs found tolbe in good "greem"nt withnthesfiducial values.

, , , , 6681&ambold">Fig.l13A.C real"zations in which thescoldest peaksis asncotdl"s os coldernthan that observ"d,n"sna funct=on ofnSMHW filternscale for Comm"autr (red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)ss

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The probabil"ty that thescoldest peaksssen on thesskynis consistentlwithnthesadoptedsfiducial cosmology ss evsluatedn"sna funct=on ofnboth filternshape "aunsize by -2u=ting thesnumor" of sim"aations withn Msup>6681&amimg-inl"ne">, , A.ssThesres"aus obt5"nedlfor thesSMHW filternaressumm"riz"dlin Fig.l, , gudoa>. Consistentlbehaviour islssen when thesGAUSS "aunSSG84sfilters are "ppliedssTheserror bars aupruth=t thess"mpltng unceatainty dueltolthesfinitesnumor" of real"zations, "aun"aenasterm"nedlu&ing anbootstraplmen=1dssAs the filters overlaplsubstantially,sdiuthoent pss="s "relhighly coraul"teussThesassnck CMB m"ps are consistentlwithnthesexpectat=ons of " stautstically isohropicsGau"&iunsmoutlclThesmost signiftca=tlasvias=on,sfound atnan effect=ve filternb"auwidthngivtn byn6681&amsimple-math"> = 21.or">A., is attributableltola singlesaug=on on thesskyn– thesCotdlSpot.

6681&amsec3"> 4.5s4ssPeakslocations asna funct=on ofnscale

, , , , 6681&ambold">Fig.l14A.Aels in thestop row &howna Lamor"tlpaoject=onnof thesnorth pole, thesusual full skynMollweiau project=on, "aun" Lamor"tlpaoject=onnof thessouth polessTheslower p>Ael &how<he peaksheights (in perch=tile of thespeaksdistribut=on on theshorizo=tal axis) asna funct=on ofnfilternscale (on thesvtrticalsaxis, in log"rithmicsscale), trunc"teuntollargornscales for 668rity. Circles aupruth=t peaksl(nodes of the graph) coloured according toltheirsperch=tile level, "aunscaled according tolkerntl siz". Bssck linesnaupruth=t edges connect=ng peaksl"tldiuthoent scales (according tolalminimal distance measure). Thescom" datas connecteuntolthescoldest "aunhottest peaks atnanynscale "relhighlighted by thicker edges, "aun"aennavysblue "aundarksreunin colour. Note that there "relthick linesnthat donnotltouch thes0 "aun1sperch=tiles in thesplot view. Thosesedges are connecteuntolextrem" perch=tile values, but at scales sm"ller than thosesshown in thesplot. ThesCotdlSpot isnaupruth=tedlby thesconnecteunnodes that havesthe sm"llest perch=tiles except for thescoarsest scale in thesplot view.

A="2">Open withnDEXTER

The appl=cat=onlof asfilternkerntl of vas=ablelsize tolalmaplextends iu snto what cap>bssconsidssed a “m"atiscale s">ce”,nsuchnthat features on diuthoent scales are aupruth=tedlby a one-p"aamsternfamily of smoothed m"psssThis technique ss often used for feature astect=on "aunmathematicalsmorphology "aalysis. Heae, we introduce asmorphologicalsauscript=on of tempesature m"ps based on thespeaksconnecteune"& graph in m"atiscale s">ce, "aun"pply thss technique tola stautsticals"aalysis of thesassnck CMB a-15ssLikesmost morphologicals"aalyses, iu ss equally appl=cableltolintrinsically non-Gau"&iun m"ps, but here wesfocus on thesGau"&iunsrandom fr">C stautstics and attempt to uaueast"aunwhat features of thesCMB tempesature m"p are autponsiblenfor the CotdlSpot.

To construct asm"atiscale rupruth=tation, westrace theslocation of thespeaks in thessmoothed, whitened CMB m"p a" thessmoothing scale ss vas=edssAs the smoothing scale sncaeases, peaks mergon"aunthe totalspeakscou=tsaucaeasesssLinking closest peaksneighboursnin position-scale s">ce, from thesfinest tolthescoarsest rutolut=on,lpaoduces an acycl=c graph that aut"ps"aat"<he peaks“mergorntree”lhistory a" thesscale ss vas=edssAssumm"rynof all the peakspositions "aunCDF r"aks for thesSSG84sfilter kerntl on scales r"aging from 6681&amsimple-math">120′69<">A.stol6681&amsimple-math">1200′69<">A.sFWHM islshown in Fig.l, , 14ssThespeaksl"rs aupruth=tedlby discs of vasying size (reflect=ng thesfilternscale) "auncolour (reflect=ng thespeakstempesature r"ak), withnpeaks atnall scaleslpaojected ontola singlesm"pssTheslower p>Ael &how<he peakslinkage graph on thescoarsernscales;nfor the stautsticals"aalysis 81nfilternscalesl"rs used, log-s">ced from 6681&amsimple-math">120′69<">A.stol6681&amsimple-math">1200′69<">A.ssPeakslof thessame type (i.e., maxima to m"xima "aunminima to minima) "aeslinkeuntolthesclosest peakson thescoarsernscale according tolaldistance measure, Msup>6681&amsimple-math">ds2 + df2A.,nwh"aen Msup>6681&amsimple-math">ds2A. islthe menric on thesun=t tph"ae, "aun6681&amsimple-math">df2A. islthe duuthoencelof peakstempesature r"aks (but only ifnthat distance is within asprudsterm"nedlfract=onnof thesfilternscale FWHM).

, , , , 6681&ambold">Fig.l15A. A="2">Open withnDEXTER

The res"auing peakslinkage graph islthen "aalysed for 6onnecteune"&ssThessrmplest quantif=ablelmeasure islthe node-augreendistribut=on, shown in Fig.l, , 15 for SMICAssThesnode-augreendistribut=on is highly peaked atn2; thss pop"aationlco"autpondsltola singlespeaksbeing tr>ced acro"& m"atiplelscale&ssPre-whitening effect=vely ascoraul"tes thesGau"&iunsm"p acro"& diuthoent scales, so that thesres"auing nodesaistribut=on &how&la sizeablelpop"aationlof augreen0 "aun1snodes. When co5"areuntoltheslinkage graphs derived from thessim"aationlenremble, the node-augreendistribut=on of thespeakslinkage graph asrived from assnck CMB a-15nis consistent, withn" slight excess in nodescou=ts of augrees 5 "aun6.

6681&amsec"> 5ssAaomalies in thesmicrowave sky

The previousssect=on established theslsck of evi="nce for signiftca=tlnon-Gau"&iun"ty in the assnck a-15. Heae we considss sevtralsimportantsaaomalies that weresoriginally astected in thesWMAP sky m"ps, "aunl"ternconfirmed i= thes"aalyses auscribsd in a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>. Manynof these are connecteuntolevi="nce for a vioaationlof isohropy, os tola pauthor"dldirect=on, i= thesCMBssTests that involve dipolarspower asymmenry,neither diauctly os vialmeasures of airect=onality, are collecteuntogether in Sect. , , 6doa>ssI= thss sect=on we considss only thosesaaomalies notldiauctly aul"teuntoldipolarspower asymmenry.

Thesmicrowave sky ss intrinsically stautstically un"sohropicsdueltolour mot=on withnautpectltolthesCMB re"t faamsssThesres"auing Dopplernboosting effect, introducedlin Sect. , , 1, was astected in thes2103sassnck a-15 ( a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Cotlaboaas=onlXXVIIs2104). For com"letene"&, Appss=ix a href="/"autcles/aa/full_html0.106/10/sup>, , B aupeats thes"aalysis with the assnck full misr"on a-15 set, thoughnbased only on thesfull veloc"ty esuimator (6681&amsimple-math">β.or">A.), which islthe sum of thesmoduaationl"aunthe abhorationlcontribut=onsssHowevtr, si="a theseffects of Dopplernboosting "relnow included in thessim"aations used for that aaalysis, this constttute&la consistency check for thisnaulease. Morelimportantly,ssi="a both thesa-15 "aunsim"aations now include theseffect, iu ss notlnece"&"rynto considss deboost"dld-15 ip>manynof the studiesnauport"dlh"ae, unlikesin a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa> (althoughnonesexcept=on in Sect. , , 6.4>makes usenof unboost"dlsim"aations tolsearch for thesfrequency-depss="nt signaturenof the effect in thesSEVEM-100, SEVEM-143, "aunSEVEM-217 sky m"ps)ssHowevtr, welnote that some c"relmust be taken dueltolthesabsencelof thesabhorationlcontribut=on in thessim"aationsssI=deed, thisnleadsltolthesslightly, but notlalarmingly, low PTE for 6681&amsimple-math">β.r/b>||69.or">A. in Appss=ix a href="/"autcles/aa/full_html0.106/10/sup>, , BssHowevtr, welnotsexpectnanynimpact on thesres"aus pauth=tsdlin thss sect=on.

Befoae pruth=ting our res"aus, we auturnltolthesisrue of asposteriorilco"auct=on, which p"autcle physicists autho tolaslco"auct=ng for thes“look-elsewh"aeneffect”l(LEE)ssSi="a there "relmanyntests that cap>bsspesformed on the a-15ntollook for a vioaationlof stautsticalsisohropy, wesexpect some tolind=cate astect=ons at, for ex"mple, roughlyn6681&amsimple-math">3σA.nlevels,ssi="a evtn " stautstically isohropicsCMB sky ss a real"zation of "n uauealying stautsticalsprocess co"autponding tolmanynindepss="nt random vas=ables. Howevtr, inlthesabsencelof an exist=ng theoaeticalsfaamswork (i.e., a physicalsmodel) tolprudictnsuchnaaomalies, iu ss diutic"aultolinterprutltheirssrgniftca=ce. It i<hen nece"&"ry, "aunequally ch"lleaging, toladdress thesquestion of hownoften suchnastect=ons wouldlbe found for stautstically isohropicsGau"&iunssktes. Unfortunasely, iu ss notlalway&lclear how tolanswer thss question.

Theae will alway&lbe asaugreenof subject=v"ty when deciding exactly how tolassess thessrgniftca=ce of these types of features in the a-15ssAs an ex"mple, onescouldlargue that theslargo-scale dipole moduaationlsignal wessee is coming speciftcally from supes-Hubble modes, in which -assspesforming 5n LEElco"auct=on for dipole moduaationlthat couldlhavesbeen ssen on sm"ll scales (6681&amsimple-math"> ≳ 100A.) wouldlnotlmake senre. Models for suchna supes-Hubble mod"aationlexist and an ex"mple was ex"mined inn, , assnck Cotlaboaas=onlXX (.116)doa>,nthesconclur"on being that thesmodel couldlonly explain p"au of the aipole moduaationl"aunthat thesalloweunp"au was pesfuctly consistentlwithncosmicsvas=a=ce.

In thss p"per, wesadoptla paagmatic approach. When therelss alclear mech"n"sm for doing so, wesattempt to co"auct for thes“m"atiplic"ty of tests”,nor thespossiblenway&lin which an anomalouslsignal might havesbeen astected but was not,n"sna consequencelof any asposterioril(a-15-driven)schoiceslmadesin search=ng for itssI= suchncases, " strong depss="ncelof thessrgniftca=ce on thesco"auct=on wouldlind=cate that welshouldlbe caut=ous aboutnthesunco"aucted res"au. When suchnaa obvioussco"auct=on ss notlpossible, we clearly asscribslthe menhodology "pplied tolthesa-15 "aunius limitationsssWith this 5pproach, wesalso aucognize that aay stautsticals"ssessm"nt ss p"rtially subject=ve, including thosesthat pur" rt to co"auct for thesLEE.

Althoughnmanynof the observ"dleffects auscribsd in this 5aunthesnext sect=on may elude theoaeticalsprudict=on today, we cont=nueltolhighlight them si="a therelss alreallpossibil"ty that thessrgniftca=ce of onesor moae might sncaease atnanl"terna-1e, perh"ps when polar"zation a-15 "rs included in thes"aalysis, "aunleadltolnewlinsights snto early Un=verse physicsssAltern"t=vely,ssuchnobservations may diauctly mot=vate the construction of models that cap>make prudict=onssfor features that cap>bsssought in newla-15 setsssThis ss p"rtic"aarly thes-asssfor upomalies on theslargost ang"aar scales, which may havesa speciftc connect=onltolinfaation.

6681&amsec2"> 5s1. Vas=a=ce,sskewne"&, kurtosis

Previouss"aalyses of thesWMAP a-15 ( a href="/"autcles/aa/full_html0.106/10/sup>, , Monteserín et al. .108; , , Cruzset al. .101; , , Gruppuso et al. .103) havesauport"dlthat thesvas=a=ce of thesCMB sky ss lower than that of sim"aations based on thes6681&amsimple-math">Λ69<">A.CDMsmoutlcl a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa> confirmed this, "aunprop sed aspossiblenexplanation of thes5ppaoent inco5"atibil"ty of the observ"dlvas=a=ce withn" fiducial cosmologicalsmodel that hassbeen asterm"nedlfrom thessame a-15 setssSpeciftcally, whilst thesmap-based vas=a=ce ss dominatedlby -2ntribut=onslfrom largo ang"aar scales on thessky, the cosmologicalsp"aamsternfitsl"rs aul"t=velylinsenr"tiveltoltheseslow-orutr 6681&amsimple-math">A.-modes, "aun"aeninsteadllargoly aominatedlby scales co"autponding tol6681&amsimple-math">> 50A.ssTherefoaenthesvas=a=ce of thesm"p appears tolbe anomalous, si="a therelss aldearthnof largo-ang"aar-scale power co5"areuntolthesmoutlcl /p>

In Sect. , , 4.1, we again confirmed the pruth=ce of low vas=a=ce sn the a-15ssHeae, we extend thes"aalysis tolinvestigate which ang"aar scales are autponsiblenfor the low vas=a=ce by "pply=ng thesun=t vas=a=ce esuimator to lower rutolut=onscom" data-sep"aased m"ps, speciftcally thosesfrom 6681&amsimple-math">Nsids69 = 102469<">A.stol6681&amsimple-math">Nsids69 = 16.or">A., withnthesco"autponding commonlm"sks, "aunthen co5"ar=ng thesres"aus withnthosesdeterm"nedlfrom 1000lMCnsim"aations. Thesres"aus "relshown in Fig.l, , 16.

, , , , 6681&ambold">Fig.l16A.top p>Ael),sskewne"& (ch=tae p>Ael),s"aunkurtosis (bottom p>Ael) obt5"nedl"tldiuthoent rutolut=onslfrom thesComm"autr (red), NILC (or"ago), SEVEM (green), "aunSMICA (blue) sky m"ps.

A="2">Open withnDEXTER

All of the com" data-sep"aas=onlmen=1ds that welconsidss yr">Csvtry consistentlres"aus which ind=cate an incaeasingly anomalousllow vas=a=ce at lower rutolut=ons, withntheslower-tail paobabil"ty aeach=ng alminimum value of 0.5% "t 6681&amsimple-math">Nsids69 = 16A.ssWenthen considss the impact of aspossiblenlook-elsewh"aeneffect by evsluat=ng thesminimum lower tail paobabil"ty of eachssim"aationlirautpective of thes6681&amsimple-math">Nsids6969<">A.srutolut=onsat which it occurs. By co5"ar=ng thesdistribut=on of these values withnthau of the a-15, we inferlthat thespaobabil"ty ss slightly weakeneuntola value of aboutn1%. Thesesres"aus "relco5"atible withn" lsck of power on largo ang"aar scalesssSi="a thesvas=a=ce esuimator is heavily weighted towardsllow-6681&amsimple-math">A. modes, this h"snan incaeasingnimpact on thesobserv"dlvas=a=ce when going from high to lowsrutolut=onssky m"ps. Conversely, the skewne"& "aunkurtosis are consistentlwithnthessim"aations, "lthoughntherelss some ind=cat=onlof asweaksscale-depss="ncel(albeit at lowssrgniftca=ce).

Wesalso investigate the stabil"ty of thesres"aus "t 6681&amsimple-math">Nsids69 = 16A. withnautpectltolthespossiblenpruth=ce of rutidualsfoaegrouauslby -2nsidss=ng twosadditionallm"sks obt5"nedlby extend=ng thesedge of thes6681&amsimple-math">Nsids69 = 16A. commonlm"sklby 56681&amsimple-math">°69<">A. "aun96681&amsimple-math">°69<">A., auducing thesusablelskylfract=onnfrom 58%ltol48% "aun40%, autpectivelyssWenthen ro-apply thesun=t vas=a=ce esuimator to the low rutolut=onscom" data-sep"aased m"pslwithntheselm"sks "aundeterm"ne thesvas=a=ce,sskewne"&, "aunkurtosis values (see Tablel, , 12).

, 6681&ambold">Tablel12A.6681&amsimple-math">Nsids69 = 16A. com" data-sep"aased m"pslobt5"nedlwithnthescommonlm"skl"auntwo extend"d versions thereofss

The res"auslfrom 48% of thessky aeveallthau only 1ssim"aationlin 1000lis found tolbe moae anomalousl(i.e., exhibit lower vas=a=ce) than thesobserv"dlm"pssInsaddition, both thesskewne"& "aunkurtosis become moae co5"atible withnthes6681&amsimple-math">Λ69<">A.CDMsmoutlclWithnthesmoae aggressive m"sk, theslower-tail paobabil"ty slightly sncaeases again. Howevtr, givtn theslimiteunnumor" of pixtlssinvolved in thes"aalysis, thss shift may be aul"teuntoltheseffects of s"mple vas=a=ce.

Ovtrall, our res"aus may be explainedlby thespruth=ce of allow-vas=a=ce anomaly in thesprimordial CMB signal – thesstabil"ty of theslow-vas=a=ce srgniftca=ce argues againstsfoaegrouau contaminat=on being autponsiblenfor the lsck of observ"dlpowerssThis ss reinforcedlby thesaucaease in vas=a=ce when aug=ons closestolthescommonlm"sklborutrs,nwh"aenfoaegrouau rutiduals are most likely tolbe observ"d,n"renomitted from thes"aalysis.

6681&amsec2"> 5s2. N-pss=" coraul"t=onnfunct=on upomalies

6681&amsec3"> 5s2.1. Lsck of largo-angle coraul"t=ons

We firstnaeassess theslsck of coraul"t=onnssen in thes2-pss=" ang"aar coraul"t=onnfunct=on ut largo ang"aar sep"aas=ons asnauport"dlin Sect. , , 4.3, "aunpreviously uoted for both WMAP 5aunthes2103sassnck tempesature m"ps ( a href="/"autcles/aa/full_html0.106/10/sup>, , Bennett et al. .103; , , Copi et al. .105). Wesattempt to quantify thss lsck of structurs using thesstautsticnprop sed by , , Spergolset al. (.103)doa>: 6681&amimg-equation">, , 6681&amhabel-eq">(30)A.A.wh"aen Msup>6681&amsimple-math">Ĉ269(θ)A. islour esuimate of thes2-pss=" coraul"t=onnfunct=on. Generally, thesuppho limit on thesintegralshassbeen taken to co"autpondltola sep"aas=onlangle of 606681&amsimple-math">°69<">A., possibly (as uoted by , , Copi et al. .109) mot=vatedlby thesCOBE-DMR 4-year res"aus (, , Hinshaw et al. 1996). Inspect=onnof thestop p>Aelnof Fig.l, , 2doa> suggests that the assnck 2-pss=" funct=on lies closestolzero betwsen 6681&amsimple-math">81°69<">A. "aun6681&amsimple-math">170°69<">A., but for 6onsistency withnpreviousswork we co5"ute the stautsticn Msup>6681&amsimple-math">S1 / 26969<">A., for 6681&amimg-inl"ne">, , A.ssThesres"aus aae pruth=t"dlin Tablel, , gudoa>. We fiaunthat thesd-15 ipdeed &howna lsck of coraul"t=ons on largo ang"aar scales, withn" srgniftca=ce consistentlwithnthat fouau by , , Copi et al. (.115)doa> (althoughnnote that the senre of thes6681&amsimple-math">pA.-values diuthos betwsen the p"pers).

, 6681&ambold">Tablel13A.6681&amsimple-math">S1 / 26969<">A. "aun6681&amimg-inl"ne">, , A. stautstics for the i>assnck fiducial 6681&amsimple-math">Λ69<">A.CDMsmoutl ut least asnlargo as thesobserv"dlvalues of the stautsticnfor the i>assnck .115 tempesature CMB m"ps withnautolut=onsp"aamstern6681&amsimple-math">Nsids69 = 6469<">A., esuimated using thesComm"autr, NILC, SEVEM, "aunSMICA m"ps.

Possiblencr"ticisms of thes6681&amsimple-math">S1 / 26969<">A. stautsticninclude that it hassbeen assrgned asposteriorilto testlfor a lsck of largo-angle coraul"t=ons, "aunthat it does notlaccou=tsfor the high augreenof coraul"t=onnbetwsen binsl"tldiuthoent ang"aar scalesssWe -ap>address thesesconcerns, "t least in p"au,lby -2nsidss=ng asmoduuied version of the commonly used "aunwell uaueastood Msup>6681&amsimple-math">χ2A. stautsticnused in previoussstudiesssInsorutr to testlthessame hypothesis asnthes6681&amsimple-math">S1 / 26969<">A. stautsticn– that there "relno coraul"t=ons on scales largornthan some ang"aar cut-offn– we donnotlctnan avtraged 2-pss=" funct=on when co5"ut=ng thes Msup>6681&amsimple-math">χ2A., i.e., ws use " stautstic defined asn6681&amimg-equation">, , 6681&amhabel-eq">(31)A.A.wh"aen Msup>6681&amsimple-math">imin6969<">A., Msup>6681&amsimple-math">imax6969<">A. denote the index of the binslco"autponding tolthesminimum "aunmaximum value of the sep"aas=onlangless Msup>6681&amsimple-math">θmin6969<">A. "aun6681&amsimple-math">θmax6969<">A., autpectivelyssIn this 5aalysis, wesadoptl Msup>6681&amsimple-math">θmin69 = 61°69<">A. "aun6681&amsimple-math">θmax69 = 181°69<">A..n6681&amsimple-math">Mij.or">A. is the covas=a=ce matrixngivtn bynEq. (, , 8), esuimated using MCnsim"aationslco"autponding tolthesfiducial 6681&amsimple-math">Λ69<">A.CDMsmoutl. Thesres"aus "relshown in Tablel, , gudoa>. Thessrgniftca=ce level of thes5nomaly is slightly sm"ller for the Msup>6681&amimg-inl"ne">, , A. stautstic than that asrived withn Msup>6681&amsimple-math">S1 / 26969<">A.ssWe note that this stautsticnis closely aul"teuntolthes Msup>6681&amsimple-math">A(x)A. measure prop sed by , , Hajiuns(2007).

A further potentialncr"ticism of thes6681&amsimple-math">S1 / 26969<">A. stautsticnaul"tes tolthesasposteriorilchoice of 606681&amsimple-math">°69<">A.nfor the sep"aas=onlangle that asl"ne"tes thesinterest=ng aug=on oflbehaviour of the coraul"t=onnfunct=on. Wa thereforelconsidss the generaliz"dlstautsticn Msup>6681&amsimple-math">S(x)A. "aunco5"ute itslvalue for "ll values of Msup>6681&amsimple-math">xA., both for the a-15 "aunfor the sim"aations. Then, for eachsvalue of Msup>6681&amsimple-math">xA., wendeterm"ne thesnumor" of sim"aations withna higher value of Msup>6681&amsimple-math">S(x)A., "aunhe=ce snfss the most signiftca=tlvalue of the stautsticn5aunthessep"aas=onlangle that itlco"autpondsltossHowevtr, si="a suchnaa "aalysis ss senr"tiveltolthesLEE, wendefinena global stautstic to evsluate thestruessrgniftca=ce of the res"au. Speciftcally, we aupeat thespaoceduaenfoa eachssim"aation, "aunsearch for theslargost paobabil"ty srautpective of thesvalue of Msup>6681&amsimple-math">xA.sat which it occurs. Thesfract=onnof thesespaobabil"ties higher than thesmaximum paobabil"ty fouaunfor the a-15 definesna global 6681&amsimple-math">pA.-valuessAs ssen in Tablel, , gudoa>, this co"autpondsltolvalues of orutr 98% for "ll of thesCMB esuimates.

Thespreviouss"aalyses essentially testlhownconsistentlthesobserv"dl2-pss=" coraul"t=onnfunct=on a-15nis withn" lsck of coraul"t=ons on largo ang"aar scales, in p"auic"aar for sep"aas=onlangless Msup>6681&amsimple-math">θ> 61°69<">A..nAnconventionall Msup>6681&amsimple-math">χ2A. stautsticnallows us to testlthesconsistency of thss quantity withnthesprudict=onssof thes6681&amsimple-math">Λ69<">A.CDMsmoutl. In this -ass, the stautsticnis defined asninnEq. (, , 7), except that welconstr>in thesco5"utations tolthosesbinslthat co"autpondltolthesintervals defined byn6681&amsimple-math">θ< 61°69<">A. "aun6681&amsimple-math">θ> 61°69<">A..nThesres"aus of thesesstudiesn"relshown in Tablel, , 14ss

, 6681&ambold">Tablel14A.6681&amsimple-math">χ2A. stautsticnfor the i>assnck fiducial 6681&amsimple-math">Λ69<">A.CDMsmoutl ut least asnlargo as thesobserv"dlvalues of the stautsticnfor the i>assnck .115 tempesature CMB m"ps withnautolut=onsp"aamstern6681&amsimple-math">Nsids69 = 6469<">A., esuimated using thesComm"autr, NILC, SEVEM, "aunSMICA m"ps.

Thes"aalysis for 6681&amsimple-math">θ< 61°69<">A. ind=cates that the observ"dl2-pss=" funct=on ss algood match tolthesmeann2-pss=" funct=on prudictedlby thes6681&amsimple-math">Λ69<">A.CDMsmoutl. Moreovtr, for 6681&amsimple-math">θ> 61°69<">A. the res"aus suggest that thespaoblem is that thesfitlof the a-15ntolthesmoutl is toolgood, "aunthis ss evtn moae pronouncedlfor "a "aalysis sn thesfull sep"aas=onlangle r"ago.

Ovtrall, thestests ind=cate an unusually good fitlof the observ"dl2-pss=" funct=on both tolzero andltolthesprudict=onssof thes6681&amsimple-math">Λ69<">A.CDMsmoutllfor "aglessabovts6681&amsimple-math">61°69<">A..nThisspaoblem may be aul"teuntolthesf>ctnthat thestheoaeticalsvas=a=ce for the best-fitlmoutl is largornthan thesobserv"dlvalue ut largo scales, so that thessim"aations based on this model that havesbeen used in "ll of thesstautsticalstests may ovtresuimate the vas=a=ce of thes2-pss=" funct=on.

6681&amsec3"> 5s2.2ssHemitph"aicalsasymmenry

We now turnltolanaeassessm"nt of thes5symmenry betwsen the real-s">cen6681&amsimple-math">NA.-pss=" coraul"t=onnfunct=onsnco5"uted on hemitph"aesnauport"dlpreviously for the WMAP ( a href="/"autcles/aa/full_html0.106/10/sup>, , Eriksen et al. .105) "aunassnck .113 tempesature m"ps ( a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>). Wesinitially focus thes"aalysis on theshemitph"aesndeterm"nedlsn theseclipt=c coordinate faams for which theslargost asymmenry was observ"dssHowevtr, welalso carrynoutlthesco"autponding calc"aations in other aulevantlrethoencelfaamss,ssuchnas thosesdefinedlby thesDopplernboost (DB,ssee Sect. , , 6.4, Appss=ix a href="/"autcles/aa/full_html0.106/10/sup>, , B, "aun, , assnck Cotlaboaas=onlXXVIIs2104)n5aunthesaipole moduaationl(DM,ssee Sect. , , 6.2) airect=onsssWe uselthessame configuaas=ons of thes6681&amsimple-math">NA.-pss=" funct=onsnas auscribsd in Sect. , , 4.3ssHowevtr, h"aenthesfunct=onsnarelnotsavtraged ovtr thesfull sky "aundepss= on alchoice of speciftc direct=on, so they constttute tools for studying stautsticalsisohropy r"ther than non-Gau"&iun"ty (, , Fhor"ira 4-636 Magueijo 1997).

Aslin Sect. , , 4.3, wes"aalyse thesCMB esuimates atnanautolut=onsof Msup>6681&amsimple-math">Nsids69 = 6469<">A. "aunquantify theirs"greem"nt withnthesfiducial cosmologicalsmodel u&ing an Msup>6681&amsimple-math">χ2A. stautstic.nThesres"aus asterm"nedlfrom thes i>assnck .115 tempesature a-15nfor the eclipt=c hemitph"aesn"relshown in Fig.l, , 17doa>ssIf welconsidss that the 6681&amsimple-math">χ2A. stautsticnitself -ap>actn"sna measure of fluctuationllevel, then "symmenry betwsen the two measured hemitph"aesncap>bssquantif=edlby thesaas=o of the corautponding 6681&amsimple-math">χ2A. valuesssThespaobabil"ties of obt5"ning values of thes6681&amsimple-math">χ2A. stautsticnor aas=o for the i>assnck fiducial 6681&amsimple-math">Λ69<">A.CDMsmoutl ut least asnlargo as thesobserv"dlvalues "relgivtn in Tablel, , 15ssSi="a we donnotlhavesanysprudict=onssconcerning thesbehaviour of algivtn hemitph"ae, inlthes-asssof thes6681&amsimple-math">χ2A. aas=os we paovi=" thesco5"lem"nt"rynpaobabil"ties of thes2-tail"dlstautstic.

Thessrgniftca=ce levels of thes3- "aun4-pss=" funct=onsninlthesnorthern hemitph"aenarelnominally vtry high, exceeding 99.9% for the pseudo-coll"ps"dl3-pss=" funct=on. Howevtr, proper interprutationlrequires that onesaucognize that thes"aalysis ss affectedlby a posteriorilchoices for the smoothing scale aau ruthoencelfaamssdefining theshemitph"aesssThis typtcally leadsltolaa ovtresuimation of the srgniftca=ce of the res"ausssAccou=t=ng for suchneffects requires the aupetition of thes5aalysis for "ll possiblenruthoencelairect=ons aau also for d-15 "t other autolut=ons. Unfortunasely, becausenof co5"utational limitations, suchnaa extend"d "aalysis ss notlpossible for these higher-orutr stautstics. Nevtrthele"&, thesobserv"dlproperties of thesassnck a-15 are consistentlwithnalclear lsck of fluctuationsninlalairect=on towardslthesnorth eclipt=c polessHowevtr, thes6681&amsimple-math">χ2A.-aas=o stautsticnind=cates a slightly sm"ller srgniftca=ce level for the asymmenry,nnotsexceeding 99% for "nynof the 6681&amsimple-math">NA.-pss=" funct=ons.

The res"auslfor the Msup>6681&amsimple-math">NA.-pss=" coraul"t=onnfunct=onsndeterm"nedlsn thesDB aau DMsrethoencelfaamsslfor thesSMICA m"p "relshown in Fig.l, , 18n5aunthespaobabil"ties aae pruth=t"dlin Tablel, , 16. Note that thespositiveshemitph"aenfor the eclipt=c ruthoencelfaamssco"autpondsltolthe southern hemitph"aenin thespreviousstable. Whilst theslargost asymmenry is ssen in eclipt=c coordinates, " substantialnasymmenry is pruth=t also for thesDMlairect=onssThis cap>bssexplainedlby thesf>ctnthat thesDMlairect=on is more closely al"gnedlwithnthessouth eclipt=c pole (withn" sep"aas=onlof arouau 476681&amsimple-math">°69<">A.) than thesDB airect=on is. For thesDB airect=on we donnotlfiaunaay signiftca=tlasymmenry. The equivalentlres"aus for Comm"autr, NILC, "aunSEVEM are consistentlwithnthosesshown h"ae.

, , , , 6681&ambold">Fig.l17A.6681&amsimple-math">NA.-pss=" coraul"t=onnfunct=onsndeterm"nedlfrom thes Msup>6681&amsimple-math">Nsids69 = 6469<">A.assnck CMB .115 tempesature esuimates aaunthescorautponding means esuimated from 1000lsim"aations. Res"aus "relshown for thes2-pss=", pseudo-coll"ps"dl3-pss=" (uppho left and right p>Aels, autpectively), equil"teraln3-pss=", "aunconnecteunrhomb=c 4-pss=" funct=onsn(lower left and right p>Aels, autpectively). Coraul"t=onnfunct=onsn"relshown for thes"aalysis pesformed on northern (blue) "aunsouthern (red)shemitph"aesndeterm"nedlsn theseclipt=c coordinate faams. Thessolid, a-shed, dot-a-shed, "aundotted linesnco"autpondltolthesComm"autr, NILC, SEVEM, "aunSMICA m"ps, autpectivelyssNote that theslinesnlie on top of eachsother. Thesshad"dld-rkl"aunlight grey aug=ons ind=cate, for ruthoence, thes68% "aun95% confi="ncelaug=ons, autpectively,ndeterm"nedlfrom thesSMICA sim"aationsss

A="2">Open withnDEXTER

, 6681&ambold">Tablel15A.6681&amsimple-math">χ2A. stautsticnand ras=o of Msup>6681&amsimple-math">χ2A. of thes6681&amsimple-math">NA.-pss=" funct=onsnshown in Fig.l, , 17doa>nfor the i>assnck fiducial 6681&amsimple-math">Λ69<">A.CDMsmoutl ut least asnlargo as thesobserv"dlvalues of the stautsticnfor the i>assnck .115 CMB m"ps esuimated on northern "aunsouthern eclipt=c hemitph"aes.

, , , , 6681&ambold">Fig.l18A.6681&amsimple-math">NA.-pss=" coraul"t=onnfunct=onsndeterm"nedlfrom thes Msup>6681&amsimple-math">Nsids69 = 6469<">A.assnckSMICA CMB .115 tempesature esuimates aaunthescorautponding means esuimated from 1000lsim"aations. Res"aus "relshown for thes2-pss=", pseudo-coll"ps"dl3-pss=" (uppho left and right p>Aels, autpectively), equil"teraln3-pss=", "aunconnecteunrhomb=c 4-pss=" funct=onsn(lower left and right p>Aels, autpectively). Coraul"t=onnfunct=onsn"relshown for thes"aalysis pesformed on neg"t=ve (blue) "aunpositives(red)shemitph"aesndeterm"nedlsn theseclipt=c (solidslines), Dopplernboost (DB,sa-shedslines), "aundipole moduaationl(DM,sdot-a-shedslines) coordinate faamss. Thesshad"dld-rkl"aunlight grey aug=ons ind=cate thes68% "aun95% confi="ncelaug=ons, autpectivelyss

A="2">Open withnDEXTER

, 6681&ambold">Tablel16A.6681&amsimple-math">χ2A. stautsticnand ras=o of Msup>6681&amsimple-math">χ2A. of thes6681&amsimple-math">NA.-pss=" funct=onsnshown in Fig.l, , 18nfor the i>assnck fiducial 6681&amsimple-math">Λ69<">A.CDMsmoutl ut least asnlargo as thesobserv"dlvalues of the stautsticnfor the SMICA m"p on hemitph"aesndefinedlby theseclipt=c ( i>firstn6olumn),sDopplernboost (DB,s i>secoauncolumn),s"aundipolar moduaationl(DM,sthiruncolumn)srethoencelfaamss.

Insconclur"on,nthescoraul"t=onnfunct=onsnfor the i>assnck .115 tempesature a-15 are consistentlwithnthe res"aus pruth=t"dlin a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>. Speciftcally, we observ" that thesnorthern hemitph"aencoraul"t=onnfunct=onsn"rs aul"t=velylfeaturele"& (both thes3- "aun4-pss=" funct=onsnlie vtry closestolzero),nwh"aeas thessouthern hemitph"aenfunct=onsnexhibit a level of structurs consistentlwithnGau"&iunssim"aationsss

6681&amsec2"> 5s3. Constr>ints on quadrupolar moduaation

The most natural extension of the c681& of stautstically un"sohropicsmodels that we havesconsidss"dlpreviously involves thesquadrupolar moduaation of an initially stautstically isohropicsCMB sky m"pssNonastect=on of alcorautponding quadrupolar power asymmenrynis curoently claim"dssAn initial BipoSHs"aalysis of thesWMAP 7-year a-15 ( a href="/"autcles/aa/full_html0.106/10/sup>, , Bennett et al. .101) fouaunevi="ncelof corautponding non-zero tpectra, Msup>6681&amimg-inl"ne">, , A. "aun6681&amimg-inl"ne">, , A., in eclipt=c coordinatesssHowevtr, , , Hanson et al. (.110)doa> demonstr>t"dlthat thessignal couldlbe attributeuntolan inco5"lete taeatm"nt of beam asymmenries in the a-15, "aunthis was subsequently confirmed in a name="InR11">, , Bennett et al. (.113)doa>. Thescorautponding "aalysis of thesassnck .113 d-15 ipd=cated 6onsistency withnstautsticalsisohropy ( a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Cotlaboaas=onlXXIIIs2104).

Heae, we paoceed further "aunconsidss the quadrupolar moduaation of thesprimordial power tpectrum "s suggested by , , Ackerman et al. (.107): 6681&amimg-equation">, , 6681&amhabel-eq">(32)A.A.Givtn suchna spectrum, thesCMB tempesature fr">Csss expecteuntolexhibit a coraul"t=onnbetwsen 6681&amsimple-math">aℓm.or">A. "aun6681&amimg-inl"ne">, , A. withn Msup>6681&amsimple-math">Δ = 0,2A.ssTherefoae, thesBipoSHscoeuticiautsn6681&amimg-inl"ne">, , A. "aun6681&amimg-inl"ne">, , A. "re senr"tiveltol6681&amsimple-math">g26i>M.or">A.. In the limit ofsweaksun"sohropy, , , Kim 4-636 Komatsu (.113)doa>nprop sed an opuimal esuimator for 6681&amsimple-math">g26i>M.or">A.: 6681&amimg-equation">, , 6681&amhabel-eq">(33)A.A.wh"aen Msup>6681&amsimple-math">a.or">A. is the CMB d-15 vector in harmon=c s">cen"aun6681&amsimple-math">C.or">A. is itslcovas=a=ce matrix, "aun6681&amimg-equation">, , 6681&amhabel-eq">(34)A.A.Heae, 6681&amsimple-math">⟨ (C-1a.rup>∗)ℓm(C-1a)m′<9 ⟩ g26i>M = 0.or">A. is the meannfr">Cssn thes"bsencelof the quadrupolar moduaation. Observation-speciftc issues suchnas inco5"lete sky covtrage, inhomogeneoussnoiss, aau asymmenric beams will res"auninlalnon-zero meannfr">C, which -ap>bssesuimated for the i>assnck d-15 u&ing sim"aationsssDueltolthesotherwisespaohibitivesco5"utational cost, wesadoptla diagonal 5pproximation for the inverse of the covas=a=ce matrix: 6681&amimg-equation">, , 6681&amhabel-eq">(35)A.A.wh"aen Msup>6681&amsimple-math">C.or">A. "aun6681&amsimple-math">N.or">A. "aenthessignal "aunnoiss power tpectra autpectivelyssUncert5"nties aae co5"uted by "pply=ng thesesuimator to sim"aationsss

, 6681&ambold">Tablel17A.ints on the quadrupolar moduaation,ndeterm"nedlfrom thesComm"autr, NILC, SEVEM, "aunSMICA foaegrouau-cleaned m"ps.

Tablel, , 17doa>npruth=ts res"auslfrom "a "aalysis of thesassnck a-15 using thesextend"d commonlm"sk, UTA76, "aunlimit=ng thesr"ago of m"atipolesltol6681&amsimple-math">2n≤ n≤ 1.1069<">A.ssWhen including d-15 "t higher 6681&amsimple-math">A.-values,ssim"aations &hownevi="ncelfor largo stautsticalsuncert5"nties in the recovtreun6681&amsimple-math">g26i>M.or">A.lvalues that aae anconsequence of thesm"ny holeslin thesmask aul"teuntolpss=" sourcss. Therefoae, imposing this limit 6681&amsimple-math">n≤ 1.1069<">A. does notlsigniftca=tly uffect the constr>ining power of thes5aalysisssWenthen esuimate the "mplitude of the quadrupolar moduaation using thesaul"t=onn6681&amsimple-math">g26or/b> = (1 / 5 ∑ M | g26i>M | 2)1 / 269

.or">A.. Dueltolthesnature of the esuimator, which ss necsssas=ly positive, the esuimat=on is biased. For an unbiased assessm"nt, we esuimate the meann"aunst"auarundeviation of 6681&amsimple-math">g26or/b>69<">A.nfrom sim"aationsssWe fiaunnonevi="ncelfor quadrupolar moduaation of thesprimordial power tpectrumssHowevtr, thesasrived limitsnallow us to impose tight constr>ints on stautstically un"sohropicsinfaation"rynmodels,ssuchnas thosesincluding vector fr">Cs dus=ng infaation.nAncom">A=onsp"ptr, , , assnck Cotlaboaas=onlXX (.116), containsl" moae co5"lete discu"&ion on the theoaeticalsimpl=cat=ons of thss constr>intss

6681&amsec2"> 5.4. Pss="-"ar=tysasymmenry

The CMB un"sohropynfr">Csdefinedlon thessky, 6681&amimg-inl"ne">, , A., may be divided into symmenric, 6681&amimg-inl"ne">, , A., aau antisymmenric, 6681&amimg-inl"ne">, , A., funct=onsnwithnautpectltolthesch=tae of the sph"ae, aslpreviously auscribsd in a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>. These funct=onsnhavesevtn aau odd "ar=ty, "aunthusnco"autpondltoltph"aicalsharmon=csnwithnevtn aau odd 6681&amsimple-math">A.-modes, autpectivelyssOn thesvtry largo scaleslco"autponding tolthesSachs-Wolfespl"teau of the tempesature power tpectrum (6681&amsimple-math">2n≤ n≤ 3069<">A.), thesUniverse &houldlbe "ar=tysneutral withnnonp"auic"aar "ar=tysprethoencelexhibitedlby thesCMB fluctuationsssHowevtr, an odd "ss="-"ar=tysprethoencelhaslpreviously been observ"dlin thesWMAP d-15 releases (, , Laau 4-636 Magueijo .105a,, , b; , , Kim 4-636 Naselsky .110a,, , b; , , Gruppuso et al. .101) aaunthesassnck .113 res"ausssHeae, we investigate the "ar=tysasymmenry in thes2115 tempesature m"ps "t 6681&amsimple-math">Nsids69 = 32A.ssWelconsidss the following esuimator: 6681&amimg-equation">, , 6681&amhabel-eq">(36)A.A.wh"aen Msup>6681&amsimple-math">D+<9(max69).or">A. "aun6681&amsimple-math">D−<9(max69).or">A. "relgivtn by 6681&amimg-equation">, , 6681&amhabel-eq">(37)A.A.6681&amimg-inl"ne">, , A.is the total numor" of evtn (6681&amsimple-math">+69<">A.) or odd (6681&amsimple-math">−<9<">A.) m"atipoleslincludedlin thessum upltol6681&amsimple-math">max69.or">A., aau 6681&amimg-inl"ne">, , A. is the tempesature ang"aar power tpectrum co5"uted u&ing anquadrat=c maximum likelihood (QML)sesuimator ( a href="/"autcles/aa/full_html0.106/10/sup>, , Gruppuso et al. .101). Thes6681&amsimple-math">( + 1) / (2π)A. f>ctor in Eq. (, , 37)neffect=velylflattens thesspectrum "crossnthes6681&amsimple-math">A.-r"ago of thesSachs-Wolfespl"teau (upltol6681&amsimple-math"> = 5069<">A.)ninlal6681&amsimple-math">Λ69<">A.CDMsmoutl.

Figuael, , 19doa>npruth=ts thesr"tio, Msup>6681&amsimple-math">RTT69

(max69).or">A., for thes2115 com" data-sep"aased m"ps, together withnthe distribut=on aeterm"nedlfrom thesSMICA MCnsim"aationslwhich serv"sn"sna rethoencelfor the expecteunbehaviour of the stautsticnin a par=ty-neutral Universe. Thesdistribut=onsnfor the other CMB m"ps "relvtry simiaar. Thesfour com" data-sep"aas=on producus "relin good "greem"nt, ipd=cating "a odd-"ar=tysprethoencel"t vtry largo scaleslfor the m"atipolesr"ago considss"dlin this testss

, , , , 6681&ambold">Fig.l19A.6681&amsimple-math">RTT69

(max69).or">A. for Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue) aeterm"nedl"t 6681&amsimple-math">Nsids69 = 32A.ssThesshad"dlgrey aug=ons ind=cate thesdistribut=on of the stautsticnasrived from thesSMICA MCnsim"aations, withnthe dark,nlighttr, anunlight grey bandslco"autponding tolthes1, 2, anun3 Msup>6681&amsimple-math">σA. confi="ncellevels.

A="2">Open withnDEXTER

Figuael, , 20doa> shows the lower-tail paobabil"ty for the a-15 "snco5"areuntolsim"aationsl"sna funct=on of Msup>6681&amsimple-math">max69.or">A.. Thesres"aus "relin good "greem"ntlwithnthosesin a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>. The cleaned CMB m"ps yr">Csgenerally consistentlpaofileslwhich signify "a "aomaloussodd-"ar=tysprethoencelin thesm"atipolesrug=on Msup>6681&amsimple-math">max69 = 2069<">A.–30. The minimum in theslower-tail paobabil"ty occursl"t 6681&amsimple-math"> = 28A. co"autponding tolasvalue of 0.2% for NILC, SEVEM, "aunSMICA, "aun0.3% for Comm"autr, , 4.

Asla firstnattempt to quantify "ny a posteriorileffects in thessrgniftca=ce levels, welconsidss hownm"ny MCnsim"aationslappear in theslower tail of the MCndistribut=on withn" paobabil"ty equal to, or lower than, 0.2%, for ut least ones Msup>6681&amsimple-math">max69.or">A.svalue ovtr a speciftc r"ago. For 6681&amsimple-math">max69.or">A.sin the r"ago 3 Msup>6681&amsimple-math">−<9<">A.50, the total numor" of sim"aated m"pslwithnthisspaopertynis less thann20 ovtr 1000lMCnm"ps, imply=ng that, evtn -2nsidss=ng the LEE, "a odd-"ar=tysprethoencelis observ"d withn" lower-tail paobabil"ty of less thann2%ss

, , , , 6681&ambold">Fig.l2069<">A. A="2">Open withnDEXTER

6681&amsec2"> 5.5. Mirror-"ar=tysasymmenry

For the i>assnck .113 d-15 release, welstudieunthespaoperties of thestempesature a-15 atnanautolut=onsof Msup>6681&amsimple-math">Nsids69 = 16A. uauea retlect=on withnautpectltolaspl"neltolsearch for mirror symmenries. Suchna symmenry might benconnecteuntolnon-nrivial topologies (, , Starobinsky 1993; , , Stevtns et al. 1993; , , de Oliveira-Costa et al. 1996). In a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Cotlaboaas=onlXXIIIs(.114), wesauport"dlevi="ncelfor "a "atisymmenryspl"ne, withn" perpss=ic"aar airect=on givtn by 6681&amsimple-math">(l,b)s=s(.64°,−17°).or">A., Howevtr, thespaobabil"ty of the res"aus was slightly depss="nt on thesmethod of foaegrouau clean=ng, withn" 6681&amsimple-math">pA.-value r"ag=ng from 0.5% for Comm"autr-Rultr to 8.9% for SMICAssThessamssdirect=on w"snalso fouau in thesWMAP 7-year a-15 ( a name="InR43">, , Finelli et al. .102), "aunis close tolthat asterm"nedlfor the aipole moduaationlin thes i>assnck .113 d-15 release ( a href="/"autcles/aa/full_html0.106/10/sup>, , aCISgudoa>), suggesting possible connections betwsen the two direct=onal "aomalies.

We now paoceed tolre"aalyse thesstauus of mirror symmenries using thes i>assnck .115 full mi"&ion tempesature a-15 atnboth Msup>6681&amsimple-math">Nsids69 = 16A. "aun6681&amsimple-math">Nsids69 = 32A.ssIn orutr tolavoid possible bias introducedlby thesuse of the Galact=csmask, , 5 the res"aus "relasrived from thesfull-sky Comm"autr, NILC, "aunSMICA m"ps auscribsd in Sect. , , 2doa>. For SEVEM, " customiz"dlm"p is firstnproducedlby inp5"nting "boutl3% of thesm"p along thesGalact=cspl"nelu&ing andiffu&ive inp5"nting techniquessThis is then smootheuntolthes5ppropriate lower autolut=onslfor further "aalysisssFollowing , , Finelli et al. (.112), wesconsidss the esuimators in thespixel domain givtn by: 6681&amimg-equation">, , 6681&amhabel-eq">(38)A.A.wh"aenthessum is ovtr alll Msup>6681&amsimple-math">Npix69.or">A.HEALPixspixels, 6681&amimg-inl"ne">, , A. is the CMB tempesature un"sohropynmeasured at thespixel definedlby thesunit vector 6681&amimg-inl"ne">, , A., aau 6681&amimg-inl"ne">, , A. is the oppositssdirect=on withnautpectltolthespl"neldefined byn6681&amimg-inl"ne">, , A., i.e.,n6681&amimg-equation">, , 6681&amhabel-eq">(39)A.A.Note that we expects6681&amsimple-math">S+<9

A. tolbe sm"ll if the "ss="s on oppositsssidss of thesmirror arelneg"t=ves of eachsother, aau 6681&amsimple-math">S−<9

A. tolbe sm"ll when they "aenthessams.

, 6681&ambold">Tablel1869<">A.6681&amsimple-math">S±<9

A. stautstics of the com" data-sep"aased m"psl"t 6681&amsimple-math">Nsids69 = 16A. "aun6681&amsimple-math">Nsids69 = 32A.ss

, , , , 6681&ambold">Fig.l2169<">A.6681&amsimple-math">S+<9

A. (top p>Ael)saau 6681&amsimple-math">S−<9

A. (bottom p>Ael)sstautstic.nThesvtrticalslinesn&hownthe minimum value for the esuimator co5"uted "t 6681&amsimple-math">Nsids69 = 32A. for Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue) m"ps.nThesgrey "aea shows the samssquantity co5"uted from 1000lsim"aateunSMICA m"ps.

A="2">Open withnDEXTER

We co5"ute thesssquantitieslfor eachsof thes3072 (12288) airect=ons defined atnautolut=ons6681&amsimple-math">Nsids69 = 16A. (32), aau allow thes6681&amsimple-math">jA. aau 6681&amsimple-math">kA. ind=cesltolrun ovtr alllof the "ixelslof the low-autolut=onsfull-sky m"ps.nWe pesform the samss"aalysis on 1000lFFP8nsim"aationslaaunstoaenthesminimum value of Msup>6681&amsimple-math">S±<9

A. for eachsof these tolco5"ute paobabil"ties. Thesres"aus "relsummariz"dlin Tablel, , 18n5aunFig.l, , 21doa>.

We confirm that thesfull mi"&ion i>assnck tempesature a-15 atn6681&amsimple-math">Nsids69 = 16A. exhibits the most "aomaloussmirror aatisymmenrysin the airect=on 6681&amsimple-math">(l,b)s=s(.64°,−17°).or">A., consistentlwithnthe res"au from thes.113 nominal mi"&ion a-15, withn" paobabil"ty which r"agoslfrom 1.6% for SEVEM tol2.9% for Comm"autrssThis is within 406681&amsimple-math">°69<">A. of thesprethored airect=on i="ntif=edlby thesaipole moduaationl"aalysis sn Sect. , , 6.2. Thescorautponding res"aus "t 6681&amsimple-math">Nsids69 = 32A. yr">Cs5pproximatelylthe samssdirect=on, 6681&amsimple-math">(l,b)s=s(.64°,−16°).or">A., withn" slightly incaeas"dlprobabil"ty, r"ag=ng from 0.8% for SEVEM tol1.9% for Comm"autrss

We also note that thesCMB pattern exhibits a mirror symmenrysin the airect=on 6681&amsimple-math">(l,b)s=s(.61°,48°).or">A., consistentlwithnthat fouau in thesWMAP 7-year a-15 ( a href="/"autcles/aa/full_html0.106/10/sup>, , Finelli et al. .102), "aunclose tolthat i="ntif=edlby thessoaar aipole (, , assnck Cotlaboaas=onlVIIIs2106). Howevtr, thessrgniftca=ce of the symmenryspattern is less thannsn thes"ntisymmenrics-ass.

Thss extension of the "aalysis tolhigher autolut=onsthannsn our previousswork shows that thes"atisymmenryspaopertyndoes notlseem tolbe confineuntoltheslargost ang"aar scales, although we havesnotsattempteuntolcorauct for "nyna posteriorilchoices madensn thes"nalysisssThesdetail"dlconnection of thss "atisymmenryspaopertyntoltheslow-vas=a=ce "aunhemitph"aicalsasymmenry observ"d on theso scaleslremainsl"n open issuess

6681&amsec2"> 5.6. Localspeaksstautstics

, , , , 6681&ambold">Fig.l2269<">A.6681&amsimple-math">70°69<">A. radiussdiscssch=taedlon thespositives"aunneg"t=ve asymmenry airect=ons determ"nedlfrom thesSMICA CMB tempesature m"p in Sect. , , 6.2. From top tolbottom, thesplotsnco"autpondltolm"pslfiltss"dlwithn" GAUSS kernel of 6681&amsimple-math">40′<9<">A. FWHM, "a SSG84lfiltss of 6681&amsimple-math">500′<9<">A. FWHM, "ad "a SSG84lfiltss of 6681&amsimple-math">800′<9<">A. FWHM, autpectivelyss

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Localsextrema or peaks, as introducedlin Sect. , , 4.5.3, -ap>bssemployeuntolsearch for localiz"dl"aomalieslon thesCMB sky by examining howntheir stautsticalspaoperties varysin patchesl"sna funct=on of location.n

Initially, wesconsidss a further test for "symmenry by examining the aiuthoences in thespeaksdistribut=on when divided according tolori"nt"t=on withnautpectltolaspreviously specifted asymmenry airect=onssIn p"auic"aar, welselect the peaksnboth inlalaisc of radiuss6681&amsimple-math">70°69<">A. ch=taedlon 6681&amsimple-math">(l,b)s=s(.25°,−18°).or">A. (thespositivesairect=on of thesaipole defined in Sect. , , 6.2 for SMICA)saau inlthes-orautponding "atipodalsaisc, then construct the empiaicalspeaksheight CDFs tolbe co5"areunwithnthe full-sky mediunsFFP8ndistribut=on, as shown in Fig.l, , 22doa>. For m"pslfiltss"dlwithn" 406681&amsimple-math">′<9<">A. FWHM GAUSS filtss thesdistribut=on of the peaksnfor the positive-airect=on aisc is sn general "greem"ntlwithnthesfull sky res"au, whilenthat for the neg"t=ve-airect=on is marginally aiuthoent. Moreovtr, thisspattern of behaviour is ssen ovtr a numor" of filtssing scales, both for the KSndeviation from thesmediunsfull-sky sim"aateunCDFs, "aunthesspaead of extremallvalues when co5"aring positives"aunneg"t=ve aug=ons.nWe also findlthat thespaoperties of thesneg"t=ve aisc uffect the p-value res"aus for asfull sky KSntest on a-15lfiltss"dlwithn"a SSG84lfiltss of 6681&amsimple-math">500′<9<">A. FWHM, "s ssen in Sect. , , 4.5.3.

We can then extend the "aalysis for the 406681&amsimple-math">′<9<">A. GAUSS-filtss"dla-15lby -2nsidss=ng the vas=ationlin thespeaksstautsticalspaoperties for asset of discs, eachsof which ss ch=taedlon aspixel definedl"t 6681&amsimple-math">Nsids69 = 256A.ssThessimplest stautstics tolconsidss aaenthespeaksnumor" countsssWentherefoaelconsidss discssof 6681&amsimple-math">30°69<">A. diumeter "aunco5"ute thespeakscounts for eachsdisc. These aaenthen co5"areuntolthes-orautponding peakscount CDFs determ"nedlfrom sim"aations, "aunthesuppho- "aunlower-tail paobabil"ties aae 81&"gnedlby -2unt=ng thesnumor" of sim"aationslaboves"aunbelow thesobserv"d counts at thessamsslocation.nThesssquantitieslcan then be visualiz"dlin thesform of Msup>6681&amsimple-math">Nsids69 = 256A. sky m"ps.nThesderived Msup>6681&amsimple-math">−log 10(UTP).or">A. m"pslfor eachscom" data-sep"aas=on method "relshown in Fig.l, , 23. Whilenwe findlthat thestotal counts of peaksnfor the sky covtrage definedlby thescommonlm"sk ss consistentlwithnsim"aations, signiftca=tlaug=onallvas=ationlis ssenssIndeed, the p-value for cert5"n aisc locations daops tol0.1% (i.e.,nthe sky counts exceed anything ssen in sim"aations). Howevtr, onesneeds tolaccount for thes" posteriorilselect=on of signiftca=tlaug=ons in the aeterm"nation of thestruessrgniftca=cessIt &houldlalso besnot"dlthat aug=onallvas=ations of thesUTP "re setn at simiaar levels when intpecting thespeak-count stautstics m"ps aurived for r"auomly selecteunrealizations of thessim"aationsssMoreovtr, thessrgniftca=ce of suchnpeak-counting "aomalieslis augrad"dlwithnlargos disc diumeters, "aunbecomes insigniftca=tlfor counts onnthesfull sky.nThus, no signiftca=tl"aomaliesl-ap>bssclaim"dnfor the peak-count stautstics of thesassnck a-15.

A powerfullnon-p"aamenricstest ofnstautsticalsisohropy isspaovided by thestwo-s"mple KS-deviation betwsen the full sky empiaicalspeaksheight CDF Msup>6681&amsimple-math">Fn(x)A. (set Eq. (, , 28)) "ad "a empiaicalspeaksheight CDF Msup>6681&amsimple-math">Fn′<9(x)A. asrived from a subs"mple of thesaistribut=on, ag5"n aefinedlby thespeaksnwithin discssof 6681&amsimple-math">30°69<">A. diumeter "s defined above.nThestwo-s"mple KS-deviation 6681&amimg-equation">, , 6681&amhabel-eq">(40)A.A.for asp"auial sky reg=on shaaesns"mples betwsen the two CDFs, "aun-ap>bsscalc"aateunextremelyleuticiautlylu&ing r"aksstautstics according tol6681&amimg-equation">, , 6681&amhabel-eq">(41)A.A.wh"aen Msup>6681&amsimple-math">rA. aau 6681&amsimple-math">r′<9<">A. asnote the r"akssof asvalue withnindex 6681&amsimple-math">iA. in the full set of 6681&amsimple-math">nA. aau aesnricteunset of 6681&amsimple-math">n′<9<">A. s"mples, autpectivelyssM"ps of thesuppho tail paobabil"ty aaenthen determ"nedlby co5"arison withnthe equivalentlquantitieslco5"uted from sim"aations; Msup>6681&amsimple-math">−log 10(UTP).or">A. m"psl"relshown in Fig.l, , 24. Thesmajor"ty of the selecteunlocations are consistentlwithnthe full-sky aistribut=on, thusnipd=cating thenstautsticalsisohropy of thesassnck m"ps.nThesmost paom"ne=tlfeature in eachsof theslocal KS-deviation m"psl"ppearsssouth of the Galact=csch=tae aau may be 81&ociateunwithn" cold reg=on crossing thesGalact=cspl"nessHowevtr, aslwithnthe peakscounts, itn-apnotsbe interpreted "s stautstically unomalousss

, , , , 6681&ambold">Fig.l2369<">A.6681&amsimple-math">−log 10(UTP).or">A. for peak counts in thes i>assnck 406681&amsimple-math">′<9<">A. GAUSS-filtss"dltempesature a-15,nwh"ae eachspixel encodes thespaobabil"ty asterm"nedlfor a 306Msup>6681&amsimple-math">°69<">A. diumeter disc ch=taedlon itss

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, , , , 6681&ambold">Fig.l2469<">A.6681&amsimple-math">−log 10(UTP).or">A. for thestwo-s"mple KS-deviation wh"ae eachspixel encodes thespaobabil"ty asterm"nedlfor a 306Msup>6681&amsimple-math">°69<">A. diumeter disc ch=taedlon it, "snco5"uted from thes i>assnck 406681&amsimple-math">′<9<">A. GAUSS-filtss"dltempesature a-15ss

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6681&amsec2"> 5.7.nThesCold Spot

, , , , 6681&ambold">Fig.l2569<">A.Top: tempesature patchsch=taedlon thesCold Spot.s i>Bottom: peak mergos taeenwithin thesCold Spot reg=on. Thesfiguaelshows a reg=on ch=taedlon thesCold Spotnlocation sn gnomon=c paoject=on, withn"ll thespeaksnia SSG84-filtss"dlm"pslwithnFWHM r"ag=ng from 6681&amsimple-math">80′<9<">A. tol6681&amsimple-math">1.10′<9<">A. ovtrlaidlon thessamssplotssThessize of the colous"dlcircleslis prop rt=onal tolthesfiltssing scale. Thescolour corautpondsltolthespeak value, normaliz"dlin unitssof 6681&amsimple-math">σA. for a givtn filtss scale. In both p>Aels thesa-15 are from thesSMICA CMB m"p at full autolut=on.

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Sincelitssdiscovtry in thesWMAP first-year a-15 ( a href="/"autcles/aa/full_html0.106/10/sup>, , Vielva et al. .104), thesCold Spot, ch=taedlat Galact=cscoordinates 6681&amsimple-math">(l,b)s=s(.10°,−57°).or">A.lhaslbeen one of thesmost extensivelylstudieunlargo-scale CMB unomalies. In thes2113 release ( a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Cotlaboaas=onlXXIIIs2104),s i>assnck confirmed thes"p"are=tly uaomaloussnature of this feature in tempesature, in terms of thes"aea of thesSMHWscoeuticiautsnonl"ag"aar scales of 6681&amsimple-math">≈ 10°69<">A. on thessky; thes2115 release h"snalso confirmed this feature (set Sects. , , 4.5.2n5aun, , 4.5.3). ThesCMB tempesature un"sohropies aaouau thesCold Spotnas observ"d bynassnck "relshown in thestop p>Ael of Fig.l, , 25. Thespeak mergos taeenwithin thesCold Spot reg=onlis pruth=t"dlin theslower p>Ael of thesfiguael5aunpaovides a m"atiscale view of itssstructure (set Sect. , , 4.5.4 for details).

The aobustness of thesaetect=on of thes"aomaliesldiscu"&"dlin this p"ptrlis alnon-nrivial issuessFor the p"auic"aar -ass of thesCold Spot, this haslbeen review"d byn, , Vielva (.110), "ad "ddrut&"dlin detail by , , Cruz et al. (.106), pay=ng speciftc attent=onstolthesimpactsof asposteriorilchoicesssIn p"auic"aar, theslatterlstudy focu"edlon thesoriginalstest that ind=cateunthespaesencelof this feature on thessky, confirming ansrgniftca=ce betwsen 1% "ad 2%ssAn altern"t=ve aaalysis of thessrgniftca=ce ba"edlon twonstautsticalstestslwithnaiuthoent levels of conservativtness w"snmadenby , , McEwen et al. (.105), paoviding values of 0.1% "ad 4.7%, autpectivelyssThenstautsticalssrgniftca=ce of the Cold Spot w"snquesuion"d byn, , Zh"ag 4-636 Hutss"r (.110) who fouau " lowssrgniftca=ce afterlpesforming anstudy ba"edlon aiuthoent kernelsssAsldiscu"&"dlin moae detail by , , Vielva (.110), this res"au -ap>also besinterpreted "s evi="ncelthat notsall kernels arelnecsssas=ly suitablelfor the aetect=on of arbitrarysnon-Gau"&iunsfeatures.

The possibil"ty that thesCold Spotnarisoslfrom instrum"ntalssystemat=cs ( a href="/"autcles/aa/full_html0.106/10/sup>, , Vielva et al. .104) or foregrouau residuals ( a name="InR82">, , Liu 4-636 Zh"ag .105; , , Cruz et al. .106) haslbeen largoly rejecteussHowevtr, sevtrallnon-st"auarunphysicalsmechun"smsnhavesbeen prop sed as possible expl"nationsssThesssinclude thesgravitational effectnproducedlby " coll"psing cosmicstexture (, , Cruz et al. .107), theslinear "aunnonlinear ISW effectncau"edlby " void in theslargo-scale structure (e.g.,n, , Tomita .105; , , Inoue 4-636 Silk .106; , , Rudnick et al. .107; , , Tomita 4-636 Inoue .108; , , Finelli et al. .106), " cosmicsbubble collis=on within thesetern"l infaationlframework (, , Czech et al. .100; , , Feeney et al. .101; , , McEwen et al. .102), "auna localiz"dlvers=on of thesinhomogeneoussreheating scenario within thesinfaation"rynp"aadigm (, , Bueno Sa=chezs2104).

Sincelthesother scenariosslack "dditional evi="nce, thesvoid hypothesis wouldlseem tolbe thesmost pl"usible, depss=ing on thessiz"s, density contrasts, "aunpaofileslassumeu inlthes-o5"utations, somesof which arelnot inlagreem"ntlwithneither observationl(, , Cruz et al. .108) or curoent Msup>6681&amsimple-math">NA.-bodylstudies (, , Cai et al. .100; , , Watson et al. .104). Howevtr, , , Szapudi et al. (.115)nhavesrece=tly aetecteuna largo void in thesWISE-2MASS galaxy catalogue al"gnedlwithnthe Cold Spot, withn"a esuimateunradiussof arouau 6681&amsimple-math">200nh-1<9

A. Mpc, "a avtraged density contrast of 6681&amimg-inl"ne">, , A., aau ch=taedlon asaedshift of 6681&amsimple-math">zn≈ 0.1569<">A.. Largo voidslwithnsimiaar churacteristics arelnot unusual in thesst"auarun6681&amsimple-math">Λ69<">A.CDMsmoutl (, , Nadathur et al. .104). In f>ct, Msup>6681&amsimple-math">NA.-bodylsim"aationslpredictn"boutl20 suchnvoidslin theslocal Universe (6681&amsimple-math">z< 0.569<">A.). Howevtr, , , Zibins(.114)n5aun, , Nadathur et al. (.114)nind=cate that thesexpecteunsrgnalsauentoltheslinear "aunnonlinear ISW effectsncau"edlby this structure is notllargo enough tolexpl"in thestempesature aecrem"ntl81&ociateunwithnthesCold Spot.s /p>

The newsassnck a-15 release allows us tolfurther exploaenthesstautsticalsnature of thesCold Spot.sTwo previoussstudies (, , Zh"os2113; , , Gurzadyan et al. .104)nhavesclaim"dninconsistencies of thesinternalspaoperties of the Cold Spot withnthesGau"&iunshypothesis, which wesau-"ddrut& h"aessIn p"auic"aar, welconsidss the small-scale fluctuationslwithin alaisc-like reg=onlof radiuss6681&amsimple-math">≈ 25°A.ss

Sevtrallstautsticalsquantitieslare co5"uted from thesfull-autolut=onstempesature m"ps within thesCold Spot reg=on. This is divided intolasch=taalsaisc of diumeter 16Msup>6681&amsimple-math">°69<">A. suroouauedlby " set of 13lconch=tatc annuli withnch=taalsradii <">cedlin steps of "boutl26Msup>6681&amsimple-math">°69<">A., thusnallowing us tolbuildlang"aar paofileslfor the mean, vas=a=ce, skewness, aau kurtosisssThese aaenthen co5"areuntolspecializ"dlCMB realizations, generated "s followsssA set of Gau"&iunsCMB skieslis sim"aateunusing thesFFP8nrethoencelspectrum, aau convolv"d withn" Gau"&iunsbeam of 56681&amsimple-math">′<9<">A. FWHMssAslfor the FFP8nsim"aationslthemselv"s, these m"psl"relautcaled, "sndiscu"&"dlpreviously. Onlylthose that contain alspotnas extremenas thesCold Spotnatnanscale Msup>6681&amsimple-math">R = 310′<9<">A. ia SMHWs<">cel"relautained, "au these aaenrotateunsuchnthat eachssim"aateuncold spotnis relocateuntolthes5ctual position of thesCold Spotn(this ensures that thesnoisespaoperties "reli="nticalsfor both a-15 aaunsim"aations). This select=on criterion corautpondsltoltheschuracteristic that originally ind=cateunthespaesencelof thesCold Spotnin thesobserv"d sky.nAsla finallstep,lfor eachsremaining CMB sim"aation alnoisesrealizationlis added, consistentlwithneachscom" data-sep"aas=on method.

The aes"aus "relpruth=t"dlin Fig.l, , 26doa>. Focusing on thespaofile of thesmean value, itnis "p"are=t that theslargost deviations from thessim"aationslappear on scales arouau 156681&amsimple-math">°69<">A., which corautpondsltola hot ring structure, "s ssen in Fig.l, , 25 "aunpaeviously discu"&"dlin , , Cayón et al. (.105)n5aun, , Nadathur et al. (.114). Noutce that on thessmallest scales thesmean paofile isnalso somewhat asviantlwithnautpectltolthessim"aations, but this may be connecteuntolselect=on bias, sincelwe are considss=ng CMB sim"aations containing anspotnthat is ut least "sncoldnas thesCold Spot. Howevtr, if welconsidss the distribut=on of the paofileslco"autponding tolthescoldest spots instead of thesspots as extremenas thesCold Spotn(removing thesbias at thessmallest scales)nthen the aes"aus do notschuago substantially (set below).n

In orutr tolquantify possible deviations from Gau"&iun"ty, we aeterm"ne thespaobabil"ty of finding " 6681&amsimple-math">χ2do

A. value largos thannthat of thesa-15 for eachsstautstic, "s summariz"dlin Tablel, , 19. Thes6681&amsimple-math">χ2do

A. value for the a-15 isnco5"uted u&ing aa esuimate of the covas=a=ce matrix betwsen aiuthoent radiallscales determ"nedlfrom thesCold Spotnsim"aations (1000lfor eachscom" data-sep"aas=on method), "au then co5"areuntolthestheoaeticals6681&amsimple-math">χ2do

A. distribut=on withn13 degrees of freedom. Thesres"aus ind=cate that thesang"aar paofilelfor the meanlis poorly aescribsd by thessim"aations, of which less thann1% "re fouau tonhavesalhigher 6681&amsimple-math">χ2do

A. thannthe a-15 (when considss=ng the distribut=on co"autponding tolthescoldest spot thisspaobabil"ty becomes 5pproximatelyl2%)ssWenhavescheckedlthat this auv=ationlis notsobviously 81&ociateunwithn" p"auic"aar sub-r"ago of "ng"aar scales, imply=ng that thesmean paofile isnaaomaloussovss the full a"ago considsseussConversely, the aadiallpaofileslof the higher-orutr momautsnare co5"atiblelwithnthesGau"&iunssim"aationsssTheslatterlaes"aus "relthen in contradict=on withn" simiaar aaalysis (u&ing discssinstead of s=ngs)nby
, , Zh"os(.113) for thesWMAP 9-year a-15. Howevtr, itl"ppearssthat this may be relateuntolthescriterial"ppli"dnfor the select=on of thesGau"&iunssim"aations u"edltoldefine thesnull hypothesisssIn p"auic"aar, , , Zh"os(.113) u"edlthescoldest pixel insreals<">cel"s a meansltoli="ntifylthose sim"aationslthat &houldlbelautained, "s opposeuntolthesexistence of cold spots as extremenas thesCold Spotnselecteunin thesSMHWscoeuticiaut m"p at Msup>6681&amsimple-math">R = 310′<9<">A.. Sincelitlis notsimplicitnthat suchna tempesature extremum is necsssas=ly 81&ociateunwithn"n extendeuncold reg=on, p"auic"aarly one defined in wavelets<">ce, thessim"aations u"edlby , , Zh"os(.113) did notscontain features co5"arableltolthesnature of thesCold Spot.sThss expl"ins why thesCold Spotnseemeuntolbenaaomalousswhen looking at thessmall-scale fluctuationsss

, , , , 6681&ambold">Fig.l2669<">A.leftltolright: mean, vas=a=ce, skewness, aau kurtosislang"aar paofileslco5"uted for r=ngs atnaadii 6681&amsimple-math">θA. ch=taedlon thesCold Spotnposition for Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)ssThesexpecteunvalue obtained from thessim"aationslis aunot"dlby thesblack dasheunline "au the dark "au lightsgrey aug=ons repruth=t the 6681&amsimple-math">1σA. aau 6681&amsimple-math">2σA. intervals.

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, 6681&ambold">Tablel1969<">A.6681&amsimple-math">χ2do

A. stautstic of thes"ag"aar paofileslof thesesuimators shown in Fig.l, , 26doa> largos thannthose determ"nedlfrom thesa-15.

In conclus=on, itl"ppearssthat onlylthe mean tempesature paofile of thesCold Spotnshouldlbelconsidsseunaaomalousswhen co5"areuntolCMB cold spots that aae 81 stautstically extreme. Alllother measures of itssinternalsstructure are consistentlwithnexpectationsss

Asla finallremark, we note that theshigh-p81&sfiltssing curoently 8ppli"dntolthesassnck CMB poaarizationlm"pslsevtrelyllimits the possibil"ty of conducting targoteunaaalysesltoldiscrim"nate betwsen aiuthoent possible origins of the Cold SpotssFor example, no poaarizationlsrgnalswouldlbelexpecteuninnthose moutlsnproducing sscond"rynun"sohropies auentolasgravitational effect,nwh"ae"s a speciftc pattern mightsbelexpecteuninnasbubble collis=on scenario ( a href="/"autcles/aa/full_html0.106/10/sup>, , Czech et al. .100)ssAppropriate testslwilllbelpursueuninnfuture work, onceltheslargo-scale poaarizationla-15 are availabless

6681&amsec"> 6. Dipole moduaationl"ad airect=onal"ty

In this sect=on, we examinesisohropy vioaationlrelateuntolaipolar "symmenry, vas=oussforms of which havesbeen not"dlsincelthesearly WMAP releases (, , Eriksen et al. .104a)ssWe pesform alnon-exhau"t=ve sssieslof testslinnansattemptntolnarrow down thesnature of thesasymmenry (on thesassumpt=onsthatlitlis notssimply " stautsticalsfluke)ssFirst, we willlbsiefly aescribs somessimiaaritieslaad aiuthoences betwsen the testslthat aae importa=tlfor making aspaoper co5"arison of thesaes"ausss

All thestestslinnthis sect=on shaae in commonlthe fitting of asaipole. This is done either by fitting for a aipole explicitly in a m"p of power on thessky (Sects. , , 6.1n5aun, , 6.5), bysemploying Baye&iunstechniquessin pixel <">celfor asspeciftc moutl (Sect. , , 6.2), or by measur=ng the coupling of Msup>6681&amsimple-math"><9<">A. tol6681&amsimple-math"> ± 169<">A. moutslin thesCMB covas=a=ce matrix (Sects. , , 6.3, , , 6.4, "ad , , 6.6). The aiuthoences arisolfrom hownthe fitted dipolesnare co5bined, which determ"nes the speciftc form of asymmenry that thestest is sensitivesto.

The testsl-ap>bssdivided intoltwoncategori"s, amplitude-ba"edl"ad airect=on-ba"ed. Sect=ons , , 6.1ntol, , 6.4 are all sensitivestolthesamplitude of asaipole moduaation. Speciftcally, Sect. , , 6.1nlooks for aipole moduaationlin thespixel-to-pixel vas=a=ce of thesa-15, whilenSects. , , 6.26681&amsimple-math">−69<">A., , 6.4 all search for aipole moduaationlof thes"ag"aar power spectrum. The aistinct=on betwsen these two 5pproacheslis mainly one of Msup>6681&amsimple-math"><9<">A. weightingss

Sect=ons , , 6.5 "ad , , 6.6 both examinesaspects of airect=onal"ty in thesa-15, wheaenthesairect=ons are extracted from aipole fits but co5binedlin diuthoent ways. Sect=on , , 6.5 fits for aipoleslin b"aunpower (withnsimiaar res"aus for vas=a=ce) "ad onlyluses the airect=on informat=on, whilenSect. , , 6.6 weights eachsdipole equally ucross all scales andluses the amplitude informat=on aslwell.

The aiuthoences betwsen the 5pproacheslof these sect=ons shouldlbelkeptnin mindlwhen co5"aring their aes"ausssFor example, although Sects. , , 6.5 "ad , , 6.6 both look for a airect=onallsrgnalsin thesa-15, they aaenopt=miz"dlfor aiuthoent forms of deviations from stautsticalsisohropyssIt is therefoaelunsurpris=ng that they aar=ve at aiuthoent res"ausssHowevtr, thessrgnalsfouau in Sect. , , 6.5, if notssimply " stautsticalsfluke, ss constrainedlby thesres"aus of Sect. , , 6.6.

Regarding thesimpactson the aipolar moduaationlres"aus of theslacklof thes"beraas=on contribut=on tolthessim"aations, we note the followingssIn general thes"aalyseslare either sensitivesonlyltollargo "ng"aar scales, orsonlylclaim possible detect=ons on suchnscales, wheaentheseffectnof "beraas=on willlbelnegligible "aunhencelthesconclus=onslare unlikelyltolchuagossA possible exceptionlis inlrelat=on tolthesres"aus of Sect. , , 6.5, wheaenclaimslare maden"boutleffectsnextending out tol6681&amsimple-math">max69 = 1500<9<">A.. It is pl"usible that theseffectsnof "beraas=on couldlst"auntolbecome importa=tlon theso scalesss

6681&amsec2"> 6.1. Vas=a=ce "symmenry

Thesstudy of power asymmenry via theslocal vas=a=ce of thesCMB fluctuationslw"s first pesform"d byn, , Akrami et al. (.114)nfor the i>assnck 2113 "aunWMAP 9-year tempesature a-15ssThe 5pproach w"snmotivat"dlby itsnconceptual aau implem"nt"t=onallsrmplicity,litssdirectly intuitivesinterpretat=on, "aunby virtue of being defined in pixel <">ce, " usefullco5"lem"nt"r"ty tolother mostly harmon=c-ba"edlmethodsssThenstautstic w"snco5"uted ovss patcheslof diuthoent siz"s "aunpositions on thessky, "aunco5"areunwithnthe values obtained from stautstically "sohropicssim"aationsssIt w"snfouau that none of thes1000lavailablessim"aationslhauna largor vas=a=ce asymmenry thansthatlesuimateunfrom thesa-15. This suggosteunthespaesencelof asymmenry atnanstautsticalssrgniftca=ce of ut least 6681&amsimple-math">3.3σA.,nwithn" prethos"dlairect=on 6681&amsimple-math">(l,b)s≈ (212°,−13°).or">A.lsn good "greem"ntlwithnother studies. In this sect=on, we aevisitnthe vas=a=ce asymmenry aau aeportlthesres"aus of thes"aalysis for the i>assnck 2115stempesature m"ps at full autolut=on, Msup>6681&amsimple-math">Nsids69 = 2048A.ss

Thes"aalysis proceeds "s followsssWesconsidss a set of discs of vas=ousssiz"s ch=taedlon thespixelssof asHEALPix m"p aefinedlby asspeciftc Msup>6681&amsimple-math">Nsids69A. valuessFor eachssky m"p,nwe first remove thesmonopole "ad aipole com" datas from thesm"sk"d sky "au then co5"ute thesvas=a=ce of thesfluctuationslon algivtn aisc using onlylthe unm"sk"d pixels. This yields "slocal-vas=a=ce map at thesHEALPix autolut=onsof interost.nWe also esuimate thesexpecteunavtrage "au vas=a=ce of thesvas=a=ce on eachsdisc from thessim"aationslaau then 6681&amsimple-math">10%69<">A. of thes"aea is unm"sk"d, although our aes"aus "relaobust ag5"nstltheschoice of this valuessThe co5"uted local-vas=a=ce amplitudes "relthen u"edltolco5"are thesa-15lwithnstautstically "sohropicssim"aationsssNote that we use onlylthe dipole "mplitudes of theslocal-vas=a=ce mapsltolmeasure thessrgniftca=ce of the asymmenry; thesamplitudes of higher m"atipoleslwerelshown by , , Akrami et al. (.114)ntolbe consistentlwithnstautstically "sohropicssim"aations "ad wentherefoaeldo notsconsidss themlin thespruth=t p"ptr.n

In , , Akrami et al. (.114), thessensitiv"ty of the method tolthesdisc size w"sn81&ut&"dlusing both stautstically "sohropics"ad "a"sohropicssim"aationsssThe free p"aameners, i.e.,nthe numor" aaunsize of the discs, werelthen fixedlby these sim"aationsssIt w"snfouau that for 3072 patcheslch=taedlon thesset of pixelssdefinedl"t 6681&amsimple-math">Nsids69 = 16<9<">A., thensim"aateunasymmenry srgnalslwerelnotsaetecteunwhen either vtry small (6681&amsimple-math">rdisc69< 4°69<">A.) or vtry largo (6681&amsimple-math">rdisc69> 16°69<">A.) discs werelu"ed.

, , , , 6681&ambold">Fig.l2769<">A.Uppho p>Ael: 6681&amsimple-math">pA.-values for vas=a=ce asymmenry measurednas thesnumor" of sim"aationslwithnlocal-vas=a=ce dipole "mplitudes largos thannthose inthos"dlfrom thesa-15,l"s a funct=on of disc radiussfor the four com" data-sep"aas"dlm"ps, Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue), "aunfor unfiltss"dl"aunhigh-p81&-filtss"dl-asssssFor the filtss"dl-ass, thesComm"autrscurve ss covss"dlby thesSMICA curve for small (6681&amsimple-math">rdisk69 ≤ 869<">A.) disks, "aunby thesSEVEM curve for largo disks (6681&amsimple-math">rdisk69> 869<">A.).s i>Lower p>Ael: local-vas=a=ce dipole airect=ons for the SMICA m"p. Thescolours,l"s ind=cateunby thescolourbar, co"autpondntolaiuthoent values of theshigh-p81&sfiltssnch=taalsm"atipolel6681&amsimple-math">069A.ssThessize of a m"rker disc corautponds, from small tollargo, tolthessize of the disc u"edlin thes"aalysis, namely 46681&amsimple-math">°69<">A., 126Msup>6681&amsimple-math">°69<">A., 206Msup>6681&amsimple-math">°69<">A., "aun706Msup>6681&amsimple-math">°69<">A.. The aipole airect=ons from thesComm"autr, NILC, "aunSEVEM com" data-sep"aas=on methods are consistentlwithnthe -ass shown h"aessTheslow- Msup>6681&amsimple-math"><9<">A. "aunWMAP-9sairect=ons are i="nticalstolthose in Fig.l, , 35.

A="2">Open withnDEXTER

The form"rseffectnis auentolasco5binationlof theslowsnumor" of pixelssper disc "ad "a insuuticiaut numor" of discs tolcovss the "ntiressky when 6681&amsimple-math">Nsids69 = 16<9<">A.nrethoencelgrids are u"ed. Howevtr, itlhaslrece=tly been shown by , , Adhikas= (.115)nthat u&ing a largos numor" of small discs (by incae"sing Msup>6681&amsimple-math">Nsids69A. tol32, 64, 128, "aun256, depss=ing on thesdisc size) in orutr tolcovss the "ntiressky allows theslocal-vas=a=ce method tolaetect theslargo-scale aaomaloussasymmenry aslwellnas thesDoppler boost srgnalsfrom thes i>assnck 2113 a-15,l"t ansrgniftca=ce of 6681&amsimple-math">> 3.3σA..n, , Fantaye (.114)nh"s demonstratedlthat thesDoppler boost srgnals-ap>bssdetecteunatn" simiaar level of signiftca=ce u&ing needletsb"aup81&sfiltssing of thesa-15, even withnlargo discs,nwh"nssim"aations are aeboosted. Here, in contrast tolthes2113 "aalysis, we use m"ps which contain Doppler boosting,sfor both sim"aations "ad a-15,l"au therefoaelweldo notsaetect anysDoppler boost srgnalswhen u"ing a largo numor" of small discsss

Theslowsobserv"d signiftca=ce levels when largo discs are u"ednis auentolthescosmicsvas=a=ce as&ociateunwithntheslargost-scale mouts. Motivat"dlby thes"aalysis of , , Fantaye (.114),l"au in orutr tol"ddrut& this issue, we also pesform aaalyseslu"ing a Butterworthnhigh-p81&sfiltss, 6681&amimg-equation">, , 6681&amhabel-eq">(42)A.A.ch=taedlat m"atipolesl6681&amsimple-math">069 = 569<">A., 10, 15, 20, aau 6681&amsimple-math">30<9<">A.. In "ddition, the filtssing of lowsm"atipoleslallows us tolostablishlthescontribut=on of suchnmoutsltol"ny aetecteunasymmenryss

Here, ba"edlon thes"aalysis of , , Akrami et al. (.114), we aesnrict our "aalysis tolthose disc sizes for which 3072 discs,nco"autponding tolan 6681&amsimple-math">Nsids69 = 16<9<">A.nm"p,ncovss the "ntiressky, i.e.,ntolthesr"ago 46681&amsimple-math">°69<">A.–906Msup>6681&amsimple-math">°69<">A.. Consistentlaes"aus -ap>bssobtained by choosing other values of Msup>6681&amsimple-math">Nsids69A. for a givtn disc size paovidedlthat thes"ntiressky ss covss"dlby thesdiscsssHere, for srmplicity,lwe worknwithnthessamss Msup>6681&amsimple-math">Nsids69A. (6681&amsimple-math">= 16<9<">A.) for all disc sizesss

Our res"aus for the measurednamplitude of thesvas=a=ce "symmenry, co5"areuntolthesvalues from thessim"aations, aslwellnas thesco"autponding aipole airect=ons, "relshown in Fig.l, , 27. Thes6681&amsimple-math">pA.-values "relgivtn for aiuthoent disc sizes "au in terms of thesnumor" of sim"aationslwithnlocal-vas=a=ce dipole "mplitudes gae"tos thannthe ones measurednfrom thesa-15. Note that sincelthesdiscs withnaiuthoent sizes u"edlin our "aalysis are correlateu, thessrgniftca=ce levels are also correlateussFor this reason we choosentolshownthe 6681&amsimple-math">pA.-values "s a funct=on of disc size instead of co5bining them intolassinglesnumor". Moreovtr, itlshouldlbelnot"dlthat thessrgniftca=ce values wespruth=t h"ae do notsincorpoaasel"ny correct=ons tolaccount for the choice of p"aameners "dopted dur=ng method calibrat=on, speciftcallylthe dipole "mplitudes "ad airect=ons for the "a"sohropicssim"aations that were u"edntolfixlthesr"ago of disc sizes "ad numor" of patches.n

It -ap>bssssen from thesuppho p>Ael of Fig.l, , 27 that for the unfiltss"dlm"p thessrgniftca=ce of the power asymmenry dropsnquickly when we incae"selthesdisc size tolradii gae"tos thann166681&amsimple-math">°69<">A.. This is nonlongss the -ass, howevtr, when the lowest m"atipolesl"re filtss"dloutssFor example, when the filtss scale is set tol6681&amsimple-math">069 = 569<">A., i.e.,nwhen the vtry lowsm"atipoleslwhich arelaffect"dlmost by cosmicsvas=a=ce arelsupprut&"d, thesvas=a=ce "symmenrylis autecteunatnthe 6681&amsimple-math">3σA. level for all disc sizes, "s shown in Fig.l, , 27. Tablel, , 20 pruth=tsnthe 6681&amsimple-math">pA.-values of thesvas=a=ce "symmenrylu"ing 86Msup>6681&amsimple-math">°69<">A. discs aaunfor vas=oussvalues of Msup>6681&amsimple-math">069A.ssOur res"aus shownthatsvas=a=ce "symmenrylis autecteunwithn" remarkablessigniftca=ce for all disc sizesnwhen vtry lowsm"atipolesl"re filtss"dloutssIn "ddition, the vas=a=ce asymmenry amplitude slowly aecae"ses withnincae"sing Msup>6681&amsimple-math">069A., "s ssen in thesuppho p>Ael of Fig.l, , 28doa>. For 6681&amsimple-math">069 ≳ 2069<">A., thendipole "mplitude becomes too small "ad wenfind nonsigniftca=tsvas=a=ce "symmenry. It is interosting tolnot", howevtr, that thesdipole airect=ons fouaunfor largo Msup>6681&amsimple-math">069A. are closely al"gnedlwithnthosenfouaunfor Msup>6681&amsimple-math">069< 20<9<">A..

, 6681&ambold">Tablel20<9<">A.6681&amsimple-math">pA.-values for the vas=a=ce asymmenry measurednby 86Msup>6681&amsimple-math">°69<">A. discs for the four com" data-sep"aas"dltempesature m"ps aad aiuthoentshigh-p81&sfiltssnscaless

Theslowho p>Ael of Fig.l, , 27 shows thesdipole airect=ons wenfind u&ing diuthoent disc sizes "au aiuthoent filtssnscales for SMICA. The aipole airect=ons for the Comm"autr, NILC, "aunSEVEM com" data-sep"aas"dlm"ps are vtry simiaar tolthose shownssThe 5symmenry direct=ons fouaunh"ae are consistentlwithnthose determ"nedlby other aaalyseslin this p"ptr.n

In thesuppho p>Ael of Fig.l, , 28doa>,lwe shownthe local-vas=a=ce dipole "mplitudes for the 86Msup>6681&amsimple-math">°69<">A. discs as a funct=on of the -h=taalsm"atipolelof theshigh-p81&sfiltss, Msup>6681&amsimple-math">069A.ssIn theslower p>Ael of thessamssfiguaelwe show,l"s an example, the mean-6681&amsimple-math">°69<">A. discs for the Comm"autrscom" data-sep"aas=on method. The pixelssof the map arelgivtn in terms of theslower-l"aunuppho-tail paobabil"ties of thesvalues from thesa-15lco5"areuntolthesvalues from thessim"aations. The m"ps for NILC, SEVEM, "aunSMICA are vtry simiaar. The numerical values of theslocal-vas=a=ce dipole "mplitudes "ad airect=ons for the Comm"autrsmethod arelgivtn in Tablel, , 21; thesvalues for the NILC, SEVEM, "aunSMICA methods are simiaar.

, , , , 6681&ambold">Fig.l28<9<">A.Uppho p>Ael: local-vas=a=ce dipole "mplitude for 86Msup>6681&amsimple-math">°69<">A. discs as a funct=on of the -h=taalsm"atipolelof theshigh-p81&sfiltss, Msup>6681&amsimple-math">069A., for the four com" data-sep"aas=on methods, Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue). The grey aug=ons, from dark tollight,nco"autpond, autpectively, tol6681&amsimple-math">1σA., Msup>6681&amsimple-math">2σA., aau 6681&amsimple-math">3σA. pho-h=tileslfrom thes1000lFFP8nsim"aationslprocess"dlby thesComm"autrsmethod.s i>Lower p>Ael: mean-6681&amsimple-math">°69<">A. discs and for the Comm"autrscom" data-sep"aas=on method; eachspixel is givtn in terms of theslower-l"aunuppho-tail paobabil"ty of the measurednvalue onsthatlpixel co5"areuntolthesvalues from thessim"aations. The pixelsssn grey co"autpondntolthe -h=taes of thes86Msup>6681&amsimple-math">°69<">A. discs on which thesnumor" of unm"sk"d pixels in thesfull autolut=on map is lower thannour threshold. The black curve suphoposeunon thesmap ind=cates thesbouau"rynof thesopposing hemisph"aeslalong thesasymmenry axis. It is clear that theslargost fract=on of 6681&amsimple-math">>69<">A.95%loutliers (red pixels)llielon thespositives"mplitude hemisph"ae of theslocal vas=a=ce dipole, whilenthe 6681&amsimple-math"><69<">A.5%loutliers (blue pixels)laaenon thesopposite hemisph"ae. Thesco"autponding m"ps for NILC, SEVEM, "aunSMICA are vtry simiaarntolthe one shown h"aes

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6681&amsec2"> 6.2. Dipole moduaation:spixel-ba"edllikelihood

, 6681&ambold">Tablel21<9<">A.

In , , PCIS13 wespruth=t"dl"as"aalysis of thesap"are=t "a"sohropicsdistribut=on of largo-scale power in thes i>assnck 2113 tempesature a-15 within thesp"aamenatc framework aefinedlby , , Gordons(.107)n5aun, , Hoftuft et al. (.109), who introduc"dl"asexplicit aipole moduaationlfieldltolmoutl potentiallhemisph"aical power asymmenryssThe following isna airectnupdate of thatn"aalysis using thes i>assnck 2115sCMB a-15 at 6681&amsimple-math">Nsids69 = 32A., autaining thes2113 commonlm"sk tolexplicitly test for consistencynwithnthesearlier study. Alllaes"aus "relfouau tonbe in excelle=t "greem"ntssIn thesfollowing, wentherefoaelonlylconsidss a smoothing scale of 6681&amsimple-math">5°A. FWHM as a repruth=tativesexample. This is theshighest "ng"aar autolut=on accessible for an 6681&amsimple-math">Nsids69 = 32A. m"p.

Recall first thesbasicsd-15 moutl "dopted in thesaipole moduaationl"pproach: sather thannassuming thesCMB skyntolbena stautstically "sohropicsGau"&iunsfield, we allowsfor an "dditionalsaipole moduaation,sres"auing in ala-15 moutl of thesform 6681&amsimple-math">d = BMs + n69<">A., wh"ae 6681&amimg-inl"ne">, , A. isnaa offset aipole fieldlm"atiply=ng "a innatnsically "sohropicssignals6681&amsimple-math">s69<">A.nwithn" dipolelof amplitude 6681&amsimple-math">αA. poinning towards somesprethos"dlairect=on 6681&amimg-inl"ne">, , A.. 6681&amsimple-math">B69<">A. dunot"s convolut=on withn"a instrumh=talsbeam, aau 6681&amsimple-math">n69<">A. dunot"s instrumh=talsnoise. Additionally, we moutl the power spectrum of thesuauerly=ng stautstically "sohropicsfieldlin terms of a two-p"aamener amplitude–tilt moutl of thesform 6681&amimg-inl"ne">, , A., wh"ae 6681&amimg-inl"ne">, , A. isnthesbost-fit i>assnck 2115s6Msup>6681&amsimple-math">ΛA.CDM spectrum ( a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Collaboaas=on XI 2116). The two p"aameners 6681&amsimple-math">qA. aau 6681&amsimple-math">nA. -ap>accommodate a aeficit in power at lows Msup>6681&amsimple-math"><9<">A. "s co5"areuntolthesbost-fit cosmology that wouldlotherwisolcae"tona tens=on withnthesuauerly=ng stautstically "sohropicsmoutl "au aes"aulin thes"aalysis measur=ng asco5binationlof both asymmenry aau power mismatch.

, , , , 6681&ambold">Fig.l29<9<">A.Top: marginallconstraints on thesdipole moduaationl"mplitude, "snder=vednfrom i>assnck 2115stempesature observationslatna smoothing scale of 6681&amsimple-math">5°A. FWHM for Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)ssThesplotscorautpondsldirectly tolFig.l32 of , , Pssnck Collaboaas=on XXIII (.114). The Comm"autr, SEVEM, "aunSMICA posteriors coincide "lmost pesfectly both internally, "aunwithnthe -o"autponding SMICA 2113 posterior, shown as a dasheunblack l"ne.s i>Bottom: co"autponding m"rginalltwo-dimens=onallconstraints on theslow- Msup>6681&amsimple-math"><9<">A. power spectrum amplitude "ad tilt, Msup>6681&amsimple-math">(q,n).or">A., aefinedlrelat=vestolthesbost-fit i>assnck 2115s6Msup>6681&amsimple-math">ΛA.CDM moutls

A="2">Open withnDEXTER

In thesabsencelof any aipole moduaation,s6681&amsimple-math">α = 069<">A., thentotalsa-15lcovas=a=ce matrix is givtn by 6681&amsimple-math">C = BS"so69BTdo

+ N.or">A., wh"ae 6681&amsimple-math">S"so69
.or">A. isnthesstandard stautstically "sohropicsCMB covas=a=ce matrix givtn by the power spectrum,s6681&amsimple-math">C<9
A., 6681&amsimple-math">N.or">A. isnthesnoise covas=a=ce matrix,l"au the co"autponding likelihood is givtn by thesusual exprut&ion for alm"ativas=atesGau"&iunsdistribut=on. Withnaipole moduaation,sthis generalizes straightforwardly tol6681&amsimple-math">C = BMS"so69MTdo


+ N.or">A., withntheslikelihood givtn by 6681&amimg-equation">, , 6681&amhabel-eq">(43)A.A.Figuael, , 29n5aunTablel, , 22 summariz"sthis fivt-dimens=onalllikelihood in terms of m"rginallp"aameners for eachsof thesfoursassnck CMB m"ps, as evaluated ovss the commonlm"sk using thesm"ati-dimens=onallgrid-ba"edlSnake "lgorithm (, , Mikkelsen et al. .113). Alllaes"aus co"autpondntola smoothing scale of 6681&amsimple-math">5°A. FWHM, theshighest autolut=on support"dlby an 6681&amsimple-math">Nsids69 = 32A.HEALPix grid, but,l"s in .113, we considss all smoothing scales betwsen 6681&amsimple-math">5°A. aau 6681&amsimple-math">10°A. FWHM, reaching simiaarnconclus=onslin eachs-ass: thesdipole moduaationlaes"aus der=vednfrom thes i>assnck 2115stempesature m"ps aae essentially "="nticalstolthes2113 aes"aus, withnimproved internalsconsistencynbetwsen the foursCMB m"ps auentolbetter mitigationlof systemat=c errors. The bost-fit dipole moduaationl"mplitude at 6681&amsimple-math">5°A. FWHM is 6–7% whilst theslow- Msup>6681&amsimple-math"><9<">A. power spectrum h"s an 5pproximatelyl3–5% lower "mplitude co5"areuntolthesbost-fit 6Msup>6681&amsimple-math">ΛA.CDM prudict=on. Theselaes"aus "relfullylconsistentlwithnexpectations givtn that thes i>assnck 2113 sky m"ps werelalready cosmic-vas=a=ce-limiteunon theso "ng"aar scales, "au the 2115sm"ps aiuthonfrom thes2113 m"ps at theslevel of onlyla few microkelvin ( a href="/"autcles/aa/full_html0.106/10/sup>, , Pssnck Collaboaas=on IX 2116).

6681&amsec2"> 6.3. Dipole moduaation:sQMLs"aalysis

In this sect=on we use thesQMLsesuimator introduc"dlinn, , Moss et al. (.111)n5aunaescribsdlinnAppss=ixl, , Cntol81&ut& theslevel of aipole moduaationlin oursesuimates of the CMB skynat 6681&amsimple-math">Nsids69 = 2048A.ssThe speciftc implem"nt"t=on is essentially "="nticalstolthat u&"dlinn, , Hanson 4-636 Lewis (.109), , , Pssnck Collaboaas=on XVII (.114), "ad , , Pssnck Collaboaas=on XXVII (.114), "ad exploits the factlthat dipole moduaationlof any cosmological p"aamener is equivale=t tolcoupling of Msup>6681&amsimple-math"><9<">A. tol6681&amsimple-math"> ± 169<">A. moutslin thesCMB covas=a=ce matrix tolleading orutr (seenAppss=ixl, , C).l, , Pssnck Collaboaas=on XX (.116) pruth=tsnan 5lternates"aalysis for asspeciftc "socurvature moutls

, 6681&ambold">Tablel22<9<">A.6681&amsimple-math">5°A. for all i>assnck 2115sCMB tempesature tolut=ons, "snder=vednby thesbrute-forceslikelihood givtn by Eq. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 43).

Sincelwe are interostedlin dipole moduaationlth"ae are three indepss=e=t esuimators. For oursp"autc"aar "pproach, theso "ae a real-valueu 6681&amsimple-math">m = 069<">A. "ad alco5"lex-valueu 6681&amsimple-math">m = 169<">A. esuimator, "au take thesform 6681&amimg-equation">, , A.H"ae 6681&amsimple-math"> i>Tℓm<9A. "ae 6681&amsimple-math">CA.-inverse filtss"dla-15 aau 6681&amimg-inl"ne">, , A.. We adopt thesinverse-vas=a=ce filtssnfrom a href="/"autcles/aa/full_html0.106/10/sup>, , Pssnck Collaboaas=on XVII (.114), wheaenthes5pproximate filtssnfunct=ons are also specifteussWe aefine 6681&amsimple-math">δCℓℓ + 169 ≡ dC<9/dX + dC + 169/dX.or">A., wh"ae 6681&amsimple-math">X.or">A. isnthesp"aamener moduaat"d, aau 6681&amsimple-math">Aℓm<9A. "au 6681&amsimple-math">Bℓm<9A. "ae numerical coeuticiauts (dutails -ap>bssfouau in Appss=ixl, , C).lThe factor Msup>6681&amsimple-math">f1m<9A. correctsnthesnormalizationlfor errors introduc"dlby masking: 6681&amimg-equation">, , 6681&amhabel-eq">(46)A.A.wh"ae 6681&amsimple-math">M(Ω).or">A.lss thesmaskssFinally, we correct thesdirect=on for the effectsnof inhomogeneoussnoise which is notsaccount"dlfor in the filtssing process, bysweightingnthe 6681&amimg-inl"ne">, , A.lby thesinverse of thesvas=a=ce der=vednfrom filtss"dl"aunmean-fieldlcorrectedssim"aations.

Thesphysics is readily uccessible in this esuimator:nthe 6681&amsimple-math"><9<">A.-depss=e=ce in moduaationldeterm"nedlby thesp"aamener 6681&amsimple-math">X.or">A. isnexprut&ed in thes6681&amsimple-math">δCℓℓ + 169A. factor, "au the releva=tsscales appsarsdirectly inntheslimius of thessum. Wesconsidss the esuimator ovss the r"ago 6681&amsimple-math">min69 = 2 ≤ max69A.ssThe moduaationl"mplitude aad airect=on "relthen givtn by 6681&amimg-equation">, , A.It is worthnre-emphasiz=ng that the quant"ties 6681&amimg-inl"ne">, , A., 6681&amimg-inl"ne">, , A., aau 6681&amimg-inl"ne">, , A.lare all depss=e=t on thes6681&amsimple-math"><9<">A. r"ago considssed.

, 6681&ambold">Tablel23<9<">A.6681&amsimple-math">A<9<">A.) aad airect=on of theslow- Msup>6681&amsimple-math"><9<">A. dipole moduaationlsignalsdeterm"nedlfrom thesQMLs"aalysis for the r"ago 6681&amsimple-math">s∈ [2,64]A.s

As alconseque=ce of the -h=taalslimiu theorem, for suuticiautly largo Msup>6681&amsimple-math">max69A.nthe 6681&amimg-inl"ne">, , A.s are Gau"&iun-distributedlwithnmean zero, so that the "mplitude p"aamener h"s a Maxwell-Boltzmannsdistribut=on. We fitstolthis aistribut=on for Msup>6681&amsimple-math">max69 ≥ 1069<">A. when co5"utingnthe 6681&amsimple-math">pA.-value, so as notstonbe influe=cedlby Poisson noise innthestails of the empirical aistribut=on ("ad wenhave determ"nedlthat this is a good fitstolthessim"aationslby apply=ng " KS test)ssFor the -ass of scalar "mplitude moduaationl(i.e.,n6681&amsimple-math">X = As69A.), aau 6681&amsimple-math">min69 = 269<">A., thencosmic-vas=a=ce-limiteunexpectation for the moduaationl"mplitude from stautstically "sohropicsskies is 6681&amimg-equation">, , 6681&amhabel-eq">(50)A.A.This is thescosmicsvas=a=ce for asscale-invas=a=t dipole moduaation, aau givts a moaenexplicit exprut&ion thannthe 6681&amimg-inl"ne">, , A.lscaling discus&"dlinn, , Hanson 4-636 Lewis (.109).

, , , , 6681&ambold">Fig.l30<9<">A.top p>Ael) 6681&amsimple-math">min69 = 269<">A. or (bottom p>Ael) 6681&amsimple-math">min69 = 100<9<">A.. Nonsigniftca=tsmoduaationlis fouaunonceltheslow- Msup>6681&amsimple-math"><9<">A. signalsis removeussWe emphasize that thesstautstic h"ae is cum"aative aad ap"are=t tre=dslin thescurves -ap>bssmisleadings

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The top p>Ael of Fig.l, , 30 pruth=tsnres"aus for the 6681&amsimple-math">pA.-valuesof thesfitt"dlmoduaationl"mplitude as a funct=on of Msup>6681&amsimple-math">max69A.. Note that th"ae are sevtral peaks, "t 6681&amsimple-math">s≈ 4069<">A. "ad 6681&amsimple-math">s≈ 6769<">A. (thesfocus of most attention inntheslitesature), aau 6681&amsimple-math">s≈ 240<9<">A.. Theslatter peak, whilennotspaeviously emphasized, is also pruth=t innthesWMAPnres"aus (seenFig.l15linn, , Bennett et al. .111).lIt is also interosting tolnot" thatn"lmoduaationl"mplitude is observ"d at 6681&amsimple-math">max69 ≈ 800<9<">A. thatnis somewhat lower thannwhat one wouldltypically expect for asstautstically "sohropicssky. Howevtr, thessigniftca=ce is notsat theslevel of the excess dipole moduaationl"t lows Msup>6681&amsimple-math"><9<">A. "aunwill notsbe discus&"dlfurth"a. The aip at 6681&amsimple-math">max69 ≈ 6769<">A.,nwithn" 6681&amsimple-math">pA.-valuesof 0.9–1.0%,nco"autpondsstoltheswell-known low- Msup>6681&amsimple-math"><9<">A. dipole moduaation, , 6. Tablel, , 23 pruth=tsnthe co"autponding aipole moduaationlp"aameners, which arelssen tolbe consistentlwithnpaevious studies. Note that th"nmean "mplitude expecteunfor asset of stautstically "sohropicssim"aations "t this Msup>6681&amsimple-math">max69A.nis 2.9%l(ip>66ose "greem"ntnwithnthesexpecteunvaluesauentolcosmicsvas=a=ce, Eq. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 50)).

Wenhave therefoaeldeterm"nedla phenomenological signature of moduaationlfor Msup>6681&amsimple-math"> = 269<">A.– Msup>6681&amsimple-math">6769<">A. withn" 6681&amsimple-math">pA.-valuesof 0.9–1.0%.lIf suchna signalshad been prudict"dlby asspeciftc moutl,lthen we couldl668im ansrgniftca=ce of about 6681&amsimple-math">3σA..nHowevtr, in thesabsencelof suchnan alpriori moutl,lwe cannassess hownoften we mightnfind " 6681&amsimple-math">3σA. effectnby cha=ce, givtn that it couldlhave occurred ovss any 6681&amsimple-math"><9<">A. r"ago. Sincelwe are lookingnfor aslargo-scale phenomenon, we assume that th"n"aalysis shouldlinclude the co"autponding low- Msup>6681&amsimple-math"><9<">A. moutslaaunstart "t 6681&amsimple-math">s= 269<">A.ssIn orutr tolcorrect for asposteriori effectsnwelthen adopt thesfollowing schem".

  • 1. Wescalcuaat" the moduaationlof eachssim"aationson thesscales 2– Msup>6681&amsimple-math"><9<">A., wh"ae 6681&amsimple-math">s∈ [10,max69]A.ssFor eachssim"aationswenfind the moduaationlthat givts thesmallest paobabil"ty,s6681&amsimple-math">η<9<">A. (in thessamssway that w"sndone for the a-15).

  • 2. Withnthesdistribut=on of 6681&amsimple-math">η<9<">A.s givtn by thessim"aationslwelthen co5"are this tolthesa-15. Thatnis,lwe calcuaat" the paobabil"ty that one wouldlfind oneself in alHubblelpatch withn" moduaationl"mplitude up tol6681&amsimple-math"> ∈ [10,max69]A. thatnis "s signiftca=ts"s (or moaensigniftca=tsthan) the moduaationlin thesrealsa-15.

If Msup>6681&amsimple-math">max69 = 132A. ("s chosen by , , Bennett et al. .111), the paobabil"ty of achieving " moduaationl"s largonas thes i>assnck a-15 in this r"ago is highes thann10% (seenFig.l, , 31).lThis is in agreem"ntnwithnthesfindings of the WMAPnteam (which fouaun10% aaun13% in thessamss6681&amsimple-math"><9<">A.-r"ago, using twolaiuthoent masks)ssHere, weldo notsquotona speciftc PTE for the aipole moduaationlsi=ce it depss=s on theschoice of both Msup>6681&amsimple-math">max69A.n(albeit notsso sens=tively) aau 6681&amsimple-math">min69A.n(which wenhave decidedlnotstonm"rginalize ovss). Howevtr, itlappsars tolbe the -ass that thesdipole moduaationlthat we observo is quite unremarkable. Thatnis,lGau"&iunsfluctuationslin alstautstically "sohropicsUniverse will reasonablynoften aes"aulin asdipole moduaationlwithn" co5"arablellevel of signiftca=ce tolthat pruth=t"dlh"ae.

, , , , 6681&ambold">Fig.l31<9<">A.6681&amsimple-math">s∈ [10,max69]A.ssThe verticalsl"nenco"autpondsstol Msup>6681&amsimple-math">max69 = 132A. which was u"edlasnthe ssarch limiu inn, , Bennett et al. (.111)ssThespaobabil"ty grows 5pproximatelyllogarithmically withn Msup>6681&amsimple-math">max69A.. This means that th"n"dopted paobabil"ty to exceednis fortunatelylnotsvtry sens=tivestol Msup>6681&amsimple-math">max69.or">A., aau for any reasonableschoice is "bovsn10%s

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Beyondnthis,levide=ce for aipole moduaationlis fouaun"t 6681&amsimple-math">s≈ 200<9<">A.– Msup>6681&amsimple-math">300<9<">A.,nwithn" smaller aip at 6681&amsimple-math">s≈ 500<9<">A.. Givtn that thesaipole moduaationlesuimator is alcum"aative quant"ty, itlis possible that these features are stautstically enha=ced by thesusual low- Msup>6681&amsimple-math"><9<">A. signal. To test this w"n"aalyse thesaipole moduaationlas a funct=on of Msup>6681&amsimple-math">max69A. again, withnthesaesnrict=on 6681&amsimple-math">min69 = 100<9<">A. applieu in orutr tolco5"letelylremove any low- Msup>6681&amsimple-math"><9<">A. influe=cessThesoutcomelis pruth=t"dlin Fig.l, , 30 (bottom). It is clear that evtn befoaelintroducing posterior correct=ons nonsigniftca=tsmoduaationlis fouau, ind=cat=ng that the 6681&amsimple-math">pA.-values of the features at 6681&amsimple-math">> 100<9<">A. werelindeeunexaggeaas"dlby theslow- Msup>6681&amsimple-math"><9<">A. mouuaation.

6681&amsec2"> 6.4. Bipolar sph"aical harmonics

In thesabsencelof thesassumptionlof stautstical "sohropy, the CMB two-poinn correlationlfunct=on 6681&amimg-inl"ne">, , A. -ap>bssmost generally exp"autd in thesbipolar sph"aical harmonic (BipoSH)sbasis repruth=tationlas follows: 6681&amimg-equation">, , 6681&amhabel-eq">(51)A.A.The BipoSHsbasis funct=ons, 6681&amimg-inl"ne">, , A.lare tensor producus of oruinary sph"aical harmonic funct=ons, "au the co"autponding exp"a&ion coeuticiauts are termtd BipoSHscoeuticiauts (, , Hajiuns4-636 Souradeep 2003; , , Hajiuns4-636 Souradeep 2006). The BipoSHsbasis proviutslalco5"lete repruth=tationlof any form of stautstical "sohropy vioaationlwithnthe keyn"dva=tago of sep"aas=ng thesang"aar scale-depss=e=ce of thessignalsin sph"aical harmonic m"atipoles, Msup>6681&amsimple-math"><9<">A.,nfrom thesnature of thesvioaationlindextd in thesbipolar m"atipolel<">ce by 6681&amsimple-math">LA..nConseque=tly, itlis possible tossim"ataneously determ"ne that suchna signalsis dipolar (6681&amsimple-math">L = 169<">A.), quadrupolar (6681&amsimple-math">L = 269<">A.), octopolar (6681&amsimple-math">L = 369<">A.), aaunso on, in nature "au that the power issaesnricteuntolspeciftc r"agos of ang"aar scales.

Thesesuimationlof BipoSHscoeuticiauts from CMB m"ps is alnaturallgeneralizationlof the more routinelyluauertakensesuimationlof thesang"aar power spectrum 6681&amsimple-math">Cl<9A.. To allowsa airectnconnectionstolthesang"aar power, welfurth"a introduc" asset of BipoSHsspectr5 at evtry bipolar harmonic moment, Msup>6681&amsimple-math">(L,M).or">A., habell"dlby asaiuthoencelindex 6681&amsimple-math">d<9<">A.,naefinedlas follows: 6681&amimg-equation">, , 6681&amhabel-eq">(52)A.A.wh"ae 6681&amimg-inl"ne">, , A. "relthe Clebsch-Gordonscoeuticiauts aau for baevity thesnotationl6681&amimg-inl"ne">, , A.. BipoSHsspectr5, clearly, "relthen simply asgeneralizedsset of CMB ang"aar power spectra, withnthesstandard CMB ang"aar power spectrum 6681&amimg-inl"ne">, , A.lbeing one of them, , 7. Whilen6681&amimg-inl"ne">, , A. quant"fies thesproperties of the stautstically "sohropicsp"au of the CMB fluctuations, th"n"dditionalsBipoSHscoeuticiauts quant"fy the stautstically "a"sohropicsp"au of the CMB two-poinn correlationlfunct=on.

, 6681&ambold">Tablel24<9<">A.6681&amsimple-math">A<9<">A.) aad airect=on of thesaipole moduaationlin Galact=c cooruinates as esuimated for the m"atipolelr"ago 6681&amsimple-math">s∈ [2,64]A. using asBipoSHs"aalysiss

ThussBipoSHsproviutslalmathemat=callylco5"lete aescriptionlof all possible vioaations of stautstical "sohropy in asGau"&iunsCMB skynm"p. It is then always possible tostranslates"aysspeciftc moutl for suchna signalsintoltheslang"ago of BipoSHs"adsproviutlalco5monl"pproach for the m"atiple specializedstests that have been implem"nted paeviously in this p"ptrs"adselsewh"ae. Howevtr, improving on th"n"aalysis of thes2113 i>assnck a-15, alnewsformalismsis developeu in orutr tolreliablyn"aalyse a m"sk"d sky, "s concisely describsdlinnAppss=ixl, , D. , , Aluri et al. (.115) proviutslalmoaeldetail"dlaescriptionlof th"n"pproach "aunincludes an explicit aemonstrationlof ius val"dity using sim"aations.

Initially, welaevisit thessimple phenomenological moutl of dipole moduaationlof the CMB skynfrom Sect.l, , 6.2, 6681&amimg-equation">, , 6681&amhabel-eq">(53)A.A.wh"ae 6681&amimg-inl"ne">, , A. repruth=ts the moduaated CMB sky, 6681&amimg-inl"ne">, , A. is thesuauerly=ng (stautstically "sohropic)lr"adom CMB sky, "au 6681&amimg-inl"ne">, , A. isna dipolar field. The BipoSHscoeuticiauts res"auing from suchna moduaationlarelgivtn by 6681&amimg-equation">, , A.H"ae 6681&amimg-inl"ne">, , A. co"autpondsstolthesBipoSHscoeuticiauts of thesuaknown, but stautstically "sohropic, unmoduaated CMB field, 6681&amsimple-math">m1M<9A. "ae thessph"aical harmonic coeuticiauts of thesmoduaationlfield, aau 6681&amsimple-math">C<9A. isnthesbost-fit CMB ang"aar power spectrum.

ThesBipoSHsrepruth=tationlfurth"a enables an esuimate of thesmoduaationlfield tolbe made ovsssspeciftc ang"aar scales byswiadowing aug=ons in m"atipolel<">ce in thessum ovsssm"atipoles 6681&amsimple-math"><9<">A. in Eq. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 55).lThis "dditionalsinformationlis importa=tsfor i="ntify=ng thesoriginlof the "sohropy-baeaking signal, which couldlbe eith"a cosmological or auentolsystemat=c "auefacts.

Wenpesform th"n"aalysis for the 6681&amsimple-math">Nsids69 = 2048A.scom" data sep"aased CMB m"ps withn"a apodizedsversionlof the commonlm"sk atlthat autolut=on aau aeconstruct thesmoduaationlsignalsin indepss=e=t bins of widthn Msup>6681&amsimple-math">Δs= 64A. up tol6681&amsimple-math">max69 = 51269<">A.ssTh"n"ppl=cat=onlof the commonlm"sk introduc"slalmean field biaslin thesBipoSHscoeuticiauts der=vednfrom thesa-15. This biaslis esuimated from thesFFP8ssim"aations "aunsubtractednfrom thesaer=vedncoeuticiautsssThespaocess of m"sking induc"slalcoupling betwsen the moduaationlfield "au the m"sk that aut"aus in asmodiftcat=onlof the spectral sh"pt of thesmoduaationlsignalsby thesmodift"d sh"pt funct=on (MSF; seenAppss=ixl, , D for autails)ssFurth"a, thencovas=a=ce of thesbias-subtractednBipoSHscoeuticiauts is notseasy to aer=ven"aalytically "n this -ass. To ovsscomelthis problem, we considss thesaiagonals5pproximationstolthescovas=a=ce matrix "adsesuimate it from sim"aations.

Thesaut"aus pruth=t"dlin the top p>Ael of Fig.l, , 32 ind=cats that thesdipole moduaationlsignalsis most signiftca=tsinntheslowest m"atipolelwiadow 6681&amsimple-math">s∈ [2,64]A.s Note that th"npower in thesaipole moduaationlfieldl6681&amsimple-math">m1<9 = ( | m11<9 | 2 + | m10<9 | 2 + | m1−1<9 | 2) / 369<">A. issaeaated tolthesaipole "mplitude by 6681&amimg-inl"ne">, , A.. Thesbost-fit amplitude (6681&amsimple-math">A<9<">A.) aad airect=on co"autponding tolthesaeconstruct"dlaipole moduaationlfieldlfrom thisslowest m"atipolelbin is quot"dlin Tablel, , 24<9a> for eachs-om" data-sep"aas=onlmethod. Also shown are the co"autponding res"aus for the cleanedlfreque=cy m"ps SEVEM-100, SEVEM-143, "aunSEVEM-217. As expecteunfor signalslwithn" cosmological origin, nonevide=ce for freque=cy depss=e=ce islssen.

, , , , 6681&ambold">Fig.l32<9<">A.Top: measur"dlaipole moduaationl(6681&amsimple-math">L = 169<">A.)npower in non-ovssl5pping CMB m"atipolelbins for Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue) "sndeterm"nedlfrom asBipoSHs"aalysis of thesa-15. Th"npower in thesaipole of thesmoduaationlfield isna 6681&amsimple-math">χ2A.-distributedlvas=ablelwithn3ndegrees of freedomssThe shadedlreg=ons in th"nplotsdepict, in dark-grey, grey, "aunlight-grey respectively, then1, 2, aau 6681&amsimple-math">3σA. equivale=t intervals of the aistribut=on funct=on der=vednfrom sim"aations, whilenthe soliunblack l"ne dunot"s its meaian. Signiftca=tspower in thesaipole moduaationlislssen tolbe limiteuntol6681&amsimple-math"> = 269<">A.– Msup>6681&amsimple-math">64A. aad aoes notsextendntolhighes m"atipoles.s i>Bottom: aipole moduaationlairect=on "sndeterm"nedlfrom thesSMICA m"p. Thesdirect=ons fouaunfrom thesoth"a com" data sep"aas=onlm"ps aae consistentlwithnthiss"aalysiss Thescolour"dlcircles dunot" the ch=taalsvaluesof thesm"atipolelbin u&ed in thes"aalysis, "s specifteulin thescolour baa. The low- Msup>6681&amsimple-math"><9<">A. aad WMAP-9sdirect=ons are i="nticalstolthose innFig.l, , 35.

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Sincelthes"mplitude of thesaipole moduaationlfield isnconsistentlwithnzerolwithin 6681&amsimple-math">2σA. for all of theshighes 6681&amsimple-math"><9<">A.-bins considssed, itlis plausible that thessimple moduaationlmoutl in Eq. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 53) is inadequate to aescribs the features ssen in thesBipoSHsspectr5 aaunshouldlm"nimally "llowsfor the "mplitude, 6681&amsimple-math">A().or">A., of thesaipole to aepss= on CMB m"atipole, Msup>6681&amsimple-math"><9<">A.. Althoughnthissmay appsarstolbe almoaelco5"lex moutl,lit aoes notsnecessas=ly lack motivat=on. It is readily conceivable that physical mecha=isms that cause a dipolar moduaationlof the r"adom CMB sky wouldlbe scale-depss=e=ts"adspossibly signiftca=tsonlylat lowswavenumbers. It is also intriguing tolnot" that, althoughnin most -assslthes"mplitude of thesmoduaationlaipole islssen at lowssigniftca=ce, thesairect=ons in the first foursbins, Msup>6681&amsimple-math"><32<9 ∈ [2,64]A., Msup>6681&amsimple-math"><96<9 ∈ [65,128]A., Msup>6681&amsimple-math"><160<9 ∈ [129,192]A., aau 6681&amsimple-math">224<9 ∈ [193,256]A., arelssen tolbe clustss"dltogoth"a, as shown in thesbottom p>Ael of Fig.l, , 32s Note that th"nlower srgniftca=ce of thesmoduaationlfor the m"atipolelbins at 6681&amsimple-math">> 64A. aut"aus in largos errors for their respectivesairect=ons thannthe valuesquot"dlfor the 6681&amsimple-math">s∈ [2,64]A. bin aecord"dlin Tablel, , 24<9a>.

Wenextendnours"aalysis tolcarry out thesdipole moduaationlaeconstructionlin cum"aative bins up tol6681&amsimple-math">max69 = 51269<">A.,smaking cum"aative increm"nts in the m"atipolelin steps of Msup>6681&amsimple-math">Δs= 64A.. The aut"aus of thiss"aalysis arelsummariz"dlin Fig.l, , 33.

, , , , 6681&ambold">Fig.l33<9<">A.Top: measur"dlaipole moduaationlpower in cum"aative CMB m"atipolelbins for Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue) "sndeterm"nedlfrom asBipoSHs"aalysis of thesa-15..nColour coding aslin Fig.l, , 32s Note that th"nmeasur"m"nts in cum"aative bins ind=cats alpower in excess of Msup>6681&amsimple-math">2σA. up tolm"atipolel6681&amsimple-math">max69 ~ 320<9<">A.. Thesvalueson the horizontals5xis dunot"s the m"ximumlm"atipolelu&ed in thes"aalysis, withn Msup>6681&amsimple-math">min69 = 269<">A..s i>Bottom: moduaationlaipole airect=on "snaecovss"dlfrom thesSMICA m"p. Thesdirect=ons fouaunfrom thesoth"a com" data-sep"aas=onlm"ps aae consistentlwithnthese direct=onss Thescolour-coded poinns repruth=t thesairect=ons aecovss"dlfor the speciftc Msup>6681&amsimple-math">max69A. u&ed in thes"aalysis, withn Msup>6681&amsimple-math">min69 = 269<">A..sThe low- Msup>6681&amsimple-math"><9<">A. aad WMAP-9sdirect=ons are i="nticalstolthose innFig.l, , 35.

A="2">Open withnDEXTER

As noted paeviously,las a conseque=ce of our motionlwithnrespect tolthesCMB aesnnframe, thesobserv"d CMB m"p isnexpecteuntolbe stautstically "a"sohropic,las h"s been aemonstrat"dlinn, , Pssnck Collaboaas=on XXVII (.114) aad Appss=ixl, , B. Reassusingly, inn, , PCIS13 it w"snesnablishedlthat suchna signalswouldlnotscontaminates"sdipole moduaationlsignalsup tol6681&amsimple-math">max69 ≈ 700<9<">A.. Wesnowsconfirm thesDopplersboost signalsus=ng thesBipoSHsmethodology.

An equivale=t aescriptionlof th"nDopplersboost in termslof BipoSHscoeuticiauts islgivtn by 6681&amimg-equation">, , A.wh"ae 6681&amimg-inl"ne">, , A., 6681&amsimple-math">β<9b> = v<9b>/c<9<">A. dunot"s the peculiar veloc"ty of our local aesnnframelwithnrespect tolthesCMB, aau 6681&amsimple-math">bν<9A. isnthesfreque=cy-depss=e=tsboost factor, "s discus&"dlinnmoaeldetaillinn, , Pssnck Collaboaas=on XXVII (.114).

SincelthesDopplersboost signalsh"s a freque=cy depss=e=ce, we pesform ours"aalysis on the SEVEM-100, SEVEM-143, "aunSEVEM-217lm"ps at 6681&amsimple-math">Nsids69 = 2048A., "aunadopt values of 6681&amsimple-math">bν<9 = 1.51,1.96<9<">A., aau 6681&amsimple-math">3.0769<">A.,nrespectively. Alm"nimumlvas=a=ce esuimator for Msup>6681&amsimple-math">β1M<9A., "s discus&"dlinnAppss=ixl, , D, is adopted withnthessh"pt funct=on 6681&amimg-inl"ne">, , A. replaced by thesco"autponding Dopplersboost termlgivtn in Eq. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 56). Co"autponding unboosted CMB sim"aationslwere also u&ed, innp"autc"aar tolcorrect for th"nmean field bias. Howevtr, welu&e asset of Doppler-boosted sim"aationslin orutr tolesuimate th"nerror on the aeconstruct"dlDopplersboost vector.

Sincelitlis expecteunthat th"nlowlm"atipolelmoutslof th"n6681&amimg-inl"ne">, , A. spectrum aae contaminated by thesdipolar signalsreported paeviously,lin orutr tolmonitor th"nimpact of thiss"aomalousssignalson the Dopplersaeconstructionlwe implem"nt alcum"aative "aalysis us=ng m"atipoles withn" vasy=ng 6681&amsimple-math">min69A.nfrom Msup>6681&amsimple-math">2A. tol6681&amsimple-math">64069<">A. in increm"nts of Msup>6681&amsimple-math">Δmin69 = 128<9<">A. aad a fixtd Msup>6681&amsimple-math">max69 = 1024<9<">A., , 8. The aecovss"dlDopplers"mplitudesnfrom thesthree SEVEM freque=cy cleanedlm"ps as a funct=on of Msup>6681&amsimple-math">min69A.narelshown in thestop p>Ael of Fig.l, , 34, whilenthe lower p>Ael ind=catssnthe co"autponding airect=on 6681&amimg-inl"ne">, , A.lin Galact=c cooruinates determ"nedlfrom thesSEVEM-217la-15. Tablel, , 25 aecordsnthesbost-fit "mplitudesnaad airect=onslfor Msup>6681&amsimple-math"> ∈ [640,1024]A.s

, , , , 6681&ambold">Fig.l34<9<">A.Top: "mplitude Msup>6681&amsimple-math">| β |.or">A.lof th"nDopplersboost from thesSEVEM-100, SEVEM-143, "aunSEVEM-217lm"ps for aiuthoent m"atipolelbins determ"nedlusing asBipoSHs"aalysisssThe m"ximumlm"atipolelof eachsbin is fixtd at 6681&amsimple-math">max69 = 1024<9<">A., whilen Msup>6681&amsimple-math">min69A.nis increm"ntedlfrom 6681&amsimple-math"> = 269<">A.ntol6681&amsimple-math"> = 64069<">A. in steps of Msup>6681&amsimple-math">Δs= 128<9<">A.. Thesdashedll"nenco"autpondsstolthes"ctualsdipole boost "mplitude, 6681&amsimple-math">| β<9b> | = 1.23 × 106sup>-3A..s i>Bottom: Dopplersboost airect=on 6681&amimg-inl"ne">, , A. measur"dlin Galact=c cooruinates from SEVEM-217. Thescolour"dlcircles dunot" Msup>6681&amsimple-math">min69A.nu&ed in thes"aalysis, whilen Msup>6681&amsimple-math">max69 = 1024<9<">A. is heldlfixtd.sThe low- Msup>6681&amsimple-math"><9<">A. aad WMAP-9sdirect=ons are i="nticalstolthose innFig.l, , 35.

A="2">Open withnDEXTER

, 6681&ambold">Tablel25<9<">A.6681&amsimple-math">| β<9b> |<9<">A.) aad airect=on in Galact=c cooruinates der=vednovsssthe m"atipolelr"ago 6681&amsimple-math">s∈ [640,1024]A. as evaluated from asBipoSHs"aalysiss

6681&amsec2"> 6.5. Ang"aar clustssing of th"npower aistribut=on

In thes i>assnck 2113 a-15 release welaeported a possible deviationlfrom stautstical "sohropy in the m"atipolelr"ago 6681&amsimple-math">s= 269<">A.–600, thussconfirming earliersfindings ba&ed onnthesWMAPna-15 (, , Hansenset al. .109; , , Axelsson et al. .113).lThis c68im of asymmetrynextending tolhighes m"atipoles w"snmade onlylonnthesbasis of th"n"l"gnm"nt of preferr"dlairect=ons as determ"nedlfrom m"ps of th"npower aistribut=onson thesskynfor speciftc m"atipolelr"agosssIn p"autc"aar, it w"snfouaunthat thesdirect=ons of thesaipolessfitt"dltolsuchnm"ps in the m"atipolelr"ago 6681&amsimple-math">s= 269<">A.–600lwere signiftca=tlylmoael"l"gneunthan in sim"aations. In "ddition, welshoweunthat th"naas=o of th"npower spectr5 in thestwo opposite hemisph"aes definedlby thesasymmetryn5xis for Msup>6681&amsimple-math"> = 2−600<9<">A. was notsstautstically "aomalouss("s lat"a confirmednovsssthe extendedlm"atipolelr"ago 6681&amsimple-math">s= 2−.100<9<">A. by , , Qu"autns4-636 Notari .115).l

, , , , 6681&ambold">Fig.l35<9<">A.6681&amsimple-math"> = 269<">A.ntol6681&amsimple-math">1500<9<">A. in thesSMICA m"p withnthescommonlm"sk applieu. Wesalso show th"npreferr"dlaipolar moduaationl5xis (habell"dl"s “low- Msup>6681&amsimple-math"><9<">A.”) der=vednin Sect.l, , 6.2, asswelllasnthe totalsdirect=on for Msup>6681&amsimple-math">max69 = 600<9<">A. determ"nedlfrom WMAP-9s( a href="/"autcles/aa/full_html0.106/10/sup>, , Axelsson et al. .113).lThesavtrage airect=ons determ"nedlfrom thestwo m"atipolelr"agos 6681&amsimple-math">s∈ [2,300]<9<">A. aad 6681&amsimple-math">s∈ [750,1500]<9<">A. arelshown "s bluesaau aeu aings,nrespectively. Th"nerror on the der=vedndirect=on that aut"aus from m"sking thesa-15 is "bout 6681&amsimple-math">60°A., withnonlylsmalllvas=at=ons aeaated tolbin size.

A="2">Open withnDEXTER

Here, weltest for the "l"gnm"nt in thes i>assnck 2115la-15 set. Wesadopt thes"pproach for the esuimationlof thesaipole "l"gnm"nt that w"sndescribsdlinndetaillinn, , PCIS13, albrieflsummary of which follows.

Note that th"nstautstics definedlin step 6 "bovsnco"autpond toltwo choices of what w"aelreferr"dltolas “global stautstics” inn, , PCIS13 in orutr tolasses& the degree tolwhich thessigniftca=ce of thesaet"aus depss=s on a speciftc choice for Msup>6681&amsimple-math">max69<9<">A.. Thesmean 6681&amsimple-math">pA.-value ovsss"lllavailablel Msup>6681&amsimple-math">max69A. measur"s the degree tolwhich clustssing is pruth=t ovssslargo m"atipolelr"agos indepss=e=tly of wheth"a the clustssing is stronglynfocu&ed in onengivtn airect=on. Clearly th"n6681&amsimple-math">pA.-values for aiuthoent Msup>6681&amsimple-math">max69A. are stronglynco"auaated, but if the clustssing is pruth=t onlylovsss"lsmalllm"atipolelr"ago,lthesRSnwill drop "ad th"nco"autponding 6681&amsimple-math">pA.-values will evtntually riss. Bylco5"aring thissvaluestossim"aations, weltest notsonlylwheth"a the aipole "l"gnm"nt in thesa-15 is stronges thannin stautstically "sohropicsr"adom sim"aations, but also wheth"a itlis pruth=t ovssslargor r"agos of m"atipoles thannexpecteu. Thesm"nimuml6681&amsimple-math">pA.-value will givt strong detect=ons if therenis alstrong asymmetrynovsss"llimiteunm"atipolelr"ago or weaker clustssing ovssslargor m"atipolelr"agos wh"nlthe clustssing is stronglynfocu&ed in angivtn airect=on.

For Comm"autr, NILC, "aunSEVEM,sonlyl500ssim"aations "res"vailable. Howevtr, 5000ssim"aations "res"vailable for SMICA, which "llows albetter esuimate of srgniftca=ce tolbe determ"nedlwh"nlthe probabilities obta"nedlaresvtry low.sInnthis -ass, we u&e half of thes5000ssim"aations tolcalibrate th"nstautstic (obta"nn6681&amsimple-math">pA.-values following step 5 "bovs) "ad th"nrema"ning half to aeterm"ne srgniftca=ce levels (com"utenth"nmean "adsm"nimumlovsssthes"np-values "s a funct=on of Msup>6681&amsimple-math">max69A. following step 6). Whensus=ng 500ssim"aations, itlis necessasy tolu&e the samelset of sim"aations tolcalibrate asswelllasnto obta"n probabilities. Alauaatedlissuelwithnthese aut"aus is that thislsst of sim"aations (co"autponding tolthesfirst 500sout of thes5000s"vailable for SMICA)laresobserv"d tolyield highes 6681&amsimple-math">pA.-values for the clustssing "agle duentola stautstical fluctuation. Anoth"a 9 sets of 500ssim"aations that canlbe obta"nedlfrom p"auttioning thes5000s"vailable SMICA sim"aations "ll ret"au in lower 6681&amsimple-math">pA.-values. As a conseque=ce, we observ" that aut"aus ba&ed onntheslargor number of sim"aations often givt lower 6681&amsimple-math">pA.-values thannwh"nlonlyl500ssim"aations "resu&ed.

InnFig.l, , 35 welshow the aipole direct=ons of thes15slowest 100-m"atipolelbins for the SMICA m"p. Here, the binning has been chosennfor visualizationlpurposes; innfurth"a "aalysis of thesassnck a-15 we u&e fines 6681&amsimple-math"><9<">A.-intervalsssThespaeferr"dllow- Msup>6681&amsimple-math"><9<">A. moduaationlairect=on determ"nedlin Sect.l, , 6.2 is also ind=catsd, along withnthesWMAP-9sret"au determ"nedlovsssthe r"ago 6681&amsimple-math">s= 269<">A. tol6681&amsimple-math">600<9<">A. ( a href="/"autcles/aa/full_html0.106/10/sup>, , Axelsson et al. .113).lThesobserv"d clustssing of th"naipole direct=ons islsimiaar tolthat shown in figur" 27 of a href="/"autcles/aa/full_html0.106/10/sup>, , PCIS13s Note that aiuthoences in m"sking, foaegrouaunsubtract=on, aau aesidualssystemat=c effectsnwill displace the airect=on of angivtn aipolelwithnrespect tolthespaevious "aalysisssSimiaar behaviour islssen for all of thesassnck com" data-sep"aasedlm"ps.l

, , , , 6681&ambold">Fig.l36<9<">A.6681&amsimple-math">pA.-values for the ang"aar clustssing of th"npower aistribut=on "s a funct=on of Msup>6681&amsimple-math">max69A., determ"nedlfor Comm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue), ba&ed onn500ssim"aationsssFor SMICA, th"n6681&amsimple-math">pA.-values ba&ed onn2500ssim"aations "res"lso shown (black). Thes6681&amsimple-math">pA.-values "resba&ed onnthesfract=on of sim"aationslwithn" highes RS, determ"nedlovsssthe 6681&amsimple-math"><9<">A.-r"ago up tolthe givtn Msup>6681&amsimple-math">max69A.,lco5"ared tolthesa-15. Th"naut"aus shown herenhave been marginalizedsovsssbin sizeslin the r"ago 6681&amsimple-math">Δs= 8<9<">A. tol6681&amsimple-math">Δs= 3269<">A..s /p> A="2">Open withnDEXTER

Inn, , PCIS13, we calc"aatedlth"nmean "aglelbetwsen all possible pairs of dipole airect=ons determ"nedlfrom m"ps of th"nlocal power in m"atipolelbins of size Msup>6681&amsimple-math">Δs= 1669<">A..sH"ae weltest th"npossible biaslarising from suchna choice by considssing bin sizeslbetwsen Msup>6681&amsimple-math">Δs= 8<9<">A. aad 6681&amsimple-math">Δs= 3269<">A. in steps of 2.sThe lowssslimits"voids signiftca=tsbin-to-binlcoupling in th"npower spectra for smaller binnings, whilst th"nuppssslimitsexcludes -assslwh"re theael"re anninsuuticiaut number of der=vedndipolessfrom which thesmean "aglelcanlbe calc"aated, thislleading tolpoor stautstics. In "dditionstosshowing aus"aus for eachsbin size, we "lso calc"aate the vas=a=ce-weight"dlmean of the power spectra ovsss"lllbin sizesl(the 6681&amsimple-math">C<9A. for a givtn bin size islweight"dlby 6681&amimg-inl"ne">, , A. wh"ae 6681&amsimple-math">Nb69A. is the bin size).sInnthis way, we marginalizesovsssbin sizeslto obta"n local power spectr5 aau thereby thesRS for eachssingle m"atipole.

Figur" , , 36sshows th"n6681&amsimple-math">pA.-values for the aiuthoent com" data-sep"aasedlm"ps, der=vedn"sndescribsdlinnstep 5 "bovs. Welseenthat th"naut"aus ba&ed onn500ssim"aations for NILC, SEVEM,s"aunSMICA arelin goodn"greem"nt.sThe Comm"autrsaut"aus arelless consistent, but thissmay besaeaated tolthesfact that com" data sep"aas=onlw"snpesformsdlindepss=e=tly for the half-missionltolut=ons,lin contrast tolthesoth"a methods, wh"re com" data-sep"aas=onltolut=ons w"aelobta"nedlfrom thesfulllmissionla-15 onlyssFor SMICA, wesalso show 6681&amsimple-math">pA.-values ba&ed onn2500ssim"aations.sTheselmoael"ccurate aut"aus show lower 6681&amsimple-math">pA.-values,s"aunmay ind=cats that those determ"nedlfrom onlyl500ssim"aations "resnotssuuticiautlylstable. Note also that for Msup>6681&amsimple-math">< 100<9<">A. th"np-values "resnotsconsistentlwithnthe detect=on of anlow- Msup>6681&amsimple-math"><9<">A. asymmetry/moduaation, as setn by oth"a methods "n this papes. Howevtr, for Msup>6681&amsimple-math">< 100<9<">A., theael"re vtry few bins aad th"nvas=a=ce of thesRS might therefoaelbs toolhigh for this effect tolbe visible.

Inn"greem"nt withnthesconclusionslin , , PCIS13, allargo degree of al"gnm"nt islssen at least tol6681&amsimple-math">max69 ≈ 600<9<">A.. Howevtr, in contrast tolthesearliersaut"aus wh"re the 6681&amsimple-math">pA.-values start"dlincreasing systemat=cally for Msup>6681&amsimple-math">max69> 1000<9<">A., the curoent 6681&amsimple-math">pA.-values rema"n lowsfor Msup>6681&amsimple-math">max69> 750<9<">A.. Thesfulllcom" data-sep"aasedlm"ps which have highes autolut=on aaunsensitiv"ty "resu&ed for the curoent analysis, instead of thessingle-freque=cy foaegrouau-cleanedlm"p (SEVEM-143)nu&ed in a href="/"autcles/aa/full_html0.106/10/sup>, , PCIS13s Wesnotenthat th"naut"aus for the updasedlSEVEM-143lm"p aae consistentlwithnthesearliersanalysis, bothnwithn"nd without correctionlfor the Dopplersmoduaation. Note also that the SMICA aut"aus withnimprovednstautstics (ba&ed onn2500ssim"aations) generally show lower p-values thannthe co"autponding res"aus ba&ed onn500ssim"aationsss

, 6681&ambold">Tablel26<9<">A.

Tablel, , 26spruth=tsnthesfract=on of sim"aationslwithn" lower mean/m"nimuml6681&amsimple-math">pA.-value than in thesa-15 for a number of diuthoent casss. Thestablelshows probabilities for SMICAlwithndiuthoent bin sizesl(showing onlylevtry second bin size sincelthese aae co"auaated), asswelllasnfor the res"aus marginalizedsovsssbin sizes. Wesalso show aut"aus for the aiuthoent com" data-sep"aasedlm"ps, res"aus ba&ed onnhalf-sing cross-spectra instead of half-missionlcross-spectra, aau aes"aus using asaiuthoent Msup>6681&amsimple-math">A.-weighting scheme, speciftcally Msup>6681&amsimple-math">(2s+ 1)C<9A. instead of Msup>6681&amsimple-math">(s+ 1)C<9A., the formsr being asmeasur" of th"nvas=a=ce of thestempesature fluctuations. Thestablelind=catssnprobabilities ofs"pproximately 0–2% for most of these casss, althoughnaut"aus for the smallest bin size showlm"ch less signiftca=tsaut"aus.lThis couldlbe duentolth"nstrong antico"auaationslbetwsen adjacent bins fouaunfor this bin size in those Galact=c 6681&amsimple-math">Nsids69 = 169<">A. patches withnvtry sm"lllavailablelskynfract=onssFor thesoth"a bin sizes, these co"auaationslaae m"ch weakers Note that m"ay of thessigniftca=ces ba&ed onnm"nimuml6681&amsimple-math">pA.-value aresonlyluppssslimits.lThis is duentolth"nfact that th"nlimiteunnumber of sim"aations in some casss aut"aus in the lowsst m"nimuml6681&amsimple-math">pA.-value being zero. Whensthesm"nimumlp-value in thesa-15 is zero, welshow the pesch=tago of sim"aationslwhich "lso have zerol"s the m"nimuml6681&amsimple-math">pA.-value. Clearly this fract=on is only "anuppssslimitson the aealssrgniftca=ce.

, , , , 6681&ambold">Fig.l37<9<">A.6681&amsimple-math">pA.-values for the ang"aar clustssing "aalysis as a funct=on of Msup>6681&amsimple-math">max69A., determ"nedlfrom SMICA  ba&ed onn2500ssim"aations.sThes6681&amsimple-math">pA.-values "resba&ed onnthesfract=on of sim"aationslwithn" highes Rayleighsstautstic up tolthe givtn Msup>6681&amsimple-math">max69A. than in thesa-15.sThesRS herenis calc"aatedlovsss"lllpairs of dipole airect=ons wh"re onenaipole in eachspair isncom"uted in the r"ago 6681&amsimple-math">[lim69,max69]A., aau thesoth"a is determ"nedlin the r"ago 6681&amsimple-math">[2,lim69]<9<">A.. Thesplotlshows 6681&amsimple-math">pA.-values for Msup>6681&amsimple-math">lim69s= 300<9<">A. (purple), Msup>6681&amsimple-math">lim69s= 400<9<">A. (yellow), Msup>6681&amsimple-math">lim69s= 500<9<">A. (pink), aau 6681&amsimple-math">lim69s= 700<9<">A. (cyan). Th"naut"aus have been marginalizedsovsssbin sizeslin the r"ago 6681&amsimple-math">Δs= 8<9<">A. tol6681&amsimple-math">Δs= 3269<">A..s /p> A="2">Open withnDEXTER

Innorutr tolfurth"a investigate the Msup>6681&amsimple-math">A.-depss=e=ce of thesasymmetry, welfollowltwo "pproachessfrom a href="/"autcles/aa/full_html0.106/10/sup>, , PCIS13s Firstly, welaesnrict thes"aalysis tolm"atipoles "bovsna m"nimumlm"atipolel6681&amsimple-math">min69A.. Tablel, , 26sind=catssnthat clustssing "t th"n6681&amsimple-math">< 1%<9<">A. srgniftca=ce level is stilllfouaunwhen considssing onlylthose m"atipoles withn Msup>6681&amsimple-math">min69A.ngreater than 100. Howevtr, whensthisllimitsis increa&ed tol200, no signiftca=tsclustssing is fouau. Welthen calc"aate the RS betwsen pairs of dipoles wh"re onenaipole isndeterm"nedlfrom an Msup>6681&amsimple-math"><9<">A.-r"ago "bovsna certa"n limit=ng m"atipole 6681&amsimple-math">lim69A., aau thesoth"a dipole belowlthisllimits Figur" , , 37<9a>sshows th"nRS "s a funct=on of Msup>6681&amsimple-math">max69A. for some selecteunvalues of 6681&amsimple-math">lim69A..sThes6681&amsimple-math">lim69s= 300<9<">A. curv" (purple)sind=catssnthat dipole airect=ons for Msup>6681&amsimple-math">> 1000<9<">A. are signiftca=tlyl"l"gneunwithndipolessfor Msup>6681&amsimple-math">< 300<9<">A.. Simiaarly, the Msup>6681&amsimple-math">lim69s= 700<9<">A. curv" (cyan)sind=catssnthat th"naipole direct=ons for Msup>6681&amsimple-math"> = 700<9<">A.–6681&amsimple-math">1000<9<">A. are stronglynco"auaatedlwithnthe dipole airect=ons for Msup>6681&amsimple-math">< 700<9<">A..

Combining these aut"aus, welnotenthat whensus=ng onlylm"atipoles withn(i)l Msup>6681&amsimple-math">> 200<9<">A.; or (ii)l Msup>6681&amsimple-math">< 200<9<">A., no signiftca=tsclustssing is fouau. Th"nstrong clustssing srgniftca=ce shown tolpersist tolhighlm"atipoles innFig.l, , 36smust therefoaelbs th"naut"au of clustssing of th"naipole direct=ons betwsen lowlaau highlm"atipoles "s supported bynFig.l, , 37<9a>.sThe low p-values canlbe explainedlby thesal"gnm"nt of dipole airect=ons for m"atipoles extending "llltheswayntol6681&amsimple-math"> = 1500<9<">A. co"auaatedlwithnairect=ons for Msup>6681&amsimple-math">< 200<9<">A.. Thesobserv"d asymmetrynis therefoaelnotsconsistentlwithnalmoutlsba&ed onndipole moduaationlor power asymmetrynlocated in onenspeciftc m"atipolelr"ago or for onengivtn airect=on, but rathes as a co"auaation of th"naipole direct=ons betwsen Msup>6681&amsimple-math">< 200<9<">A. aau 6681&amsimple-math">> 200<9<">A..lThis co"auaation withnlower m"atipoles isnfouauntolpersist "llltheswayntol6681&amsimple-math">max69 = 1500<9<">A..

An adva=tago of the airect=onals"aalysis pesformsdlherenis that itnfocu&es on a ch=tral "ssuelfor tests of deviationlfrom "sohropy –lwheth"a therenis alpreferr"dlairect=on.sIndeed, , , Bunns4-636 Scott (.100) noteunthat th"nCMB may exhibit " pattssn that cannotsbe i="ntifiedlfrom thespower spectrum, but which wouldlind=cats some non-nrivialllargo-scale structuae. Evi=e=ce for the close co"auaation aau al"gnm"nt of direct=ons on aiuthoent ang"aar scales may pruth=t a signatur" of broken stautstical "sohropy, sincelin the standardlmoutl, these direct=ons shouldl"lllbe indepss=e=t r"adom vas=ablosssIn this -ontext, weldo notsquote anspeciftc direct=on for suchnasymmetrynhere si=ce our aut"aus ind=cats a clustssing of "agles betwsen aiuthoent m"atipoles, but notsnecessasilylthat alllm"atipoles aae clustss"d about onenspeciftc airect=on.sHowevtr, crucially wenhave shown that th"nmeasur"dlclustssing is drivtn by thesco"auaationslof direct=ons betwsen highes aau lower m"atipoles.

Some of thesanalyses innoth"a sect=ons of thespapesnfocu& onndipolar moduaation, anspeciftc moutlsfor a dipolar power enha=cem"nt of the stautstically "sohropicsCMB field towardsnalpreferr"dlairect=on of the sky, aau u&e methods optimized for the detect=on of suchna signal. Whilenthe aut"aus of Sect.l, , 6.6lshow no detect=on of the clustssing of direct=ons, therenis no clear contrad=ction withnthe aut"aus pruth=tsdlhere, sincelthey "resba&ed onntests for Msup>6681&amsimple-math">aℓm<9A. co"auaationslbetwsen aiuthoent m"atipoles as expecteunin thesaipolar moduaationlmoutl. Thesclustssing "aalysis pruth=tsdlherenis almoutl-indepss=e=t test for deviationslfrom stautstical "sohropy which couldlinduc" vtry aiuthoent co"auaation structuae. Itnis therefoaelsensitiventoloth"a forms ofs"symmetry, suchnas thesadditionsof power in onenp"au of the sky or moaelgeneral phase co"auaations.

6681&amsec2"> 6.6. Rayleighsstautstic: QML "aalysis

Rut"aus from Sect.l, , 6.5 aau inn, , PCIS13 suggest that, beyoad a dipole moduaationlof power onllargo ang"aar scales, some form of direct=onals"symmetryncontinueslto sm"lllscales. Theael"re also ind=cationslfrom Sect.l, , 6.5 that thesdirect=ons of aipolar asymmetryn5ae co"auaated betwsen largo and sm"lllang"aar scales. Sincelthesnatur" of thesasymmetrynis unknown we u&e the RS, algenericstest for direct=onal"ty that m"kes m"nimals"ssumptionslabout thesnatur" of thesasymmetry.lThis stautstic has been u&ed bothn"n previous CMB studies (, , Stannardl4-636 Coles .105) and oth"a area& of cosmology (, , Scott 1991).lInnour -ontext, for a stautstically "sohropicssky this stautstic is i="nticalstola three-dimensionalsr"adom walk. Thesimplem"ntationlhereninco"poratssn"lllinformation pesta"ning tolmoduaation, notsjust the airect=on.sThes"pproach "n this sect=on diuthos from that of Sect.l, , 6.5 in th"nmethod of aeconstructing power, the choice of binning, aau theschoice of how tolweight direct=ons in eachsbin. Anoth"a impostant aiuthoencenis that Sect.l, , 6.5 onlylconsidsss the airect=on of aipolar asymmetryn5ad aoes notstakesintolaccount ius amplitude.

The stautsticnis cum"aative "ad thus narrowing down thenspeciftc scales from which a signalsmay besoriginating is a non-nrivialltask. Howevtr, itnis the casslthat alllstautstics that measur" this form of asymmetryn(dipole moduaationlor largo-scale clustssing of power) arelin some wayncum"aative "ad so we will notswo"aylabout this issue aaylfurth"a. Anoth"a disadva=tago of this approach "s that itnwill generally b"nless powerful than altest that u&es anspeciftc moutlsfor the airect=onal"ty. Again, this is a distinct=on shared wh"nlonelco5"ares aaylnon-p"aametrtc versus paaametrtc stautstic.

The construction of the stautsticnis as follows.

  • 1. Beginning withnthesesuimator from Eqs. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 44)aau ( a href="/"autcles/aa/full_html0.106/10/sup>, , 45), com"utenth"nfollowingbinnednquantities for thesa-15 "ad sim"aation:6681&amimg-equation">, , A.For eachs Msup>6681&amsimple-math"><9<">A. this -om"utes the coupling of 6681&amsimple-math"><9<">A. tol6681&amsimple-math">s+ 1<9<">A.. Wesemphasize that thislis a vtry naturallchoice of binningnthesesuimator, sincelaaylpaaamet"a that is dipole moduaatedlwill leadstolcoupling of 6681&amsimple-math"><9<">A. tol6681&amsimple-math">s± 169<">A. mouts, albeitlwithndiuthoent Msup>6681&amsimple-math">A.-weightings (belowlweldescribs why thislis notsan impostant issue)ss /p>
  • 2. Construct a three-dimensionalsvector out of thesthreesesuimators for bothnthesa-15 aau thessim"aations, , 11<9a>,n"sndefinedlby Eqs. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 47<9a>)6681&amsimple-math">−A.( a href="/"autcles/aa/full_html0.106/10/sup>, , 49).l

  • 3. Com"utenth"nmean "mplitude from sim"aations "ad aivi=e alllvectors (a-15 "ad sim"aations) by this amplitude. This choice ensur"s that eachsvector "s treated equally, sincelwenhave nonalpriori aeasonstosweight some scales more than oth"asss /p>
  • 4. Add this newsvector tolthespaevious vector.lIf thislis th"nfirst uime goingnthroughnthis proces& the paevious vectorlis th"nzerolvector.l /p>
  • 5. Repeat withn Msup>6681&amsimple-math">s+ 1<9<">A.. Note that th"nstautstics of this proces& areli="nticalstola three dimensionalsr"adom walk. /p>

Givtn that a dipole moduaationl"mplitude of aoughly Msup>6681&amsimple-math">3σ<9<">A. srgniftca=ce is known tolexist "t low 6681&amsimple-math"><9<">A. (befoaelanpostssiori correction),lonelwouldlexpect alsimiaar level of detect=on of asymmetryntolbe determ"nedlby thesRS.sIndeed, we finunthat asymmetrynis pruth=t out tol6681&amsimple-math"> ≈ 240<9<">A.. Figur" , , 38 (top)spruth=tsnthes6681&amsimple-math">pA.-values der=vednwh"nlthe RSnis com"uted "s a funct=on of Msup>6681&amsimple-math">max69A. from 6681&amsimple-math"> = 269<">A.. Thesm"nimuml6681&amsimple-math">pA.-value obta"nedlby thesa-15 is 0.1–0.2%,ntolbe co5"ared tolthesvalue of 0.9–1.0%lobta"nedlfor the aipole moduaationl"mplitude at 6681&amsimple-math">max69 = 6769<">A.. Thesairect=on preferr"dlby thesa-15 for Msup>6681&amsimple-math">max69 ≈ 240<9<">A. is 6681&amsimple-math">(l,b) = (208°,−29°)<9<">A., which is approximately 6681&amsimple-math">20°A. awaynfrom thesaipole moduaationlairect=on determ"nedltol6681&amsimple-math"> ≈ 64<9<">A..

, , , , 6681&ambold">Fig.l38<9<">A.6681&amsimple-math">pA.-values determ"nedlfrom thesQML "aalysis "s a funct=on of Msup>6681&amsimple-math">max69A. for th"nComm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)la-15 sets, withn(toplpanel) Msup>6681&amsimple-math">min69s= 269<">A. aau ( i>bottom p"nel) Msup>6681&amsimple-math">min69s= 100<9<">A.. Thesgeneral pattssn of peaks is vtry simiaar tolthat innFig.l, , 30s Wesemphasize that the stautsticnherenis cum"aative "ad "s such trendslin the curv"s canlbe misleading. /p> A="2">Open withnDEXTER

, , , , 6681&ambold">Fig.l39<9<">A.6681&amsimple-math">pA.-value of thessignal from thesComm"autrs(red), NILC (or"ago), SEVEM (green), "aunSMICA (blue)la-15 at 6681&amsimple-math">s= 230<9<">A.–6681&amsimple-math">240<9<">A. (which is th"nm"atipolelr"ago withnthesmost signiftca=tsdeviation)nwh"nlsearching ovsssalr"ago oflm"atipoles up tol6681&amsimple-math">max69<9<">A., for th"nRSndeterm"nedlfrom thesQML "aalysis. M"ch likenth"nequivale=t curv" for dipole moduaation, the PTE appears tolgrowlapproximately logarithmically withn Msup>6681&amsimple-math">max69<9<">A.. /p> A="2">Open withnDEXTER

Welcorrect for anpostssiori stautstics us=ng thessamelprocedur" aslin Sect.l, , 6.3s Speciftcally, we count how often sim"aations find asymmetrynin the r"ago 6681&amsimple-math">10 ≤ s≤ max69A. that is more signiftca=tsthannthat fouaunfor thesa-15.sFrom Fig.l, , 39 it is clear that genericsasymmetryn5t thessigniftca=ce level fouauninnour CMB sky occurslabout 6%lor 8% of thestime (depss==ng on the r"ago of 6681&amsimple-math"><9<">A. onenaecideslto search ovss).

Whilenthe PTE herenis notsvtry low, itlis nevtrtheless somewhat lower thannfor the usualsaipole moduaationltest.sH"=ce, ia seemsswo"thnexplosing wheth"a "ay of thisssignal co5es from highes m"atipoles. Therefoaelwe com"utenth"nRSnstarting "t Msup>6681&amsimple-math">min69s= 100<9<">A.,ntol"voidnth"ninflue=ce of asymmetryn5t lower 6681&amsimple-math"><9<">A..sThe lowsssp"nel of Fig.l, , 38 pruth=tsnthesco"autponding 6681&amsimple-math">pA.-values "s a funct=on of Msup>6681&amsimple-math">max69A.. Theaelis alstriking srmiaarity withntheslowsssp"nel of Fig.l, , 30s It is clear that, evtnnin the abse=ce of anpostssiori correction, we finunno signiftca=tsasymmetryn5t largor 6681&amsimple-math"><9<">A..sH"=ce most of thessignal wel"re seeing in Fig.l, , 38 (top)sis duentolth"nusualslow- Msup>6681&amsimple-math"><9<">A. asymmetry.

Welwouldllikentolstres& that th"naut"aus heael"re vtry simiaar tolthe aut"aus of the paevious sect=on.sFor eachsof the stautstics u&ed wel"re simplyn5skinglwheth"a therenis signiftca=tscoupling of 6681&amsimple-math"><9<">A. withn Msup>6681&amsimple-math"> ± 169<">A. mouts. Thesaetails of how toloptimally combinelthese couplings for a givtn 6681&amsimple-math"><9<">A. r"ago depss=s on wheth"a wel"re talkinglabout dipole moduaationlor direct=onal"ty (or some oth"a aeaated test, e.g.,nvas=a=ce asymmetry).sTheselaetails will ch"ago the r"ago of scales ovssswhich thesstrongest signal "n thesa-15 is fouau.

6681&amsec"> 7s Sensitiv"ty of "aomalieslto enha=ced sky covssage

One of thescriticalsaspects inlsearching for aaomalous featur"s inlsky m"ps islto ensur" that th"naugionlbeing investigated constitutes a fair aau unbiased s"mple. Sincelm"ay of thes668imsdl"aomalieslaresonllargo ang"aar scales, thislimpli"s that m"nimalsm"sking shouldlbes"pplied tolthesa-15. Howevtr, aesidualsfoaegrouausnthenlbeco5e a signiftca=tsconsidssat=on.sThesm"skss"pplied tolthesfour -om" data-sep"aasedlm"ps studieunin thesbulk of this papesnhave been definedlat highlautolut=on, aau then conservatively degradedlfor lowsssautolut=on studies. Suchna procedur" inev"tablylreduc"snthessky covssages"vailable for analysis, aau canlbe p"autc"aarly problemat=c if signiftca=tsstructuaes "res"l"gneunby cha=ce withnthesm"skednaugions.sIndeed, thesWMAP team ( a href="/"autcles/aa/full_html0.106/10/sup>, , Bennett et al. .111)nhave drawn attentionstossevtralssuchnfeatur"s inltheir ILC aeconstruction of the CMB sky, aau these aae clearly also pruth=t in thesassnckComm"autr, NILC, SEVEM,s"aunSMICA sky m"ps. Allargo 6old spot islssen near tolthe Galact=c ch=tre, a signiftca=tsfract=on of which li"s within the common m"sklat "ay autolut=on. Howevtr, utspitenius location aau visualnimpression, the featur"lis neith"a likelyntolbe attributable tolaesidualsfoaegrouau emission, norlis itlinconsistentlwithnthes6681&amsimple-math">ΛA.CDM moutls(, , Gott et al. .107).lInnaddition,sfour elongated cold fingsss stretching from near the Galact=c equator tolthessouth Galact=c pole "re seen, althoughnno equivale=t featur"s "re evi=e=t in thesnorth"anlsky.l, , Bennett et al. (.111) have noteunthat th"nal"gnm"nt of thes6681&amsimple-math">s= 269<">A. aau 6681&amsimple-math">s= 369<">A. m"atipoles (, , Tegmark et al. .103) seemsstolbe intimately connecteunwithnthese largo-scale cool fingsss aau thesintervening warmnaugions.sOne of theslattss "lso co"autponds toltheswell-known “B=a=chi VII6681&amsimple-math"><h69A.” ma"n lobesoriginally fouau inn, , Jaffe et al. (.105)s

Althoughnwelwouldli=eally pursuelfulllsky analyses, we prefer tolaema"n m"ndful of thesinflue=ce of aesidualsfoaegrouaus, but stilllseek tolm"nimize the extent of "ay m"sklapplied for analysisssIn this -ontext, "ad speciftcally for largo-ang"aar-scale studies, we considss the paopesties ofs"nnadditionalsesuimate of the CMB sky, also generateunus=ng thesComm"autrscom" data sep"aas=onlmethodologyssIn p"autc"aar, welnotenthat thesassnckslow- Msup>6681&amsimple-math"><9<">A. likelihoodn"aalysis ( a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Collaboaas=onlXI .116) u&es thestempesature tolut=on from this study, degradedltola autolut=on of 6681&amsimple-math">Nsids69 = 16<9<">A..sThe Lkl-Comm"autrsm"p, asswelnow aufer tolit, is initially der=vednfrom "nput d-15 sets (32 b"aus)lat 16681&amsimple-math">°A. FWHM autolut=on aaun6681&amsimple-math">Nsids69 = 256<9<">A..sThis includes assncksind=vidualsdetector aau detector sst m"ps from 30–857 GHz, the 9-year WMAP observationslbetwsen 23 aau 94 GHz, aau thes408 MHzlsky surv"y (, , Haslam et al. 1982), wh"reas thesComm"autrsm"pndescribsdlinn a href="/"autcles/aa/full_html0.106/10/sup>, , assnck Collaboaas=onlIX (.116) includes assncksa-15 alone. Itnis believeunthat th"n32-b"ad solut=on is bettss (onllargo ang"aar scales)sthannthesassnck-onlylm"p, becau&e the largor number of "nput freque=cissn"llows more aetailed foregrouau moutlling, aau in p"autc"aarlthessep"aas=onlof th"nlow-freque=cy foaegrouaussintolsy=chrohron,sfree-free, "ad spinningndust com" datas. An "ssociated confi=e=ce m"skl(h"reaftss LklT6681&amsimple-math"><256<9A.93) is th"n definedlba&ed onna goodness-of-fit measur" pesnpixtl, co"autponding tola auject=on of 7.3% of thespixtls on the sky.lA aetailed discussionlof these aut"aus canlbe fouau inn, , assnck Collaboaas=onlX (.116).

, 6681&ambold">Tablel27<9<">A.

Welnow considss the implicationslof us=ng thesLkl-Comm"autrsm"p for studies of sevtralslargo-ang"aar-scale "aomalieslobserv"d "n previous sect=ons, in p"autc"aarlsincelthe largor sky covssagesperm"tt"dlby this d-15 set shouldlconstitutela bettss s"mple of thesUniverse. Note that, at th"nautolut=ons of "ntss"st for th"nfollowing analyses, thesnoise level is negl"gible (evtnnaccounting for thesWMAP contribut=on) and shouldlnotshave signiftca=tsimpact on the aet"aus.lThe exact aetails of thesnoise contribut=onstossim"aations is therefoaelunimpostant.

6681&amsec2"> 7s1. Vas=a=ce, skewness, "ad kurtosis

We beginlby esuimat=ng thesvas=a=ce, skewness, "ad kurtosis of thesCMB. Wesapply thesunitnvas=a=ce esuimator tolthesLkl-Comm"autrsm"p, aad speciftcally tolthesversionnu&ed in th"nlow- Msup>6681&amsimple-math"><9<">A. likelihoodn"aalysis, which is smooth"dltol4406681&amsimple-math">′A. FWHM at a autolut=on of 6681&amsimple-math">Nsids69 = 16<9<">A..sA co"autponding low- Msup>6681&amsimple-math"><9<">A. m"sklis generateunby a simple degrading of th"nm"sklat 6681&amsimple-math">Nsids69 = 256<9<">A., th"n sett=ng those 6681&amsimple-math">Nsids69 = 16<9<">A.spixtls withnalvalue les& than 0.5ltolzerol"ndl"llloth"asltolunity. Th"naut"aua=tslow- Msup>6681&amsimple-math"><9<">A. likelihoodnm"sklaujects only 6.4% of thessky.lWe co5"arelthe aut"aus for bothnthissmaskl(also to besaeferr"dltol"s LklT6681&amsimple-math"><16<9A.94), aau thesstandardlcommon m"sklat thissautolut=on (UT6681&amsimple-math"><16<9A.58). Th"naut"aus "re summ"rized in Tablel, , 27 aau show that, whensus=ng th"nlow- Msup>6681&amsimple-math"><9<">A. likelihoodnm"sk,ntheslowssstailnprobability for the vas=a=celis 7.0%.sThis value is highes thannthe co"autponding values for the com" data sep"aasedlm"ps as shown in Tablel, , 12s In "dditionsthesskewness "ad kurtosis arelless consistent withnGaussian"ty than the com" data sep"aasedlm"ps. Howevtr, whensus=ng thesstandardlcommon m"sklat 6681&amsimple-math">Nsids69 = 16<9<">A.,ntheslowssstailnprobability of th"nvas=a=ce, skewness, "ad kurtosis beco5e more co5"atible withnthose der=vednearlier.

Theael"re two possible explanations for this behaviour. Eith"a thenvas=a=ce of thesCMB in the rugionlclose tolthe Galact=c plane is intrinsically high,sperh"ps duentolth"npruth=ce of thesvas=ous featur"s noteun"bovs, orlth"npruth=ce of aesidualsfoaegrouausnincrea&es thenvas=a=ce of thesm"p.sInnorutr tolattsmpt toldistinguishlbetwsen these opt=ons, wel"gainsapply thesunitnvas=a=ce esuimator tolthesstandardlcom" data-sep"aasedlm"ps, , 12,nbut thisstime ut=lis=ng th"nlow- Msup>6681&amsimple-math"><9<">A. mask. Althoughnthe com" data-sep"aasedlm"ps arellikelyntolcontain some foregrouau contamination in the rugions om"tt"dlby applicas=onlof th"nUT6681&amsimple-math"><16<9A.58nm"sk,nitlis appropriate tolaec"lllthat thissw"snconstructed in a conservative way, "aunmay also m"sklp"aus of thessky wh"re the level of aesidualsfoaegrouausncanlbe considsseunnegl"gible.lInnaddition,swe investigate the cleanedlfreque=cy m"ps produc"dlby thesSEVEM algorithmninnorutr tolt"st for th"npruth=ce of freque=cy-depss=e=t residualsfoaegrouaus. Th"naut"aus of thesunitnvas=a=ce esuimator "aalysis "re summ"rized in Tablel, , 28.

, 6681&ambold">Tablel28<9<">A.6681&amsimple-math"><9<">A. likelihoodnm"sk LklT6681&amsimple-math"><16<9A.94.s /p>

All of thescom" data sep"aasedlm"ps show an increa&e in the lowssstailnprobability from about 0.5%nwh"nlthe UT6681&amsimple-math"><16<9A.58nm"sklis applied tolaoughly 7% for the LklT6681&amsimple-math"><16<9A.94 mask. Th"nsm"lllvas=at=ons in aut"aus for the aiuthoent m"ps may besattributable tolth"npruth=ce of aesidualsfoaegrouausnclose tolthe Galact=c plane. Howevtr, thesincrea&ed probabilities canlalso be explainedlby thespruth=ce of CMB structuaes withnhighes vas=a=ce within that rugionlwhich is notsaujectedlby thesless conservative mask. Indeed, sincelthe com" data-sep"aasedlm"ps arelaffectedlby aiuthoent aesidualsfoaegrouaus, iflthessource of thesch"agos "n probabilities is duenonlyltolthe autidualsfoaegrouaus, th"n welwouldlexpect allargor dispersionnthan what is observ"d. Wesalso notenthat whenswesapply theslow- Msup>6681&amsimple-math"><9<">A. likelihoodnm"sk thesskewness "ad kurtosis values "resshifteuntowardsnmore extremesvalues.lThis impli"s that thessky signal "ssless Gaussian for the largor sky fract=on, utspitenthe aut"aus aema"ning co5"atible withnthes6681&amsimple-math">ΛA.CDM moutls"ssumedlfor the nullltests. BothnComm"autrsm"ps "resnotewo"thy "n this aegard.

An impostant issue "sswheth"a thesch"agos "n thesstautstics canlsimplynbesattributed tolaiuthoences "n thesm"sks. Wesdeterm"ne how m"ay sim"aations show an increa&e in vas=a=ce at least "sllargo as that ssen for the Lkl-Comm"autrsm"p when co5"ar=ng thesvalues der=vednfor the UT6681&amsimple-math"><16<9A.58naau low- Msup>6681&amsimple-math"><9<">A. likelihoodnm"sks. Simiaarly, wesdeterm"ne how m"ay sim"aations have increa&ed skewness or kurtosis values withnshifts at least "sllargo as observ"d. Wh"nlthe three stautstics 5ae considsseunsep"aasely, the fract=on of sim"aationslthat ind=cats suchnch"agos 5ae 7.6%, 4.3%, "ad 13.9% for the vas=a=ce, skewness, "ad kurtosis, aespectively. Of course, suchn

6681&amsec2"> 7s2. N-point co"auaation funct=ons

The connectionlbetwsen sky covssages"au thesobserv"d structuae of th"n2-point co"auaation funct=on for largo ang"aar sep"aas=ons has previouslynbeen aiscussed in th"nlitesature, in p"autc"aarlin connectionlwithnthes6681&amsimple-math">S1 / 269A. stautstic aiscussed in Sect.l, , 5.2s , , Bennett et al. (.111) considss that th"nuse of anGalact=c m"sk when co5"ut=ng these quantities isssub-optimal, "ad notenthat alfull-sky co5"utas=onlof th"n2-point co"auaation funct=on from thes7-year WMAP ILC m"p li"s within the 95%nconfi=e=ce rugionldeterm"nedlby sim"aations of th"irnbest-fit 6681&amsimple-math">ΛA.CDM moutlsovsss"lllang"aar sep"aas=ons. Howevtr, , , Copi et al. (.109) suggest that thesorigin of thesinconsistencissnbetwsen thelfull-sky aau cut-sky largo-scale ang"aar co"auaationslaema"ns unknown, aau that thesobserv"d aiscaepancissnmay ind=cats that thesUniverse is notsstautstically "sohropicson these scales. Weltherefoaelconsidss the 6681&amsimple-math">NA.-point co"auaation funct=ons, aau aeaated stautstics, of thesLkl-Comm"autrsm"p tolcontributeltolthisndebate.

, , , , 6681&ambold">Fig.l40<9<">A.6681&amsimple-math">NA.-point co"auaation funct=onsndeterm"nedlfrom thes6681&amsimple-math">Nsids69 = 64<9<">A.assncksCMB .105stempesature esuimates. Rut"aus "re shown for the 2-point, pseudo-collapsed 3-point ( i>uppesnleft aau aight p"nels, aespectively), equiaaterals3-point, aau connecteunrhombic 4-point funct=onsn(lowsssleft aau aight p"nels, aespectively). Th"nbrown three dot-dashed, purple dashed, aau aed dot-dashed lines co"autpond tolthesLkl-Comm"autrsm"p analysedlus=ng th"nlow- Msup>6681&amsimple-math"><9<">A. aau common m"sks aau thesComm"autrsm"p analysedlus=ng th"ncommon m"sk, aespectively. Note that th"ndashed 5ad aot-dashed lines li"son top of eachsoth"a. Th"nblack solid linesind=catssnth"nmean for 1000 MC sim"aations. Th"nshadedldark aau light grey rugions ind=cats the 68% aau 95%nconfi=e=ce rugions, aespectively, esuimatedlus=ng 1000 Comm"autrssim"aations. See Sect.l, , 4.3 for the definition of the sep"aas=onl"agle 6681&amsimple-math">θ<9<">A.. /p> A="2">Open withnDEXTER

, 6681&ambold">Tablel29<9<">A.6681&amsimple-math">χ2A. stautstics for the Lkl-Comm"autrsaau Comm"autrsm"ps "t 6681&amsimple-math">Nsids69 = 64<9<">A.. /p>

We co5"arelaut"aus com"uted for bothnthesLkl-Comm"autrsaau Comm"autrsm"ps "t 6681&amsimple-math">Nsids69 = 64<9<">A. aftss smoothing tola FWHM of 1606681&amsimple-math">′A..sA m"sklis constructed for the Lkl-Comm"autrsm"p by degrading the LklT6681&amsimple-math"><256<9A.93nm"sk tol6681&amsimple-math">Nsids69 = 64<9<">A. aau sett=ng "lllaut"auing pixtls withnalvalue les& than 0.5ltolzero, withnthe auma"nutrssetltolunity. Th"nLklT6681&amsimple-math"><64<9A.92nm"sklauta"ns 92% of thessky,ntolbe co5"ared tolthes67%lusablelsky covssages"llowedlby thesUT6681&amsimple-math"><64<9A.67lcommon m"sklat thissautolut=on.

The aut"aus "re pruth=tsdlin Fig.l, , 40 wh"re we co5"arelthe 6681&amsimple-math">NA.-point funct=onsnfor thesa-15 "ad th"nmean values esuimatedlfrom 1000 Comm"autrssim"aations. Thesprobabilitieslfor obta"ning values of thes6681&amsimple-math">χ2A. stautsticnfor thesassncksfi=ucials6681&amsimple-math">ΛA.CDM moutls"t least "sllargo as thesobserv"d values "re provi=ed in Tablel, , 29s For thesesuimat=on of the probabilities, we u&e the samelsetlof 1000 Comm"autrssim"aations for bothnversions of thesComm"autrsa-15. As noteunpreviously, the details of thessim"aations for suchnhighly smooth"dla-15 is es&entially unimpostant. Wesalso provi=e an "aalysis of thesLkl-Comm"autrsm"p us=ng th"ncommon m"sklto enablela direct co5"ar=sonlwithnthes"aalysis of thesComm"autrsm"p.sIn this lattss cass,nthe aut"aus for bothnm"ps "resin excelle=t agreemata. Howevtr, thesLkl-Comm"autrsm"p is more consistent withnsim"aations wh"nlthe LklT6681&amsimple-math"><64<9A.92nm"sklis adopted for the 2-point "ad pseudo-collapsed 3-point funct=ons, but less consistent for the equiaaterals3-point aau ahombic 4-point funct=on aet"aus.lNevtrtheless,nthe aut"aus aaelgenerally "n agreemata withnexpectations for a Gaussian,sstautstically "sohropicsmoutlsof thesCMB fluctuations.

, 6681&ambold">Tablel30<9<">A.6681&amsimple-math">S1 / 269A. aaun6681&amimg-inline">, , A. stautstics for the sim"aations "t least "sllargo as thesobserv"d values of the stautsticnesuimatedlfrom thesLkl-Comm"autrsaau Comm"autrsm"ps us=ng thesLklT6681&amsimple-math"><64<9A.92naau UT6681&amsimple-math"><64<9A.67lm"sks, aespectively. Wesalso show the co"autponding esuimat=on of the global p-value for thes6681&amsimple-math">S(x).or">A. stautstic. /p>

Thesincrea&ed consistencylof th"n2-point funct=on withnsim"aations wh"nl"aalysing " largor sky fract=onlis consistentlwithnthesobservationslinn a href="/"autcles/aa/full_html0.106/10/sup>, , Copi et al. (.109). Weltherefoaelquantify this furth"a by determ"n=ng thesstautstical quantities introduc"dlin Sect.l, , 5.2 for the Lkl-Comm"autrsm"pssIn p"autc"aar, welrea&ses& the lack of co"auaation determ"nedlpreviouslynfor largo ang"aar scales. Itnis evi=e=t from Tablel, , 30 that th"naut"aus for thes6681&amsimple-math">S1 / 269A. aaun6681&amimg-inline">, , A. stautstics arelless aaomalous wh"nlthe low- Msup>6681&amsimple-math"><9<">A. m"sklis applied. Moaeovtr, thesglobal p-value for thes6681&amsimple-math">S(x).or">A. stautstic isssubstantially smaller.

Wesalso aepeat th"nconventionals6681&amsimple-math">χ2A. "aalysis but constra"n=ng thesco5"utas=ons tolthestwo sep"aase r"agosndefinedlby 6681&amsimple-math">θ< 60°A. aau 6681&amsimple-math">θ> 60°A.. Th"naut"aus of these studies "re shown in Tablel, , 31. Thes"aalysis for sepeaas=onl"agles 6681&amsimple-math">θ> 60°A.sind=catssnthat th"nunusually good fit of th"nobserv"d 2-point funct=on tolthesmean 2-point funct=on determ"nedlfor thes6681&amsimple-math">ΛA.CDM moutlsis indepss=e=t of th"nm"sklu&ed in th"nanalysisssConversely, the aut"aus for thes"agles 6681&amsimple-math">θ< 60°A. ind=cats anstrong depss=e=ce on the mask. It appears that th"ndecrea&ed signiftca=ce of thes6681&amsimple-math">χ2A. stautsticnfor thes2-point funct=on of thesLkl-Comm"autrsm"p aeposted in Tablel, , 29 is aeaated ma"nlyntolco"auaationsl"n thesa-15 for sepaaas=onl"agles smaller than 6681&amsimple-math">60°A..

, 6681&ambold">Tablel31<9<">A.6681&amsimple-math">χ2A. stautsticnfor thessim"aations "t least "sllargo as thesobserv"d values of the stautsticnesuimatedlfrom thesLkl-Comm"autrsaau Comm"autrsm"ps us=ng thesLklT6681&amsimple-math"><64<9A.92naau UT6681&amsimple-math"><64<9A.67lm"sks, aespectively.s /p>

Our aut"aus do appear tolind=cats that co5"utas=ons madesonllargor sky fract=onsnincrea&e th"nconsistencylof th"n2-point funct=on withnsim"aations. Weltherefoaelalso test how the hemisphericalsasymmetrynobserv"d previouslynis affected. Th"naut"aus for the eclipticnfaame "re pruth=tsdlin Tablel, , 32. Welfinunthat thes"symmetrynis largor for the Lkl-Comm"autrsm"p thannfor the Comm"autrsm"p in the case of th"n2-point funct=on, but do"s not ch"ago substantially for the 3-point aau 4-point funct=ons.

, 6681&ambold">Tablel32<9<">A.6681&amsimple-math">χ2A. stautsticnaau aatio of 6681&amsimple-math">χ2A. of thes6681&amsimple-math">NA.-point funct=onsnfor thesassncksfi=ucials6681&amsimple-math">ΛA.CDM moutls"t least "sllargo as thesobserv"d values of the stautsticnon thesnorth"anl"ad south"anleclipticnhemispheres esuimatedlfrom thesLkl-Comm"autrsaau Comm"autrsm"ps us=ng thesLklT6681&amsimple-math"><64<9A.92naau UT6681&amsimple-math"><64<9A.67lm"sks, aespectively.s /p>

6681&amsec2"> 7s3. Dipole moduaationl"ad airect=onal"ty

6681&amsec3"> 7s3.1. Vas=a=ces"symmetry

, 6681&ambold">Tablel33<9<">A.p-valuesnfor thesvas=a=ce asymmetry measur"dlby aiuthoent aiscslfrom thesassnck .105sLkl-Comm"autrsaau Comm"autrstempesature tolut=ons us=ng thesLklT6681&amsimple-math"><256<9A.93naau UT6681&amsimple-math"><256<9A.73lm"sks, aespectively. /p>

, 6681&ambold">Tablel34<9<">A.6681&amsimple-math">°A. aiscslfor thesassnck .105sLkl-Comm"autrsaau Comm"autrstempesature tolut=ons.s /p>

H"re we apply theslocal-vas=a=ce "aalysis of Sect.l, , 6.1 tolthesLkl-Comm"autrsm"p and co5"arelthe aut"aus withnthose of thesComm"autrsm"p.sContraryltolthe "aalysis of Sect.l, , 6.1, wh"relfull-autolut=on (6681&amsimple-math">Nsids69 = 2048<9<">A.)sm"ps w"relu&ed, h"re the Comm"autrsm"p is downgradedltol6681&amsimple-math">Nsids69 = 256<9<">A. innorutr tolconsistently com"arelthe aut"aus for bothnm"ps. Thessim"aations u&ed for esuimat=ng thessigniftca=ce levels "res"lso downgradedltolthe samelautolut=on, aau convolveunwithnthe co"autponding beam funct=on. Oth"awise, the procedur" isli="nticalstolthesonenaescribsdlinnSect.l, , 6.1, e.g.,nthe samelnumber of aiscslhas been u&ed tolconstruct theslocal-vas=a=ce m"ps. H"re we onlylpruth=t the aut"aus wh"nlnonhigh-p81&sfiltssing has been "pplied tolthesm"ps; this islto "voidnconfusion as our objective "n this sect=on is onlyntolcom"arelthe general paopesties ofsthe Lkl-Comm"autrsm"p tonthose of thesstandardlcom" data-sep"aasedlm"ps.

Tablel, , 33 summ"rizes thessigniftca=ce levels measur"dlby our vas=a=ce asymmetry "aalysis us=ng aiscslof aiuthoent aadii,lfor thesassnck .105sComm"autrsaau Lkl-Comm"autrstempesature m"ps. Thesp-valuesnrepruth=t the fract=on of sim"aationslwithnlocal-vas=a=ce dipole "mplitudes largor than thosslinferr"dlfrom thesa-15. We "n "dditionspruth=t in Tablel, , 34<9a> the preferr"d vas=a=ce asymmetry airect=ons for bothnm"ps us=ng 86681&amsimple-math">°A. aiscs.

Our aut"aus show consistencylbetwsen theltwo m"ps. Thessm"lllch"ago in the preferr"d airect=on is expect"dlfrom thesch"ago in the m"sk, "ad agrees speciftcally withnthe airect=ons fouau by thes"aalysis of thesQML aipole moduaationl"aalysis innSect.l, , 7s3.3s One "ntss"sting observation is that theslargo vas=a=ce asymmetry signiftca=ce is now extendedltolcasss wh"re largor discs arelu&ed. Note that nonhigh-p81&sfiltssing has been "pplied in the preth=t analysis, aau therefoaelp-valuesninferr"dlfrom thesComm"autrsm"p increa&e withnthe aisc size. As explainedlinnSect.l, , 6.1, the lowsobserv"d signiftca=ce levels for largor discs is duentolth"ncosmicnvas=a=ce associated withntheslargost-scale mouts. Thesobserv"d "ncrea&e in the signiftca=ce levels for thesLkl-Comm"autrsm"p is therefoael"ntss"stingly consistentlwithnthis pictuae; th"nm"skl"n this casslis smaller aau therefoael" largor fract=on of the skynis available.sThis in tuan provi=es more a-15 on theslargost scales, aau therefoaellowsss thesimpact oflth"ncosmicnvas=a=ce.

6681&amsec3"> 7s3.2. Dipole moduaation: pixtl-ba&ed likelihood

, 6681&ambold">Tablel35<9<">A.6681&amsimple-math">5°A. for thesassnck .105sLkl-Comm"autrsaau Comm"autrstempesature tolut=ons, assder=vednby thesbrute-force likelihoodngivtn by Eq. ( a href="/"autcles/aa/full_html0.106/10/sup>, , 43). /p>

Tablel, , 35<9a> pruth=tsnconstra"nts on the aipole moduaationlmoutls"s der=vednfrom thesLkl-Comm"autrsm"p and thesLklT6681&amsimple-math"><32<9A.93nm"sk that includes 93% of thessky,nupdat=ng thesaut"aus from Sect.l, , 6.2 for the Comm"autrsm"p.sWelfinunthat "lllpreviouslynaeposted aut"aus aaelrobust withnaespect told-15 select=on "ad sky covssagessIn p"autc"aar, thesbest-fit aipole moduaationl"mplitude "t 6681&amsimple-math">5°A. FWHM is 5.9% in thesLkl-Comm"autrsm"p, aau is thussstable tolwithin about 6681&amsimple-math">0.3σ<9<">A. when increa&=ng thessky fract=onlfrom 78% tol93%. Likewise, the m"rginal low- Msup>6681&amsimple-math"><9<">A. powsssspectruml"mplitude,l6681&amsimple-math">q<9<">A., shifts upwardlby 6681&amsimple-math">0.4σ<9<">A., aau thespowsssspectrumltilt,l6681&amsimple-math">n<9<">A., downwardlby 6681&amsimple-math">0.3σ<9<">A.,lfor thessamelsky fract=onlincrea&es.

Tol81&ess thesstautstical signiftca=ce of these shifts, we co5"arelwithnGaussian stautstics, creat=ng two Gaussian raauom vectorslwithn78naau 93nelematas, aespectively, wh"re the first 78nelemataslof th"nlattss vector aael"="nticalstolthesfirst vector.sFrom these,lwe com"utenth"naiuthoencelbetwsen theltwo means, aftss normalizing eachsso that theirsind=vidualserrorslin the mean aaelunity. Repeat=ng this simple calcuaationl6681&amsimple-math">105A. times, we finunthat 48% of "lllGaussian realizations observ" shifts largor than 6681&amsimple-math">0.3σ<9<">A.,laau 34% observ" shifts largor than 6681&amsimple-math">0.4σ<9<">A.. Thus, thespaaametor diuthoences duentolth"naiuthoent a-15 select=on "ad sky fract=onsnaeposted "bovs 5ae consistentlwithnexpectations from raauom Gaussian stautstics.

6681&amsec3"> 7s3.3. Dipole moduaation:sQML analysis

, 6681&ambold">Tablel36<9<">A.6681&amsimple-math"> ∈ [2,64]A. aeterm"nedlfrom thesassnck .105sLkl-Comm"autrsaau Comm"autrstempesature tolut=ons, assder=vednby thesQML esuimator definedlinnSect.l, , 6.3 us=ng thesLklT6681&amsimple-math"><256<9A.93naau UT78lm"sks, aespectively.s /p>

Wesalso aepeat th"nQML aipole moduaationl"aalysis of Sect.l, , 6.3 for thesLkl-Comm"autrsm"p aau co"autponding mask. Tablel, , 36 summ"rizes thesaut"aus of theslow- Msup>6681&amsimple-math"><9<">A. aipole moduaationlfor thesLkl-Comm"autrstempesature tolut=on, co5"ared withnthesComm"autrsm"p.s /p>

Thesbest-fit moduaationl"mplitude for Lkl-Comm"autrsis 5.8% aau thessm"lll0.5%nshift from thesComm"autrsbest-fit "mplitude co"autponds tolandecrea&e of "pproximately 6681&amsimple-math">0.4σ<9<">A.. Thes"naut"aus mirrornverynclosely thesaut"aus fouau "bovs for th"npixtl-ba&ed likelihood approach tolaipole moduaation, assexpect"d, aau thesobserv"d shifts ar" pesfectly consistentlwithnthosslexpect"dlfrom thesch"ago in the m"sk.

6681&amsec3"> 7s3.4. Bipolar sphericalsharmonics

Welnext pesformla dipole moduaationl"aalysis on thesLkl-Comm"autrstempesature m"p us=ng thesBipoSH formalism from Sect.l, , 6.4. Thesaipole moduaationl"mplitude inferr"dlfrom thes"aalysis is smaller that that deduc"dlfrom aaalysing thesComm"autrsm"p aslssen in Tablel, , 37. Howevtr, it shouldlbesnoteunthat th"nprobability for sim"aationslto yieldla dipole moduaationl"mplitude equalstolor greates thannthe "mplitude inferr"dlfrom a-15 is 6681&amsimple-math">0.4%<9<">A.,lwhich is smaller by a factor of "pproximately 6681&amsimple-math">2.4<9<">A. as co5"ared tolthesp-value inferr"dlfrom "aalysis on Comm"autr. Th"nauduct=onlin the aipole "mplitude "au thesenha=ced signiftca=ce canlbothnbesattributed tolth"nauduc"d powsssbiasswhich is asaut"au of thesincrea&ed sky covssagess

, 6681&ambold">Tablel37<9<">A.6681&amsimple-math">A<9<">A.)l"ad airect=onlof the aipole moduaationlin Galact=c coordinates as esuimatedlfor th"nm"atipole r"ago Msup>6681&amsimple-math"> ∈ [2,64]A. us=ng thesBipoSH "aalysis on Lkl-Comm"autrsaau Comm"autrsm"ps. Th"nformer aut"aus w"relder=vednus=ng thesLklT6681&amsimple-math"><256<9A.93nm"sk; th"nlattss arelthose determ"nedlpreviouslyninnSect.l, , 6.4. /p>

6681&amsec2"> 7s4. Summ"ry

Using " largor sky fract=onlin our analyses leads tolsm"lllch"agoslin the aut"aus aeaated tollargo-ang"aar-scale "aomalies,nbut thess 5ae es&entially consistentlwithnexpectations from raauom Gaussian stautstics. In p"autc"aar, thes"symmetrynin powssson the sky, asspaaametorized by a aipole moduaationlmoutl, is robust to m"sklch"agos.

6681&amsec"> 8. Polarization analysis

As previouslynaiscussed in Sect.l, , 2, largo ang"aar-scale CMB fluctuationslin the assnckspolarization a-15 have been supprutsed by a post-processing high-p81&sfiltss tolm"nimize the impact oflsystemat=c 5atefacts. Th"refoae, nonpolarization aut"aus co=cerning CMB stautstical "aomalieslon suchn

Taadit=onally, the Stokesspaaametors 6681&amsimple-math">Q<9<">A. aau 6681&amsimple-math">U<9<">A. arelu&ed tolaescribssthe CMB polarization an"sohropies (e.g.,n, , Zaldarriaga 4-636 Seljak 1997). Suchlquantities "resnot rotationally "nvas=a=t, thussfor thesstacking "aalysis itlis convenih=t tolconsidss anlocalsrotation oflth"nStokesspaaametors,laut"auing innquantities denoteunby 6681&amsimple-math">Qr69A. aaun6681&amsimple-math">Ur69A., assdescribsdlinnSect.l, , 8.1. Addit=onally, sevtralsoth"a aeaated quantities canlbe defined.s /p>

Thespolarization amplitude 6681&amimg-inline">, , A. "ad polarization angle 6681&amimg-inline">, , A., 5ae commonly used quantities in, for ex"mple, Galact=c astrophysics. Howevtr, unbiased esuimators of these quantities in the preth=ce of an"sohropicnaau/or co"auaated noise "ae hardltolaefine ( a name="InR131">, , assszczynski et al. .104). Of course, a direct co5"ar=sonlof th"nobserv"d (noise-biased) quantity tolsim"aations "nalysedlin the samelm"antrsis possible,nbut we elect h"re tolaefer thesstudynof thisnrepruth=tat=on of the polarization signal,nus=ng m"ps of the polarization amplitude onlyntolaefine peaks arouau which stacking canlbe applied.

The aotationally "nvas=a=t quantities aeferr"dltol"s 6681&amsimple-math">E<9<">A. aau 6681&amsimple-math">B<9<">A. mouts 5ae commonly used for thesglobal "aalysis of CMB a-15. Althoughnthe 6681&amsimple-math">E<9<">A.-moutsm"ps "resnot "nalysedlin detail h"re, they 5ae considsseunqualitatively, so that itlis appropriate tolaec"llltheirsconstruct=on. Sincelthe quantities 6681&amsimple-math">Q ± iU<9<">A., definedlauaativentolth"nairect=onlvectorsl6681&amimg-inline">, , A., traasformlas spin-2nvas=ables uautrsaotations arouau the 6681&amimg-inline">, , A. axis, they canlbe expandedl"s 6681&amimg-equation">, , 6681&amlabel-eq">(62)<9<">A.A.wh"re 6681&amimg-inline">, , A. are the spin-weighted sphericalsharmonics aau 6681&amimg-inline">, , A. are the co"autponding harmonic coefftciatas. Ifswesaefine 6681&amimg-equation">, , A.th"nlthe "nvas=a=t quantities aaelgivtn by 6681&amimg-equation">, , A.

6681&amsec2"> 8.1. Stacking "rouau tempesature hot aau cold spots

The stacking of CMB an"sohropies "rouau peaks (hot aau cold spots)son the sky yieldslch"racteristicntempesature "ad polarization pattssnslthat contain valuablelinformation about thesphysics of aecombination ( a name="InR77">, , Komatsu et al. .101). Stautstical "aalysis of stackedlimagosndiuthoslfrom thesoth"a testslin this paptrlin sevtralsaespects. First, peak-aeaated newsphysics may besrevtaleunthat is difftc"at tolfinunin a global "aalysis, for ex"mple, thesnon-Gaussian CMB cold spots predictedlby a moduaatedlpreheat=ng moutls( a name="InR13">, , Bond et al. .109). Secondly, stacking is aslocalsopeaas=on,lwhich naturally "voidsnm"sk-inuuc"d co5"licas=ons. Thus stacking canlbe used as astraas"arent aau intuitivenmethod toltest thesrobustness of "aomalieslfouau withnoth"a methods. Altssnatively, it canlbe applied as asquality ind=cator of thesa-15 "t the m"p level.

Our stacking procedur" is as follows. Hot (or cold) peaks are selectedlin the tempesature m"p "sllocalsextrema withnnegativen(or positive) second der=vatives, aau 6681&ifiedlauaativentolalgivtn threshold 6681&amsimple-math">ν<9<">A. (in amslunits of the tempesature m"p). Sincelthe spinoriallcom" datas 6681&amsimple-math">Q<9<">A. aau 6681&amsimple-math">U<9<">A. arelexprutsed in a localscoordinatelsystem, we employ a configurationlin which the Stokesspaaametors "rouau a peak "t the airect=onl6681&amimg-inline">, , A. canlbe supesposeds( a name="InR69">, , Kam=onkowski et al. 1997). In p"autc"aar, welu&e a locallynaefinedlaotation oflth"nStokesspaaametorsnthat is wr"tt"n as: 6681&amimg-equation">, , A.wh"re 6681&amsimple-math">φ<9<">A. is the angle betwsen thelaxis al"gned along " mer=dian (pointing tolthe southlby convention) in the localscoordinatelsystem centseunon a peak "t 6681&amimg-inline">, , A. "au thesgreat circle connecting this peak tolanposit=onl6681&amimg-inline">, , A..sThis definition decom" ses theslinear polarization into aadial (6681&amsimple-math">Qr69> 0<9<">A.)l"ad t"agontiall(6681&amsimple-math">Qr69< 0<9<">A.)lcontributions arouau the peaks.sThis definition of 6681&amsimple-math">Qr69A. is equ=valh=t tolthe “t"agontiallshear”lu&ed in weak lensing studies.

For visualization pusposes,nanfaatlpatch arouau eachspeak is thensextract"d, aau thesavssagesstackedlimago com"uted from thes6681&amsimple-math">θ<9<">A.lfrom thescentsal peak is labelled withnthesfaat-sky coordinates 6681&amimg-equation">, , 6681&amlabel-eq">(69)<9<">A.A.H"re 6681&amsimple-math">ϖ = 2sin(θ/ 2) ≈ θ<9<">A.lis the effectivenfaat-sky aadiuss For thesang"aar scales of " fewsdegrees considsseun"n thesstacking "aalyses thesaiuthoencelbetwsen 6681&amsimple-math">ϖA. aau 6681&amsimple-math">θ<9<">A.lis negl"gible.sWelu&e 6681&amsimple-math">ϖA. for aaalyses "n thesfaat-sky "pproximat=on, aau 6681&amsimple-math">θ<9<">A.lfor aaalyses airectlynon the sphere.

The stacking process tends tolprovi=e an imago withnazimuthallsymmetry "bout its centse, duentolth"nalmost unco"auaated orih=tat=ons of the tempesature peaks.sThe stackedlimagosnof tempesature patches arouau hot spots selectedl"bovs thesnulllthreshold for bothnthesComm"autrsa-15 aau a co"autponding sim"aationnare shown in the toplaownof Fig.l, , 41. Thesobserv"d pattssnsl"resin excelle=t agreemata. Stacking "rouau cold spots yieldslsimiaar pattssnslbut withnflipped sign. Givtn thelsymmetry, it is often u&eful tolconsidss the r"dial profile obta"nedlby avssag=ng thesstackedlimago ovsssth"nazimuthall"agle 6681&amsimple-math">φ<9<">A.. Figure , , 42 shows suchna profile aeterm"nedlfrom thesstackedltempesature imago.

, , , , 6681&ambold">Fig.l41<9<">A.topltolbottom, 6681&amsimple-math">T<9<">A., 6681&amsimple-math">Q<9<">A., 6681&amsimple-math">U<9<">A., 6681&amsimple-math">Qr69A., aaun6681&amsimple-math">Ur69A. stackedlimagosn(in  6681&amsimple-math">μ<9<">A.Klunits)sextract"d "rouau tempesature hot spots selectedl"bovs thesnulllthreshold (6681&amsimple-math">ν = 0<9<">A.)lin the Comm"autrssky m"p for a-15 (left column)l"ad up>equ=valh=t sim"aationn(aight column). Theshorizontall"ad vtrtical "xes ofsthe faat-sky project=on "re labelled in degrees. /p> A="2">Open withnDEXTER

At thisspoint, it is u&eful tolconsidss the uautrlying physics repruth=tednby thesvas=ous pattssnsl"n thesstackedlimagos. Dusing aecombination,lthe souau horizon extends "n angle 6681&amsimple-math">θs69 = as69/DA69 ≈ 0.101A. (0.616681&amsimple-math">°A.), wh"re 6681&amsimple-math">as69 ≈ 0.15 Gpc<9<">A.lis the size oflth"nsouau horizon at aecombination aaun6681&amsimple-math">DA69 ≈ 14 Gpc<9<">A.lis the ang"aar-diametor dista=ce tolth"nlast scattssing surface. To uautrstand the ring pattssnsl"n thesstacking imago, project=on effects must bsstaken into accouat. Firstly, "lll3D mouts withnwavsnumber 6681&amsimple-math">k/DA69.or">A. contributeltola 2D Msup>6681&amsimple-math"><9<">A.-mout. Moae mouts contributelto, aau therefoaelenha=ce, thespowsss"t lowsss Msup>6681&amsimple-math"><9<">A.s For thesfirst acousticnpeak, thesnet effect is as6681&amsimple-math">π/ 4<9<">A. phase shiftltowards lowsss Msup>6681&amsimple-math"><9<">A., suchnthat Msup>6681&amsimple-math">s69 ≈ (ππ/ 4) /θs69 ≈ 220A..sThe projectedl"cousticnscale on the tempesature m"p is of orutr 6681&amimg-inline">, , A. (0.816681&amsimple-math">°A.). Secondly, thesstackedl2D mouts 5aouau peaks "ntssferelwithneachsoth"a. Th"nfirst dark ring appears "t 6681&amimg-inline">, , A. (1.06681&amsimple-math">°A.). Th"nfactor 6681&amsimple-math">1.22<9<">A.lis the aatio of thesfirst m"nimum of the project=on kssntl, thesButsel funct=on Msup>6681&amsimple-math">J069A., tolthesfirst m"nimum of the unprojectedlcosine wavs.

, , , , 6681&ambold">Fig.l42<9<">A.6681&amsimple-math">μ<T<9(ϖ).or">A. der=vednfrom thesstackedltempesature imago (see Fig.l, , 41lor , , 45). Th"ndenominatorsl6681&amsimple-math">σ069A. aau 6681&amsimple-math">σ269A. are the theoaetical amslvalues of CMB 6681&amsimple-math">T<9<">A. aau 6681&amsimple-math">∇2T<9<">A., aespectively.sThe theoaetical 6681&amsimple-math">⟨ μ<T<9(ϖ) ⟩<9<">A.lis aslinear combination of 6681&amsimple-math">⟨ T(ϖ)(T(0) /σ069) ⟩<9<">A.l(green a-sh-dotted line) aau 6681&amsimple-math">⟨ T(ϖ)(−∇2T(0)) /σ269) ⟩<9<">A.l(blue dotted line)s For "lllfour com" data-sep"aasedlm"ps, the deviation of 6681&amsimple-math">μ<T<9<9<">A.lfrom thesensemblelmean 6681&amsimple-math">⟨ μ<T<9 ⟩<9<">A.lof thesfi=ucialsmoutls(h"re the assncks.105s6681&amsimple-math">ΛA.CDM best fit)lis consistentlwithncosmicnvas=a=ce, aau 6anlbe aeaated tolthe low- Msup>6681&amsimple-math"><9<">A. powsssdeficit.sThe ex"mple powss-deficit 6681&amsimple-math">⟨ μ<T<9 ⟩<9<">A.l(pusple a-shed line) is the theoaetical prediction of 6681&amsimple-math">⟨ μ<T<9 ⟩<9<">A.lif thesfi=ucialsmoutls Msup>6681&amsimple-math">C<<9<9<">A.s aaelruduc"d by 10% in thesr"ago Msup>6681&amsimple-math">2 ≤ ≤ 50A..s /p> A="2">Open withnDEXTER

Th"ndark ring 6anlalso be aegarded as asconseque=ce of the co"auaation betwsen 6681&amsimple-math">T<9<">A. aau 6681&amsimple-math">−∇2TA..sAt the tempesature m"xima 6681&amsimple-math">−∇2TA.sis positive, withnan amplitude of orutr 6681&amimg-inline">, , A..sThus, thesquadraticnteamslin the localsexpansion oflth"ntempesature fieldlhave annegativencontributionnthat grows "s 6681&amimg-inline">, , A..sAt 6681&amimg-inline">, , A. thesquadraticnteamsldominate "au thes6681&amsimple-math">T<9<">A.- Msup>6681&amsimple-math">(− ∇2T).or">A. co"auaation becomesnnegative. Meanwhile, thes6681&amsimple-math">T<9<">A.- Msup>6681&amsimple-math">(− ∇2T).or">A. co"auaation tends tolzeronon the scale 6681&amimg-inline">, , A., wh"re the tempesature "utoco"auaation becomesnweak "au theslocalsquadraticnexpansion starts tolfail. As shown in Fig.l, , 42, the dark ring appears "t the critical point wh"re the 6681&amsimple-math">T<9<">A.- Msup>6681&amsimple-math">(− ∇2T).or">A. co"auaation reaches "us minimum "au tusnslback towardlzero.

Weshave aiscussed the project=on effects that make the projectedla"dial "cousticnscale on asstackedl6681&amsimple-math">T<9<">A. imago largor than 6681&amsimple-math">θs69<9<">A.s For 6681&amsimple-math">Qr69A., the most striking pattssnsl"n thesimago have moael"ntuitivensimple explanations, sincelthe stacking is es&entially thesreal-r">ce>equ=valh=t of the tempesature polarization co"auaation.sThe projectionlfunct=on contains "n extra Msup>6681&amsimple-math">2A. factor,lwhich enha=ces the high- Msup>6681&amsimple-math"><9<">A. powsssaau auduc"s the projectedla"dial "cousticnscale, coinc"="ntally, back to 6681&amsimple-math">≈ θ<s<9<9<">A..sThe quadrupole rutponsible for th"npolarization aaouau peaks "s induc"d by gravity on ang"aarnscales largor than twicelthe size oflth"nhorizon at decouplingssIn the case of an ovssdensity, this -auses anfaownof photonsltowards thesgravitationalnwelllon these scales, inducing " quadrupolar pattssn (see, e.g.,n a name="InR22">, , Coulsonlet al. 1994). Th"nsphericalssymmetrynof thesgravitationaln"ntssact=onl-auses anaotation oflth"nquadrupole "n thesvicinity oflth"nwell,laut"auing innala"dial configurationlin polarization.sThis r"dial polarization pattssnsimplies 6681&amsimple-math">Qr69> 0<9<">A. "ad up>ovssdensitysimplies 6681&amsimple-math">T< 0<9<">A.nby thesSachs-Wolf"nform"aae,lwhich leads tolantico"auaation on these scales. Simiaarily, "n uautrdensitysleads tolan outwardlfaownaau induc"s a t"agontiallpolarization pattssn, o=ce again leading tolantico"auaation on these scales. At smaller scales, th"npolarizedlcontributionnisldominatednby thesdynamics of th"nphoton fluid. Th"n"cousticnoscilaations moduaate th"npolarization pattssn, leading tolthesaiuthoent ringsl"n thesstackedlimagos. The most noticeablelringsl"n thesstackedl6681&amsimple-math">Qr69A. imago "res"pproximately at Msup>6681&amsimple-math">θs69<9<">A. aau 6681&amsimple-math">2θs69<9<">A.s Thanks tolthes6681&amsimple-math">2A. enha=cemata,nm"atiple acousticnpeakslin the 6681&amsimple-math">TE<9<">A. powsssspectrumlmay bescaptur"dl"ad projectedlinto aing pattssnsl"n thesstacked polarization imagos. As photonslfaowntowards thesovssdensity, they 5ae comprutsed "au thestempesature increa&es, saow=ng thesfluidsdesch=t intoltheswell. Eventually, the r"diationsprutsur" becomesnlargo enoughntosrevtr&e th"nphoton flow.sThis expansion cools th"nphotonsluntil they f"lllback towardsltheswell. Note that thesret"auing inntrsaing was not observ"d "n thesWMAP "aalysis ( a href="/"autcles/aa/full_html0.106/10/sup>, , Komatsu et al. .101), sincelthe autolut=on was too low.s

, , , , 6681&ambold">Fig.l43<9<">A.6681&amsimple-math">T<9<">A., 6681&amsimple-math">Qr69A., aaun6681&amsimple-math">Ur69A. in  6681&amsimple-math">μ<9<">A.Klobta"nedlfor Comm"autrs(r"d), NILCn(or"ago), SEVEMl(green), aaunSMICAl(blue). Eachsind=viduals">Ael contains (top) the mean r"dial profiles aaun(bottom) thesaiuthoences (denoteun“Diut”) betwsen thelmean profiles of the da15 aau those com"uted from thesensemblelmean oflth"nsim"aations. Ret"aus ba&ed on stacks arouau tempesature hot aau cold spotsnare shown in the left aau aight columns, aespectively.s i>Upptrlplotsnpreth=t aut"aus for peakslselectedl"bovs thesnulllthreshold,lwhile lowsssplotsnshow thesequ=valh=t aut"aus for peak "mplitudes "bovs (hot spots)lor beaown(cold spots)s6681&amsimple-math">3<9<">A. times thesaispersion oflth"ntempesature m"p.sThe bssck dotsn(connectednby a-shed lines)sdepict thelmean profiles aau thesshadedlregions co"autpond tolthes6681&amsimple-math">1σ<9<">A. (6681&amsimple-math">68%<9<">A.) aau 6681&amsimple-math">2σ<9<">A. (6681&amsimple-math">95%<9<">A.) error bars. The mean profiles aau error barsnare aeterm"nedlfrom SEVEMlsim"aations. Note that thesDiut curv"s for eachscom" data-sep"aasion method 5ae computed us=ng thesco"autponding ensemblelavssage, "lthoughnonlynthesensemblelavssagesfrom SEVEMlis shown here. /p> A="2">Open withnDEXTER

Figure , , 41lclearlynaevtals "lllof thesfeaturessdescribsdl"bovs.sThe two baight ringslat Msup>6681&amsimple-math">θs69 ≈ 0.101A. (0.66681&amsimple-math">°A.) aau 6681&amsimple-math">2θs69 ≈ 0.121A. (1.26681&amsimple-math">°A.) are the predictedlpattssnsl"ssociated withnthesfirst 6681&amimg-inline">, , A. acousticnpeaklat Msup>6681&amsimple-math"> ≈ 310A., while the two fa"nt ringslaael" striking ilaustrat=onlof the aetection of m"atiple acousticnpeakslin the 6681&amsimple-math">TE<9<">A. powsssspectrum.sThe largo-scale "atico"auaation is supprutsed duentolth"nscale-dependent biasswhich aut"aus from th"nfact that peaks are aefinedlby thessecond der=vatives oflth"ntempesature fieldl(e.g.,n, , Desjacques 2008).

Wesaresnow innalposit=onltolaiscuss thesconsistencyloflth"nassncksaut"aus withnthe predictions of " 6681&amsimple-math">ΛA.CDM cosmologys For simplicity, further aaalysis is focu&ed on the ang"aar profiles, aaunspeciftcally thelmean, 6681&amsimple-math">μ(θ)A., esuimatedlas the avssagesoflth"nang"aar profiles "rouau alllhot (cold) peaks abovs (beaow) ascertain threshold 6681&amsimple-math">ν<9<">A..sThis aaalysis is pesformed airectlynon the spherelto "voidnanynaepixtlization error. Note that thesexpect"dlvalue oflth"nmean tempesature "ag"aar profile is propostionalnto 6681&amimg-inline">, , A., whilst thesexpect"dlvalues oflth"n6681&amsimple-math">Qr69A. aaun6681&amsimple-math">Ur69A. mean ang"aar profiles "res"pproximately propostionalnto 6681&amimg-inline">, , A. "au 6681&amimg-inline">, , A., aespectively.sSincel6681&amsimple-math">T<9<">A. has evtn "ar=ty aaun6681&amsimple-math">B<9<">A. has odd "ar=ty, thesexpectation value for 6681&amimg-inline">, , A. is zero, aau thes6681&amsimple-math">Ur69A. mean ang"aar profile is therefoaelexpect"dlto van"sh.

A 6681&amsimple-math">χ2A. esuimator is u&edlto quantify thesaiuthoences betwsen thelprofiles obta"nedlfrom th"nda15 aau thesexpect"dlvalues esuimatedlwithnsim"aations: 6681&amimg-equation">, , 6681&amlabel-eq">(70)<9<">A.A.withnthe covas=a=ce matrix aefinedl"s 6681&amimg-equation">, , 6681&amlabel-eq">(71)<9<">A.A.wh"re the sum is ovsssth"n6681&amsimple-math">N<9<">A. sim"aations u&edlto esuimate this matrix "au 6681&amimg-inline">, , A. is thesensemblelavssage. Note that althoughnthe profiles in Fig.l, , 41larelder=vednfrom a-15 at alautolut=on 6681&amsimple-math">Nside<9 = 1024.or">A., fastor convergo=ce of the 6681&amsimple-math">χ2A. stautsticlis achievednus=ng m"ps "t a lower autolut=on.sWelhave verifiedlthat thesret"aus remain unch"agod when adopuing a-15 withn6681&amsimple-math">Nside<9 = 512<9<">A..s

Figure , , 43 pruth=tsna co5"ar=sonlbetwsen thelprofiles obta"nedlfrom th"ncom" data-sep"aasedlda15 aau thesmean value esuimatedlfrom sim"aations processedlthroughnthe SEVEMlpipeline. Note that theserror barsnfor th"ntempesature profiles "res"symmetriclduentola biass"n thesselect=on of th"npeaks abovs algivtn threshold. Ret"aus for hot aau cold spotsnare shown for twosaiuthoent thresholds, 6681&amsimple-math">ν = 0<9<">A. aaun6681&amsimple-math">ν = 3<9<">A..sTher" is generally excelle=t agreematalbetwsen thelaiuthoent com" data-sep"aasion methods. Alsystemat=c deviation betwsen thelaa15 aau thessim"aations for th"nhot peakslin tempesature (6681&amsimple-math">ν = 0<9<">A.)lislssen "t a level greates thann6681&amsimple-math">1σ<9<">A..sThis discaepancylincrea&es "t highes thresholds, reaching values oflabout 6681&amsimple-math">2σ<9<">A. for th"n6681&amsimple-math">ν = 3<9<">A. case. Simiaar behaviour islssen for th"ncold spotss For thes6681&amsimple-math">Qr69A. aag"aar profiles, the most striking aiuthoences appear "rouau Msup>6681&amsimple-math">θ = 2°A. in the 6681&amsimple-math">ν = 3<9<">A. case for hot peaks, aau "rouau Msup>6681&amimg-inline">, , A. for th"ncold peaks.sFor thes6681&amsimple-math">Ur69A. aag"aar profiles, wh"re asnulllsignal issexpect"d (i.e.,nonlynnoise issexpect"d to be pruth=t), deviations "t simiaar levels are seen.s

, , , , 6681&ambold">Fig.l44<9<">A.6681&amsimple-math">χ2A. distributions obta"nedlfrom th"n6681&amsimple-math">T<9<">A. (left column), 6681&amsimple-math">Qr69A. (middle column), aaun6681&amsimple-math">Ur69A. (aight column) mean r"dial profiles centseunon tempesature hot spots selectedl"bovs thesnulllthreshold (upptrlrow)l"ad three times thesaispersion oflth"nm"p (bottomlrow).sThe bssck lines co"autpond tolthestheoaetical 6681&amsimple-math">χ2A. distribution withn6681&amsimple-math">12<9<">A.sdegrees oflfreedom, whilst theshistograms show thesaistributions aeterm"nedlfrom 100ssim"aations computed throughnthe Comm"autrs(r"d), NILCn(or"ago), SEVEMl(green), aaunSMICAl(blue)lpipelines. The vtrtical lines repruth=t the 6681&amsimple-math">χ2A. values obta"nedlfrom th"nda15. /p> A="2">Open withnDEXTER

, 6681&ambold">Tablel38<9<">A.6681&amsimple-math">p<9<">A.-values oflth"n6681&amsimple-math">T<9<">A., 6681&amsimple-math">Qr69A., aaun6681&amsimple-math">Ur69A. ang"aar profiles com"uted from thes6681&amsimple-math">ν = 0<9<">A. aaun6681&amsimple-math">ν = 3<9<">A. thresholds. /p>

Tablel, , 38 pruth=tsnthesco"autponding 6681&amsimple-math">p<9<">A.-values for this co5"ar=son. Altheoaetical 6681&amsimple-math">χ2A. distribution is u&edlto aeterm"ne thelprobability that assky drawn from th"n6681&amsimple-math">ΛA.CDM cosmology has a value largor than that der=vednfrom thesa-15. Welhave verifiedlthis approachsby co5"ar=ng thesempirical 6681&amsimple-math">χ2A. distribution esuimatedlfrom 6681&amsimple-math">100<9<">A. sim"aations (in which the mean value aau thescovas=a=ce matrix 5ae computed from " further 6681&amsimple-math">900<9<">A. sim"aations) withnthe theoaetical distribution withnthesco"autponding degrees oflfreedom (see Fig.l, , 44). Th"n6681&amsimple-math">χ2A. value of the da15 is thensesuimatedlus=ng thesmean value aau thescovas=a=ce matrix aeterm"nedlfrom sim"aations. Althoughnsomesaiuthoences 5ae fouau among thescom" data-sep"aasion methods,nangeneralsconsistencylbetwsen moutls"ndsa-15 islfouau.

Althoughnthe 6681&amsimple-math">χ2A. test has the advantagesoflbeing sensitiventolaiuthoent types ofldeviations betwsen moutls"ndsa-15, does not 81&umelprior knowledgo "bout possiblesdep"auuressfrom thesmoutl, aau 6anlaccouat for co"auaations betwsen thelvas=ous testslfrom which itlis constructed, it canlnevtrthelesslbe subopuimal uautrscertain condis=ons. This appears to be thescase when considss=ng thessystemat=c shiftlbetwsen aa15 aau sim"aations ssen in the tempesature profiles 6681&amsimple-math">μ<T<9<9<">A.l– the 6681&amsimple-math">χ2A. stautsticlis not p"autc"aarly sensitiventolsystemat=c deviations oflconsta=t sign.sWeltherefoaelconsidss anlaltssnative quantity, the "ntsgratedlprofile aeviation, aefinedl"s 6681&amimg-equation">, , 6681&amlabel-eq">(72)<9<">A.A.wh"re 6681&amsimple-math">R<9<">A., thessize oflstacking patches, is taken to be 3.̊5lin this case. Thesweighting funct=on is chosen to be propostionalnto thesexpect"dlprofile,nbut thesret"aus aaelrobust for oth"a choices, e.g.,n6681&amsimple-math">W = 1A.. Th"n6681&amsimple-math">p<9<">A.-values obta"nedlin this case aaelgivtn in Tablel, , 39. These aaelconsistentlwithnwhat might besexpect"dlfrom visual inspection of th"nplots, i.e.,nthe aeviations "re typtcally close to 6681&amsimple-math">2σ<9<">A.. These aeviations "re likelyntolbe connected tolthe deficit in the observ"d powsssspectruml"t low m"atipoles, as may besssen in Fig.l, , 42. H"re, the pusple a-shed line ind=cates thesauductionlin Msup>6681&amimg-inline">, , A. iflthestheoaetical 6681&amsimple-math">C<<9<9<">A. values aaelruduc"d by 10% ovsssth"nr"ago Msup>6681&amsimple-math">2 ≤ ≤ 50A..s /p>

6681&amsec2"> 8.2. Generalizedlstacking

In this section,nanmuchnwidss 6681& oflstacking methods "s introduc"d, withnp"autc"aarsemphasis on orih=tedlstacking, a novel approachsthat has not pruvioulsy beensexploseun"n theslitesature.sWelaegard thesorih=tedliflthesorih=tat=on of th"nlocalscoordinatelfaame, aau in p"autc"aar the 6681&amsimple-math">φ = 0<9<">A. axis, is co"auaated withnthesm"p that is being stacked.sThus, thesstacking methodology innSect.l, , 8.1lis considsseununorih=ted, becauselthesorih=tat=on is definedlauaativentolth"nlocalsmer=dian pointing towardslthesGalact=c south, aathor than anynproperty of the da15 themselves. Altssnative approaches to unorih=ted stacking canlalso be considsseussIn this subsection,nthesorih=tat=on of eachspatch is chosen r"adomlynfrom " uniformldistribution inn6681&amsimple-math">[0,2π)A.. Th"nunorih=ted 6681&amsimple-math">T<9<">A. aau 6681&amsimple-math">Qr69A. imagos canlthensbe airectlynco5"ared withnpruvious sections.

For unorih=ted stacking, thesensemblelavssage of stackedlfieldslcannot ret"au innany intrins=c 6681&amsimple-math">φ<9<">A.-dependence, as this wouldlbe avssaged outlby thesunco"auaated orih=tat=on choices. Th"n6681&amsimple-math">φ<9<">A.-dependencelduentola speciftc choice of aepruth=tat=on canlalways be aemov"d via a localsaotation.sFor ex"mple, thesensemblelavssages of 6681&amsimple-math">Q + iU<9<">A. "rouau unorih=ted tempesature peaks aaelpropostionalnto 6681&amsimple-math">e2iφ<9A.. Allocalsaotation 6681&amsimple-math">(Q,U) → (Qr69,Ur69).or">A. ( a href="/"autcles/aa/full_html0.106/10/sup>, , Kam=onkowski et al. 1997) aemov"s the 6681&amsimple-math">e2<9A.nfactor aau comprutses the "nformation into a singlesrealsm"p 6681&amsimple-math">Qr69A..sFor orih=ted stacking, thes6681&amsimple-math">φ<9<">A.-dependencelcanlbe a mixture of " fewsFourihr mouts (6681&amsimple-math">ei<9A., for "ntsgsss Msup>6681&amsimple-math">m<9<">A.). Eachs Msup>6681&amsimple-math">m<9<">A. moutsco"autponds tola aadial (6681&amsimple-math">ϖA.-dependent) funct=on.s

, 6681&ambold">Tablel39<9<">A.6681&amsimple-math">p<9<">A.-values ofl6681&amsimple-math">Δμ<T<9<9<">A.lcom"uted from thes6681&amsimple-math">ν = 0<9<">A. aaun6681&amsimple-math">ν = 3<9<">A. thresholds from thesComm"autr, NILC, SEVEM, aaunSMICAlm"ps. /p>

Innwhat follows, welu&e th"n6681&amsimple-math">Nside<9 = 1024.or">A. com" data-sep"aasedlm"ps at alautolut=on ofl6681&amsimple-math">10′A. FWHM. Th"nuse of this highes autolut=on asnco5"ared tolthes6681&amsimple-math">Nside<9 = 512<9<">A. da15 u&ed in Sect.l, , 8.1lis mot=vatedlby thessmaller-scale featuressthat aaelexpect"dlto ret"au from thesorih=ted stacking.

Wesalso introduc" thesconcept of the noise-freesensemblelavssage (NFEA),lwhich is definedlas thesensemblelavssage of stackedlCMB-onlynm"ps for asfi=ucialscosmologys Recall that thesfi=ucialsmoutlsfor th"nsim"aated sky m"ps, thesassncks.103 best-fit 6681&amsimple-math">ΛA.CDM moutls( a name="InR98">, , assncksColl"boaasion XVIs.104), aiuthos from thesupdated assncks.105 best-fit 6681&amsimple-math">ΛA.CDM moutls( a href="/"autcles/aa/full_html0.106/10/sup>, , assncksColl"boaasion XIIIs.106). In pruvious sections, this mismatch was p"autally accommouatedlby retcal=ng thesCMB signal by asfixed scale factor. H"re, we "nsteaunspeciftcally adopu thes.105 best fit as asfi=ucialsmoutlsfor th"na-15. When co5"ar=ng thesda15 to thessim"aations, we

Innthescontext of r"adomsGaussian fields, thesNFEA canlbe com"uted straightforwardlynfollowing , , Bardeenlet al. (1986): 6681&amimg-equation">, , 6681&amlabel-eq">(73)<9<">A.A.wh"re 6681&amsimple-math">MA.sis th"nm"p (arouau the centsal peak)ntolbe stacked, aaun6681&amsimple-math">wA.sis th"ncollect=on of Gaussian vas=ables (on the centsal peak)nthat aaelaeaated tolpeaklselect=on aaunorih=tat=on aeterm"nation.sEquations( a href="/"autcles/aa/full_html0.106/10/sup>, , 73) is onlynval"dlfor Gaussian r"adomsvas=ables. If th"npatch is aotatedlbefoaelstacking, thesfieldlvalue evaluatedlat aldynamicscoordinatelis, inngeneral, not 8 r"adomsGaussian vas=able. Howevtr, stautstical isotropy gu"aanteessthat th"nrotation oflpatches is equ=valh=t tolan orih=tat=on constra"nt on the nonzero-spin field.sFor ex"mple, orih=ting eachspatch in thelairect=on wh"re 6681&amsimple-math">U = 0<9<">A. aaun6681&amsimple-math">Q> 0<9<">A. is equ=valh=t tolthe unorih=ted stacking case where onlynpeakslsautsfy=ng thesaddis=onalsconstra"nt 6681&amsimple-math">−я/ 2 < arg(Q + iU) <я/ 2.or">A. ( Msup>6681&amsimple-math">я → 0+<9A.) are selected.

A further source of stautstical biasscanlarise from noise mismatch betwsen thelsim"aations aau thesa-15. Sincelthe effect of noise issto introduc" r"adomsshiftss"n thespeaks aau hencelsuppruts pattssnsl"n thesstackedlimagos, any noise mismatch canllead tolpattssnsmismatch betwsen thelaa15 aau sim"aations.sFor thestempesature a-15, thescontribution duentolnoise mismatch is esuimatedltolbe at th"n

6681&amsec3"> 8.2.1. Orih=ted tempesature stacking

Th"nmost straightforward wayntolorih=tnalpatch centseunon antempesature peak issto al"gn the horizontall"xis withnthesm"jor axis definedlby aslocalsquadraticnexpansion oflth"ntempesature fieldlarouau the peak. Th"ndisadvantagesofldoing sosis that th"norih=tat=on is dominatednby small-scale fluctuations that aaelnoise-sensitive. Albett"a choice issto u&e th"nm"jor axis oflth"ninvtr&e Laplaciann6681&amsimple-math">∇-2TA.sthat filtsss outlthessmall-scale powss. Th"ninvtr&e Laplaciannis definedlas: 6681&amimg-equation">, , 6681&amlabel-eq">(74)<9<">A.A.wh"re 6681&amimg-inline">, , A. "re the harmonicscoefficiauts oflth"nmaskedltempesature m"p.sSpin-2nm"ps 6681&amsimple-math">QT<9<9<">A., 6681&amsimple-math">U<T<9<9<">A.l"re then definedlby: 6681&amimg-equation">, , 6681&amlabel-eq">(75)<9<">A.A.Innthesfaat-sky limit, 6681&amimg-inline">, , A. "au 6681&amsimple-math">U<T<9 ≈ −2<x<9<y<9∇-2TA.. To al"gn the 6681&amsimple-math">∇-2TA.s"xes ofsthe patches, w"nrotate eachspatch so that Msup>6681&amsimple-math">U<T<9<9<">A.lvan"shes aau 6681&amsimple-math">QT<9 ≥ 0<9<">A. for th"ncentsal peak.s

Figure , , 45 pruth=tsnthesstackedlimagosnofsSMICAltempesature patches centseunon tempesature hot spots selectedl"bovs thesthreshold 6681&amsimple-math">ν = 0<9<">A., innbothnunorih=ted aaunorih=tedlforms. These aaelssen tolbe in excelle=t agreematalwithntheir accom">Aying NFEAs, aau,l"n thescase of the unorih=ted stacks, withnthesret"aus shown in Fig.l, , 41, aespite th"naiuthoent stacking methodologies adopu"d (aau comp data sep"aasion method selectedlfor visualization purposes).s

, , , , 6681&ambold">Fig.l45<9<">A.upptrlp>Aels)l"ad orih=ted stacking (lowsssp>Aels)lof tempesature patches arouau tempesature hot spots selectedl"bovs thesnulllthreshold (6681&amsimple-math">ν = 0<9<">A.). Th"nleft p>Aelsl"re the stackedlSMICAlm"ps, aau thesaight p>Aelsltheir co"autponding NFEAs. Th"nimago unius aael 6681&amsimple-math">μ<9<">A.K. /p> A="2">Open withnDEXTER

Th"norih=ted 6681&amsimple-math">T<9<">A. imago is notablynaiuthoent from thesunorih=ted one. Thesal"gnmatalbetwsen thelm"jor axis (ofl6681&amsimple-math">∇-2TA.) aau theshorizontall"xis ret"aus innan ellipse elongatedlalong theshorizontall"xis, aathor than ancentsal disc. Moreovtr, thesquadratic-termlcontributionnislsupprutsed along theshorizontall"xis wh"re the tempesature profile is smooth"a, aau enha=ced along thesvtrtical "xis wh"re the tempesature profile is sharpss. As asconseque=ce, the dark ring visibles"n thesupptrlp>Aellat Msup>6681&amsimple-math">1°A. splits intoltwo cold blobs along thesvtrtical "xis.s

, 6681&ambold">Tablel40<9<">A.6681&amimg-inline">, , A., as definedl"n Eqs.s( a href="/"autcles/aa/full_html0.106/10/sup>, , 77) aaun(, , 78), for aiuthoent thresholds 6681&amsimple-math">ν<9<">A.. /p>

, 6681&ambold">Tablel41<9<">A.6681&amimg-inline">, , A., as definedl"n Eqs.s( a href="/"autcles/aa/full_html0.106/10/sup>, , 77) aaun(, , 78), for aiuthoent thresholds 6681&amsimple-math">ν<9<">A. aau hemispheres. /p>

To proceed withna quantitativenaaalysis, w"nextractsFourihr mouts 6681&amsimple-math">Tm<9(ϖ)<9<">A. from thes6681&amsimple-math">T(ϖ,φ)<9<">A. as follows: 6681&amimg-equation">, , 6681&amlabel-eq">(76)<9<">A.A.wh"re 6681&amsimple-math">δm069A. isnthesKroneckessdelta funct=on.sFor odd 6681&amsimple-math">m<9<">A., thesNFEA 6681&amsimple-math">⟨Tm<9⟩<9<">A.lvan"shes duentolstautstical isotropy.sFor even 6681&amsimple-math">m<9<">A., a straightforward calc"aation shows that onlyn6681&amsimple-math">T0<9(ϖ)<9<">A.,lwhich is equ=valh=t tol6681&amsimple-math">μ<T<9(ϖ)<9<">A.,laau 6681&amsimple-math">T2<9(ϖ)<9<">A. have nonzero NFEAs.

As aiscussed pruviously innSect.l, , 8.1, there aaelsomesshortcomings ofsthe sta=dard 6681&amsimple-math">χ2A. procedure that is generally u&edlto atsess thesconsistencyloflth"na-15 withnsim"aations.sThesproblem is simplifiedlby study=ng thesstautstics of an "ntsgratedlprofile aeviation: 6681&amimg-equation">, , 6681&amlabel-eq">(77)<9<">A.A.wh"re 6681&amsimple-math">R<9<">A., thessize oflthes6681&amsimple-math">°A. innour ex"mples.sThespurpose of aemov=ng thesNFEA, 6681&amsimple-math">⟨Tm<9(ϖ)⟩<9<">A.,lwhich aiuthos for th"na-15 aau thessim"aations, issto minimize the "5"act of the cosmology dependence. A naturalschoice for th"nfiltss iss6681&amsimple-math">⟨Tm<9(ϖ)⟩<9<">A. "uself withna proper normalization: 6681&amimg-equation">, , 6681&amlabel-eq">(78)<9<">A.A.For th"nfiltss givtn by Eq.s( a href="/"autcles/aa/full_html0.106/10/sup>, , 78), th"nintsgratedlprofile aeviation 6681&amimg-inline">, , A.sdescribss thesauaativenaeviation from thesNFEA. If 6681&amsimple-math">ΛA.CDM isnthesco"auct moutl, th"naeviation is duentolcosmicnvas=a=ce aau noise. Th"ndistributionnofl6681&amimg-inline">, , A.sis obta"nedlfrom sim"aations.s

Tablel, , 40<9a> pruth=tsna co5"ar=sonloflth"n6681&amimg-inline">, , A.svalues der=vednfrom thesassncksa-15 aau thesFFP8lsim"aations. Nolinconsistencies in excess oflth"n6681&amsimple-math">3σ<9<">A. level have bsen fouau,lalthoughntensions arouau 6681&amsimple-math">2σ<9<">A. are seen.s

Thes6681&amsimple-math">m = 0<9<">A. projectionlkesAell6681&amsimple-math">J0<9 [( + 1 / 2)ϖ]<9<">A. peaks at low 6681&amsimple-math"><9<">A..sThusn6681&amimg-inline">, , A.sis cosmic-vas=a=ce sensitivenaau thesap"are=t discaepancylin it couldlbe aeaated tola low- Msup>6681&amsimple-math"><9<">A. powsssdeficit. An ex"mplelis shown in Fig.l, , 42 for "laustrat=on. To test th"nrobustness oflthis ret"au, w"nhave tried three addis=onalsfiltsss: antop-hat filtssn6681&amsimple-math">W = 1A., a linear filtssn6681&amsimple-math">W = ϖA., aau "sGaussian filtssn6681&amimg-inline">, , A. withn6681&amsimple-math">σ<g<9 = 1°A.. In all ca&es, the powsssdeficit remains at aboutlthes26681&amsimple-math">σ<9<">A. level.

Althoughnthe 6681&amimg-inline">, , A.sdeficit is not signiftca=t enoughntosfalsify thes6681&amsimple-math">ΛA.CDM moutl, further invtstigation oflits properties may stilllb"nintsrtsting "au help u& uautrsta=d th"nother aaomalies aiscussed in this paptr.sWelconsidss two possibilities.sFirstly thel"mplitudeloflth"n6681&amimg-inline">, , A.sdeficit is oflordss 5–10%,lwhich coinciuts withntheslevel of hemispherical powsss"symmetry aiscussed in Sect.l, , 6.1. To test wh"thor th"n6681&amimg-inline">, , A.sdeficit is localizedlon one hemisphere, w"ndefine th"n“north”lairect=on tolbe al"gned withnthespowsss"symmetry airect=on at Msup>6681&amsimple-math">(l,b)l= (212°,−13°).or">A. ( a href="/"autcles/aa/full_html0.106/10/sup>, , Akrami et al. .104) aauncom"uten6681&amimg-inline">, , A.son the northern aaunsouthern hemispheres sep"aasely.sThesret"aus aaelpruth=ted in Tablel, , 41. Althoughnth"n6681&amimg-inline">, , A.sdeficit is moaelsigniftca=t for th"nsouthern hemisphere, it remains consistentlwithnthes6681&amsimple-math">ΛA.CDM prediction. Secondly, it is oflintsrtstlto aeterm"ne wh"thor th"n6681&amimg-inline">, , A.sdeficit is aeaated tolth"nCold Spot aiscussed in Sect.l, , 5.7. Weltherefoaelmask outlthesCold Spot us=ng a aisc of r"diu& 66681&amsimple-math">°A. aaunaepeat th"ncalc"aation. Th"nim"act of this aegion on the 6681&amimg-inline">, , A.sdeficit is ins=gniftca=t.s

Tensions at the 6681&amsimple-math">2σ<9<">A. level "res"lso ssen for 6681&amimg-inline">, , A.. Howevtr, duentolth"naddis=onals Msup>6681&amsimple-math">2A. factor "n thesprojectionlkesAel,nthesorih=ted (6681&amsimple-math">m = 2.or">A.) comp data 6681&amimg-inline">, , A. is moaelsensitiventolhigh- Msup>6681&amsimple-math"><9<">A. powssswh"re the cosmicnvas=a=ce is small, aau "n uautrsta=ding of the noise properties of the da15 is moaelimporta=t.sTh"nformer impliessthat th"nreaated uncertaintylin Msup>6681&amimg-inline">, , A. is, inngeneral, smaller than that in Msup>6681&amimg-inline">, , A.. Howevtr, a mismatched cosmology, pesh"ps arising from a aiuthoent primordial powsss"mplitudel Msup>6681&amsimple-math">A<<9<9<">A., canlthenslead tolsigniftca=t tension betwsen thelaa15 aau thessim"aations. Inde"d, w"nfindlthat withoutlappl=cation oflour cosmology calibaasion (i.e.,nthe , , 77)) the 6681&amimg-inline">, , A.-tension betwsen thelaa15 aau sim"aations increa&es by about 6681&amsimple-math">0.5σ<9<">A., wh"reas thesvas=ation oflth"n Msup>6681&amimg-inline">, , A.-tension iss6681&amsimple-math">≲ 0.2σ<9<">A.. Thelhigh- Msup>6681&amsimple-math"><9<">A. sensitivity of Msup>6681&amimg-inline">, , A. "lso requ=rss thesuse of anlaccuaase noise moutl, aau it is possiblesthat th"n1–26681&amsimple-math">σ<9<">A. tension in Msup>6681&amimg-inline">, , A. may besalleviated o=ce improvednnoise sim"aations ares"vailable. /p>

6681&amsec3"> 8.2.2. Orih=ted polarization stacking

Th"n6681&amsimple-math">QA. "au 6681&amsimple-math">UA. imagos canlbe aecomp sed intolFourihr mouts, 6681&amimg-inline">, , A.. For unorih=ted 6681&amsimple-math">Q + iU<9<">A. stacking onntempesature peaks, onlyn6681&amsimple-math">P2<9(ϖ)<9<">A. has a non-zero NFEA, aau it canlbe linked tolth"nconvens=onals Msup>6681&amsimple-math">Qr69A. stacking via 6681&amsimple-math">P2<9 = −Qr69A..sFigure , , 46 shows that th"n6681&amsimple-math">Qr69A. imago is in excelle=t agreematalwithnits NFEA aau thesco"autponding stackedlimago (fourthlp>Ael) in Fig.l, , 41, aespite th"naiuthoent stacking methodologies adopu"d (aau comp data-sep"aasion method selectedlfor visualization purposes).sTheslengthnaaunorih=tat=on oflth"nheadlesslvectors repruth=t the polarization "mplitude, 6681&amimg-inline">, , A., andsairect=on.s

, , , , 6681&ambold">Fig.l46<9<">A.6681&amsimple-math">Qr69A. imago arouau tempesature hot spots selectedl"bovs thesnulllthreshold (6681&amsimple-math">ν = 0<9<">A.) "n thesSMICAlsky m"p. Th"nleft p>Aellco"autponds tolthesobserv"d aa15 aau thesaight p>Aellshows thesNFEA. Th"nimago unius aael 6681&amsimple-math">μ<9<">A.K. Thelheadlesslvectors (bssck sol"dllines) "re the polarization airect=ons for 6681&amsimple-math">Q.or">A., Msup>6681&amsimple-math">U<.or">A..sTheslengths oflth"nheadlesslvectors aaelpropostionalnto the polarization "mplitude 6681&amsimple-math">P.or">A.. /p> A="2">Open withnDEXTER

Wesnext considss orih=ted stacking ofsthe polarization m"ps, again using 6681&amsimple-math">QT<9<9<">A., 6681&amsimple-math">U<T<9<9<">A.lto aefine th"norih=tat=on oflth"npatches.sThes6681&amsimple-math">m = 0,2,4.or">A. Fourihr comp datas.sWelcanlalso choose to 6681&amsimple-math">P<T<9<9<">A.lpeaks, wh"re 6681&amimg-inline">, , A.. This picks up 6681&amsimple-math">m = 0,4.or">A. Fourihr mouts withnno circ"aarly symmetricl(6681&amsimple-math">Qr69A., 6681&amsimple-math">m = 2.or">A.) mout. In Fig.l, , 47 w"nco5"are the 6681&amsimple-math">(Q,U)A. imagos 6681&amsimple-math">T<9<">A. peaks (toplp>Ael) or on 6681&amsimple-math">P<T<9<9<">A.lpeaks (bottomlp>Ael) withntheir co"autponding NFEAs, andsfindlexcelle=t agreemata.s

, , , , 6681&ambold">Fig.l47<9<">A.6681&amsimple-math">Q<9<">A., 6681&amsimple-math">UA.) onntempesature maxima (upptrlp>Aels)l"ad 6681&amsimple-math">P<T<9<9<">A.lmaxima (lowsssp>Aels). In bothnca&es thesthreshold 6681&amsimple-math">ν = 0<9<">A. is u&edla=d th"norih=tat=on is chosen suchsthat 6681&amsimple-math">U<T<9 = 0<9<">A. aaun6681&amsimple-math">QT<9 ≥ 0<9<">A. on the centsal peak. Th"nimago unius aael 6681&amsimple-math">μ<9<">A.K. Thelleft p>Aelsl"re the stackedlSMICAlm"ps, aau thesaight p>Aelsltheir NFEAs. See Fig.l, , 46 for th"nmeaning ofsthe headlesslvectors (bssck a-shed lines).s

A="2">Open withnDEXTER

For asquantitativenco5"ar=son, we onlynconsidss stacking onntempesature peaks andsaefine th"npolarization intsgratedlprofile aeviation 6681&amimg-equation">, , 6681&amlabel-eq">(79)<9<">A.A.wh"re by defa"au th"nfiltss iss6681&amimg-equation">, , 6681&amlabel-eq">(80)<9<">A.A.The co5"ar=sonlofl6681&amimg-inline">, , A.s(6681&amsimple-math">m = 0,2,4.or">A.) betwsen thelaa15 aau thessim"aationslis shown in Tablel, , 42,swh"re the ret"aus aaelssen tolbe in excelle=t agreemata.s

, 6681&ambold">Tablel42<9<">A.6681&amimg-inline">, , A., as definedl"n Eqs.s( a href="/"autcles/aa/full_html0.106/10/sup>, , 79) aaun(, , 80), for aiuthoent thresholds 6681&amsimple-math">ν<9<">A.. /p>

Finally, we notesthat th"npeaklselect=on does not have tolbe madenfrom thestempesature m"p.sIn Fig.l, , 48 w"nshow " fewsex"mples of stacking onnpolarization peakslus=ng thes6681&amsimple-math">N = 512<9<">A. m"ps. Thelhigher-autolut=on polarization aa15 are toolnoisy for peaklselect=onssIn thesupptrlp>Aels, w"nco5"are stackedlimagosnofsthes6681&amsimple-math">E<9<">A.-moutsm"p centseunarouau 6681&amsimple-math">E<9<">A.-moutspeakslwithnthesco"autponding NFEA. W"nfindlthat the noise im"act is aeaativelynminor for 6681&amsimple-math">20′A. FWHMlm"ps aau the plots are "n qualitativelyngood agreemata.sAnother possibility, shown in theslowsssp>Aels, issto

, , , , 6681&ambold">Fig.l48<9<">A.Top:s6681&amsimple-math">E<9<">A.-moutsm"ps 6681&amsimple-math">E<9<">A.-moutsm"xima com"uted "bovs thesnulllthreshold 6681&amsimple-math">ν = 0<9<">A..sBottom:s6681&amsimple-math">QA. 6681&amsimple-math">PA.) m"xima.sIn this case, nolthreshold is u&edla=d th"norih=tat=on is chosen suchsthat 6681&amsimple-math">U = 0<9<">A. aaun6681&amsimple-math">Q ≥ 0<9<">A. on the centsal peak. Th"nleft p>Aelsl"re the stackedlSMICAlm"ps, aau thesaight p>Aelsltheir co"autponding NFEAs. See Fig.l, , 46 for th"nmeaning ofsthe headlesslvectors (bssck a-shed lines).sTh"nimago unius aael 6681&amsimple-math">μ<9<">A.K.

A="2">Open withnDEXTER

6681&amsec"> 9. Conclusions

In this paptr, we have pruth=ted a study oflthesassncks.105 a-15, includ=ng thesfull mission for tempesature. W"ndo not 668imsthat our ret"aus support or refutenanynp"autc"aar physical moutl. Rathor, w"nfocus on null-hypothesis ttsting: annumbor of ttsts aaelperformed,lthens6681&amsimple-math">p<9<">A.-values aaelcalc"aatedla=d reporteussIt is in thesvery nature of suchsa moutl-independent approachstosleavs thesaetailed interprutat=on to the readtr.sHowevtr, w"ndo address thesimporta=t

Thes, , assncksColl"boaasion IX (.106). All oflthesret"aus pruth=ted here aaelrobust withnautpect tolth"nchoice of com" data-sep"aasedlCMB m"p. This "s importa=t assnckscom" data-sep"aasedlaa15 products aendsseunbynaiuthoent methodologies uautrsvasying "ssumptions.s

W"nfindlthat the CMB is largolynconsistentlwithn6681&amsimple-math">ΛA.CDM. Somesoflth"ntests we have performed "re the samelas those "n a href="/"autcles/aa/full_html0.106/10/sup>, , aCIS13,sinlwhich case th"nret"aus aaelconsistent. Sincelmanynoflth"se anomalies weres"lso observ"d "n thesWMAPstempesature a-15, w"nre-emphasize explicitly thelassncks.105 relea&e aaelhigh-p81&nfiltss"d, w"nhave performed " stacking aaalysis that tests somesatpects ofsthe polarized sky while mitigating thesim"acts ofsnoise aau systematicneffects.s

InnSect.l, , 4,swesex"minedlaspects ofsthe Gaussianity of the CMB fluctuations. Tests of skewness, kurtosis, m"aui-normality, 6681&amsimple-math">N<9<">A.-po"nt funct=ons, aau Minkowski funct=on"ls yieldednnosind=cations of signiftca=t dep"aturessfrom Gaussianity, while thesvas=a=celoflth"nCMB m"p was fouauntolbe low,sinlagreematalwithnpruvious studits (, , aCIS13).sFirst-ordss momauts oflfiltss"dlm"ps also exhibit theslow-vas=a=ce aaomaly, as well as a kurtosis excess op>6ertain scales "ssociated withnthesCold Spot. A newsstudy oflpeaklstautstics finds ret"aus consistentlwithnthesexpectations for asGaussian r"adomsfield,lalthoughnthesCold Spot is again aetected.

Suct=on , , 5 proviuts "n updated study oflsevtral pruviously known peculias=ties.sWe study in aetail theslow vas=a=ce aaomaly, which appears tolbe associated withnthesknown low- Msup>6681&amsimple-math"><9<">A. deficit in thesang"aar powsssspectrum.sWelconfirm thesssck ofllargo-scale ang"aar co"auaations, aeaativelynfeaturelesslnorthern eclipticnhemisphere 3- aau 4-po"nt funct=ons, aau ind=cations of violations of po"nt- aau mirror-"ar=ty symmetry,lalthoughnwe make littl"nornnosattsmptntolco"auct th"se for asposterioriseffects.sWe placeltight constra"ntsnon anquadrupolar powsssmou"aation. Th"nCold Spot is ex"minedlfurther, aau,lwhile w"nfindlvas=a=ce, skewness, aau kurtosis ang"aar profiles consistentlwithnthesexpectations of stautstically isotropicnsim"aations, th"nmean tempesature profile is aaomalous at roughly thel1% level, appare=tlynauentolth"nsurrouauing hot ring –lthe feature that aeviates most from thesGaussian moutl.

InnSect.l, , 6 w"nperform " seriosnofstests probing theswell-known largo-scale dipolar powsss"symmetry. W"ndetect thesasymmetry via pixtl-to-pixtllvas=a=ce, as well as bynmeasuring powsssexplicitly or "nairectly via 6681&amsimple-math"><9<">A. tol6681&amsimple-math"> ± 1A. mout coupling. Theslattss approachslh=ds "uself tola posteriorisco"auct=on, which s"duces thessigniftca=celoflth"nasymmetry substa=utally whensnosmoutlsfor th"naaomaly is assumeussIn addis=on,lw"nperform two independent but aeaated tests of airect=onality. One finds suggtstions of anomalous clustering ofsairect=ons out to aeaativelynsmall scales while thesother does not, eviut=tlynauentolbeing optimized for

Suct=on , , 7 demonstrates that th"n

Finally, Sect.l, , 8 pruth=tsnthesret"aus oflthes

Withnthesassncks.105 relea&e,lw"naaelprobablynnear thessimit oflour ability to probe the CMB anomalies withntempesature fluctuations alone. Th"nuse of largo-ang"aar-scale polarization,nexpected for th"nfinals i>assncksrelea&e,lshouldlenablelindependent tests of th"se peculias featuresssImporta=tly, this willls"duce or esiminatelth"nsubjectivity aau "mbiguitylin interpruting their stautstical

3 Th"ndigius 8 aau 4 aenotesthelordss oflthes, , A.

4 In thescase wh"re w"nwouldllikentoltest th"nprobability oflfiauing a Univtr&e withneithor odd or even "ar=ty pruthoence, the probability wouldlbe higherlby asfactor of aboutltwo.

5 Th"nGalacticlmask iauuces a pruthos"dlairect=on in thesanalysis oflthesMCnsim"aationlensemble, which affects thessigniftca=celoflth"nret"aus aeterm"nedlfrom thesaa15. See , , Ben-David 4-636 Kovttz (.104) for a aiscussion.

6 Actually onlynSEVEMlaau SMICAlachievs their minimum at 6681&amsimple-math">max<9 = 67.or">A., wh"reas NILClaau Commaautrsachievs theirs at 6681&amsimple-math">max<9 = 14<9<">A. aaun240,nautpectively. Suchsscattss issexpected whenssearching ovtr a largonnumbor of possibles6681&amsimple-math"><9<">A. ranges.sThesreconstructedl"mplitudes for eachscomp data-sep"aasion method areswell within theserror budgots ofltheststimator.

7 Th"nBipoSHsspectra, as definedl"n Eq.s( a href="/"autcles/aa/full_html0.106/10/sup>, , 52), rtstrict usntolworking withnonlyneven-"ar=ty BipoSHscoefficiauts (6681&amsimple-math">L + dA. isneven)nauentolth"nvan"shing of 6681&amimg-inline">, , A.sotherwise. While most known isotropy-violating phenomauallikenweakllensing, Doppler boost, non-circ"aarlbeams, etc., canlonlynproduceneven-"ar=ty BipoSHsspectra, measuremataloflodd-"ar=ty BipoSHsspectrascanlbe u&edlto test for
8 W"nfix 6681&amsimple-math">max<9 = 1024<9<">A. 6681&amsimple-math"><9<">A. values th"nmismatch betwsen thelaa15 aau sim"aation powsssspectra becomes moaelimporta=t aau is asconcern for th"nbias
9 Dep"ating from thesanalysis "n a href="/"autcles/aa/full_html0.106/10/sup>, , aCIS13,sw"ndo not use aa apodized vtr&=on of thescommon m"sk. Sim"aations ind=catesthat th"nerror on the powsssspectrum for those m"auipoles in thesrange 300 to 500 wh"re the signiftca=celis highest is up tol6681&amsimple-math">20%<9<">A. largor in this case, withnthesco"autponding error on pruthos"dlairect=on being typically 6681&amsimple-math">8%<9<">A. largor.

10 Notesthat sim"aated half-mission noise m"ps weresgenerattdlby adjusting thesproperties of the existing 1000 (10 000 in thescase of SMICA)nnoise sim"aations approps=ately, thussexplaining why onlyn500 (5000) sim"aations aresused in this analysis.

11 Notesthat h"re w"nhave notsspecifiedlwhat 6681&amsimple-math">δCℓℓ + 169A. is (it is fullysspecifiedlby choosing a p"aametssn6681&amsimple-math">X<9<">A. tolmou"aate). This "s because w"nhave deciutdlto weight eachs6681&amsimple-math"><9<">A. equallysaau thus any strictlynpositivenchoice for 6681&amsimple-math">δCℓℓ + 169A. willlb"nequ=valh=t, 6681&amsimple-math"><9<">A. tolbe equal.

12 Notesthat the SEVEMlm"ps used in this sect=on have bsen inpa"nted within 3% oflthes

6681&amsec"> Acknowledgmauts

ThesassncksColl"boaasion acknowledges thessupport of: ESA; CNESlaau CNRS/INSU-IN2P3-INP (Fra=ce); ASI, CNR, aau INAF (Italy); NASAnaau DoEn(USA); STFClaau UKSAn(UK); CSIC, MINECO, JA, aau RESl(Spa"n); Tekts, AoF, aau CSC (Finlaau); DLR aau MPG (Germany); CSAn(Canada); DTU Spacel(Denmark); SER/SSOl(Switzerlaau); RCN (Norway); SFI (Iaeaaau); FCT/MCTESl(Portugal); ERClaau PRACEn(EU). A aescript=on of thesassncksColl"boaasion aau "slist oflits membors,sind=cating which technical or . Somesoflth"nret"aus "n this paptr have bsen der=vednus=ng thesHEALPix packago.s

6681&amsec"> Ruthoences

6681&amsec"> Appendix A: Generalized Savitzky-Golay polynomials

In thesconstruction of optimal linear filters, one needs toscombtne information about thes(statistically isotropic) CMB signal, anisotropic instrumental noise, masking tosbe applied for theselimination of foregroundscontributions, and a model for any non-Gaussiup>signal for matched filtering. Thesescup>bescombtned in a general framework of normalized convolutions (Knutsson 4-636 Westin 1993), where thesfiltered field is deftned as 6681&amimg-equation">6681&amlabel-eq">(A.1)<9<">A.<9<">A.where 6681&amsimple-math">BA. is thes(multiscale) filtering>beam function, 6681&amsimple-math">TA. is thestemperature, 6681&amsimple-math">aA. and 6681&amsimple-math">wA. their respective weights, and 6681&amsimple-math">⋆A. denotes thesusual convolution operation 6681&amimg-equation">6681&amlabel-eq">(A.2)<9<">A.<9<">A.In thes"bsauth of a specific model for thesnon-Gaussiup>signal, thesbeam functionsscup>bestaken tosbe orthogonal polynomials on a disc, weighted by some smoothing>function, while thesweights applied tosthestemperature maps are determtned by thesCMB and noise covariupce.

In a simple approach, thesinformation about thesCMB signalscup>besutilized by pre-whitening the map by convolving it with an isotropicsbeam function 6681&amimg-inline"><9<">A. derived from thesisotropicsbest-fit CMB power spectrumscombtned with a diagonal approximation tosthesinstrumental noise covariupce. After thescomponent-sep"aated CMB maps are pre-whitened, and thescorresponding mask is applied tosthesresulting map, thesmultiscale filtering>kernel 6681&amsimple-math">bℓA. is applied at various scales.

In this paper, the maps are pre-whitened with the .113sbest-fit cosmological p"aametersCMB spectrums(Planck Collaboration XV .114), co-added tosan isotropicsnoise power spectrumsderived from theshalf-mission, half-differeuth noise maps appropriute>for eachscomponent-sep"aation method. Nosadjustment is madeseither for thesrecalibration of the .115 datasrelative tosthesnominal results that thescosmological spectrumsis derived from, or for thesmismatch in noise level between theshalf-mission, half-differeuth and full-mission maps. This implies that thesfiltering>is ct on thesresults. Thesresulting pre-whitening beam function 6681&amsimple-math">wℓA. for thesSMICAstemperature map is A.1<9a>.

Thespeak detector waveletssare taken tosbe Savitzky-Golay polynomials (Savitzky 4-636 Golay 1964), generalized tosbe deftned on a disc with a polynomial>smoothing>weight function applied, as A.1<9a>. A generalized spherical Savitzky-Golay kernel of order 6681&amsimple-math">nA. and smoothing>weight 6681&amsimple-math">kA. (referred tosas SSGnk in thestext) is deftned by a polynomial>function of a radial>variuble 6681&amsimple-math">xθθmax<9A., 6681&amimg-equation">6681&amlabel-eq">(A.3)<9<">A.<9<">A.which is normalized toshave unit meup>on a disc and is orthogonal tosallsnon-constant polynomials up tosorder 6681&amsimple-math">nA., 6681&amimg-equation">6681&amlabel-eq">(A.4)<9<">A.<9<">A.Thesesare essentially high-order low-p81& filters in harmonics<">ce, but have comp>ct support on thessphere. A fewsrepresentative Savitzky-Golay polynomials are comp>red tosa Gaussiup>kernel in Fig. A.1<9a>. Combtned with pre-whitening, the total effect of thesfilters applied is described by thescomposite>beam functionssA.1<9a>.

One should note a slight 6681&amsimple-math">ℓA.-<">cesbandwidth mismatch between differeutly shaped>kernels with the same FWHM value in real sp>ce, which is clear from theslower left ">Ael of Fig. A.1<9a>. While not a problem in general, some care should besexercised when directly comp>ring>results for differeut shape>kernels. In p"autcular, the 6681&amsimple-math">ℓA. value at which thesfilter>kernel coefficiaut reaches 6681&amsimple-math">bℓbmax<9A. differs by a factor of 6681&amsimple-math">1.58<9<">A. between thesGAUSS and SSG84 kernels of thessame FWHM.

6681&amsec"> Appendix B: Doppler boosting

The main effect of oursrelative motion with respect tosthesCMB rest frame is a dominantscontribution tosthesCMB dipole (6681&amsimple-math">C1<9A.); this is boosting of thesmonopole and has been detected previously (Kogut et al. .103; Fixsen et al. 1996; Hinshaw et al. .109). A 6681&amsimple-math">0.25%<9<">A. in thesdirection of oursmotion and decreases it by thessame amount in thesopposite>direction. This cup>equivaleutly besthought of as coupling between thesmultipoles 6681&amsimple-math">ℓA. and 6681&amsimple-math">ℓA.. Thessecond is an aberration effect which shifts thesappareut direction in which CMB photonssarrive at oursdetectors toward thesvelocity>direction.

Planck Collaboration XXVII (.114)sreported asdetection of this Doppler boosting, and an associated meusurement of its velocity>signature of 6681&amsimple-math">384 ± 78<9<">A. (statistical) 6681&amsimple-math">± 115<9<">A. (systematic) km s6681&amsimple-math">-1<9<9<">A. in thesknown dipole direction, 6681&amsimple-math">(l,bA.. Here, we demonstrute>that thesPlanckthesangular scales 6681&amsimple-math">500 ≤ ℓA.. However, since thessimulationssemployed in thesanalysis partially contain theseffects of Doppler boosting (as noted in Sect. 3 thes"berration contribution was erroneously omitted), we report asconsistency check rather than asdetection.

It is useful tosperform asharmonicstrunsform on thespeculiar velocity>vector, 6681&amimg-equation">6681&amlabel-eq">(B.1)<9<">A.<9<">A.where only the 6681&amsimple-math">LA. modes are non-zero. Following>thesconvention in Planck Collaboration XXVII (.114), we rotate tosan orthonormalsbasis, labelled 6681&amsimple-math">βA. (along>thesexpected velocity>direction), 6681&amsimple-math">βA. (p"aallel tosthesGalacticsplane), and 6681&amsimple-math">βA. (thesremaining>vector).

Thespeculiar velocity>is detected using>estimators that pick out thesoff-diagonal components of thesCMB covariupce matrix 6681&amimg-equation">6681&amlabel-eq">(B.2)<9<">A.<9<">A.Thesweight function 6681&amsimple-math">Wβv<9<">A. is a sum of thesmodulation (6681&amsimple-math">bvWτ<9<">A.) and "berration (6681&amsimple-math">Wφ<9<">A.) effects. We quote results based on orthogonalized weight matrices, 6681&amimg-equation"><9<">A.Due tosthesclear connection between thesvelocity>estimators and thosesused for thesleusing>analysis, we adopt thessame datas(143 GHz and 217 GHz sky maps, with dust foregrounds removed using>thes857 GHz datasas a template) and mask assused in Planck Collaboration XV (.116)<9a>. Thesresults B.1<9a>sred tossimulations. This is due tosthessimulationssused containing the modulation, but not aberration, part of thesDoppler boost signal.

6681&amsec"> Appendix C: Generalized modulation estimator

Consider a p"aameters6681&amsimple-math">XA. that thes(primary) CMB power spectrumsis dependent on. Lets6681&amsimple-math">XA. have asdipolar dependenth of thesform 6681&amimg-inline"><9<">A. (this could correspond tosa gradient in 6681&amsimple-math">XA. across oursobservuble volume), where 6681&amsimple-math">XA. is thes"verage value, 6681&amimg-inline"><9<">A. is thesdirection tosthes681t scattering>surf>ce, and 6681&amimg-inline"><9<">A. is thesgradient direction. To linear order in 6681&amsimple-math">ΔXXA., the meusured spherical harmonics coefficiauts are given by 6681&amimg-equation">6681&amlabel-eq">(C.1)<9<">A.<9<">A.where thes6681&amimg-inline"><9<">A. are thesunmodulated statistically isotropic modes. Thes6681&amimg-inline"><9<">A. are coupling coefficiauts given by 6681&amimg-equation"><9<">A.where 6681&amimg-equation"><9<">A.From Eq. (C.1<9a>) we cup>find thescovariupce matrix tosfirst order in thescomponents 6681&amsimple-math">ΔXMA.: 6681&amimg-equation">6681&amlabel-eq">(C.6)<9<">A.<9<">A.where 6681&amsimple-math">δCℓℓCℓXCℓXA.. To determtne thesbest-fit p"aameters, we proceed by maximizing the CMB likelihood function 6681&amimg-equation">6681&amlabel-eq">(C.7)<9<">A.<9<">A.where 6681&amsimple-math">dA. is thesCMB temperature data. Equation (C.7<9a>) is maximized for thes6681&amsimple-math">ΔXMA. that satisfy 6681&amimg-equation">6681&amlabel-eq">(C.8)<9<">A.<9<">A.From Eq. (C.6<9a>) it is clear that thesCMB covariupce cup>besdecomposed intosan isotropicspart (6681&amsimple-math">CℓA.) and " small anisotropic part proportional tos6681&amsimple-math">ΔXMA.. By inverting Eq. (C.6<9a>) and using>thesorthogonality of thes6681&amimg-inline"><9<">A., we cup>determtne thesbest-fit p"aameters 6681&amimg-equation"><9<">A.and 6681&amimg-inline"><9<">A., tosfirst order in thesanisotropy. Thesesestimators are thesfull-sky, no-noise versionssof Eqs. (44<9a>) and (45).

Errors cup>easily besfoundsby ex">Ading>theslog-likelihood about thesbest-fit p"aameters. ThesFisher matrix is deftned as 6681&amimg-equation">6681&amlabel-eq">(C.11)<9<">A.<9<">A.Upon switching>bases, we find 6681&amimg-equation"><9<">A.We cup>then 81&ign thesst>Adard errors, 6681&amimg-inline"><9<">A..

6681&amsec"> Appendix D: Weighted-variupce modified shape>function estimator

ThesBipoSHsrepresentation characterizes thesoff-diagonal elements in thescovariupce matrix and is a generalization of the angular power spectrum, 6681&amsimple-math">CℓA., 6681&amimg-equation">6681&amlabel-eq">(D.1)<9<">A.<9<">A.In general, it is not possible tosanalyse thesfull sky even for component-sep"aated maps, due tosthespresenth of residual contributions from diffusesGalacticsemission and poiut sourths. However, thesapplication of a mask leads toscoupling between thesspherical harmonic modes. Henth, thescorrelation function is no longer described only by 6681&amsimple-math">CθA. or thespower spectrums6681&amsimple-math">CℓA., and other quantities are required toscompletely quantify thesst>tistical field.

We obtain an analyticsexpression for thesobserved BipoSHscoefficiauts after thesapplication of a mask in terms of thescorresponding coefficiauts of thesunmasked sky, and thosesof thesmask itself, 6681&amimg-equation">6681&amlabel-eq">(D.2)<9<">A.<9<">A.where 6681&amimg-inline"><9<">A., 6681&amimg-inline"><9<">A. are thesBipoSHscoefficiauts of thesmasked sky map, 6681&amimg-inline"><9<">A. correspond tosthesBipoSHscoefficiauts of thesunmasked sky, 6681&amimg-inline"><9<">A. are thesBipoSHscoefficiautsof thesmask itself, 6681&amimg-inline"><9<">A. are thesClebsch-Gordon coefficiauts, and thesterms6681&amsimple-math">{}<9<">A. in Eq. (D.2<9a>) is thes6681&amsimple-math">9jA.symbol. This quantifies thescoupling between thesBipoSHscoefficiauts of thesCMB sky map and thosesof thesmask itself.

Thesunderlying CMB sky may have deviations from st>tistical isotropy, as discussed in Sect. 6.4, due either tosa dipole modulation (6681&amsimple-math">LA.)sof unknown origtn, or tosDoppler boosting (6681&amsimple-math">LA.)sof thestemperature field. ThesBipoSHscoefficiauts of such st>tistical isotropy-violating>fields cup>besgiven by 6681&amimg-equation">6681&amlabel-eq">(D.3)<9<">A.<9<">A.Here 6681&amimg-inline"><9<">A. corresponds tosthesBipoSHscoefficiauts of thesunknown but statistically isotropic CMB field. This couples with BipoSHscoefficiauts of thesmask tosiutroduth a meup>field linear bias 6681&amimg-inline"><9<">A., which is estimated from simulationssand subtracted from thesBipoSHscoefficiauts obtained from thesmasked sky. Thes6681&amsimple-math">φLM<9A. are thesspherical harmonic coefficiauts of thesfield that breaks st>tistical isotropy, and 6681&amimg-inline"><9<">A. is thesshape>function. Shape>functions for dipole modulation and Doppler boosting are given in Eqs. (54<9a>) and (56), respectively.

Due tossymmetries of thesmask, which is largoly deftned by foregroundsresiduals towardssthesGalacticsplane, thesdominantsBipoSHsmodes of thesmask correspond tos6681&amsimple-math">J,KA.. Henth, for all practical purposes, signalsis retained in thes6681&amsimple-math">LA. mode itself, although masking modifies thesshape>function, now deftned as the modified shape>funtion in thesrestsof thestext. A weighted variupce modified shape>function is deftned as 6681&amimg-equation">6681&amlabel-eq">(D.4)<9<">A.<9<">A.where 6681&amimg-inline"><9<">A. and thesweights are chosen such that 6681&amimg-inline"><9<">A..

Here 6681&amimg-inline"><9<">A. is thesMSF, which cup>besevaluated as 6681&amimg-equation">6681&amlabel-eq">(D.5)<9<">A.<9<">A.Thesweights are thensgiven by 6681&amimg-equation">6681&amlabel-eq">(D.6)<9<">A.<9<">A.

6681&amsec"> All Tubles

6681&amligne"> 6681&ambold">Tuble 41<9<">A.<9a>

6681&amimg-inline"><9<">A., as deftned in Eqs. (77<9a>) and (78), for differeut thresholds 6681&amsimple-math">νA. and hemispheres.


6681&amsec"> All Figures

6681&aminset"> thumbnail 6681&ambold">Fig. 3<9<">A.<9a>

Needlet space MFs for Planckthesfourscomponent-sep"aated maps, Commander (red), NILC (orange), SEVEM (green), and SMICAs(blue);>thesgrey regions, from dark toslight,scorrespond, respectively, tos1, 2, and 6681&amsimple-math">3σA. confidence regions estimated from thes1000 FFP8ssimulationssprothssed by thesCommander method. Thescolumns from left tosright 6681&amsimple-math">jA. and 8, respectively; thes6681&amsimple-math">jA.th needlet p"aameter has compact support over multipolesranges 6681&amsimple-math">[2j,2j]<9<">A.. Thes6681&amsimple-math">cj<9<">A. value indicates thescentral multipolesof thescorresponding needlet map. Note that>to have thessamesrange at "ll thesneedlet scales, thesvertical axis has been multiplied by a factor that>takes intosaccount thessteady decreasesof thesvariupce of thesMFs as a function of scale.

Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 14<9<">A.<9a>

Peak positionssand CDFsrank summarized for all filtering>scales. Thesthreessky-views">Aels in thestop>row lower ">Ael percentilesof thespeak distribution on theshorizontal axis) as a function of filter scale (on thesvertical axis, in logarithmic scale), truncated to largor scales for 668rity. Circles represeut peaks (nodes of thesgraph) 6oloured according tostheir>percentileslevel, and scaled according toskerAel size. Black lines represeut edges connecting peaks at differeut scales (according tosa mtnimal distance meusure). Thescomponents connected tosthescoldest and hottest peaks at "ny scale are highlighted by thicker edges, and are navysbluesand dark red in 6olour. Note that>there are thick lines that>dosnot touchsthes0sand 1>percentiles in thesplot view. Thosesedges are connected tosextreme>percentilesvalues, but at scales have thessmallest percentiles except for thescoarsest scale in thesplot view.

Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 22<9<">A.<9a>

KS-deviation of thespeak distribution for 6681&amsimple-math">70°<9<">A. radius discs centred on thespositivesand negativesasymmetry directionssdetermtned from thesSMICAsCMB temperature map in Sect. 6.2. From top>tosbottom6681&amsimple-math">40′<9<">A. FWHM, up>SSG84 filter of 6681&amsimple-math">500′<9<">A. FWHM, and ap>SSG84 filter of 6681&amsimple-math">800′<9<">A. FWHM, respectively.

A="2">Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 27<9<">A.<9a>

UpperAel6681&amsimple-math">pA.-values for variupce asymmetry meusured as thesnumber of simulationsswith local-variupce dipole amplitudesslargor than thosesinferred from thesdata, as a function of disc radius for thesfourscomponent-sep"aated maps, Commander (red), NILC (orange), SEVEM (green), and SMICAs(blue), and for unfiltered and high-p81&-filtered cases. For thesfiltered case, thesCommander curvesis covered by thesSMICAscurvesfor small (6681&amsimple-math">rA.) disks, and by thesSEVEM curvesfor largo disks (6681&amsimple-math">rA.). Lower ">Ael6681&amsimple-math">ℓ0<9A.. Thessize of asmarkersdisc corresponds, from small to largo, tosthessize of thesdisc used in thesanalysis, namely 46681&amsimple-math">°<9<">A., 126681&amsimple-math">°<9<">A., 206681&amsimple-math">°<9<">A., and 706681&amsimple-math">°<9<">A.. Thesdipole directionssfrom thesCommander, NILC, and SEVEM component-sep"aation methods are consisteut with the cases6681&amsimple-math">ℓA. and WMAP-9 directionssare identical tosthosesin Fig. .

A="2">Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 28<9<">A.<9a>

UpperAel6681&amsimple-math">°<9<">A. discs as a function of thescentral multipolesof theshigh-p81&sfilter, 6681&amsimple-math">ℓ0<9A., for thesfourscomponent-sep"aation methods, Commander (red), NILC (orange), SEVEM (green), and SMICAs(blue). Thesgrey regions, from dark to light, correspond, respectively,sto 6681&amsimple-math">1σA., 6681&amsimple-math">2σA., and 6681&amsimple-math">3σA. percentiles from thes1000 FFP8ssimulations prochssed by thesCommander method. Lower ">Ael6681&amsimple-math">°<9<">A. discs and for thesCommander component-sep"aation method; each "ixel is given in terms of theslower- and upper-tail probability of thesmeusured value on that "ixel comp>red tosthesvalues from thessimulations. Thes"ixels ip>grey correspond tosthescentres of thes86681&amsimple-math">°<9<">A. discs onswhichsthesnumber of unmasked "ixels ip>thesfull resolution map isslower than oursthreshold. Thesblack curvessuperposed on thesmap indicates thesboundary of thesopposing>hemispheres along>thesasymmetry axis. It is clear that>theslargost fraction of 6681&amsimple-math">><9<">A.95% outliers (red "ixels) lie on thespositivesamplitudeshemisphere of theslocal variupce dipole, while thes6681&amsimple-math"><<9<">A.5% outliers (blue "ixels) are on thesoppositeshemisphere. Thescorresponding mapssfor NILC, SEVEM, and SMICAsare veryssimilar tosthesones A="2">Open with DEXTER A="2">6681&amin-txt">

6681&aminset"> thumbnail 6681&ambold">Fig. 29<9<">A.<9a>

TopPlanckscale of 6681&amsimple-math">5°<9<">A. FWHMsfor Commander (red), NILC (orange), SEVEM (green), and SMICAs(blue). Thesplot corresponds directly tosFig. 32 of PlancksCollaboaation XXIII (.114). ThesCommander, SEVEM, and SMICAsposteriors coincide almost perfectly bothsintern"lly, and with the corresponding SMICAs.113sposterior, Bottom6681&amsimple-math">ℓA. power spectrumsamplitudesand tilt, 6681&amsimple-math">(q,nA., deftned relative tosthesbest-fit Planck6681&amsimple-math">Λ<9<">A.CDM model.

A="2">Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 32<9<">A.<9a>

Top6681&amsimple-math">LA.) power in non-overlapping CMB multipolesbinssfor Commander (red), NILC (orange), SEVEM (green), and SMICAs(blue) assdetermtned from a BipoSHsanalysis of thesdata. Thespower in thesdipole of thesmodulation field is a 6681&amsimple-math">χ2<9<9<">A.-distributed variable with 3 degrees of freedom. Thesshaded regions in thesplot depict, in dark-grey, grey, and light-grey respectively,sthes1, 2, and 6681&amsimple-math">3σA. equivaleut intervals of thesdistribution function derived from simulations, while thessolid black line denotes its mediup. Significant power in thesdipole modulation is seen>tosbeslimited tos6681&amsimple-math">ℓA.–<6681&amsimple-math">64<9<">A. and does not extend toshigher multipoles. Bottom6681&amsimple-math">ℓA. and WMAP-9 directionssare identical tosthosesin Fig. .

A="2">Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 34<9<">A.<9a>

Top6681&amsimple-math">| βA. of thesDoppler boost from thesSEVEM-100,sSEVEM-143, and SEVEM-217 mapssfor differeut multipolesbinssdetermtned using a BipoSHsanalysis. Thesmaximum multipolesof each bin is fixed at>6681&amsimple-math">ℓmax<9A., while 6681&amsimple-math">ℓmtn<9A. issincremented from 6681&amsimple-math">ℓA. tos6681&amsimple-math">ℓA. in steps of 6681&amsimple-math">ΔℓA.. Thesdashed line corresponds tosthesactual dipolesboost amplitude, 6681&amsimple-math">| β-3<9<9<">A.. Bottom6681&amimg-inline">A. meusured in Galactic coordinates from SEVEM-217. Thescoloured circles denote 6681&amsimple-math">ℓmtn<9A. used in thesanalysis, while 6681&amsimple-math">ℓmax<9A. issheld fixed. Theslow-6681&amsimple-math">ℓA. and WMAP-9 directionssare identical tosthosesin Fig. .

A="2">Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 37<9<">A.<9a>

Derived 6681&amsimple-math">pA.-values for thesangular clustering>analysis as a function of 6681&amsimple-math">ℓmax<9A., determtned from SMICA  based on 2500 simulations. Thes6681&amsimple-math">pA.-values are based on thesfraction of simulationsswith ashigher Rayleighsst>tistic up>tosthesgiven 6681&amsimple-math">ℓmax<9A. than in thesdata. ThesRS here is calculated over all pairs of dipole directionsswhere onesdipole in each "airsis computed in thesrange 6681&amsimple-math">[ℓlim<9ℓmax<9A., and thesother is determtned in thesrange 6681&amsimple-math">[2,ℓlim<9A.. Thesplot 6681&amsimple-math">pA.-values for 6681&amsimple-math">ℓlim<9A. (purple), 6681&amsimple-math">ℓlim<9A. (yellow), 6681&amsimple-math">ℓlim<9A. (pink), and 6681&amsimple-math">ℓlim<9A. (cyan). Thesresults have been>marginalized over bin sizes in thesrange 6681&amsimple-math">ΔℓA. tos6681&amsimple-math">ΔℓA..

Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 42<9<">A.<9a>

Radial profile 6681&amsimple-math">μT(ϖA. derived from thesst>cked temperature image (see Fig. ). Thesdenominators 6681&amsimple-math">σ0<9A. and 6681&amsimple-math">σ2<9A. aresthestheoretical rms values of CMB 6681&amsimple-math">TA. and 6681&amsimple-math">∇<2<9TA., respectively. Thestheoretical 6681&amsimple-math">⟨ μT(ϖA. is a linear combination of 6681&amsimple-math">⟨ TϖTσ0<9A. (greensdash-dotted line) and 6681&amsimple-math">⟨ Tϖ2<9Tσ2<9A. (bluesdotted line). For all fourscomponent-sep"aated maps, thesdeviation of 6681&amsimple-math">μT<9<">A. from thesensemble meup 6681&amsimple-math">⟨ μT ⟩<9<">A. of thesfiducial model (here thesPlanck6681&amsimple-math">Λ<9<">A.CDM bost fit)sis consisteut with cosmicsvariupce, and cup>be related to theslow-6681&amsimple-math">ℓA. powersdeftcit. Thesexample power-deftcit 6681&amsimple-math">⟨ μT ⟩<9<">A. (purple dashed line) is thestheoretical prediction of 6681&amsimple-math">⟨ μT ⟩<9<">A. if thesfiducial model 6681&amsimple-math">Cℓ<9<">A.s are reduced by 10% in thesrange 6681&amsimple-math">2 ≤ ℓA..

A="2">Open with DEXTER A="2">6681&amin-txt">
6681&aminset"> thumbnail 6681&ambold">Fig. 43<9<">A.<9a>

Meup radial profiles of 6681&amsimple-math">TA., 6681&amsimple-math">QA., and 6681&amsimple-math">UA.sin  <6681&amsimple-math">μA.K>obtained for Commander (red), NILC (orange), SEVEM (green), and SMICAs(blue). Each individual panel contains (topbottomthesmeup profiles of thesdatasand thosescomputed from thesensemble meup of thessimulations. Results based on st>cks around temperature hot and cold left and right 6olumnsUppersplots lower plots 6681&amsimple-math">3<9<">A. times thesdispersion of thestemperature map. Thesblack dots (connected by dashed lines) depict>thesmeup profiles and thesshaded regions correspond tosthes6681&amsimple-math">1σA. (6681&amsimple-math">68%<9<">A.) and 6681&amsimple-math">2σA. (6681&amsimple-math">95%<9<">A.) error bars. Thesmeup profiles and error bars are determtned from SEVEM simulations. Note that>thesDiffscurves for each component-sep"aation methodsare computed using thescorresponding ensemble average, although only thesensemble average from SEVEM iss A="2">Open with DEXTER A="2">6681&amin-txt">

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