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Mon.Not.R.Astron.Soc.000,1–9(2008) Printed3November2012 (MNLATEXstylefilev2.2) Integrated Sachs-Wolfe tomography with orthogonal polynomials Gero Ju¨rgens⋆1 and Bjo¨rn Malte Scha¨fer2 1Institutfu¨rtheoretischeAstrophysik,Zentrumfu¨rAstronomie,Universita¨tHeidelberg,Albert-Ueberle-Straße2,69120Heidelberg,Germany 2 2AstronomischesRecheninstitut,Zentrumfu¨rAstronomie,Universita¨tHeidelberg,Mo¨nchhofstraße12,69120Heidelberg,Germany 1 0 2 3November2012 n a J ABSTRACT Topic of this article are tomographic measurements of the integrated Sachs-Wolfe effect 7 withspecificallydesigned,orthogonalpolynomialswhichprojectoutstatisticallyindependent 2 modesofthegalaxydistribution.ThepolynomialsarecontructedusingtheGram-Schmidtor- ] thogonalisationmethod.ToquantifythepoweroftheiSW-effectincontrainingcosmological O parameters we perfom a combined Fisher matrix analysis for the iSW-, galaxy- and cross- C spectraforwCDM cosmologiesusingthesurveycharacteristicsofPLANCK andEUCLID. Thesignalto noiseratio hasalso beenstudiedforothercontemporarygalaxysurveys,such . h asSDSS,NVSSand2MASS.Forthecross-spectraourtomographicmethodprovidesa16% p increase in the signalto noise ratio and an improvementof up to 30% in conditionalerrors - o onparameters.Includingallspectra,themarginalisederrorsapproachaninversesquare-root r dependencewithincreasingcumulativepolynomialorderwhichunderlinesthestatisticalin- t dependenceoftheweightedsignalspectra. s a [ Keywords: cosmology:large-scalestructure,integratedSachs-Wolfeeffect,methods:ana- lytical 1 v 8 7 7 1 INTRODUCTION ducethelocalvarianceeffectandgained7percentinthesignalto 5 noiseratioforthecross-spectra. . The integrated Sachs-Wolfe (iSW) effect is one of the secondary 1 In this work we aim to formulate a tomographic approach anisotropies ofthecosmic microwavebackground (CMB).Time- 0 with help of an orthogonal set of weighting polynomials, which evolving gravitational potentials in the large-scale structure gen- 2 is similar to a former application to weak lensing spectra eratetemperaturefluctuationsintheCMB(Sachs&Wolfe1967). 1 (Schaefer&Heisenberg 2011). The orthogonality of the polyno- : TheiSW-effectisavaluabletoolforinvestigatingdarkenergyand v non-standard cosmologies since it issensitive to fluids with non- mialswillgenericallyleadtoadiagonal signal covariancematrix i andwillthereforeprovidecumulativestatisticalindependentmea- X zeroequationof state(Crittenden&Turok1996).Forthisreason surementswithincreasingpolynomialorder. itsdetectionisofparticularrelevanceforcosmologyandthenature r The article has the following structure: In Section 2 we a ofgravity(Lueetal.2004;Zhang2006)evenifitssignalstrength provide introductory information about dark energy cosmologies, isverylow. Since the iSW-effect is generated in time-evolving poten- CDMpowerspectraandlinearstructuregrowthwithinthesecos- mologies (Sections2.1-2.3).Wealso introduce a galaxy distribu- tial wells for photons on their way from the last scattering sur- tion function (Section 2.4) and give a short introduction to the face to us, it will be strongly correlated with the galaxy den- iSW-effect (Section 2.5). The orthogonal polynomials are moti- sity field. Therefore, the cross-spectrum will provide valuable additional cosmological information. The iSW effect has been vatedandconstructedinSection3.1and3.2,alsotheirmostimpor- tantpropertiesarediscussed(Section3.3).InSection4wediscuss measured in such cross-correlation studies (Boughnetal. 1998; howtomographywithorthogonalpolynomialscanimprovestatis- Boughn&Crittenden 2004; Vielvaetal. 2006; McEwenetal. ticalconstraintsoncosmologicalparameters.Aftercalculatingthe 2007;Giannantonioetal.2008).However,duetothelineofsight noisecontributions(Section4.1)weperformaFishermatrixanal- integration,adetaileddistanceresolutionoftheprocessescannot ysis(Section4.2)anddiscusssignaltonoiseratiosandstatistical bewithdrawnfromthesespectra. errors(Section4.3-4.4).TheresultsaresummarisedinSection5. A former approach correlated large scale structure observa- tionsfromvarioussurveywiththeCMBanisotropiestostudythe The reference cosmological model used is a spatially flat iSW-effectasafunctionofredshiftandtoformulateareliablelike- wCDMcosmologywithGaussianadiabaticinitialperturbationsin lihoodformulationforparameterconstraints(Hoetal.2008).Also thecolddarkmatterdensityfield.Thespecificparameterchoices recently,Frommertetal.(2008)presentedanoptimalmethodtore- are Ωm = 0.25, ns = 1, σ8 = 0.8, Ωb = 0.04 and H0 = 100hkm/s/Mpc,withh=0.72.Thedarkenergyequationofstate issettow= 0.9andthesoundspeedisequaltothespeedoflight, − ⋆ [email protected] cs=c. c 2008RAS (cid:13) 2 GeroJu¨rgens andB.M. Scha¨fer 2 COSMOLOGYANDISW-EFFECT In contrary to the pressureless dark matter component the baryons inside a dark matter halo can loose energy via radiative 2.1 Darkenergycosmologies coolingandformstars.Becauseofthisdifferentbehaviour,strictly In spatially flat dark energy cosmologies with the matter density speaking,onecannotdeducethefractionalperturbation∆n/ n in h i parameterΩ ,theHubblefunctionH(a)=dlna/dtisgivenby themeannumberdensityofgalaxies n fromthedarkmatterover- m h i densityδ=∆ρ/ρ.Inaverysimpleway,however,thelinearrelation H2(a) H2 =Ωma−3+(1−Ωm)a−3(1+w), (1) betweenthetwoentities, 0 ∆n ∆ρ with a constant dark energy equation of state parameter w. The =b , (8) n ρ value w 1 corresponds to the cosmological constant Λ. The h i h i relationb≡etw−een comoving distance χand scale factor a isgiven is a good approximation in most cases and was proposed by Bardeenetal. (1986). The bias parameter b can gener- by ally depend on scale (Lumsdenetal. 1989), time (Fry 1996; 1 1 Tegmark&Peebles 1998) as well as the galaxies luminosity and χ=c da , (2) a2H(a) morphology.Forsimplicitywesetthegalaxybiastounitythrough- Za inunitsoftheHubbledistanceχ =c/H . outthispaper,b 1.Anestablishedparametrisationoftheredshift H 0 ≡ distributionn(z)dzofgalaxiesis z 2 z β 1 z 3 2.2 CDMpowerspectrum n(z)dz=n exp dz with = 0 Γ (9) 0 z − z n β β Thelinear CDMdensity power spectrum P(k) describesthe fluc- 0!  0!  0 ! tuation amplitude of the Gaussian homogeneous density field δ, wichwasintroducedbySmailetal.(1995)andwillalsobeusedin hδ(k)δ∗(k′)i=(2π)3δD(k−k′)P(k),andisgivenbytheansatz tghailsaxwyorska.mTphleepzaram=et1e.r4z006izsreiflaβte=dt3o/t2h.eFmineadlilayn,trheedsΓh-iffutnocftitohne med 0 P(k) knsT2(k), (3) (Abramowitz&Stegun1972)determinesthenormalisationparam- ∝ etern . withthetransferfunctionT(k).Inlow-Ω cosmologiesT(k)isap- 0 m proximatedwiththefitproposedbyBardeenetal.(1986), ln(1+2.34q) T(q) = 2.5 TheintegratedSachse-Wolfe(iSW)effect 2.34q 1+3.89q+(16.1q)2+(5.46q)3+(6.71q)4 −1/4(4,) Duetoitsexpansionouruniversehadcooleddownsufficientlyto × allow theformation of hydrogen atoms at aredshift of z 1089 ≃ wherethewavhenumberk=qΓisrescaledwiththeshapepiarame- (Spergeletal.2003).Fluctuationsinthegravitationalpotentialim- terΓ(Sugiyama1995)whichassumescorrectionsduetothebaryon posed a shift in the decoupled photons which were emitted in densityΩ , the (re)combination process (Sachse-Wolfe effect). This primary b anisotropy canbeobservedinthecosmicmicrowavebackground Γ=Ωmhexp−Ωb1+ √Ω2mh. (5) l(aCrMgeBs)cainlesfoarrmouonfdtietsmmpeeraantutreemflpuecratutuarteioTnsCM∆BT=/T2C.M72B6≃K1(0F−ix5soenn The spectrumP(k)is normalised to the variance σ8 on the scale 2009). R=8Mpc/h, Besides this, photons are subjected to several other effects on their way to us, which lead to secondary anisotropies 1 σ2 = dkk2P(k)W2(kR) (6) (Aghanimetal.2008),ofwhichonlythemostimportantonesare R 2π2 mentioned here: Gravitational lensing (Bartelmann&Schneider Z withaFouriertransformedsphericaltophatfilterfunction,W(x)= 2001), Compton-collisions with free cluster electrons (Sunyaev- 3j (x)/x. j (x)isthesphericalBesselfunctionofthefirstkindof Zeldovich effect, Zeldovich&Sunyaev 1980) and with elec- 1 ℓ orderℓ(Abramowitz&Stegun1972). trons in uncollapsed structures (Ostriker-Vishniac effect, Ostriker&Vishniac 1986) and gravitational coupling to linear time-evolving potential wells (integrated Sachs-Wolfe effect, 2.3 Structuregrowthwithclusteringdarkenergy Sachs&Wolfe1967, whichwillbesubjectofthiswork). Assuming acompletely transparent space, i.e.vanishing op- Linear homogeneous growth of the density field, δ(x,a) = ticaldepthduetocomptonscatteringτ (η) = 0,thetemperature D (a)δ(x,a = 1), is described by the growth function D (a), opt + + fluctuationsτ(θˆ)generatedbytheiSW-effectcanbeexpressedby whichisthesolutiontothegrowthequation(Turner&White1997; thelineofsightintegral(Sachs&Wolfe1967) Wang&Steinhardt1998;Linder&Jenkins2003), d2 1 dlnH d 3 τ(θ) ∆TiSW = 2 χHdχa2H(a) ∂ Φ(θχ,χ), (10) da2D+(a)+ a 3+ dlna daD+(a)= 2a2 Ωm(a)D+(a). (7) ≡ TCMB c3 Z0 ∂a ! reachingouttothelimitofNewtoniangravity.UsingthePoisson equation wecan writethis integral in termsof the dimensionless 2.4 Galaxydistribution potentialφ=Φ/χ2 =∆ 1δ/χ2 fromthedensityfieldδ: H − H Galaxiesformwhenstrongpeaksinthedensityfielddecouplefrom theHubbleexpansionduetoself-gravity.Thesesocalledprotoha- τ(θ)= 3Ωm χHdχa2H(a) d D+ φ(θχ,χ). (11) c da a los approximately undergo an elliptical collapse (Moetal. 2007; Z0 Shethetal.2001). Heuristically,theeffectoriginatesfromanunbalancebetweenthe c 2008RAS,MNRAS000,1–9 (cid:13) iSW-effect withorthogonalpolynomials 3 photon’sblue-shiftwhenenteringatimevaryingpotentialwelland thered-shiftexperiencedattheexit. 100 Theeffectvanishesidenticallyinmatterdominateduniverses Ω =1,sincethenD /aisaconstant.Therefore,anon-zeroiSW- m + signal will be an indicator of a cosmological fluid with w , 0. 10−1 Aftertheradiationdominatederaitwillthusbeavaluabletoolfor investigatingdarkenergycosmologies. SincetheinverseLaplacianwhichsolvesforthepotentialin χ) thePoissonequationintroducesak 2term,smallscalefluctuations p(i 10−2 − willbequadraticallydamped. Forthisreason theiSW-effectpro- videsasignalonlargescalesandwillbenegligibleaboveℓ 100. ≈ In order toidentify the sources of the effect it is sensible to 10−3 investigatethecrosscorrelationoftheiSWamplitudewiththeline i=0 i=1 ofsightprojectedgalaxydensityγ: i=2 i=3 γ(θ)=b Z0χHdχn(z)ddχz D+δ(θχ,χ). (12) 10−4 i=4500 co10m00ovingdis1t5a0n0ceχ[M2p00c0/h] 2500 3000 We obtain the dimensionless observables γ and τ from a line of sight integration of the two dimensionless source fields δ and φ Figure1.Orthogonalpolynomials pi(χ),i=0...4,asafunctionofcomov- ingdistanceχ.Thelowestorderpolynomialisshowninblue,thehighest weightedbyfunctionswhichcarryunitsofinverselength. order in green. The construction was performed with the Gram-Schmidt Ifoneisinterestedinrathersmallscalesonecanapproximate algorithmatmultipoleorderℓ=100. thespherelocallyasbeeingplaneandperformaFouriertransform γ(ℓ)= d2θγ(θ)e−i(ℓ·θ) . (13) field.However,duetothefactthatboththecross-correlationspec- Z trumandthegalaxyspectrumarelineofsightintegratedquantities, Clearly, there isno directional dependence, γ(ℓ) = γ(ℓ),and one non-linear effectsof parameters onthesignals could beaveraged candefinethespectrumCγγ(ℓ): outandvaluabletomographicalinformationwouldbelost. Tomographicalmethodssplitupthesignalfromdifferentdis- hγ(ℓ)γ∗(ℓ′)i=(2π)2δD(ℓ−ℓ′)Cγγ(ℓ) (14) tancesandarethereforeabletoincreasethesignaltonoiseratioand Theobservable τ canbetransformed inanalogous way. Withthe thesensitvitywithrespecttocosmological parameters. Incaseof twoweightingfunctions thegalaxyspectrathisimpliesthatadditionalcovariancesbetween thedifferentspectrahavetobetakenintoaccount. H(z) Wγ(χ) = n(z) c D+(z) For a direct tomograpphy in the line of sight integration of the iSW signal the knowledge of the large scale structure poten- H d D W (χ) = 3Ω a2 + (15) tialwouldbenecessary.Areconstructionofthepotentialfromthe τ m c da a galaxy,however,wouldnotreachtherequiredaccuracy. wecannowderivethespectra(Limber1953), Tocircumventthisissueweperformtomographyinthegalaxy χH dχ signalandcross-correlatethesewiththeiSWsignal.Inthecourse C (ℓ) = W2(χ)P (k=ℓ/χ) γγ χ2 γ δδ of this we are able to withdraw tomographical information also Z0χH dχ from the iSW signal. We use specifically designed polynomials Cτγ(ℓ) = χ2 Wτ(χ)Wγ(χ)Pδφ(k=ℓ/χ) for a distance weighting of the galaxy distribution. Defining the Z0 weightedgalaxycovariancesasascalarproductofthepolynomials χH dχ C (ℓ) = W2(χ)P (k=ℓ/χ) . (16) willleadtostatisticallyindependentgalaxyspectraoncethepoly- ττ χ2 τ φφ Z0 nomialsareorthogonalised. Thisnonlocalbinningofthegalaxies Thepowerspectracanberelatedtothedensitypowerspectrum: leadsto adiagonalisation of thegalaxy signal covariance matrix. Thepolynomialscanthenalsobeusedfortomographicalmeasure- P (k) P (k) P (k)= δδ , P (k)= δδ . (17) mentsintheiSW-galaxycross-correlations. φφ (χ k)4 δφ (χ k)2 H H The multiplication factors k 2 and k 4 tilt the spectra to smaller − − 3.2 Constructionoforthogonalsetsofpolynomials values for increasing mutipole order ℓ and show once again the iSW-effecttobealargescalephenomenon. Weighting the given galaxy distribution function n(χ) = n(z)dz/dχ = n(z)H(z)/c with a polynomial p(χ) modifies the i galaxyweightingfunctionto 3 TOMOGRAPHYWITHORTHOGONAL H(z) POLYNOMIALS Wγ(i)(χ)= pi(χ)Wγ(χ)= pi(χ)n(z) c D+(z). (18) 3.1 Motivation Forthepolynomials p(χ)and p(χ)werequireorthogonality i j Measurements of the iSW-effect provides integrated information p,p =0for(i, j) (19) h i ji aboutthestructureformationhistoryofouruniversesincethelast withrespecttothefollowingscalarproductforthepolynomials: scattering surface. Cross-correlation withthe galaxy density field increases the signal to noise ratio significantly and the spectrum χH dχ p,p S(ij)(ℓ) W(i)(χ)W(j)(χ)P(k=ℓ/χ). (20) isnoiselessduetouncorrelatednoiseintheCMBandthedensity h i ji≡ γγ ≡ χ2 γ γ Z0 c 2008RAS,MNRAS000,1–9 (cid:13) 4 GeroJu¨rgens andB.M. Scha¨fer 0 1 i=0 1.5 i=1 i=2 2 i=3 i=4 3 1 +orderi1 45 −5 −1χ[()]H 0.5 mial 6 χp()i no χ) oly 7 −10 W(γ 0 p 8 −0.5 9 10 −15 −1 1 2 3 4 5 6 7 8 9 10 30 100 300 1000 3000 5000 polynomialorderi+1 comovingdistanceχ[Mpc/h] Figure2.Numericalaccuracyfortheorthogonalityrelationhpi,pjiatℓ= Figure3.WeightedgalaxyefficiencyfunctionWγ(i)(χ),i=0...4,asafunc- 20inlogarithmicrepresentation.Theaccurayimposesalimitonthenumber tionofcomovingdistanceatmultipoleorderℓ=20. ofincludedpolynomials. roughlyat thepositionswheretheprevious polynomial reachesa The necessary properties for a scalar product are obviously full- filled( p,p > 0, p,p = 0 p 0andlinearity).Weuse local maximumor minimum, which intuitivelyindicatestheir or- h i ii h i ii ⇔ i ≡ thogonality. theGram-Schmidtproceduretoconstructorthogonalpolynomials AsonecanseeinFig.2orthogonalityisfulfilleduntilnumer- outofthefamilyofmonomials icallimitationsbecomesignificantatapolynomialorderofq 9. χ i Theinreasingnumerical deviationsfromtheorthogonalityco≈ndi- p′i(χ)= χ , (21) tion( p,p =0fori, j)isduetotheiterativemethod,whichcu- node! h i ji mulateserrorsthroughout theprocess.Thisimpliestheaccuraccy where χ sets the position of the node of the first polynomial, node toshrinkfrom10 15fori=0to10 3fori=8.Thisisawellknown whichisinourcasesettothemedianvalueoftheredshiftdistri- − − disadvantage of the Gram-Schmidt orthogonalisation method, es- bution. However, a change inχ iscompletely absorbed inthe node pecciallywhendealingwithfunctionsasopposedtovectors,since coefficientandhasnoinfluenceonthepolynomials.Startingwith thereistlargernumericalnoiseintheevaluationofthescalarprod- thezero-orderpolynomial ucts.However,aswewilllatersee,itisnotnecessaryforourap- p0(χ)= p′0(χ)≡1, (22) plicationtogotoevenhigherorders. InFig.3theweightedgalaxyefficiencyfunctionsW(i)(χ)are thepolynomialsareconstructediteratively, γ depicted,whicharemodifiedbythepolynomialsp(χ)atanangular i pi(χ)= p′i(χ)−Xij−=10 hhpp′ij,,ppjjii pj(χ). (23) owscridatlheerouootfftℓtohme=opg1or9lay.pnThoyhm,eWicaalγ(0sh)e(iχei)ra==rch0Wyγre(aχfte)tr.hsOetnoheitghcheandwieseatiasginhlcyteinogebnsfdeurnvoceftittohhnee Theprocedurehastobeperformedforeverymultipoleℓ.Theindex functions,whereoneafteranotherapproacheszero. ℓofthepolynomials pi(χ)hasbeenomittedforclarity.Asonecan The modified spectraC(ii)(ℓ) andC(i)(ℓ) are shown in Fig. 4 γγ τγ see,thezero-orderscalarproductisequaltothegalaxyspectrum: and Fig. 5, respectively. The drop in amplitude is mainly an ef- p ,p =S (ℓ). (24) fectoftheabsenceofnormalisation,whileonecaninfactobserve h 0 0i γγ slight differences in shape. However, these differences are small, Therefore, the unweighted case is already contained in the first sincethepolynomials only mildlydepend on themultipole order weightingfunction.Finally,wecanweightalsothetracerdensity ℓ.Therefore,theoverallshapeofthespectraisstilldominatedby modesγ(ℓ)themselveswithapolynomial pi(χ) thezero-order spectraC(00)(ℓ)andC(0)(ℓ),respectively.Thanksto γγ τγ χH theorthogonalisationthesespectranowprovidestatisticallyinde- γ(i)(ℓ)= dχW(i)(χ)δ (25) γ pendentinformation.Inthenextsectionweaimtocombinesingals Z0 fromthegalaxy distribution,theiSW-effectandthecross-spectra forwhichageneralizedversionofthewellknownexpressionfor toinvestigatestatisticalboundsoncosmologicalparameters. thecovarianceholdsincaseofhomogeneousandisotropicrandom fields: hγ(i)(ℓ)γ(j)∗(ℓ′)i=(2π)2δD(ℓ−ℓ′)Sγ(iγj)(ℓ) (26) withS(ij)(ℓ) δ . 4 STATISTICS γγ ∝ ij Thissectionaimstoconnectcosmicvarianceandstatisticalnoise withtheiSW-signalanditscross-correlationsintoameaningfulsta- 3.3 Propertiesoforthogonalpolynomials tisticalformulation. Inthecourse of thisweconstruct covariance InFig.1theorthogonalpolynomialsareshownuptoapolynomial matricesforthepolynomial-weightedspectra.Statisticalerrorson order of i = 4. They show an increasing number of zero points cosmologicalparametersareestimatedintheFisher-matrixformal- c 2008RAS,MNRAS000,1–9 (cid:13) iSW-effect withorthogonalpolynomials 5 10−2 whichreducestoaPoissonianresultinthecaseofwmbeingeither 0or1: 1 σ2 = with n= w . (28) ww n n ℓ() Xn (ii)Cγγ10−5 Thecountedquantityninourcaseisdefinedasthemeandensityof +ℓℓ(1)π2 wgahliacxhieisspcehrasrtaecrtaedriisatnic,afolrfowrhtihcehEwUeCwLilIlDsugbasltaixtuytesunrv=ey4.0C/aorncsmidiner2-, a ctr ingtwodifferentsetsofweightingfactorsw andv ,wegeneralise e m n p s thestandarddeviationto angular 10−8 iiii====0123 σ2wv= w1 v wmvm , (29) i=4 m m n n m i=5 X i=6 whichwPillinthPecontinuumlimitbeacrossvarianceweightedwith i=7 i=8 twodifferentpolynomials.Forthecontinuumlimitthetransition 10−11 10 30 100 300 1000 3000 multipoleorderℓ ... n dχn(χ)... (30) → Figure4.Puregalaxy-galaxyspectraSγ(iγi)(ℓ),i = 0...8,weightedwithor- Xm Z thogonalpolynomials pi(χ),asafunctionofthemultipoleorderℓ.Sγ(0γ0)(ℓ) is performed which conserves the total number count n due to (blue)referstothenon-tomographic spectrum Sγγ(ℓ).Onecanseeade- theunitnormalisedgalaxydistributionfunctionn(χ).Thediscrete creaseinamplitudeforincreasingmultipoleorderℓ. weightingsetsw andv arethenrepresentedby p(χ)and p(χ) m n i j sothatthenoisecovarianceinthecontinuouscasereads 10−7 N(ij)(ℓ) σ2 = 1 dχn(χ)pi(χ)pj(χ) . (31) γγ ≡ ij n dχnR(χ)pi(χ) dχn(χ)pj(χ) The noise term N(Rij)(ℓ) still depenRds on ℓ since the polynomials γγ ℓ) 10−9 are constructed for each multipole order separately. We omit the (i)(Cτγ ℓ-dependenceofthepolynomials pi(χ)forclarity.Eqn.(31)moti- +ℓℓ(1)π2 vatesthefollowingchoiceofnormalisationforourpolynomials: pectra 10−11 pi(χ)← dχpn(i(χχ))p(χ) . (32) s i ar i=0 ul i=1 InthisnoRrmalisationthegalaxynumbernoisereads ang 10−13 ii==23 i=4 N(ij)(ℓ)= 1 dχn(χ)p(χ)p(χ), (33) i=5 γγ n i j i=6 Z i=7 i=8 whilethegalaxyspectrumcanbewrittenas 10−15 10 30 m10u0ltipoleorde30r0ℓ 1000 3000 S(ij)(ℓ)= χH dχW(i)(χ)W(j)(χ)P(k=ℓ/χ) (34) γγ χ2 γ γ Figure5.Galaxy-iSW cross-spectraC(i)(ℓ),i = 0...8,weighted withor- Z0 τγ Thelimitationinpolynomial order due toincreasing noise inthe thogonalpolynomials pi(χ),asafunctionofthemultipoleorderl.Cτ(0γ)(ℓ) polynomialsp(χ)canalreadybeillustrated:Sincen(χ)isaslowly (blue)referstothenon-tomographicspectrumCτγ(ℓ). i varying function the rapid oscillations of high order polynomials willdrivethevaluesoftheintegrals dχn(χ)p(χ)tosmallernum- i ism.Furthermore,weinvestigatethesignalstrengthofthedifferent bers and will therefore increase theR noise in pi(χ). We point out spectraandtheirdependenceonthenumberofpolynomialsused. that fororder zerothenon-tomographic caseisrecovered, giving thestandardPoissonianexpressionforthenoiseN =1/nandthe 00 integratedgalaxyspectruminthesignalpartS(00)(ℓ)=S (ℓ). γγ γγ 4.1 Variancesofgalaxynumbercounts While the orthogonalisation procedure leads to a diagonal galaxy signal covariance, the noise part will not be diagonal any Forforecastingstatisticalerrors,weneedtoderiveexpressionsfor more: N(ij) , 0fori , j.Incontrarytothismethod,atraditional thesignalcovarianceandnoise.Wewillstartfromadiscreteformu- γγ binning inz would lead toa diagonal noise contribution and off- lationwithasetofweightingcoefficientsw forthecountedgalaxy m diagonalsinthesignalpart. numberm.Clearly,theweightingcoefficientw willdependonthe m distanceoftherespectivegalaxy.Later,wewillgeneralisethefor- malismtothecontinuous case, inwhichtheweightingprocedure 4.2 Fisheranalysis isperformedbythepolynomialsp(χ).Thestandarddeviationσ i ww ofaweightedgalaxycountwithweightingcoefficientswmisgiven InordertousebothiSWsignalsandgalaxyspectrainourFisher by analysis,wenowdefineanextendeddatavector 1 σ2ww= mwm nwn Xm,n wmwnδmn (27) x(ℓ)= γτ(i)((ℓℓ)) ! . (35) P P c 2008RAS,MNRAS000,1–9 (cid:13) 6 GeroJu¨rgens andB.M. Scha¨fer The total covariance matrix, C(ℓ) = S(ℓ)+ N(ℓ), for these data 10 vectorsisblock-diagonalduetotheindependenceoftheℓ-modes: Eachblock 3 C (ℓ) C(j)(ℓ) C(ℓ)= ττ τγ (36) C(i)(ℓ) C(ij)(ℓ) τγ γγ ! 1 consistsofasignalpart 2)C n S(ℓ)= CSττ(γiτ)((ℓℓ)) SCγτ((iγγjj))((ℓℓ)) ! , (37) (∂trlqh 0.3 q=0 q=1 whereS(ij)(ℓ) δ byconstruction,andanoisecontribution 0.1 q=2 γγ ∝ ij q=3 q=4 N (ℓ) 0 q=5 N(ℓ)= τ0τ Nγ(γij)(ℓ) ! . (38) 0.03 qqcr==o67ss withpolynomialorders0 6 i,j 6 q.Duetouncorrelatednoisein gal & cross 0.01 theCMBandthegalaxydensityfieldthenoiseofthecross-spectra 10 30 100 300 1000 3000 multipoleorderℓ vanishes.TheCMBpartconsistsofthepureiSWsignal S (ℓ)= χH dχW2(χ)P (k=ℓ/χ) (39) Figure6.Sensitivity tr(∂hlnC(ℓ))2oftheFishermatrixwithrespectto ττ χ2 τ φφ theHubbleparameterqhasafunctionofthemultipoleorderℓ,andcumu- Z0 lative polynomial order q. Sensitivities are shown with derivatives of all withPφφ(k)=Pδδ/(χHk)4whilethenoisecanbesplitintothepri- spectra taken into account (solid lines) in comparison to the case where mary CMB fluctuations CCMB(ℓ) and an instrumental noise term onlythecross-spectrawereconsideredinthesignalpart(dashedlines).For C (ℓ)ofPLANCK(PlanckCollaborationetal.2011): thecovariancethesurveypropertiesofEUCLIDhavebeenassumed. beam Nττ(ℓ)=CCMB(ℓ)+w−1 exp −∆θ2ℓ2 , (40) withthebeamsize∆θ = 8.7(cid:16)7 10−4(cid:17)andthesquaredpixelnoise 30 × w 1 = 0.2µK/T (Knox1995).Thenoiselesscrossspectraare − CMB formedbyonemodifiedweightingfunctiononly: 10 χH dχ C(i)(ℓ)= W (χ)W(i)(χ)P (k=ℓ/χ) (41) 3 τγ χ2 τ γ δφ Z0 withPδφ(k) = Pδδ/(kχH)2.Wepointoutthatonlythegalaxypart 2C(cid:1) 1 ofthesignal covariancewasdiagonalised byour method. Conse- n l quently,thecross-spectraCτ(iγ)(ℓ)aretheonlyoff-diagonalentriesin ∂ns(cid:0)0.3 thecovariancematrix. trq 0.1 qq==01 The likelihood for observing these Gaussian-distributed q=2 modes x(ℓ) for a given parameter set p is defined as (Tegmark, q=3 0.03 q=4 1997): q=5 q=6 1 1 0.01 q=7 L(x(ℓ)|p)= (2π)N detC(ℓ) exp −2xT(ℓ)C−1(ℓ) x∗(ℓ) . (42) cgraol s&s cross ! 0.003 10 30 100 300 1000 3000 DefiningpthedatamatrixasDij(ℓ) = xi(ℓ)xj(ℓ)withhDi=C multipoleorderℓ andusingtherelation,lndet(C) = trln(C)onecanwritetheχ2- functional exp( χ2/2), with help of the logarithmic likeli- Figure7.Sensitivity tr ∂nslnC(ℓ)2oftheFishermatrixwithrespectto hoodL Lln∝: − theinitial slopeoftheqpower spectrum ns asafunction ofthe multipole (cid:0) (cid:1) ≡− L orderℓ,andcumulative polynomial orderq.Sensitivities areshownwith χ2 = 2L=tr lnC+C−1D . (43) derivatives ofallspectratakenintoaccount(solidlines)incomparisonto − ℓ the case where only the cross-spectra were considered in the signal part Xh i (dashedlines).Forthecovariance thesurveypropertiesofEUCLIDhave Each multipole ℓ provides (2ℓ+1) independent m-modes. If we beenassumed. interprete asaBayesian probability, the local behaviour of the L likelihoodfunctionaroundthepointofmaximumlikelihoodisde- terminedbytheHessematrixofLatthispoint: wewillworkinthelimit∂S /∂p ∂N /∂p andthereforene- ∂2L ij µ ≫ ij µ (C−1)µν≡ ∂p ∂p (44) glect the noise dependence onthe cosmological parameters. This µ ν approximationiswelljustifiedinourcase. TheFisherinformationmatrixisthengivenastheexpectationvalue Nextwehavealookattheratioofthesensitivitiesofthespec- ofthisquantitysummedoverallmultipoleordersℓ: tra with respect to cosmological parameters and the covariance. This quantity equals the contribution of a certain multipole ℓ to ∂2L 2ℓ+1 ∂ ∂ F = = tr lnC(ℓ) lnC(ℓ) . (45) therespectiveFishermatrixdiagonalelement: µν ∂p ∂p 2 ∂p ∂p * µ ν+ Xℓ µ ν ! Foreach ℓ the(2ℓ+1)/2 m-modesprovide statisticallyindepen- tr ∂lnC(ℓ) 2= 2 dFµµ . (46) dent information. In the course of our Fisher matrix calculations s ∂pµ ! r2ℓ+1 dℓ c 2008RAS,MNRAS000,1–9 (cid:13) iSW-effect withorthogonalpolynomials 7 100 30 10 3 2C(cid:1) 3 n l Ωm 1 ∂ trq(cid:0)0.3 qq==01 Σ 1 q=2 q=3 0.1 q=4 q=5 0.03 qqcr==o67ss 0.3 NEUVSCSLID SDSS gal & cross 2MASS 0.01 10 30 100 300 1000 3000 q=1 multipoleorderℓ q=8 0.1 10 30 100 300 Figure8.Sensitivity tr ∂ΩmlnC(ℓ)2oftheFishermatrixwithrespectto multipoleorderℓ thematterdensityparqame(cid:0)terΩmasaf(cid:1)unctionofthemultipoleorderℓ,and Figure9. Cumulative signal tonoiseratio Σdepending onthe multiple- cumulativepolynomialorderq.Sensitivitiesareshownwithderivativesof orderℓforthesurveycharacteristics of2MASS(dotted),SDSS(dashed- allspectratakenintoaccount(solidlines)incomparisontothecasewhere dotted),EUCLID(dashed)andNVSS(solid).Shownistheimprovement onlythecross-spectrawereconsideredinthesignalpart(dashedlines).For betweencumulativepolynomialorderq=1(blue)andq=8(green). thecovariancethesurveypropertiesofEUCLIDhavebeenassumed. galaxy surveys the most important survey parameters are the sky InFig.6-Fig.8weshowthesesensitivitiesinsolidlinesforthefull coverage f and the median redshift z . In Fig 9 the signal informationfromgalaxyspectra,cross-spectraandiSW-effectin- sky med tonoiseratiosforthesurveycharacteristicsofEUCLID,2MASS cludedfortheparametersh,n andΩ ,respectively.Atzeroorder s m (z 0.1,Afshordietal.2004),SDSS(z 0.5,Bielbyetal. theyallexhibitacertainℓ-rangeatwhichthecovarianceisinsen- med ≈ med ≈ 2010) and NVSS (z 1.2, Boughn&Crittenden 2005) are sitivetovariationsoftherespectivecosmologicalparameter.Nat- med ≈ shown. Clearly, alsothe signal tonoise ratioincreases for higher urally,angularscalesinthevicinityofthiszeropointdonotcon- polynomial orders due to the diagonal structure of the signal co- tributemuchsensitivitytotheFishermatrix.Thiseffectiscuredif variance.Wefindanimprovementof 16%inthesignaltonoise weincludeallpolynomials06i6q.Thecombinationofmultiple ≈ ratiobetween cumulative polynomial order q = 1and q = 8. As lineofsight-weightedmeasurementsliftsthesensitivitiesatthese expected, at a multipole order of a few hundred the cumulative pointscontinuouslywithincreasingnumberofinvolvedpolynomi- signalstrengthsaturatesasaresultofthePoissoniank 2 damping alsuntiltheeffectsaturates. − termintheiSW-effect.Theactual realisationofthematterdistri- Formultipoleordersℓreachinghighervalues(ℓ 3000)the ≈ bution in the observed universe introduces a systematic noise in sensitivity starts to drop rapidly. On these small scales the noise the iSW detections known as local variance. Using the so called contribution begins to dominate and the Fisher matrix ceases to optimalmethodonecandecreasethisbiasbyworkingconditional growfurther. onthelargescalestructuredataandgain7%insignaltonoisera- In dashed linesthe sensitivities are shown if only the cross- tio(Frommertetal.2008).Thetomographicalmethodpresentedin spectraareincludedinthederivatives.Againthesensitivitiesgrow ourworkshouldbealsoapplicabletothereconstructedlargescale with increasing cumulative polynomial order q, although in this structureusedintheoptimalmethod.Therefore,acombinationof case the zero order sensitivity does not suffer from any singular thesetwomethodswouldbesensible. effects. Characteristic properties of the iSW-effect are recovered showing it to be a large scale effect due to the k 2 proportional- − ityoriginatinginthePoissonequation.Abovemoltipolesofabout 4.4 Statisticalerrors l 100 the information provided by the cross spectra becomes ≈ negligible. Clearly, the cross-spectra Fisher matrix is most sensi- The Crame´r-Rao inequality introduces a lower limit on the tivetothematterdensityparameterΩ ,whichshowsthestrongest marginalizedstandarddeviationoftheestimatedcosmologicalpa- m increaseinsensitivityforincreasingcumulativepolynomial order rameters.Thesearegivenbythediagonalelementsoftheinverse q. Fishermatrix: 1 σµ> (F−1)µµ 2 . (48) 4.3 Signaltonoiseratio In Fig(cid:16). 10 the(cid:17)se errros are depicted for the five cosmological pa- Asignal’spowertoconstraincosmologicalparametersismostre- rametersΩ ,σ ,h,n andw.Theplotfollowstheevolutionofthe m 8 s liablyquantifiedbythesignaltonoiseratio errorswithincreasingnumberofincludedpolynomialsq.Whilefor Σ2 = fsky 2ℓ2+1tr C−1(ℓ)S(ℓ) 2 . (47) swmitahlldpiffoelyrennotmcihaalraocrdteerrisstitchsefCorraemace´hr-Rpaaroamerertoerrs,tdheecirmeapsreovraepmidenlyt Xℓ (cid:16) (cid:17) slowsdownforhigherorderpolynomialsandassumesacharacter- Besidesthephysicalprocessthesignaltonoiseratiostronglyde- istical behaviour. This behaviour can approximately be described pends on the characteristics of the survey at hand. In the case of bythe inverseroot of thepolynomial order σµ ∝ 1/√q. A simi- c 2008RAS,MNRAS000,1–9 (cid:13) 8 GeroJu¨rgens andB.M. Scha¨fer 0.255 100 Ω σm8 Ωm 0.25 h ns w 0.245 0.795 0.8 0.805 σ8 0.805 σsµ 10−1 σ8 0.8 or err 0.795 zed 0.7 0.h72 0.07.474 ali argin 10−2 h0.72 m 0.7 0.995 1 1.005 ns 1.005 ns 1 10−3 0.995 1 3 5 7 9 −0.91 −0.9 −0.89 cumulativepolynomialorderq+1 w Figure 10. Lower limits onthe marginalized statistical errors σµ onthe Figure12.The2-dimensional 1σ-errorellipses forthecosmological pa- estimates of the cosmological parameters Ωm (circles), σ8 (squares), h rameters Ωm,σ8,h,ns andwfromEUCLIDusingtomographywithor- (lozenges),ns(triangles,pointingup)andw(triangles,pointingdown)de- thogonal polynomials are shownin this plot. The1σconfidence regions rivedfromtheCrame´r-Raoinequality,asafunctionofthecumulativepoly- decrease insize withincreasing numberofincluded polynomials, reach- nomialorderq.TheFishermatrixwasderivedincludingthederivativesof ingfromq = 2(blue)toq = 8(green).Theellipsesareevaluated witha allspectraCγ(iγi)(ℓ),Cτ(iγ)(ℓ)andCττ(ℓ).Again,theEUCLIDsurveycharac- maximummultipoleorderofℓmax=3000. teristicshavebeenused. For studying the improvement provided by the cross-spectra, we Ω plottheconditionalerrorsasafunctionofcumulativepolynomial 0.14 σm8 order q inFig.11. Here,only the derivatives of thecross-spectra h ns weretakenintoaccountintheFishermatrixcalculation.AgainΩm 0.12 w issubjectedtothestrongestimprovement,itsconditionalerrorde- creasesby 30%.Incontrasttothemarginalizederrorstheevolu- σµ 0.1 ≈ s tionoftheconditionalerrorsdoesnotshowa1/√qbehaviourbut error 0.08 rathersaturatesatpolynomialorderofq≈5. al Finally, we are interested in the 2-dimensional marginalized on logarithmiclikelihoodχ2 aroundthefiducialmodel p diti 0.06 m fid con χ2 = pµ−pµ,fid (F−1)µµ (F−1)µν pµ−pµ,fid (50) 0.04 m p p (F 1) (F 1) p p ν− ν,fid ! − νµ − νν ! ν− ν,fid ! for which the 1σ-error ellipses are depicted in Fig. 12. Starting 0.02 withq=2,wehavecombineduptoninepolynomials.Besidesthe expectedshrinkingoftheellipsesforhigher numbersofincluded 0 1 3 5 7 9 polynomials, itisinterestingtoseehow thedegeneracies slightly cumulativepolynomialorderq+1 change their orientations in the course of tomographic improve- Figure11.Conditionalstatisticalerrorsσµ,conontheestimatesofthecos- ment.Thisisverylikelyduetodistancedependenciesofthesignal mologicalparametersΩm(circles),σ8(squares),h(lozenges),ns(triangles, sensitivities. pointingup)andw(triangles,pointingdown).TheFishermatrixwasde- rivedincluding thederivatives ofthecross-spectraC(i)(ℓ)only,EUCLID τγ surveycharacteristicshavebeenused. 5 SUMMARY In this paper a tomographic method for measuring iSW-galaxy larcharacteristicwasfoundintheapplicationofthismethodtothe cross-spectra and galaxy spectra has been presented. It has been weaklensingshearspectra(Schaefer&Heisenberg2011).Clearly, the cosmological parameter Ω profits most of the tomographic carriedoutbysight-weightingoftheiSW-effectandthetracerden- m sityfieldwithspecificallyconstructedorthogonalpolynomials. method, which was already indicated by its sensitivity improve- (i) TheGram-Schmidt orthogonalisation procedure hasbeen mentdiscussedinSection4.2.Goingtoevenhigherordersisdif- used to construct orthogonal polynomials in order to diagonalise ficultduetocumulativeerrorsintheGram-Schmidtorthogonalisa- the weighted galaxy signal covariance. The method projects out tionmethod. statistically independent signal contributions at the price of off- If we areinterested inhow asingle cosmological parameter diagonals in thenoise part. It differs fromtraditional tomograph- canbeconstrainedassumingthatallotherparametersarefixed,we icalapproaches,forinstancefrommosttomographicaltechniques have to study the conditional errors. These can be obtained from in weak lensing measurements, in which the noise part is diago- theinversediagonalelementsoftheFishermatrix nalised.Duetocumulativenumericalerrorswithincreasingpoly- σµ,con =(Fµµ)−12 . (49) nomialorder,thismethodislimitedtoorderi≈8. c 2008RAS,MNRAS000,1–9 (cid:13) iSW-effect withorthogonalpolynomials 9 (ii) The improvement of the signal to noise ratios with cu- FrommertM.,EnsslinT.A.,KitauraF.S.,2008, MNRAS,391, mulativepolynomialorderwasinvestigatedforthegalaxysurveys 1315 2MASS (Afshordietal. 2004), SDSS(Bielbyetal. 2010), NVSS FryJ.N.,1996,ApJL,461,L65 (Boughn&Crittenden2005)andEUCLID.Thesignaltonoisera- Giannantonio T., Scranton R., Crittenden R. G., Nichol R. C., tioforthecross-spectraonlyhasbeenimprovedby16%atacumu- Boughn S. P., Myers A. D., Richards G. T., 2008, Phys. Rev. lativepolynomialorderofq=8. D,77,123520 (ii)AFisher-matrixanalysiswasusedtoforcasthowwellcos- Ho S., Hirata C. 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