The kinetic SZ tomography with spectroscopic redshift surveys

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by  Jiawei Shao

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed25January2011 (MNLATEXstylefilev2.2)
The kinetic SZ tomography with spectroscopic redshift surveys
Jiawei Shao1,2⋆, Pengjie Zhang1 , Weipeng Lin1, Yipeng Jing1, Jun Pan3
1 1KeyLaboratoryforResearchinGalaxiesandCosmology†,ShanghaiAstronomicalObservatory,NandanRoad80,Shanghai,200030,China
1 2GraduateSchooloftheChineseAcademyofSciences,19A,YuquanRoad,Beijing,China
0 3ThePurpleMountainObservatory,2WestBeijingRoad,Nanjing210008,China
2

n
a 25January2011
J

4
2 ABSTRACT
The kinetic Sunyaev Zel’dovich effect (kSZ) effect is a potentially powerful probe to the

] missing baryons. However, the kSZ signal is overwhelmed by various contaminations and
O
the cosmologicalapplicationis hamperedby loss of redshiftinformationdue to the projec-
C tion effect. We proposea kSZ tomographymethodto alleviate these problems,with the aid
. ofgalaxyspectroscopicredshiftsurveys.Weproposetoestimatethelargescalepeculiarve-
h
locity throughthe 3D galaxy distribution, weigh it by the 3D galaxy density and adopt the
p
productprojectedalongthelineofsightwith aproperweightingasanestimatorofthetrue
-
o kSZtemperaturefluctuationΘ.Sincetheunderlyingdirectionaldependenceintheestimator
r Θˆ closelyresemblesthatinthetruekSZsignalΘ,Θˆ istightlycorrelatedwithΘ.Itthusavoids
t
s the problem of null correlation between the galaxy density and Θ, which prohibitsthe kSZ
a
extractionthroughtheusualdensity-CMBtwo-pointcrosscorrelationmeasurement.Wethus
[
propose to measure the kSZ signal through the Θˆ-Θ cross correlation. This approach has a

2 numberofadvantages.(1)DuetotheunderlyingdirectionaldependenceofΘˆ,itisuncorre-
v
latedwiththeprimaryCMB,thethermalSZeffectandastrophysicalcontaminationssuchas
1
the dusty star forming galaxies. Thus the Θˆ-Θ cross correlation picks up the kSZ signal in
0
3 theSZsurveywithacleanmanner.(2)Withtheaidofgalaxyredshifts,thecrosscorrelation
1 recoverstheredshiftinformationofthekSZsignalandallowsformoredetailedinvestigation
. onmissingbaryons.(3)SincethegalaxysurveysusuallyhavehighS/N,theS/NofthekSZ
4
measurementthroughtheΘˆ-Θcrosscorrelationcanbesignificantlyimproved.
0
0 WetesttheproposedkSZtomographyagainstnon-adiabaticandadiabatichydrodynam-
1 ical simulations. We confirm that Θˆ is indeed tightly correlated with Θ at k . 1h/Mpc,al-
: though nonlinearities in the density and velocity fields and nonlinear redshift distortion do
v
weakenthetightnessoftheΘˆ-Θcorrelation.We furtherquantifythereconstructionnoisein
i
X Θˆ from galaxy distribution shot noise. Based on these results, we quantifythe applicability
r of the proposed kSZ tomographyfor future surveys. We find that, in combination with the
a
BigBOSS-Nspectroscopicredshiftsurvey,thePLANCKCMBexperimentwillbeabletode-
tectthekSZwithanoverallsignificanceof 50σandfurthermeasureitsredshiftdistribution
∼
atmanyredshiftbinsover0<z<2.
Keywords: cosmology:observations–large-scalestructureofUniverse–cosmicmicrowave
background
1 INTRODUCTION 2004,2006forcensusofthebaryonbudget),while 50%ofthe
∼
baryonsremainselusivetorobustdirectdetection.
RobustevidencesfromCMBandBBNshowthatthebaryonicmat-
teraccountsfor 4%ofthetotalmatterandenergyoftheuniverse. Lookingforthese“missing”baryonsiscrucialforthevalid-
∼
However,onlyafractionofthisbaryonbudgethasbeendetectedin ityofourstandardcosmologymodel.Thestandardtheoryofhier-
thelocaluniverse, eitherintheformofstars,interstellarmedium archical structure formationmodels indicates that themajorityof
(ISM)andintraclustermedium(ICM)(refertoFukugita&Peebles baryonsexistbetweengalaxiesasthediffuseintergalacticmedium
(IGM). Numerical simulations in the standard cosmology further
suggestthatalargeportionofthisIGMisintheformofwarm-hot
intergalactic medium (WHIM) (Cen&Ostriker 1999; Dave´etal.
⋆ Email:[email protected] 2001;Cen&Ostriker2006)withtemperature105K<T< 107 K,
Email:[email protected] whichisbelievedtoresideinmoderatelyoverdensestructuressuch
†
2 Shaoet al.
asfilaments.Ononehand,thiskindintergalacticmediumisionized tivisticcorrectionsbymassiveclusters.Othercontaminationssuch
dominantlybycollisions,andistransparenttoLyαradiation,and asthedustystar-forminggalaxies(Halletal.2010)makethekSZ
thusishardtotracebyLyαforest.Theabsorptionsignaturetoward detectionevenmoredifficult.Forthesereasons,evenforCMBex-
a bright X-ray source is also too weak to be resolved by current periment as advanced as Planck, which has multiple bands over
spectrographs.Ontheotherhand,thoughWHIMemitsradiationin widefrequencyrange, itisdifficulttodetectthekineticSZeffect
theUVandsoftX-raybands,theemissionstrengthistooweakfor directly,givenitslimitedangularresolution.Secondly,likethether-
currentinstrumentstodetect.Therehavebeenobservationstrying malSZeffect,thekineticSZeffectsuffersfromsevereprojection
todetectWHIMviaabsorptionandradiationsignature,butmany effect.Itmeasurestheelectronmomentumprojectedalongtheline
of them are of low significance or misinterpreted. See (Bregman of sight, thus the redshift information of baryons is entangled in
2007)forarecentreviewofthestatusofWHIMdetections. theprojectionwithaprojection lengthof thesizeof thehorizon.
Amongthemethodsofprobingmissingbaryons,theSunyaev Thisnotonlydegradesitspowertoinfertheevolutionofmissing
Zel’dovich(SZ)effectisoneofthemostpromising.Ouruniverse baryons over thecosmic epoch, but could alsolead tolargebias.
isalmostcompletelyionizedafterz = 6,wherefreeelectronsare Forexample,thepatchyreionizationcouldcontributeasignificant
prevailing in galaxy clusters as high energy ICM and in the less fractiontothekineticSZeffect(Zahnetal.2005;McQuinnetal.
overdensefilamentarystructuresasIGM.Freeelectronswillscatter 2005; Ilievetal. 2007). Without redshift information, unless the
offCMBphotonsthroughinverseComptonscattering,andgenerate reionization process is well understood and interpreted appropri-
secondary CMB temperature anisotropies, which isknown asthe ately,theextracontributionfrompatchyreionizationcouldbemis-
SunyaevZel’dovicheffect.Therefore,theSZeffectiscontributed interpretedasthesignofmissingbaryons.It’sinterestingtonotice
by virtually all electrons. In principle, from the SZ observations, theworkofHerna´ndez-Monteagudo&Ho(2009),whichproposed
weareabletofindallelectronsandhenceallbaryons, duetothe to recover the signature of the bulk flow of the missing baryons
electricneutralityoftheuniverse. by cross correlating future CMB data sets with kSZ estimates in
ThetwomajorcategoriesoftheSZeffectarethethermalSZ galaxyclusters.
effect (tSZ),arisingthrough the thermal motion of electrons, and In this paper, we propose a kinetic SZ tomography method
thekineticSZeffect(kSZ),arisingthroughthebulkmotionofelec- toovercometheaboveobstacles.Thebasicideaistocrosscorre-
trons.TheefficiencyforgivenelectronstogeneratetheSZeffectis late the CMB observation with a galaxy redshift survey or other
proportional tothe thermal temperature for the thermal SZeffect surveysof thelargescalestructurewithsufficiently accuratered-
andisproportionaltothebulkpeculiarvelocityforthekineticSZ shiftinformationsuchasthe21cmintensitymapping(Changetal.
effect.Sincethethermaltemperatureofelectronsstronglycouples 2008).Duetothecancellationmechanismarisingfromtheunder-
to the electron density, the dominant contribution to the thermal lyingdirectionaldependenceofthekineticSZsignalasaresultof
SZeffectcomesfromthehigh-densityandhigh-temperatureICM. its vector nature, direct cross correlation between the kinetic SZ
Forexample,Whiteetal.(2002)foundthat75%ofthetotalther- effectandthegalaxynumber densityvanishes (refertoFig.1for
malSZeffectatmultipoleℓ<2000comesfromvirializedregions moredetailedexplanation).Thisdifferssignificantlyfromthetight
withgasoverdensityδ >100andHerna´ndez-Monteagudoetal. correlation between the thermal SZ effect and the galaxy density
gas
(2006)foundsimilarresults:80%ofthetSZsignalcomesfromcol- (Zhang&Pen2001;Shaoetal.2009).Onewaytocircumventthis
lapsedstructures.Sincethemissingbaryonsarelikelyinlessdense cancellationistosquarethekineticSZeffectandmeasureitscor-
regionswithlowertemperature,theabilitytofindmissingbaryons relationwithgalaxies(Dore´etal.2004;DeDeoetal.2005).
throughthethermalSZeffectislimited. Weproposeanalternativeapproachtoavoidthecancellation.
On the other hand, the kinetic SZ effect has a better poten- One can reconstruct the peculiar velocity field from the galaxy
tial to probe the missing baryons. The peculiar velocity is deter- spectroscopic redshift survey, weigh it with the observed galaxy
mined by the large scale gravitational potential and is thus only numberdensityandotherredshift-dependentfactorstoreconstruct
weaklycoupledtolocalmassconcentration. Hence,thecontribu- a weighted momentum field. This weighted momentum field has
tiontothekSZeffect isroughly proportional tothetotalmassof roughlythesamedirectionaldependenceasthetruekSZsignaland
eachbaryoncomponent,e.g.,ICMandIGM.Namely,itisanap- thusavoidsthecancellation.Wethusexpectameasurablecrosscor-
proximatelyunbiasedprobeofbaryons,regardlessoftheirthermal relationbetweenthereconstructedfieldandtheCMBmap,which
state(aslongas theyareionized). Sincethe massfractionof the extractsthekSZcomponentintheCMBtemperaturefluctuation.
missingbaryonsis 50%,wethusexpectacomparablecontribu- Thecrosscorrelationsignalshouldarisesolelyfromthecor-
∼
tion to the kSZ effect from the missing baryons. This makes the respondingredshiftrangewheregalaxiesreside,toanexcellentap-
kinetic SZ effect a promising probe of the missing baryons. For proximationforareasonablythickgalaxyredshiftbin(∆z & 0.2)
example,therehavebeenworksstudyingthecontributionofkSZ andsufficientlysmallangularscales(ℓ&10).Thusitdisentangles
effect by WHIM to the CMB anisotropies (Atrio-Barandelaetal. the kinetic SZ contribution in this redshift range from contribu-
2008;Ge´nova-Santosetal.2009). tions of any other redshifts and recovers the redshift information
The remaining question is to measure the kinetic SZ effect, of the kinetic SZ effect. It is for this reason that we dubbed this
forwhichthereareseveralobstacles.Firstofall,thesignalofthe methodasthekineticSZtomography,analogoustothethermalSZ
kinetic SZ effect is not only weak, but also lack of spectral fea- tomography and also to the well known lensing tomography. An
turestoextractfromtheoverwhelmingprimaryCMB.Thus,from immediateapplicationofthekineticSZtomographyistoseparate
the CMB measurement alone, we only expect to detect it in the thepatchreionizationfromthelatetimekineticSZeffect.Itisalso
auto power spectrum measurement at ℓ & 3000, where the pri- effectivetoeliminatecontaminationsfromtheprimaryCMB,ther-
maryCMBdampssignificantlyandthekineticSZeffectbeginsto mal SZ and other foreground contaminations, which do not have
dominate(e.g.Ma&Fry2002;Zhangetal.2004),andatν=217 thecharacteristicdirectionaldependenceandthusshouldbeuncor-
GHzfrequencyband,wherethe(non-relativistic)thermalSZeffect related with the reconstructed momentum field. Since the galaxy
vanishes. However, refer toan interesting paper by Nozawaetal. surveysusuallyhaveexcellentsignaltonoise(S/N),thecrosscor-
2006concerningtheresidualtSZeffectat217GHzduetotherela- relationmeasurement canachievemuchhigher S/Nthantheauto
ThekineticSZtomographywithspectroscopicredshiftsurveys 3
is negligible. Realistic surveys have much lower galaxy number
densityandthusshotnoiseinthegalaxydistributioninducesnon-
negligible reconstruction noise. We directly quantify it from our
simulationswithaproperscaling( 4.1).Wearethenabletofore-
§
castitsperformanceforsurveycombinationslikethePlanckCMB
experiment plustheBigBOSSspectroscopic redshift survey( 4).
§
Wediscussandconcludein 5.Wepresentmoretechnicaldetails
§
andfurtherdiscussionsinthetwoappendices.
2 THEKINETICSZTOMOGRAPHY
Bulk motions of free electrons induce secondary CMB
anisotropies, namely the kinetic Sunyaev Zel’dovich (kSZ)
effect (Sunyaev&Zeldovich 1972, 1980), with temperature
fluctuations
∆T (1+δ )v nˆ
Θ(nˆ) |kSZ = χ n¯ σ e · exp[ τ(z)]adχ
≡ T e e T c −
CMB Z
pW (z)dχ
≡ k kSZ
Z
χi+∆χi/2
Θ ; Θ = pW (z)dχ, (1)
≡ Xi i i Zχi−∆χi/2 k kSZ
where Θ is the contribution from electrons in the i-th redshift
Figure1.ThevanishingcorrelationbetweenthekineticSZeffectandthe i
bin, spanning the comoving coordinate range χ ∆χ/2 < χ <
galaxynumberdensity.Thelineofsightisfromtheobserver(O),through i− i
χ +∆χ/2.χisthecomoving radialcoordinate, cisthespeedof
thepointsB,EandA,tothelastscatteringsurface.Onegalaxyresidesat i i
pointGandweshalldiscussthevelocitydistributiongiventheexistenceof light and τ is the Thompson optical depth. p ≡ (1+δe)v is the
thisgalaxy.ThelineEGisperpendiculartothelineofsightandAE=EB. (normalized)electronmomentumandthesubscript“ ”denotesthe
k
Thevelocityatanypointalongthelineofsight,e.g.,thepointA,canbe projectionalongthelineofsight.δeistheelectronnumberoverden-
decomposed,withrespecttothepositionofthegivengalaxy,intoonecom- sity,andvistheelectronpeculiarvelocity.Theweightingfunction
ponentalongthedirectionAGandoneperpendiculartothedirectionAG. W = χ n¯ σ exp( τ)a/cmodulatesthecontributionfromeach
Theperpendicularcomponenthasequalprobabilitytobealongthedirec- redkSsZhift. TehreouTghout−the paper, we focus on the kSZ effect after
tion AC orthe direction ADand thus the net contribution to the kinetic reionizationandthussetχ = 1,anexcellentapproximation.Due
e
SZ-galaxycrosscorrelationiszero.Sotheonlyvelocitycomponentwhich
totheneutrality,δ =δ ,whereδ istheoverdensityof(ionized)
maycontributeistheonealongthedirectionAG.However,fromthesym- e gas gas
gas.
metryargument,iftheevolutioneffect(lightconeeffect)canbeneglected,
Asavector,pcanbealwaysdecomposedintoagradient(ir-
velocityatpointBhasanequalprobabilitytohaveacomponentalongthe
directionBG,withthesameamplitude.Theprojectionofthetwoontothe rotational)partpEandcurl(rotational)partpB,p≡pE+pB,where
lineofsightcancelsexactly.SothecrosscorrelationbetweenthekineticSZ ∇×pE = 0 and ∇·pB = 0 respectively. Here, the “E”and “B”
effectandthegalaxydensityvanishes. notationsareanalogoustotheelectromagneticfields.Ascanbein-
ferredfromEq.1,thecontributionfromthegradientpartislargely
canceledoutwhenintegratingalongthelineofsight,aslongasthe
correlationmeasurementofthekineticSZeffectfromtheSZsur- weightingfunctionW variesslowlyacrossacorrelationlength
kSZ
veysalone.Laterwewillshowthat,PLANCKplusBigBOSScan ofp ,whichisoftheorder100Mpc/htoday.1.ForthekineticSZ
E
detectthekineticSZeffectat 50σlevel. effect after reionization, W only changes significantlyover the
∼ kSZ
Atthebeginningstageofthiswork,Hoetal.(2009)published Hubblescale,thusthecontributionfromthegradientpartisnegli-
a work based on similar idea, whose applicability is further con- gibleandtheonlysignificantcontributioncomesfromp (Vishniac
B
firmedinthispaper.Thetwoworksarecarriedoutindependently 1987).
andthusdifferinmanydetails.WetesttheproposedkSZtomog- p ingeneralhastwosourcesofcontribution.Since p =
raphyagainstacontrolledsetof hydrodynamical simulationsand pB=(1+δ ) v+ δ v,theB-modeofpcanco∇m×efrBom
quantify the tightness of correlations between the reconstructed t∇he×B-mode ofev∇o×r from∇thee×cross talk between the density and
mapandthetruekSZsignalatvariousredshiftsover0<z<2.We
velocity.Followingthesamenotation, wecandecompose theve-
furtherinvestigateseveral complexitiessuchasredshiftdistortion
locityintoa“E”mode(gradientpart)v anda“B”mode(rotational
E
andfeedback,anddemonstratetherobustnessofthekSZtomogra- part)v .Forpurelygravitationalinteraction,thevelocity“B”mode
B
phyagainstvariouscomplexities.Bothworksconfirmthepowerof
decays, until multi-streaming and shell crossing arise due to the
thekSZtomography.
nonlinearevolution(seeforexamplechapter2ofBernardeauetal.
The paper is organized as follows. In 2, we introduce the 2002 for a discussion). Thus in the linear and weakly nonlinear
§
kSZtomographymethodanddiscussitslimitationsingeneral.We regimesagoodapproximationisv=v andtheonlycontribution
E
thentestitagainstourhydrodynamicsimulationsin 3.Therecon- to the kSZ effect comes from the cross talk between the density
§
structionandthetestaredoneinboththerealspace( 3.1)andthe
§
redshift space ( 3.2). In this section, we approximate galaxies as
§
darkmatterparticles.Soitcorrespondstoidealizedsurveysofvir- 1 This condition can be violated atthe epoch ofreionization, where the
tually infinite galaxies such that shot noise in galaxy distribution patchyreionizationcausesWtovarysignificantlyover 10h/Mpcscales.
∼
4 Shaoet al.
gradientandthevelocity.ThisisthewellknownOstriker-Vishniac ofthematterfield.Inthesameregime,namelyatlargescale,the
(OV)effect(Ostriker&Vishniac1986;Vishniac1987).Inthenon- 3Dgalaxydistributionisagoodproxyoftheunderlying3Dmat-
linear regime,v grows andcanalsocontributetothekSZeffect terdistribution.Thiscanbedescribedbyagalaxybiasδ =b δ .
B g g m
(Zhangetal.2004). Undertheabovecondition,giventheobserved3Dgalaxydistribu-
ThekineticSZtomography thatweproposerequirescombi- tionδobs(x),weareabletoobtainanestimatorofthevelocityfield,
g
nation of aSZ survey and agalaxy spectroscopic redshift survey whichinFourierspacereads
withoverlappingskycoverage.Itcontainsthreemajorsteps:
k
in•theCio-tnhstrreudcsthaift2bDinm.IadpeaΘˆlliyf,rΘoˆmshthoeul3dDbedtiisgtrhitbluyticoonrroelfagteadlawxiieths vˆ(k)=−ifHδogbs(k)k2 , (3)
i
Θ,thetruekineticSZsignal fromthisredshift bin.Acrucialin- wherekisthe3Dwavevector, f dlnD/dlnaandDisthelin-
i ≡
gredienttoguaranteeatightcorrelationistoestimatethepeculiar eardensitygrowthrate.Thesuperscript“hat”inv(vˆ)andinother
velocitythroughthe3Dgalaxydistributionanduseittoconstruct symbols (e.g. Θˆ) denotes thereconstructed quantity. Inreality, to
Θˆ .Hereafter,weoftenneglectthesubscript“i”whereitdoesnot suppress the noise and stabilize the reconstruction, we often ap-
i
causeconfusion. plysomefilterstothedensityfield,beforeapplyingEq.3.Soδogbs
CrosscorrelatethereconstructedΘˆ withaoverlappingSZsur- shouldbetreatedasthesmootheddensityfield.
vey•. The cross correlation signal, to an excellent approximation, Eq.3isabiasedestimatorofthetruepeculiarvelocityv.(1)
solely comes fromthekinetic SZinthe chosen redshift bin. It is The galaxy bias causes vˆ to be overestimated by a factor bg. (2)
this step that recovers the redshift information of the kinetic SZ Redshiftdistortioncausestheobserveddensitytodeviatefromthe
effect,eliminatesvariouscontaminationsandreducesstatisticaler- underlyingmatterdensityandthusbiasesthevelocityreconstruc-
rors. tion. (3) Nonlinearities in the density evolution causes deviation
Interpretthemeasuredcrosscorrelationsignalandreconstruct fromEq. 2. (4) Inthe nonlinear regimewhere shell crossing and
the•truekineticSZsignal.Tricksimilartotheoneadoptedinthe multi-streaminghappen, velocityvorticity(rotationalpart,orcurl
thermalSZtomography(Shaoetal.2009)canalsobeappliedhere. part)develops,whichiscompletelymissedbytheestimatorEq.3.
Theimperfectnessofthevelocityreconstructionisnotasse-
Thecurrentpaperwillfocusonthefirsttwostepsandonlybriefly
vereasitlooks,forthekineticSZtomography.Laterwewillshow
discussthethirdstep.
thattheperformanceofthekSZtomographyisinsensitivetodeter-
ministicerrorsinthevelocityreconstruction.Thusadeterministic
galaxy density and velocity bias, uncertainties in f and H, linear
2.1 ThekineticSZreconstruction
redshift distortion (the Kaiser effect) do not degrade the kSZ to-
TheprimarygoalofthispaperistoextractthekSZsignalaswellas mography.However,stochasticerrorsfromstochasticgalaxybias,
itsredshiftinformationthroughcrosscorrelatingSZsurveyswith nonlinearitiesintheevolutionofdensityandvelocitydo.Laterwe
galaxyspectroscopicredshiftsurveys.Asexplainedearly,theusual willquantifytheirimpactsandshowthemtobemoderateatrele-
two-point cross correlation between the kinetic SZ effect Θ and vantscales.
galaxyoverdensityδgvanishes(hΘδgi=0),duetothecancellation With the 3D density field δg and thereby the recovered 3D
ofpositiveandnegativevelocitiesalongthelineofsight(Fig.1). velocity field vˆ in hand, we can then reconstruct a weighted 2D
Anaturalsteptoavoidsuchcancellationistorecoverthevelocity momentummap
informationandweighthegalaxiesaccordingly.Thiscanbedone
inspectroscopicgalaxyredshiftsurveys. Θˆ dχpˆ Wˆ(z), (4)
Spectroscopic galaxy redshift surveys, such as LAMOST2, ≡ k
Z
BOSS3,BigBOSS4,SKA5,Euclid6andJDEM/ADEPT7,willmea-
sure the 3D galaxy distribution δg(x). At least part of them will wherepˆ ≡(1+δogbs)vˆ(x),andvˆ(x)istheinverseFouriertransform
overlapwithSZsurveyssuchasACT8,SPT9 andPLANCK10 on ofvˆ(k).WechoosetheweightingfunctionWˆ(z)=WkSZ(z).Thein-
the sky coverage. We are able to recover the velocity field from tegralinEq.4isoverthecorrespondingredshiftbin.Again,wecan
δ (x)andthenΘˆ,thegalaxymomentumproperlyweighted.Since alsoapplysomefilterstothedensityfield,beforetakingtheprod-
g
thedirectionofthepeculiarvelocityistakenintoaccountinΘˆ,the uctinpˆ.Thesefiltersarenotnecessarytobethesameastheones
crosscorrelation ΘˆΘ nolongersuffersfromtheusualcancellation forthevelocityreconstruction.Furthermore,thedensitymeasure-
andthus ΘˆΘ ,h0. i mentintheproductmaynotevenbethesameastheoneusedin
h i
Inthelinearregime,themassconservation(i.e.thecontinuity thevelocitymeasurement,ascorrectlypointedbyHoetal.(2009).
equation)reducesto Herewewanttoclarifyalikelyconfusingpoint.Asdiscussed
before, the kinetic SZ effect is mainly contributed by p instead
δ˙ + v=0, (2) B
m ∇· ofpE.Ontheotherhand, thereconstructed vˆ inEq.3isactually
where δm is the matter overdensity and v is the peculiar velocity pE, whichbecomesclear laterinEq.8.Itthusseemsthat there-
constructionmissesthedominantcontributiontothekineticSZef-
fectandthusshouldfailtowork.However,thereisanextrafactor
2 http://www.lamost.org/website/en (1+δ ) in the estimator Θˆ (Eq. 4). Recall that, in the OV effect
g
3 http://cosmology.lbl.gov/BOSS/ (Ostriker&Vishniac1986;Vishniac1987),itisthecross-talkbe-
4 http://bigboss.lbl.gov/index.html
tweenthedensitygradientandthecurl-freevelocitythatgenerates
5 http://www.skatelescope.org/
a curl component in p. Here, the cross-talk between the density
6 http://sci.esa.int/euclid
7 http://jdem.gsfc.nasa.gov/ andpEgeneratesaB-modeinthereconstructedpˆ,whichistightly
8 http://www.physics.princeton.edu/act/index.html correlated with the true pB on relevant scales. This explains the
9 http://pole.uchicago.edu/ reasonableperformanceofthereconstructiontechniqueandtheki-
10 http://www.rssd.esa.int/index.php?project=planck neticSZtomography.
ThekineticSZtomographywithspectroscopicredshiftsurveys 5
2.2 HowtoquantifythekSZtomographyperformance? Forthesereasons,inthemaintextwewillfocusonrtoquan-
tifytheperformanceof thekineticSZtomography andleavedis-
TheestimatorΘˆ iscertainlyimperfect.Itcanhavebothsystemati-
cussiononb intheAppendixA.
calandrandomoffsetswithrespecttoΘ.Thesedeviationsshould Θˆ
bebothscaleandredshiftdependent. Thesedeviationscanbevi-
sualizedbyaΘ-Θˆ plot.Alternatively, itcanbequantifiedbytwo 2.3 Originsofthestochasticityr,1
parameters, r and b . r describes the tightness of the Θˆ-Θ corre-
Θˆ
lation11 andb isthebiasinΘˆ withrespect toΘ.Clearly,bothr A number of approximations made in the reconstruction pipeline
Θˆ causethestochasticityintheΘ-Θˆ relation(r , 1).(1)Firstofall,
and b depend on the redshift range of galaxies used for recon-
Θˆ
asanapproximationtotheexactmassconservationequation
struction.Wewanttoquantifytheredshiftdependenceofrandb.
Thuslaterintheanalysiswewillchooseaprojectionlengthacross δ˙ + (1+δ )v=0, (8)
whichevolutionsinrandb arenegligible.Forsuchprojection,r m ∇· m
Θ
andbΘˆ arefunctionsofzandthe2Dwavevectork ,theinverseof the starting point Eq. 2 only holds where δm ≪ 1. (2) From
theperpendicularspatialseparation, ⊥ Eq. 2 to Eq. 3, we have made assumptions of linear evolution
(δ (k,z) D(z)δ (k,z)), deterministic bias between δ and δ ,
r(k ,z) PΘˆΘ(k⊥,z) , (5) anmd curl-f∝reevelocmity. Ni one of theseapproximations areg exact min
⊥ ≡ PΘˆ(k ,z)PΘ(k ,z) thenonlinearregime.(3)Evenundertheseassumptions,Eq.2and
⊥ ⊥ 3 only hold in real space. Namely, we have implicitly assumed
and p
that the observed δobs is the galaxy overdensity δ in real space
g g
(δobs = δ ). However, in reality, what we directly measure is the
P (k ,z) g g
bΘˆ(k⊥,z)≡ sPΘΘˆ(k⊥,z) , (6) tghaelarxeydsnhuifmtbdeisrtoorvteiordne,nδssit,yδδgs.iTnhreedδssh-δiftrseplaactieon(δiogsbsst=ocδhgsa)s.tDicu,edutoe
⊥ g g g g
tononlinearmappingbetweenredshiftandrealspace,nonlineari-
Laterwewillrecognizek astheperpendicularcomponentofthe
usual3Dwavevectork.T⊥hePsarethecorrespondingpowerspec- tiesinboththedensityandvelocityfields(e.g.Whiteetal.2009)
andvelocityvorticity(Carlsonetal.2009).(4)Evenifwehaveper-
tra.
fectE-modevelocityreconstruction,westillhavenohandleonthe
Thereisnoguaranteethatb isclosetounity,eveninthelin-
Θˆ B-modevelocity,whichcontributesthekineticSZeffect.(5)InEq.
earregime.Asexplainedearlier,thevelocityestimationisbiased.
Further,intheproductΘˆ (1+δobs)vˆ,δobsisalsobiasedwithre- 4,wemultiplythereconstructed velocitywith1+δogbs insteadof
∝ g g 1+δ .Thepossiblestochasticitybetweenδ andδ alsoincreases
specttoδ .LaterintheAppendixAwewillshowthecomplicated e e g
e thestochasticitybetweenΘandΘˆ.
behaviorofb .
Θˆ Withtheaidofourhydrodynamicsimulation,weareableto
However,largedeviationofb fromunitydoesnotnecessar-
Θˆ quantify the combined influence of all these factors on r, except
ilymeanpoorperformanceofthekineticSZtomography,forthree
reasons.(1)First,thesignal-to-noiseratio(S/N)ofthe ΘΘˆ mea- for the stochastic galaxy bias. Hoetal. (2009) used the halo oc-
h i cupation model and N-body simulations to produce galaxy mock
surement,issolelydeterminedbyr.TheS/Nofeachmode(Fourier
catalog.Thisapproachcapturesthegalaxystochasticityandshows
ormultipolemode)is
thatthekSZtomographyisrobustagainstit.Oursimulationshave
S 2 = P2ΘΘˆ relativelysmallboxsize(100h−1Mpc)andhencedonotallowusto
N P2 +(P +PN)(P +PN) followthesameapproach.Soweleavethisissueforfurtherinves-
(cid:18) (cid:19) ΘΘˆ Θ Θ Θˆ Θˆ tigation.
1
= , (7)
1+r 2 1+ PΘN 1+ PΘNˆ
− (cid:18) PΘ(cid:19) PΘˆ! 3 TESTINGAGAINSTHYDRODYNAMICAL
where PN and PN arethecorresponding noisepower spectra. We SIMULATIONS
Θ Θˆ
find that the b dependence drops out in the error estimation. A
Θˆ We test the kinetic SZ tomography against our hydrodynamical
rescalingΘˆ b Θˆ leavesnoeffectontheS/Nofthecrosscorre-
→ Θˆ simulations. The simulations are run with the GADGET2 code
lation measurement, since both PN,P b2 and r is unchanged
under this scaling. (2) b , 1 dΘeˆfiniΘˆtel∝y aΘffˆects the expectation (Springel 2005) in a ΛCDM cosmology with parameters: Λ =
Θ 0.732, Ω = 0.268, Ω = 0.044 h = 0.71, σ = 0.85. The box
0 b 8
value of the cross correlation. However, since in the theoretical
sizeof the simulationis L = 100h 1Mpc oneach side, in which
−
interpretation of the measured crosscorrelation, one can take the
5123darkmatterparticlesand5123gasparticlesareinitiallyseeded
galaxybias,redshiftdistortionandpossiblyothercomplexitiesinto
(SeemoredetailsaboutthesimulationinJingetal.2006;Linetal.
accountandthusavoidsystematicalerrorsinducedbyb ,1rea-
Θ 2006).Wehaveannon-adiabaticrun,inwhichgasparticlesareal-
sonably well.(3) BasedonthesametechniqueinthethermalSZ
lowedtocoolandcondenseintocollisionlessstarparticles,along
tomography(Shaoetal.2009),wecancombinethecrosscorrela-
with which SN feedback is taken into account. We also have an
tionmeasurementP andtheautocorrelationmeasurementP to
ΘΘˆ Θˆ adiabaticrunwiththesamecosmologicalparametersandthesame
obtainP = P2 /(r2P ).Thisestimationreliesonnoinformation
Θ ΘΘˆ Θ initialconditions.Wewillfocusonthenon-adiabaticrun,sinceit
ofb andthusavoidsthebiasproblem.
Θˆ hasbettercaptureonthegastrophysicsandhencebettermodeling
ofthekineticSZeffect.Unlessotherwisespecified,allsimulations
arebasedonthissimulation.Wealsoanalyzetheadiabaticrunto
11 rthatwedefinediffersfromtheonedefinedinHoetal.(2009).First, betterunderstandingthegeneralityofthekSZtomography(§3.3.2).
theirrisforthevelocityfieldinsteadofthemomentumfield,3Dinstead Wechoose inthisworkafewrepresentative redshiftstoquantify
of2D.Second,theirrisnotthecrosscorrelationcoefficient,butisactually thefeasibilityofthetomographytechnique,whichismainlychar-
analogoustor/bofthe3Dvelocityfield,inournotation. acterizedbythequantityr(k ,z).
⊥
6 Shaoet al.
Ourhydrodynamicsimulationshavedirectinformationofthe
gas momentum distribution and thus the kinetic SZ effect. How-
ever, thesimulations don’t simulategalaxies. Toproceed, weap-
proximate galaxies as simulation dark matter particles. Since the
galaxy stochasticity is likely sub-dominant (Bonoli&Pen 2009;
Baldaufetal.2010),thisapproximationisreasonabletoestimater
betweenΘˆ andΘ,althoughitindeedover-estimatesit.Asdemon-
stratedbyHoetal.(2009),thekSZtomography isrobust against
the galaxy stochasticity. So we will leave this issue elsewhere.
Since the number density of simulation particles is much higher
than that of galaxies in any realistic surveys, shot noise in the Θˆ
reconstructionisnegligible.Tousethemeasuredrforforecasting,
we have totake the shot noise inevitable inrealistic surveys into
account.Weleavethisissueuntilnextsection.
WetestthekineticSZtomographyatseveraltypicalredshifts
z = 0,0.53,1.02,2.08. We reconstruct the velocity and conse-
quentlythemomentumfieldfromthedarkmatterdistributioninthe
correspondingsimulationoutputs.Attheseredshifts,weprojectthe
3Dmomentumfieldoverasingleboxsize(100h 1Mpc)togetthe
−
2Dmomentummaps.Wethencomparethesemapswiththemaps
ofunderlyingkineticSZsignal,whichisdirectlymeasuredbymul-
tiplyingthegasdensityandvelocitydistributioninthecorrespond-
ingsimulationoutputs.Sincethesimulationshaveinformationof
peculiarvelocity,wecarryoutthereconstructionandcomparison
inbothrealspaceandredshiftspace. Figure2.TheΘˆ-Θmap(inarbitraryunit)intherealspaceatz=0,0.53,
1.02,2.08.InordertofurtherreducethescatteringofΘˆ-Θrelation,weav-
eragetherecoveredvelocityfieldineveryneighboring43=64cellstoget
3.1 Thereconstructioninrealspace theaveragedvelocityfieldvˆaswellasthedensityfieldδonamuchcoarser
partitioning,say,643cells.Weshowherethecell-to-cellcorrespondencein
3.1.1 Thereconstructionprocedure allthreeCartesiandirectionsbyrandomlyselecting512outof643couples.
Therecovered Θˆ fieldintimately followsthekSZsignalalongthediago-
Wefirstcarryoutthereconstructioninrealspace.Thereconstruc-
nalathighredshifts,whileatlowredshift,therearegreatfluctuationsdue
tion is a two-step process. The first step is to reconstruct the pe-
tonon-linear evolutionofthedensityfieldandtherebythevelocity field.
culiarvelocity.Ateachredshiftz,weconstructthematterdensity
i ThethreecomponentsinCartesiandirectionsaredenotedbydifferentpoint
field δ(x) using clouds-in-cells (CIC) scheme in a partitioning of
types,trianglesforx,squaresforyandcirclesforz.
n3 = 2563 cells,followedbyaFouriertransformtogetthematter
g
distribution in Fourier space δ(k). We then apply a Gaussian fil-
ter12 W (k) = exp( k2R2/2) to the original fieldδ(k) in order to equation (Eq. 9) is the unsmoothed density, which preserves the
G − s
wipeoffnonlinearfluctuationsonsmallscales,whichareuncorre- necessary information of the true density field in the deriving of
latedwiththelargescalevelocity. Theadopted smoothing length themomentumfield.Inpractice,weconstructateachredshiftthree
R =1.56h 1Mpcappliedinthewholecontextisaroundtheradius mapsalongthreeCartesiancoordinatedirectionsofthesimulated
s −
of atypical cluster,and wewilldiscuss theinfluenceof different box respectively. Astheycan beconsidered asthreeindependent
smoothing lengthin 3.3.1. Wethenobtain the reconstructed ve- measurements,inthefollowingfigureswhichconcernstatistics,we
§
locityvˆ(k)fromEq.3throughthissmootheddensityfield.Bythe showtheaverageresultsunlessspecified.
inverse Fourier transform, we obtain the real space velocity field Atthesametime,weobtainthetruekSZsignalinasimilar
vˆ(x). way
The second step is to project the momentum field along the v
lineofsightnˆ togetthereconstructed2Dmapatredshiftzi Θj1,j2(nˆ)= nˆ· b,j1c,j2,j3(1+δb,j1,j2,j3), (10)
j3
vˆ X
Θˆj1,j2(nˆ)= nˆ · j1,cj2,j3(1+δj1,j2,j3). (9) where δb is the nonlinear overdensity of baryons. vb is the bulk
Xj3 velocityderiveddirectlyfromthesimulationbyaveragingwithin
Sincetheweighting functionW changes littleover thatscales, thehostcell(j1,j2,j3)
kSZ
so wetreat it asa constant and omit it.The sum isover a single vmw
simulated box at the investigatedredshift, insteadof stacking the vb = i mi wi i , (11)
box in the light cone. Notice that the density term in the above P i i i
wherev andm arethecomovingpeculiarvelocityandthemassof
i i P
i-thgasparticleinthehostcell.w isasplinekernelusedtosmooth
i
12 Hoetal. (2009) populated halos with galaxies. Since in that case the thegasparticles.
galaxy density is low, and shot noise is non-negligible, they applied the
WienerfiltertoreducethePoissonnoise.Thisisalsoanecessaryprocedure
whendealing withrealdata. Inourwork, weusesimulation particles as 3.1.2 TheΘˆ-Θrelation
galaxies,whosenumberdensityismuchhigherandthustheshotnoiseis
negligible.Forthisreason,weinsteadapplygenerallyaGaussianwindow InFigure2weshowthecell-to-cellΘˆ-Θcorrespondence.TheΘˆ-Θ
functiontofilteroutsmallscalenonlinearfluctuations. datapointsscatteraroundΘˆ =Θ,meaningabiasbΘ 1.However,
∼
ThekineticSZtomographywithspectroscopicredshiftsurveys 7
Figure3.Thecrosscorrelation coefficientsrbetweenΘˆ andΘ.Thecor- Figure4.TheΘ-Θˆsrelationintheredshiftspace.ThesameroutineasFig.
relation coefficient is averaged over the three components. The two are 2isusedinderivingthisrelation.Athighredshifts,therecoveredΘˆsover-
hshigifhtlsy,wcohrilreellaetsesd,corr≃r0el.a9t,edontolasrmgeallsecralsecakl⊥es6end1shdMuepcto−1thfoernaolnl-tlhineea4rirteieds- twheeigdhesnstihtyeubnydaerflaycintogrkSZ1s+igfn/a3l,Θa,ndbetchaeusheigthheerKreadissheriftefftheectheignhhearncthees
coming into play. Nevertheless, the correlation coefficient is still tight at slopeis,since f Ω0.6(∼z)islargeratearlyepochs.Ontheotherhand,the
z=0,withr>0.6uptok 8hMpc 1. scattersinΘ-Θˆsi≃smumchlargerthanthatintherealspaceduetothelarger
⊥∼ − stochasticitiesofdensity-velocityrelationandtherealspace-redshiftspace
mapping.
this should not be over-emphasized since this only represents an
unrealisticcaseofgalaxy biasb = 1without redshiftdistortion.
g
Reconstructionsbasedongalaxysamplewithb , 1wouldresult
g
inbΘ ,1andhenceadifferentslopeoftheΘˆ-Θrelation.Redshift at z = 0, theΘˆ-Θ correlationisstillprettytight, withr > 0.6to
distortion also changes the slope of the Θˆs-Θ relation, as can be k 8hMpc 1.Wedon’tcorrectforthealiasingeffect,suchthat
−
seenfromFig.4. sm⊥a∼llscaleresultsmaybemisleading(Jing2005).Nevertheless,we
TheΘˆ-Θrelationshowsnon-negligibledispersionaroundthe doexpectgoodresultsuptoaquarteroftheNyquistwavenumber,
mean,butstillreasonablytight.Thismeansthatthestochasticityis i.e.around2hMpc 1,andit’ssafeforustoestimatethecrosspower
−
noticeable,butnotyetoverwhelming.Wealsonoticethat,thedis- spectrumuptoℓ 2000 3000.
∼ −
persiongetsstrongeratlowerredshift.Thisisnotsurprising,since
thereconstructionisbasedonlineartheoryandthusworksbetterat
higherredshift.Nonlinearitiesinthedensityfieldandvelocityfield
degradesthereconstructionaccuracy,asdiscussedin 2.3.
§ 3.2 Thereconstructioninredshiftspace
In reality, what we get from a galaxy survey is the galaxy num-
3.1.3 Thecrosscorrelationcoefficientr
ber density in the redshift space. As expected in 2, the redshift
§
Outofthetwoquantitiesconcerning tothereconstructionperfor- space distortion will induce new sources of uncertainty and fur-
mance,thecrosscorrelationcoefficientrdescribesthetightnessof therdegradesthereconstruction.Weareabletoquantifyitsimpact
Θˆ-ΘrelationandisamajormeasureofthekineticSZtomography. throughoursimulations.Duetoitsownpeculiarmotion,theappar-
b is of less importance in quantifying the kSZ tomography per- entpositionofeachdarkmatterparticlealongthelineofsightcan
Θˆ
formance,soweleaverelatedresultstotheAppendixA.Tomea- bewrittenas
surethem, weperform2DFouriertransformsofΘ(Θˆ).Θ(k ) =
v nˆ
d2x Θ(x )exp(ik x )/A,whereAistheareaofthemap.N⊥ote xs =x+ · , (12)
⊥ ⊥ ⊥· ⊥ H
here k and x are both 2D variables. Wethen obtain the power
RspectFrui⊥gm. P3Θshwo⊥iwths(t2hπa)t2δthDe(ks⊥to−chka⊥s′t)icPiΘty(ki⊥n)t=hehΘΘˆ(-kΘ⊥)rΘel(akt⊥io′)ni.is in wthheecroemnˆovdienngotpeescuthliearuvneitlovceictyto.rxsailsonpgostihtieonlininetohfesriegdhsth,ifatnsdpavceis,
general not asevere issue, even inthe stronglynonlinear regime, and from here on the superscript “s” indicates the corresponding
consistentwithFig.2.Asshownwehavestrongcorrelations,r & quantityintheredshiftspace.Asaresultofthisdisplacement,what
0.9 to k = 1hMpc 1at all redshifts, and even r & 0.8 to k weobservedisadistorteddensityfieldδs.Withδs asthestarting
−
3hMpc⊥1exceptz=0.Nonlinearitiesdodegradethereconstruct⊥ion∼, point,wecanreconstructthemomentumfieldsΘˆs throughEq.3,
−
as we see that r decreases towards low redshifts. However, even followingthesameprocedureasintherealspace.
8 Shaoet al.
Figure5.TheΘ-Θˆs relation intheredshiftspaceaftercorrecting forthe Figure6.Thecrosscorrelationcoefficientsr(k ,z)s intheredshiftspace.
Kaiser effect. It’s interesting tofindthat the slopes approach unity atall Thicklinesshowrs withKaisereffectcorrecte⊥dfor.ComparedtoFig.3,
investigatedredshifts,whilethescatterskeepnearlyunchangedcompared r(k ,z)sissuppressedthroughoutallscales,especiallyatk &1hMpc−1.
toFig.4. At⊥z . 1,thereconstruction workspoorly,withr 0.3o⊥nscalesk &
3hMpc−1.However,there’saconsiderablecorrelatio≃nstrengthrs&0.5⊥for
k .1hMpc−1atz=0.Athigherredshiftwherenonlinearityofvelocityis
n⊥otsignificant,thecorrelationstrengthisstilltightenough,e.g.withr 0.7
fork 1hMpc 1atz=2.08. ≃
⊥∼ −
3.2.1 Θ-Θˆsrelationintheredshiftspace
TheΘ-ΘˆsrelationisshowninFig.4,whichissignificantlydifferent 3.2.2 rs(k ,z)intheredshiftspace
fromtheoneinrealspace(Fig.2).Wecanseethatintwoaspects. ⊥
(1)TheaverageslopeoftheΘ-Θˆ relationchanges,fromΘˆ Θto The cross correlation coefficient r in redshift space is shown in
Θˆ aΘwitha>1.Thisiscausedbythelinearredshiftdis≃tortion Fig. 6. Compared to Fig. 3, the correlation coefficient is sup-
≃ pressed throughout all scales, especially at k & 1hMpc 1. At
(theKaisereffect),whichinduces −
k & 3hMpc 1andz . 1,thekineticSZreco⊥nstructionandthus
−
th⊥ekineticSZtomographyworkspoorlysincethereconstructedΘˆ
δs(k)=δ(k)(1+βµ2k), (13) barelyresembles thetruesignal Θ (r . 0.3).These resultsshow
unambiguouslythatredshiftdistortionisasignificantsourceofer-
where µ = nˆ kˆ is the cosine of the angle between the line of ror in the kinetic SZ tomography. It’s also worthwhile to see the
k ·
sightnˆ andthewavevectork.β= f/b .TheKaisereffectenhances changes if we correct for the Kaiser effect. They’re also shown
g
thegalaxyoverdensitybyafactor 1+β/3 > 1andthuscauses inFig.6asthicklines. Asindicated byFig.5, correctingfor the
≃
the slope a > 1. We also notices that the slope a decrease from Kaisereffectdoesnotinfluencerstoomuchonscalesofinterest.
high redshifts to low redshifts. There are two causes. First, β = Asexplainedin 2.3,thedegradationiscausedbythestochas-
§
f Ω0.6(a)(Peebles1980)(inourcasewhereb = 1)decreases ticity in the δ-δs relation. They can be induced by the nonlinear
≃ m g
withdecreasing redshift. Second, the finger of God effect caused mappingbetweentherealspacedensityandtheredshiftspaceden-
bysmallscalerandommotionsuppressestheredshiftspacegalaxy sity(e.g.Scoccimarro2004)andthestochasticitybetweentheden-
density.Thiseffectbecomesstrongeratlowerredshift.(2)Scatters sity and velocity field (e.g. Whiteetal. 2009). This stochasticity
in the Θ-Θˆs relation are significantly larger than that in the real theninducesthestochasticityinthereconstructedvelocityvˆ,with
space (Fig. 2). Stochasticitiesin the density-velocity relation and respecttotruevelocity(Eq.3).Inthemomentumreconstructionwe
realspace-redshiftspacemappingarelargelyresponsibleforthese needtomultiplythereconstructedvelocitybyanextrafactor1+δ
largerscatters. (Eq.4)toobtainthereconstructedmomentum. Sothestochastic-
It’sinterestingtoshowtheΘ-Θˆsrelationifwecorrectforthe ityinthemomentumreconstructionandthusthekineticSZrecon-
Kaisereffect.ThefigureisshowninFig.5.Wefindthattheslopes structionhasanextrasource,fromtheterm1+δ.Allthesecom-
approachunityatallinvestigatedredshifts,whilethescatterskeep plexities worsen the reconstruction, increase scatters in Θ-Θˆ and
nearly unchanged compared to Fig. 4. This is, however, not un- decreaser.
expected,sincethedeterministicKaiserformulaonlychangesthe Despite the above degradations, the reconstructed Θˆ still
amplitude and isreversible given cosmology. Thusfinger of God showsreasonablytightcorrelationwiththetruesignalΘatk .
shouldbemainlyresponsibleforthescattersoftherelation. 1hMpc 1. We will show in the next section that this correla⊥tion
−
ThekineticSZtomographywithspectroscopicredshiftsurveys 9
Figure 7. The deviations between different smoothing scales, say, 0, Figure8.Thedifferenceofthecorrelationcoefficientbetweenadiabaticand
0.78h−1Mpc,1.56h−1Mpcand3.12h−1Mpc,inbothrealspace(toppanel) non-adiabaticsimulations.Theoverallrsisinsensitivetogastrophysics,and
andredshiftspace(bottompanel).Intherealspace,smoothinglengthwith there’reonly 2-3%changesofrs.Itwillaccordinglyinfluencetheestima-
0.78h 1Mpcworksbestatz=2.08while3.12h 1Mpcisthemostsuitable torofthesig∼nal-to-noise ofthecrosspowerspectrum byseveral percent
− −
at z=0. However, in the redshift space, the enhancement due to smooth- level, however, forkSZtomography,this issufficientforthecurrentsur-
ingislesseffective, andalargersmoothinglength isbetter. Probably an veys.
anisotropicsmoothingotherthanaGaussiansmoothingshouldworkbetter.
Asamediansmoothinglength, 1.56h 1Mpciseffective atbothlow and
−
highredshift.
comprehensiveinvestigationonthisissue.Rather,wewillrestrict
totheGaussianfilterandinvestigatethedependenceofronR .We
s
strength allows for robust kinetic SZ tomography, combining the arbitrarily compare between the cases of R = 0,0.78,1.56,3.12
s
Planck CMB experiment and the BigBOSS galaxy spectroscopic h 1Mpc.Forclarity,weonlyshowthecomparisonsatz = 0and
−
redshiftsurvey,orothersurveyswithcomparablepower.
z=2.08inFig.7.
Aninterestingbehaviortonoticeisthatrsatz=0.53isworse
Thebasic(andobvious)conclusionisthat,smoothingisnec-
thanatz = 0.Thisresultisconsistentwithalargerscattersinthe
essary to suppress small scale nonlinearities and improve the re-
Θ-Θˆ plot at z = 0.53 than at z = 0, implying that stochasticities
construction. Fig. 7 (upper panel) shows general improvement in
induced by the Finger of God effect becomes the largest at that
r whensmoothing istaken, comparing tothe caseof nosmooth-
epoch.Atredshifthigherthanz 1,thereconstructionsuffersless
∼ ing (Rs = 0). For example, smoothing the density field with
from the nonlinearities and is thus stronger, e.g. with r > 0.7 at R = 1.56h 1Mpc can boost r by 30% at k = 1hMpc 1and a
k < 1h/Mpcatz=2.08.Actually,wefindinsimulationsthatthe s − ∼ −
factorof2ormoreatsmallerscales,inrealspace.
co⊥movingvelocitydispersionpeaksatz 0.6.Thiswouldinduce
∼ Gaussiansmoothingislesseffectiveinredshiftspace:theim-
thelargestanisotropiesandhencemakersatz 0.6theworst.
∼ provementinrsisoftenlessthan10%onscalesofinterest(bottom
panel, Fig. 7). The redshift space overdensity is anisotropic, so a
spherical Gaussian smoothing willnot work well.Ananisotropic
3.3 Uncertaintiesinr
filtermayworkbetterinredshiftspace.Thisiscertainlyaninter-
Intheabovesection,wequantifytheimpactofredshiftdistortion estingtechnicalissueforfurtherinvestigation.
on r and thus show that redshift distortion degrades the kinetic Arightsmoothingshouldbalancebetweensuppressingsmall
SZ reconstruction significantly. There are other factors affecting scalenonlinearities and preserving largescale signal. If R istoo
s
the reconstruction. Here, we briefly discuss the influence of fil- large,itmaywipeofftoomuchlargescaleclusteringresponsible
tersadoptedtosmooththedensityfield( 3.3.1)andgastrophysics forpeculiarvelocityandthusdegradethereconstruction(decrease
§
( 3.3.2). r).ThismaybethereasonthatR =0.78h 1Mpcworksbetterthan
§ s −
largerR ,atz=2.08intherealspace.However,overallR =1.56
s s
h 1Mpcworksreasonablywellatallredshiftsinvestigated.
3.3.1 Theinfluenceofdensitysmoothingonr −
Thesimulationdatawedealwithislikeanidealsurvey,with
TheresultsshownabovealladoptaGaussianfilterwithsmoothing negligibleshotnoise,uniformselectionfunctionandregularsurvey
lengthR =1.56h 1Mpctosmooththedensityfieldbeforevelocity boundary.Smoothingforrealdataisofcoursemuchmorecompli-
s −
reconstruction. Thereconstruction robustnessnowonder depends cated.Forexample,abigissueinrealsurveyisshotnoisedueto
onthewaytosmooththedensityfield.Wedonotaimtoperforma lowgalaxynumberdensity,especiallyinspectroscopicredshiftsur-
10 Shaoet al.
veys.Forthisissue,onecanrefertoHoetal.(2009)fordiscussion
ontheapplicationoftheWienerfilter.
3.3.2 randgastrophysics
r also depends on gastrophysics. All theresults shown above are
basedonournon-adiabaticsimulation,withradiativecooling,star
formation and supernova feedback. Although we are not able to
robustly quantify its detailed dependence on these gastrophysical
processes, we can obtain a rough estimation by comparing the
aboveresultstoouradiabaticsimulationwithidenticalinitialcon-
ditions.Therelativedifferencesofrs betweenthetwosimulations
areshowninFig.8.Wefindthat,overallrandthustheperformance
ofthekineticSZtomographyisinsensitivetothegastrophysics.r
atk < 1hMpc 1onlyvariesbylessthan2-3%.Theinfluenceof
−
⊥
gastrophysicsislargeratsmallerscales,butisstilllessthan5%up
tok =3hMpc 1.
−
⊥
Thisinsensitivitytogastrophysicshastwoimplicationsonour
kineticSZtomography.First,itisunlikelythatsomerealisticgas-
trophysical process not included in our non-adiabatic simulation
can dramatically suppress r and thus invalidates the tomography.
Second,thissignificantlysimplifiesthetheoreticalinterpretationof
thetomographyresults.Basedonthesametechniquein(Shaoetal.
2009),wecancombinethetwomeasuredcorrelations, ΘΘˆ and Figure9.Theratio PN1,2/PΘˆs oftheshotnoisepower spectrum andthe
ΘˆΘˆ ,toobtain ΘΘ arisingfromthesameredshiftbin.hTheionly reconstructedmomentumpowerspectrum.Boththe∆2 and∆2 aremuch
hunkniown quantihty inithis approach is r, which we shall calibrate smaller than the reconstructed momentum powers, wNi1th N2 tNe2rms domi-
againstsimulations.However,ifrisverysensitivetogastrophysics, nating N1 towards smaller scales. Given a small number density in cur-
rent galaxy redshift survey, the total shot noise would be comparable to
thecalibrationonrwouldbeverydifficultduetolargeuncertain-
the signal. Nevertheless, on scales of interest for kSZ tomography, say
ties in our theoretical and numerical understanding of these gas-
k . 1hMpc−1,galaxysurveyswouldprovidearelatively lesscontami-
trophysics. The insensitivity of r to gastrophysics shown in Fig. n⊥atedreconstructionofthemomentum.
8impliesthat,despiteimperfecttheoretical andnumerical under-
standingofthesegastrophysics,rcanstillbeaccuratetoafewper-
centlevelatrelevantscales.Thisprecisionsufficesforthekinetic
weomittheprojectionalongthelineofsight)
SZtomography.
Θˆs(nˆ)WΘˆs(nˆ ) (1+δs+δ )(vˆs+vˆ ) nˆ
h c c ′ i ∝ h N N ·
×(1+δs′+δ′N)(vˆs′+v′N)·nˆ′i
= (1+δs)vˆs nˆ(1+δs′)vˆs′ nˆ′
4 ERRORFORECAST h · · i
ThermeasuredabovequantifiestheperformanceofthekineticSZ +h(1+δs)vN·nˆ(1+δs′)v′N·nˆ′i
+ δ vˆs nˆδ vˆs nˆ .
tomographyforavirtuallyidealgalaxysurvey,forwhichthesimu- h N · ′N ′· ′i
lationparticlenumberdensityishighandtheshotnoiseisnegligi- The last two terms in the above equation are the leading non-
ble.However,inrealsurveys,thegalaxynumberdensityisafactor vanishing noise terms. We denote the first noise term as N
of104-105smaller,resultinginmuchlargerandthusnon-negligible (1+δs)v whilethesecondasN δ vˆs 1 ∝
shotnoise.Theshotnoiseaffectsboththevelocityandmomentum By rNandomizing the particl2e∝posNitions in the simulation, we
reconstruction.Weuseoursimulationtoquantifytheseeffectsand can directly measure δ and derive therein v δ (k)kˆ/k. Fol-
N N ∝ N
thenapplytheseresultstoforecasttheperformanceofthekinetic lowing the same procedure as in reconstructing the momentum
SZtomography.OurtargetCMBexperimentisPlanckandthetar- map,weproducetwocorrespondingmapsofthesetwoshotnoises.
getgalaxyspectroscopicsurveyisBigBOSS(Schlegeletal.2009). In order to estimate the contribution of shot noise, we compute
ThekineticSZtomographybasedonotherSZsurveyslikeSPTand the power spectrum of the two noise terms, and show the ratios
galaxysurveyslikeADEPT,EuclidandSKAisexpectedtowork P /P in Fig. 9. We can see that both the P and P noise
N1,2 Θˆs N1 N2
better. termsaremuchsmallerthanthereconstructedmomentumpowers.
AlthoughN andN termsarecomparableontheverylargescales,
1 2
theydeviatesignificantlytowardssmallerscales,withN termspre-
2
dominantlyoverweighingN terms.Onthesmallscales,N terms
4.1 Estimatingthereconstructionnoisefromgalaxy 1 2
would turnout tobe several thousandth of thereconstructed mo-
distributionshotnoise
mentumΘˆs,andtheycanbecomparable toorevendominatethe
Thediscretenessofdarkmatterparticlesinducesspuriousfluctua- signal,givenamuchsmallernumberdensityincurrentgalaxyred-
tions(shotnoise)inthedensityfield,whichwedenoteasδ ,and shiftsurveys.Nevertheless,onscalesofinterestforkSZtomogra-
N
therebyinthereconstructedvelocity,whichwedenoteasv .The phy,sayk .1hMpc 1,galaxysurveyswouldprovidearelatively
N −
correlation of the contaminated momentum Θˆ is (for simplicity, lesscontam⊥inatedreconstructionofthemomentum.
c