non‐gaussianity in the cmb: primordial and...
TRANSCRIPT
Non‐GaussianityintheCMB:primordialandsecondary
KendrickSmithUniversityofCambridgeLosCabos,Jan2009
References:
Smith,HuandKaplinghatastro‐ph/0607315SmithandZaldarriagaastro‐ph/0612571Smith,Zahn,DoreandNoltaarxiv:0705.3980Smithetalarxiv:0811.3916DvorkinandSmitharxiv:0812.1566Smith,SenatoreandZaldarriagatoappearSenatore,SmithandZaldarriagatoappear
Outline
1. Secondarynon‐Gaussianity:gravitationallensinghowarethestatisticsoftheCMBaffected?whatcosmologicalinformationisadded?
2.Secondarynon‐Gaussianity:reionizationanewstatisticwhichisolatestheepochofpatchyreionization
3.Primordialnon‐Gaussianitywhatsignalscanonelookfor?newresultsfromWMAP(firstoptimalanalysis)
Part 1: CMB lensing
CMB photons are deflected by gravitational potentials between last scattering and observer. This remaps the CMB while preserving surface brightness:
!T (!n)! !T (!n + d(!n))where is a vector field giving the deflection angle along line of sight
!n
Deflection angles Unlensed CMB Lensed CMB
d(!n)
!"+
WayneHu
To first order, the deflection field is a pure gradient:
where the lensing potential is given by the line-of-sight integral
Redshift kernel: broad peak at z ~ 2 RMS deflection: ~2.5 arcmin, coherent on degree scales (l ~ 100)
CMB acquires sensitivity to new parameters (e.g. neutrino mass)
da(!n) = !a!(!n)
!(!n) = !2" !!
0d"
#"! ! "
""!
$!("!n, #0 ! ")
CMB lensing: theory
AntonyLewis
d(!n)
CMB lensing: power spectrum
Gravitationallensingappearsintemperatureandpolarization,butpolarizationismoresensitive
B‐modepolarizationisgenerated
CMB lens reconstruction: idea
Idea: from observed CMB, reconstruct deflection angles (Hu 2001)
Lensed CMB Reconstruction + noise
!n
Deflection angles Unlensed CMB Lensed CMB
!"+
!"!n
CMB lens reconstruction: quadratic estimator
Lensing potential weakly correlates Fourier modes with
Formally: can define estimator which is quadratic in CMB temperature:
Intuitively: use hot and cold spots of CMB as local probes of lensing potential (analagous to cosmic shear: galaxy ellipticities are used as probes)
l != l!
!!(l)
Lensed CMB Reconstruction + noise
!"!n
!T (l) T (l!)"" # [$l! · (l$ l!)]!(l$ l!)
!!(l) !"
d2l1(2")2
[(l · l1)Cl1 + (l · l2)Cl2 ]T (l1)T (l2)Ctot
l1Ctot
l2
(l2 = l" l1)
CMBlensreconstruction:higher‐pointstatistics
Lensreconstructionnaturallyleadstohigher‐pointstatistics
e.g.takeCMBtemperatureT (n)! !!(l) (applyquadraticestimator)
! !C!!"
(takepowerspectrum)
defines4‐pointestimatorintheCMB
Or:takeCMBtemperature,galaxycounts
! !C!g"
g(n)T (n)! !!(l) (applyquadraticestimator)
(takecrosspowerspectrum)Defines(2+1)‐pointestimatorinthe(CMB,galaxy)fields
Canthinkofthelensingsignalformallyasacontributiontothe3‐pointor4‐pointfunction,butlensreconstructionismoreintuitive
Smithetal,0811.3916
CMBlensing:lensreconstructionvspowerspectrum
ConsidertwomethodsforextractingCMBlensingsignal:
1.measureCMBpowerspectralensingmakessomecontribution,unlensedspectraactas“noise”2.measurepowerspectrumoflenspotential(4‐pointinCMB)withinreconstructionnoise(powerspectrum)
CTT! , CTE
! , CEE! , CBB
!
C!!"
Secondmethodhasbettersignal‐to‐noiseandismoreflexible
e.g.CMBlensingdetectioninWMAP3:
N!!"
CTT! :lensingis“detected”at0.3sigma
C!!" :1sigma
C!g" :3.5sigma(g=NVSSgalaxycounts)
Smithetal,0811.3916
NVSS: NRAO VLA Sky Survey
1.4 GHz source catalog, 50% complete at 2.5 mJy
Mostly extragalactic sources: AGN-powered radio galaxies Quasars Star-forming galaxies
Well-suited for cross-correlating to WMAP lensing potential: Nearly full sky coverage Low shot noise High median redshift
(fsky = 0.8)(Ngal = 1.8! 106)(zmedian = 0.9)
WMAP3-NVSS result with systematic errors
Detection significance: fit in one large bandpower, in multiple of fiducial model
Result: 1.15 +/- 0.34, i.e. a 3.4 sigma detection, in agreement with the expected level
Smith,Zahn,Dore&Nolta,0705.3980
CMBlensreconstruction:futureprospects
C!!" Issensitivetolate‐universeobservablessuchas:
neutrinomassdarkenergyequationofstatecurvature
(forecastsforanall‐skypolarizationsatellite)
Needhigh‐resolutionCMBmeasurements:manyexperimentsonhorizon(SPT,ACT,Planck,Polarbear…)
Probeslargeangularscalesathighredshift(e.g.earlydarkenergy)
Smithetal,0811.3916
Outline
1. Secondarynon‐Gaussianity:gravitationallensinghowarethestatisticsoftheCMBaffected?whatcosmologicalinformationisadded?
2.Secondarynon‐Gaussianity:reionizationanewstatisticwhichisolatestheepochofpatchyreionization
3.Primordialnon‐Gaussianitywhatsignalscanonelookfor?newresultsfromWMAP(firstoptimalanalysis)
Part2:reionizationandtheCMB
HowdoesreionizationaffecttheCMB?Considerhomogenousreionizationfirst(i.e.,x_e=functionofzonly)
Dvorkin&Smith,0812.1566
1.Screening:overallamplitudeofCMBpowerspectrum=Ase!2!
Createsdegeneracy:hardertomeasureA(tauactsas“nuisanceparameter”)Conversely,ifdegeneracycanbebroken,canconstrainreionization(e.g.WMAP5:)! = 0.087± 0.017
2.NewpolarizationfromThomsonscatteringafterreionization:GeneratesE‐modepeakat(horizonsize)thatcanbreakdegeneracy! ! 8
3.Dopplereffect:contributestolarge‐scaletemperatureanisotropyMostlycancelsalongline‐of‐sightsincev=“puregradient”tolowestorder
ReionizationandtheCMB:patchyreionization
Considerareionizationbubblewithopticaldepthandradialvelocityvr
WhatCMBanisotropydoesthebubblegenerate?!bub
Zahnetal(2005)
!T (n) = !!bubT (n)![Q± iU ](n) = !!bub(Q± iU)(n)
!T (n) = !!bub(vr/c)
!(Q± iU)(n) = (!bub)!
610
2!
m=!2
±2Y2m(n)T2m(z,n)
1.Screening:
3.kSZ:
2.“Thomson”:
Patchyreconstruction:quadraticestimator?
Idea:ifreionizationispatchy,theopticaldepthisafield,notaconstant!(n)
Canwewritedownaquadraticestimator!!(l) (“taureconstruction”)analogoustotheestimator? (“lensreconstruction”)
Lensreconstruction“works”becauselensinggeneratesanisotropywhichlooks(heuristically)like()x(unlensedCMB):
!!(l)
Question:Doespatchyreionizationgenerateanisotropyofthesameform,withreplacedby?
!
Tlensed = Tunlensed + (!!) · (!Tunlensed) + · · ·(Q ± iU)lensed = (Q ± iU)unlensed + (!!) ·!(Q ± iU)unlensed + · · ·
! !
Anisotropygeneratedbythescreeningeffecthasform()x(unlensedCMB)=>screeningeffectis“captured”byquadraticestimatorformalism
Heuristically:amplitudeofrecombinationpeaksismodulatedbytau(regionofskywithlargertau=>lowerobservedacousticpeaks)
Patchyreconstruction:quadraticestimator?
1.Screeningeffect
!T (n) = !!(n)T (n)![Q± iU ](n) = !!(n)(Q± iU)(n)
!
Heuristically:large‐scaleE‐modefromreionizationismodulatedbytau(regionofskywithlargertau=>largerreionizationE‐mode)
GeneratesB‐modes,butmuchsmallerthanlensingatpowerspectrumlevel
Dvorkin&Smith,0812.1566
Patchyreconstruction:quadraticestimator?
2.Thomsonscattering(polarizationonly)
AnisotropygeneratedbyThomsonscatteringhasform()x(unlensedCMB)=>Thomsonscatteringis“captured”byquadraticestimatorformalism
!(Q± iU)(n) =!
610
!d!(n)
dz
2"
m=!2
±2Y2m(n)T2m(z,n)
" !(n)!0
(Q± iU)reion(n)
!
ThekSZanisotropyisaline‐of‐sightintegral:
!T (n) = !1c
!dz
d!
dzvr(z,n)
Zahnetal(2005)
kSZfromreionizationcontributestothesmall‐scaletemperaturepowerspectrum
Patchyreconstruction:quadraticestimator?
3.kSZ(temperatureonly)
AnisotropygeneratedbykSZdoesnothaveform()x(unlensedCMB)=>kSZisnot“captured”byquadraticestimatorformalism
!
Patchyreionization:Quadraticestimator
Conclusion:screeningandpolarizedThomsoneffectscanbeusedtoconstructaquadraticestimatorfor(mathisthesameaslensreconstruction,butwithdifferentl‐weighting)
Fromhere,onecanconstructestimatorsasinthelensingcase:
!!(l) = (2-point in CMB fields T,E,B)!C!!
" = (4-point in CMB fields)!C!X
" = ((2+1)-point in CMB + cross-correlation field X)
Dvorkin&Smith,0812.1566
!(n)
Quadraticestimator:exaggeratedexample
Dvorkin&Smith,0812.1566
Quadraticestimator:signal‐to‐noise
Ideasforimprovingthesignal‐to‐noise:1.cross‐correlatewithothertracersofreionization2.combinewithlensreconstructiontojointlysolvefortauandphi
Dvorkin&Smith,0812.1566
Temperature Polarization
Parameterforecastsinatoyreionizationmodel
Twoobservables:Amplitudeofthepowerspectrumisdeterminedby
LocationofpeakisdeterminedbythetypicalbubbleradiusR
Ahigh‐sensitivitysatelliteexperimentcanconstrainbothatthe~10%level
Dvorkin&Smith,0812.1566
ReionizationandtheCMB:conclusion
TwomethodsforextractingreionizationinformationfromCMB:1.kSZfromsmall‐scaletemperaturepowerspectrum2.quadraticestimator
Complementarystrengthsandweaknesses:
1.“sees”kSZ,butnotscreeningorpolarizedThomsonexperimentalsensitivityalreadyexists(SPT/ACT)probesreionizationbubblesonsmallscalespowerismixedwithothersecondaries,modelinguncertainty
2.“sees”screeningandpolarizedThomson,butnotkSZmorefuturistic,requireslow‐noisepolarizationprobestaufluctuationsonlargescalesisolatespatchysignal,mapcanbecross‐correlated,orusedfor“cleaning”
Unsolvedproblem:findstatistictoisolatethekSZsignalintemperature!
(2000 ! ! ! 4000)!!
(! = Drec/Rbub ! 300)
(2000 ! ! ! 4000)
Outline
1. Secondarynon‐Gaussianity:gravitationallensinghowarethestatisticsoftheCMBaffected?whatcosmologicalinformationisadded?
2.Secondarynon‐Gaussianity:reionizationanewstatisticwhichisolatestheepochofpatchyreionization
3.Primordialnon‐Gaussianitywhatsignalscanonelookfor?newresultsfromWMAP(firstoptimalanalysis)
Part3:Primordialnon‐Gaussianity
In“vanilla”modelsofinflation,theinitialfluctuationsareGaussianThe3‐pointcorrelationfunctionfunctioniszero:
!!(k1)!(k2)!(k3)" = 0However,moreexoticmodelscanpredictnonzerothree‐pointfunctions“Local”shape:e.g.curvatonmodel
!(x) = !G(x) + f localNL !G(x)2
!!(k1)!(k2)!(k3)" # f equil.NL
!3"
i=1
k1 + k2 + k3 $ 2ki
k3i
#"3(
$
i
ki)
!!(k1)!(k2)!(k3)" # f localNL
!1
k31k
32
+ symm."
"3
#$
i
ki
%
“Equilateral”shape:higher‐derivativeinteractions,DBIinflation
Arethesetheonlyshapestolookfor?
Thetwocubicoperatorsdonotgenerate3‐pointfunctionswhicharepreciselyequal:
!(k1, k2, k3)k31k
32k
33k
3t
!
"k3t ! kt
#
$%
i<j
kikj
&
'! 3k1k2k3
(
) +12! 8!
k1k2k3k3t
Canorthogonalizethefamilyofbispectraparameterizedby
kt = k1 + k2 + k3
!(k1, k2, k3) = (kt ! 2k1)(kt ! 2k2)(kt ! 2k3)
Effectivefieldtheoryofinflation(Cheungetal2007):
!
Anew“folded”shape
Thereisa“window”atwhereanewshapeappears
Localshape:“squeezed”triangles()containthesignal‐to‐noiseEquilateralshape:equilateraltriangles()Foldedshape:“foldedtriangles”
Considercorrelationcoefficientwithequilateralshape,asfunctionof:
Senatore,SmithandZaldarriaga(toappear)
!
!1 ! !2, !3!1 ! !2 ! !3
(!1 + !2) ! !3
! ! 5.1
AnalysisinWMAP5forthcoming
Moregeneralshapeanalysis?
Powerspectrumanalogue:onecouldeither1.estimateC_l’sinl‐bins2.estimateP(k)ink‐bands3.estimateinflationaryparameters(n_s,running,….)
Bispectrum:analogously,onecouldeither:1.estimateb_{l1,l2,l3}inlbins2.estimateF(k1,k2,k3)inkbins3.estimatefnlparameters
Sofar,only#3hasbeenperformedonWMAPdata,but#1+2wouldbepossibleusingourmachinery
….Isthereagoodtheoreticalmotivationfordoingthis?
Three‐pointfunctionintheobservedCMB
Non‐gaussianinitialconditionsfrominflation+lineartransferfunctions=non‐GaussianCMB
f localNL = 0 f local
NL = 3000
MicheleLiguori
Optimalestimator:sumovertriples(l1,l2,l3)withinversesignal‐to‐noiseweighting
!T (l1)T (l2)T (l3)" # fNL!2
!"
i
li
#
WMAPanalysis:background
f localNL = 32± 34
f localNL = 87± 30
Creminellietal(WMAP3)
Arobustdetectionwouldruleoutmostmodelsofinflation!(e.g.slow‐roll)Whichanalysisshouldbebelieved?
Yadav&Wandelt(WMAP3!!)
Komatsuetal(WMAP5)f localNL = 55± 30
Reasonforthediscrepancy:“step”atl=450
Q‐band(40GHz)
V‐band(60GHz)
W‐band(90GHz)
Yadav&Wandelt(2007)
Mustbecarefultoavoidmakingaposteriorichoices….!
UseofV+WismotivatedaprioriUseoflmax=750isnot
Problem:haveestimatesoffnlwithdifferentvaluesbutcomparablestatisticalerrors
Generalprinciple:inthissituation,either1.haveevidenceforsystematics2.estimatorissignificantlysuboptimal
e.g.(Creminellietal,WMAP3,lmax=350)(Yadav&Wandelt,WMAP3,lmax=750)
Isthisshiftbetweenandconsistentwithstatistics?
Ifso,oneshouldbeabletoreweighttheestimatorsothataddingthesmallscalesimprovesbymorethan4
Questions:1.Arethelargeandsmallscalesconsistent?(i.e.dotheygiveaconsistent?)2.AreWMAP3andWMAP5consistent?
f localNL = 32± 34
f localNL = 87± 30
!max = 350 !max = 750
!(f localNL )
Motivationforoptimalestimator:
1.smallererrorbars(forWMAP5)2.eliminatearbitrarychoices:differentimplementationsshouldagreeprecisely3.cannotgetmultipleestimateswithcomparableerrors;getsingle“bottomline”estimateforagivendataset
Optimalestimator
d! (S + N)!1d
!(f localNL ) = 21
Results:WMAP5optimalanalysis
WMAP5 optimal: !4 < f localNL < 80 (95% CL)
WMAP5 suboptimal: !11 < f localNL < 121 (95% CL)
+SDSS (Slosar et al 2008): !1 < f localNL < 61 (95% CL)
Smith,SenatoreandZaldarriaga(toappear)
Results:arethelargeandsmallscalesconsistent?
Quantifythisbydefining:
!fNL = !fNL(!max = 750)! !fNL(!max = 350)
Insimulation:!fNL = 19 (at 1!)
!fNL = 52
InWMAP3,wefind:
!fNL = 32!fNL = 20 (optimalestimator)
(suboptimal)somedisagreementw/Yadav‐Wandelt(2007):
InWMAP5,wefind:!fNL = 9 (optimalestimator)
(suboptimal)!fNL = 16withgoodagreementwithKomatsu(2008)
Smith,SenatoreandZaldarriaga(toappear)
Results:areWMAP3andWMAP5consistent?
Define:Wefind:(optimalestimator)(suboptimal)
Insimulation:,soWMAP3andWMAP5areconsistent
!fNL = !fNL(WMAP5)! !fNL(WMAP3)!fNL = !20!fNL = !25
!fNL = 16 (at 1!)
Smith,SenatoreandZaldarriaga(toappear)
Speculation:whataboutthehighsignificanceinWMAP3?
Yadav&Wandelt(2007):2.9sigma,Ourpipeline:2.3sigma,(suboptimalestimator)2.5sigma,(optimal)
Probabilityofa2.5sigmaresultbychance:1.25%Canwereally“dismiss”thehighsignificanceinseenWMAP3?
Howmanyequally“interesting”parameterscouldwehavelookedfor?
Inaddition,mighthavefoundsomethinginWMAP1‐only,orinWMAP5‐only….
Nowconsider:whathappenswhenweaddmoredata?WehaveinWMAP3Supposeisreally56;thenweexpecttogetinWMAP5SupposeWMAP3isstatisticalfluke;thenweexpectashifttowardzero….
f localNL , f equilateral
NL , r, w0,!K , (dns/d ln k), !0, !!1
!f localNL = 87
!f localNL = 69
!f localNL = 56
!f localNL = 56
f localNL f local
NL ! 56
Primordialnon‐Gaussianity:systematictests‐large‐scaleforegrounds:‐small‐scaleforegrounds:negligible‐pointsources:negligible‐three‐pointfunctionfromsecondaries:
Primordialnon‐Gaussianity:conclusions‐Systematicsseemcompletelyundercontrol!‐Firstimplementationofoptimalestimator:reducesfrom30to21‐Eliminatesarbitrarychoiceswhichcomplicateinterpretationofresults‐Wecurrentlyfindjustunder2sigma
f localNL ! few (conservativegalacticmask,aftercleaning)
f localNL ! few (largest:ISW‐lensing)
!(f localNL )
f localNL
!4 < f localNL < 81 (WMAP5)
!1 < f localNL < 61 (WMAP5+Sloan)