Consistency relations for large-scale structures: applications to the integrated Sachs-Wolfe effect and the kinematic Sunyaev-Zeldovich effect

0.17 MB·

by  Luca Alberto Rizzo

Astronomy&Astrophysicsmanuscriptno.ISW_consistency_relations_Jan15 (cid:13)cESO2017
January17,2017
Consistency relations for large-scale structures: applications to
the integrated Sachs-Wolfe effect and the kinematic
Sunyaev-Zeldovich effect
LucaAlbertoRizzo1,DavidF.Mota2,andPatrickValageas1
1 InstitutdePhysiqueThéorique,CEA,IPhT,F-91191Gif-sur-Yvette,Cédex,France
2 InstituteofTheoreticalAstrophysics,UniversityofOslo,0315Oslo,Norway
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1 January17,2017
0
2
ABSTRACT
n
a Consistency relations of large-scale structures provide exact nonperturbative results for cross-correlations of cosmic fields in the
J
squeezedlimit.TheyonlydependontheequivalenceprincipleandtheassumptionofGaussianinitialconditions,andremainnonzero

5 atequaltimesforcross-correlationsofdensityfieldswithvelocityormomentumfields,orwiththetimederivativeofdensityfields.We
1 showhowtoapplytheserelationstoobservationalprobesthatinvolvetheintegratedSachs-WolfeeffectorthekinematicSunyaev-
Zeldovich effect. In the squeezed limit, thisallows us to express the three-point cross-correlations, or bispectra, of two galaxy or

] matterdensityfields,orweaklensingconvergencefields,withthesecondaryCMBdistortionintermsofproductsofalinearanda
O nonlinearpowerspectrum.Inparticular,wefindthatcross-correlationswiththeintegratedSachs-Wolfeeffectshowaspecificangular
C dependence.TheseresultscouldbeusedtotesttheequivalenceprincipleandtheprimordialGaussianity,ortocheckthemodelingof
large-scalestructures.
.
h
p Keywords. Cosmology–large-scalestructureoftheUniverse
-
o
r
t1. Introduction non-Gaussianinitial conditions). Hence, such relations express
s a kinematic effect that vanishes for equal-times statistics, as a
a
[Measuring statistical properties of cosmological structures is uniformdisplacementhasnoimpactonthestatisticalproperties
not only an efficient tool to describe and understand the main ofthedensityfieldobservedatagiventime.
1componentsof our Universe, but also it is a powerfulprobe of Inpractice,itishoweverdifficulttomeasuredifferent-times
vpossiblenewphysicsbeyondthestandardΛCDM concordance density correlations and it would therefore be useful to obtain
9
model.However,onlargescalescosmologicalstructuresarede- relations that remain nonzero at equal times. One possibility
4
scribed by perturbative methods, while smaller scales are de- to overcome such problem, is to go to higher orders and take
0
4scribed by phenomenological models or studied with numeri- into account tidal effects, which at leading order are given by
0cal simulations. It is therefore difficult to obtain accurate pre- the response of small-scale structures to a change of the back-
.dictions on the full range of scales probedby galaxy and lens- grounddensity.Suchanapproach,however,introducessomead-
1
ing surveys. Furthermore,if we consider galaxy density fields, ditionalapproximations(Valageas2014a;Kehagiasetal.2014b;
0
theoreticalpredictionsremainsensitivetothegalaxybiaswhich Nishimichi&Valageas2014).
7
1involvesphenomenologicalmodelingof star formation,evenif Fortunately,itwasrecentlynoticedthatbycross-correlating
:weusecosmologicalnumericalsimulations.Asaconsequence, densityfieldswithvelocityormomentumfields,orwiththetime
vexact analytical results that go beyond low-order perturbation derivativeofthedensityfield,oneobtainsconsistencyrelations
Xitheoryandalsoapplytobiasedtracersareveryrare. thatdonotvanishatequaltimes(Rizzoetal.2016).Indeed,the
Recently, some exact results have been obtained kinematiceffectmodifiestheamplitudeofthelarge-scaleveloc-
r
a(Kehagias&Riotto 2013; Peloso&Pietroni 2013; ityandmomentumfields,whilethetimederivativeofthedensity
Creminellietal.2013;Kehagiasetal.2014a;Peloso&Pietroni fieldisobviouslysensitivetodifferent-timeseffects.
2014; Creminellietal. 2014; Valageas 2014b; Hornetal. In this paper, we investigate the observationalapplicability
2014, 2015) in the form of “kinematic consistency relations”. of these new relations. We consider the lowest-order relations,
They relate the (ℓ + n)-density correlation, with ℓ large-scale which relate three-point cross-correlations or bispectra in the
wave numbers and n small-scale wave numbers, to the n-point squeezed limit to products of a linear and a nonlinear power
small-scale density correlation.These relations,obtainedat the spectrum. To involve the non-vanishing consistency relations,
leading order over the large-scale wave numbers, arise from we study two observable quantities, the secondary anisotropy
the Equivalence Principle (EP) and the assumption of Gaus- ∆ ofthecosmicmicrowavebackground(CMB)radiationdue
ISW
sian initial conditions. The equivalence principle ensures that to the integrated Sachs-Wolfe effect (ISW), and the secondary
small-scalestructuresrespondtoalarge-scaleperturbationbya anisotropy∆ duetothekinematicSunyaev-Zeldovich(kSZ)
kSZ
uniform displacement while primordial Gaussianity provides a effect.Thefirstprocess,associatedwiththemotionofCMBpho-
simplerelationbetweencorrelationandresponsefunctions(see tonsthroughtime-dependentgravitationalpotentials,dependson
Valageasetal. (2016) for the additional terms associated with the time derivativeof the matter density field. The second pro-
Articlenumber,page1of9
A&Aproofs:manuscriptno.ISW_consistency_relations_Jan15
cess,associatedwiththescatteringofCMBphotonsbyfreeelec- ThesimplestrelationthatonecanobtainfromEq.(1) is for
trons,dependsonthefreeelectronsvelocityfield.Weinvestigate thebispectrumwithn=2,
the cross correlationsof these two secondary anisotropieswith
k ·k
bothgalaxydensityfieldsandthecosmicweaklensingconver- hδ˜(k,η)δ˜ (k ,η )δ˜ (k ,η )i′ =−P (k,η) 1
g 1 1 g 2 2 k→0 L k2
gence.
D(η )−D(η )
Thispaperisorganizedasfollows.Insection2werecallthe ×hδ˜ (k ,η )δ˜ (k ,η )i′ 1 2 , (2)
g 1 1 g 2 2 D(η)
consistencyrelationsoflarge-scalestructuresthatapplytoden-
sity, momentum and momentum-divergence (i.e., time deriva- where we used that k = −k −k → −k . For generality, we
2 1 1
tiveofthedensity)fields.Wedescribethevariousobservational considered here the small-scale fields δ˜ (k ) and δ˜ (k ) to be
g 1 g 2
probesthatweconsiderinthispaperinsection3.Westudythe associated with biased tracers such as galaxies. The tracers as-
ISWeffectinsection4andthekSZeffectinsection5.Wecon- sociatedwithk andk canbedifferentandhavedifferentbias.
1 2
cludeinsection6. Atequaltimestheright-handsideofEq.(2)vanishes,asrecalled
above.
2.2.Consistencyrelationsformomentumcorrelations
2. Consistencyrelationsforlarge-scalestructures
The density consistency relations (1) express the uniform mo-
2.1.Consistencyrelationsfordensitycorrelations
tionofsmall-scalestructuresbylarge-scalemodes.Thissimple
kinematiceffectvanishesforequal-timecorrelationsoftheden-
As described in recent works (Kehagias&Riotto 2013;
sityfield,preciselybecausetherearenodistortions,whilethere
Peloso&Pietroni 2013; Creminellietal. 2013; Kehagiasetal.
is a nonzero effect at different times because of the motion of
2014a;Peloso&Pietroni2014;Creminellietal.2014;Valageas
the small-scale structure between different times. However, as
2014b;Hornetal.2014,2015),itispossibletoobtainexactre-
pointed out in Rizzoetal. (2016), it is possible to obtain non-
lations between density correlations of different orders in the
trivialequal-timesresultsbyconsideringvelocityormomentum
squeezedlimit,wheresomeofthewavenumbersareinthelinear
fields,whicharenotonlydisplacedbutalsoseetheiramplitude
regimeandfarbelowtheothermodesthatmaybestronglynon-
affectedbythelarge-scalemode.Letusconsiderthemomentum
linear. These “kinematic consistencyrelations”,obtainedat the
pdefinedby
leadingorderoverthe large-scalewavenumbers,arise fromthe
equivalenceprincipleandtheassumptionofGaussianprimordial p=(1+δ)v, (3)
perturbations.Theyexpressthefactthat,atleadingorderwhere
where v the peculiarvelocity.Then, in the squeezedlimit k →
a large-scale perturbation corresponds to a linear gravitational
0,thecorrelationbetweenonelarge-scaledensitymodeδ˜(k),n
potential(hencea constantNewtonianforce)overtheextentof
small-scaledensitymodesδ˜(k ),andm small-scalemomentum
a small-size structure,the latter falls withoutdistortionsin this j
modes p˜(k ) can be expressed in terms of (n+m) small-scale
large-scalepotential. j
correlations,as
Then,in the squeezedlimitk → 0, the correlationbetween
one large-scale density mode δ˜(k) and n small-scale density n n+m
modesδ˜(kj)canbeexpressedintermsofthen-pointsmall-scale hδ˜(k,η) δ˜(kj,ηj) p˜(kj,ηj)i′k→0 =−PL(k,η)
correlation,as Yj=1 jY=n+1
n n+m n+m D(η)k ·k
× h δ˜(k ,η ) p˜(k ,η )i′ i i
hδ˜(k,η) n δ˜(k ,η )i′ =−P (k,η)h n δ˜(k ,η )i′ (cid:26) Yj=1 j j jY=n+1 j j Xi=1 D(η) k2
j j k→0 L j j
Yj=1 Yj=1 n+m (dD/dn)(η) n i−1
+ i h δ˜(k ,η ) p˜(k ,η )
n D(η)k ·k D(η) j j j j
× i i , (1) i=Xn+1 Yj=1 jY=n+1
Xi=1 D(η) k2 k n+m
× i [δ (k)+δ˜(k,η)] p˜(k ,η )i′ . (4)
where the tilde denotesthe Fouriertransformof the fields, η is k2 D i i i !jY=i+1 j j (cid:27)
the conformaltime, D(η) is the linear growth factor, the prime These relations are again valid in the nonlinearregime and for
in h...i′ denotes that we factored out the Dirac factor, h...i = biased galaxy fields δ˜ (k ) and p˜ (k ). As for the density con-
g j g j
h...i′δD( kj),andPL(k)isthelinearmatterpowerspectrum.It sistencyrelation(1),thefirsttermvanishesatthisorderatequal
isworthsPtressingthattheserelationsarevalideveninthenon- times.Thesecondtermhowever,whicharisesfromthep˜ fields
linearregimeandforbiasedgalaxyfieldsδ˜g(kj).Theright-hand only, remains nonzero. This is due to the fact that p˜ involves
sidegivesthesqueezedlimitofthe(1+n)correlationatthelead- the velocity, the amplitude of which is affected by the motion
ingorder,whichscalesas1/k.Itvanishesatthisorderatequal inducedbythelarge-scalemode.
times,becauseoftheconstraintassociatedwiththeDiracfactor The simplest relation associated with Eq.(4) is the bispec-
δD( kj). trum among two density-contrast fields and one momentum
PThe geometrical factors (ki · k) vanish if ki ⊥ k. Indeed, field,
the large-scalemodeinducesa uniformdisplacementalongthe hδ˜(k,η)δ˜ (k ,η )p˜ (k ,η )i′ =−P (k,η)
directionofk.Thishasnoeffectonsmall-scaleplanewavesof g 1 1 g 2 2 k→0 L
k ·k D(η )−D(η )
waadvisepnluamcebmeresnkt.iTwhitehrekfio⊥re,kt,haesttehremysreimnathineirdiegnhtti-chaalnadftesirdseucohf ×(cid:18) 1k2 hδ˜g(k1,η1)p˜g(k2,η2)i′ 1D(η) 2 +
Eq.(1)mustvanishinsuchorthogonalconfigurations,aswecan k 1 dD
+i hδ˜ (k ,η )δ˜ (k ,η )i′ (η ) . (5)
checkfromtheexplicitexpression. k2 g 1 1 g 2 2 D(η) dη 2 (cid:19)
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LucaAlbertoRizzo etal.:ConsistencyrelationsfortheISWandkSZeffects
For generality,we consideredherethe small-scale fields δ˜ (k ) 3. Observablequantities
g 1
andp˜ (k )tobeassociatedwithbiasedtracerssuchasgalaxies,
andthgetr2acersassociatedwithk andk canagainbedifferent Totestcosmologicalscenarioswiththeconsistencyrelationsof
andhavedifferentbias.Atequalt1imesE2q.(5)readsas large-scale structures we need to relate them with observable
quantities.We describe in this section the observationalprobes
k dlnD thatweconsiderinthispaper.Weusethegalaxynumberscounts
hδ˜(k)δ˜g(k1)p˜g(k2)i′k→0 =−ik2 dη PL(k)Pg(k1), (6) or the weak lensing convergenceto probe the density field. To
applythemomentumconsistencyrelations(6)and(10),weuse
where P (k) is the galaxy nonlinear power spectrum and we theISWeffecttoprobethemomentumdivergenceλ(morepre-
g
omittedthecommontimedependence.Thisresultdoesnotvan- ciselythetimederivativeofthegravitationalpotentialandmatter
ishthankstothetermgeneratedbyp˜ intheconsistencyrelation density)andthekSZeffecttoprobethemomentump.
(5).
3.1.Galaxynumberdensitycontrastδ
g
2.3.Consistencyrelationsformomentum-divergence
Fromgalaxysurveyswe cantypicallymeasurethegalaxyden-
correlations
sity contrast within some redshift bin, smoothed with some
Inadditiontothemomentumfieldp,wecanconsideritsdiver- finite-sizewindowonthesky,
genceλ,definedby
δgs(θ)= dθ′WΘ(|θ′−θ|) dηIg(η)δg[r,rθ′;η], (11)
∂δ Z Z
λ≡∇·[(1+δ)v]=− . (7)
∂η whereWΘ(|θ′−θ|)isa2Dsymmetricwindowfunctioncentered
onthedirectionθonthesky,ofcharacteristicangularradiusΘ,
The second equality expresses the continuity equation, that is, I (η)istheradialweightalongthelineofsightassociatedwith
g
theconservationofmatter.Inthesqueezedlimitweobtainfrom anormalizedgalaxyselectionfunctionn (z),
g
Eq.(4)(Rizzoetal.2016)
dz
n n+m Ig(η)=(cid:12)dη(cid:12)ng(z), (12)
hδ˜(k,η)Yj=1δ˜(kj,ηj)jY=n+1λ˜(kj,ηj)i′k→0 =−PL(k,η) r = η0 −(cid:12)(cid:12)(cid:12)(cid:12) η(cid:12)(cid:12)(cid:12)(cid:12)is the radial comovingcoordinate along the line of
sight, and η is the conformaltime today. Here and in the fol-
× h n δ˜(k ,η ) n+m λ˜(k ,η )i′n+m D(ηi)ki·k lowingweu0setheflatskyapproximation,andθisthe2Dvector
(cid:26) Yj=1 j j jY=n+1 j j Xi=1 D(η) k2 thatdescribesthedirectionontheskyofagivenlineofsight.The
superscript“s”inδs denotesthatwesmooththegalaxydensity
n+m n n+m g
− hδ˜(k,η) δ˜(k ,η ) λ˜(k ,η )i′ contrast with the finite-size window WΘ. Expandingin Fourier
i i j j j j spacethegalaxydensitycontrastwecanwrite
i=Xn+1 Yj=1 jY=n+1
j,i
×(dD/dη)(ηi)ki·k . (8) δgs(θ) = Z dθ′WΘ(|θ′−θ|)Z dηIg(η)
D(η) k2 (cid:27)
× dkeikkr+ik⊥·rθ′δ˜ (k,η) (13)
g
These relations can actually be obtained by taking derivatives Z
withrespecttothetimesηj ofthedensityconsistencyrelations wherekkandk⊥arerespectivelytheparallelandtheperpendicu-
(1),usingthesecondequality(7).Asforthemomentumconsis- larcomponentsofthe3Dwavenumberk=(k ,k )(withrespect
k ⊥
tencyrelations(4),theserelationsremainvalidinthenonlinear tothereferencedirectionθ =0,andweworkinthesmall-angle
regimeand for biased small-scale fields δ˜g(kj) and λ˜g(kj). The limitθ ≪ 1).Definingthe2DFouriertransformofthewindow
second term in Eq.(8), which arises from the λ˜ fields only, re- WΘ as
mains nonzeroat equaltimes. This is due to the fact that λ in-
volves the velocity or the time-derivativeof the density, which W˜Θ(|ℓ|)= dθe−iℓ·θWΘ(|θ|), (14)
probes the evolution between (infinitesimally close) different Z
times. weobtain
The simplest relation associated with Eq.(8) is the bispec-
trum among two density-contrast fields and one momentum- δgs(θ)= dηIg(η) dkW˜Θ(k⊥r)eikkr+ik⊥·rθδ˜g(k,η). (15)
divergencefield, Z Z
k ·k
hδ˜(k,η)δ˜ (k ,η )λ˜ (k ,η )i′ =−P (k,η) 1 3.2.Weaklensingconvergenceκ
g 1 1 g 2 2 k→0 L k2
D(η )−D(η ) From weak lensing surveys we can measure the weak lensing
×(cid:18)hδ˜g(k1,η1)λ˜g(k2,η2)i′ 1D(η) 2 + convergence,givenintheBornapproximationby
1 dD Ψ+Φ
+hδ˜g(k1,η1)δ˜g(k2,η2)i′D(η) dη(η2)(cid:19). (9) κs(θ)=Zdθ′WΘ(|θ′−θ|)Zdηrg(r)∇2 2 [r,rθ′;η], (16)
whereΨandΦaretheNewtoniangaugegravitationalpotentials
Atequaltimes,Eq.(9)readsas
andthekernelg(r)thatdefinestheradialdepthofthesurveyis
hδ˜(k)δ˜g(k1)λ˜g(k2)i′k→0 =−k1k2·kddlnηDPL(k)Pg(k1). (10) g(r)= ∞drsdzsng(zs)rs−r, (17)
Zr drs rs
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A&Aproofs:manuscriptno.ISW_consistency_relations_Jan15
where n (z ) is the redshift distribution of the source galaxies. 1998). Thetemperatureperturbation,∆ = δT/T,dueto this
g s kSZ
Assumingnoanisotropicstress,i.e.Φ = Ψ,andusingthePois- kinematicSunyaevZeldovich(kSZ)effect,is
sonequation,
∇2Ψ=4πGNρ¯0δ/a, (18) ∆kSZ(θ)=−Zdl·veσTnee−τ =ZdηIkSZ(η)n(θ)·pe, (26)
whereG istheNewtonconstant,ρ¯ isthemeanmatterdensity
N 0
where τ is again the optical depth, σ the Thomsoncross sec-
oftheUniversetoday,andaisthescalefactor,weobtain T
tion,ltheradialcoordinatealongthelineofsight,n thenumber
e
densityoffreeelectrons,v theirpeculiarvelocity,andn(θ)the
κs(θ)= dηIκ(η) dkW˜Θ(k⊥r)eikkr+ik⊥·rθδ˜(k,η), (19) radial unit vector pointingeto the line of sight. We also defined
Z Z
thekSZkernelby
with
I (η)=−σ n¯ ae−τ, (27)
rg(r) kSZ T e
I (η)=4πG ρ¯ . (20)
κ N 0
a
andthefreeelectronsmomentump as
e
3.3.ISWsecondaryanisotropy∆ISW n v =n¯ (1+δ )v =n¯ p . (28)
e e e e e e e
FromEq.(7)λcanbeobtainedfromthemomentumdivergence
orfromthetimederivativeofthedensitycontrast.Thesequanti- Because of the projection n · p along the line of sight, some
e
ties are not as directly measured from galaxy surveys as den- care must be taken when we smooth ∆ (θ) over some finite
kSZ
sity contrasts. However, we can relate the time derivative of sizeangularwindowWΘ(|θ′−θ|).Indeed,becausethedifferent
the density contrast to the ISW effect, which involves the time linesofsightθ′intheconicalwindowarenotperfectlyparallel,
derivative of the gravitational potential. Indeed, the secondary ifwedefinethelongitudinalandtransversemomentumcompo-
cosmic microwave background temperature anisotropy due to nents by the projection with respect to the mean line of sight
theintegratedSachs-Wolfeeffectalongthedirectionθ readsas n(θ) ofthe circularwindow,e.g. p = n(θ)·p , the projection
ek e
(Garrigaetal.2004) n(θ′)· p receives contributions from both p and p . In the
e ek e⊥
limit of small angleswe could a priorineglectthe contribution
∆ (θ) = dηe−τ(η) ∂Ψ + ∂Φ [r,rθ;η] associatedwithpe⊥,whichismultipliedbyanangularfactorand
ISW Z ∂η ∂η! vanishes for a zero-size window. However, for small but finite
∂Ψ angles, we need to keep this contribution because fluctuations
= 2 dηe−τ(η) [r,rθ;η], (21) alongthelinesofsightaredampedbytheradialintegrationsand
Z ∂η
vanishintheLimberapproximation,whichdampsthecontribu-
whereτ(η)istheopticaldepth,whichtakesintoaccountthepos- tionassociatedwith pek.
sibility of late reionization,and in the second line we assumed For small angles we write at linear order n(θ) = (θx,θy,1),
noanisotropicstress,i.e.Φ=Ψ.Wecanrelate∆ toλthrough close to a referencedirectionθ = 0. Then,the integrationover
ISW
thePoissonequation(18),whichreadsinFourierspaceas theangularwindowgivesforthesmoothedkSZeffect
−k2Ψ˜ =4πG ρ¯ δ˜/a. (22)
N 0 ∆ksSZ(θ) = Z dηIkSZ(η)Z dkeik·nr(cid:20)p˜ekW˜Θ(k⊥r)
Thisgives
k ·p˜
−i ⊥ e⊥W˜′(k r) . (29)
Θ ⊥
∂Ψ˜ 4πG ρ¯ k⊥ (cid:21)
= N 0(λ˜ +Hδ˜), (23)
∂η k2a
Here we expressed the result in terms of the longitudinal and
transversecomponentsofthewavenumbersandmomentawith
whereH = dlna/dηistheconformalexpansionrate.Integrat-
ingtheISWeffectδ oversomefinite-sizewindowonthesky, respecttothemeanlineofsightn(θ)ofthecircularwindowWΘ.
ISW Thus,whereastheradialunitvectorisn(θ) = (θ ,θ ,1),wecan
weobtainasinEq.(15) x y
definethetransverseunitvectorsasn = (1,0,−θ )andn =
⊥x x ⊥y
(0,1,−θ ),andwewriteforinstancek=k n +k n +k n.
∆IsSW(θ) = Z dηIISW(η)Z dkW˜Θ(k⊥r)eikkr+ik⊥·rθ WedenoyteW˜Θ′(ℓ) = dW˜Θ/dℓ.Thelastterm⊥xin⊥Exq.(2⊥9y)i⊥syduekto
thefinitesizeΘofthesmoothingwindow,whichmakesthelines
λ˜ +Hδ˜
× , (24) ofsightwithintheconicalbeamnotstrictlyparallel.Itvanishes
k2 foraninfinitesimalwindow,whereWΘ(θ) = δD(θ)andW˜Θ = 1,
W˜′ =0.
with Θ
e−τ
I (η)=8πG ρ¯ . (25)
ISW N 0
a 4. ConsistencyrelationfortheISWtemperature
anisotropy
3.4.KinematicSZsecondaryanisotropy∆
kSZ In this section we consider cross correlationswith the ISW ef-
ThomsonscatteringofCMBphotonsoff movingfreeelectrons fect.Thisallowsustoapplytheconsistencyrelation(9),which
inthehotgalacticorclustergasgeneratessecondaryanisotropies involves the momentum divergence λ and remains nonzero at
(Sunyaev&Zeldovich 1980; Gruzinov&Hu 1998; Knoxetal. equaltimes.
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LucaAlbertoRizzo etal.:ConsistencyrelationsfortheISWandkSZeffects
4.1.Galaxy-galaxy-ISWcorrelation whereǫ˜isastochasticcomponentthatrepresentsshotnoiseand
the effect of small-scale (e.g., baryonic)physicson galaxyfor-
Totakeadvantageoftheconsistencyrelation(9),wemustcon-
mation.Fromthedecomposition(35),itisuncorrelatedwiththe
sider three-point correlations ξ3 (in configuration space) with large-scale density field (Hamausetal. 2010), hδ˜(k)ǫ˜(k)i = 0.
oneobservablethatinvolvesthemomentumdivergenceλ.Here, Then, in Eq.(34) we neglected the term hǫ˜δ˜ (λ˜ +H δ˜ )i. In-
using the expression (24), we study the cross-correlation be- deed,thesmall-scalelocalprocesseswithingth1er2egion2θ2should
tween two galaxy density contrasts and one ISW temperature
be very weakly correlated with the density fields in the dis-
anisotropy, tant regions θ and θ , which at leading order are only sensi-
1 2
ξ (δs,δs ,∆s )=hδs(θ)δs (θ )∆s (θ )i. (30) tivetothetotalmasswithinthelarge-scaleregionθ.Therefore,
3 g g1 ISW2 g g1 1 ISW2 2 hǫ˜δ˜g1(λ˜2+H2δ˜2)ishouldexhibitafastdecayatlowk,whereas
theterminEq.(34)associatedwiththeconsistencyrelationonly
Thesubscriptsg,g ,andISW denotethethreelinesofsightas-
sociatedwiththeth1reeprobes.2Moreover,thesubscriptsgandg decaysas PL(k)/k ∼ kns−1 with ns ≃ 0.96. In Eq.(34), we also
1
recallthatthetwogalaxypopulationsassociatedwithδs andδs assumedthatthegalaxybiasbggoestoaconstantatlargescales,
can be differentand have differentbias. As we recalledg in secg1- whichisusuallythecase,butwecouldtakeintoaccountascale
dependence [by keeping the factor b (k,η) in the integral over
tion 2, the consistencyrelations rely on the undistortedmotion g
k].
ofsmall-scalestructuresbylarge-scalemodes.Thiscorresponds
to the squeezed limit k → 0 in the Fourier-spaceequations(1) Thesmall-scaletwo-pointcorrelationsh1·2i′aredominated
and(8),whichwritesmorepreciselyas bycontributionsatalmostequaltimes,η1 ≃η2,asdifferentred-
shifts would correspond to points that are separated by several
k≪k , k ≪k , (31) Hubbleradiialongthelinesofsightanddensitycorrelationsare
L j
negligiblebeyondHubblescales.Therefore,ξ isdominatedby
3
where k is the wavenumber associated with the transition be- thesecondtermthatdoesnotvanishatequaltimes.Theintegrals
L
tween the linear and nonlinear regimes. The first condition en- alongthelinesofsightsuppressthecontributionsfromlongitu-
sures that δ˜(k) is in the linear regime, while the second condi- dinalwavelengthsbelowtheHubbleradiusc/H,whiletheangu-
tionensuresthehierarchybetweenthelarge-scalemodeandthe larwindowsonlysuppressthewavelengthsbelowthetransverse
small-scalemodes.Inconfigurationspace,theseconditionscor- radiicΘ/H. Then, for small angular windows, Θ ≪ 1, we can
respondto useLimber’sapproximation,k ≪ k hencek ≃ k .Integrating
k ⊥ ⊥
overk throughtheDiracfactorδ (k +k +k ),andnextover
Θ≫ΘL, Θ≫Θj, |θ−θj|≫|θ1−θ2|. (32) k1kandk k2k,weobtaintheDiracfaDctorks(2π1k)2δD2(rk1−r)δD(r2−r).
Thisallowsustointegrateoverη andη andweobtain
The first condition ensures that δs(θ) is in the linear regime, 1 2
g
whereasthenexttwoconditionsensurethehierarchyofscales. dlnD
ξ = −(2π)2 dηb (η)I (η)I (η)I (η)
Theexpressions(15)and(24)give 3 Z g g g1 ISW2 dη
ξ3 = Z dηdη1dη2Ig(η)Ig1(η1)IISW2(η2) ×Z dk⊥dk1⊥dk2⊥δD(k⊥+k1⊥+k2⊥)W˜Θ(k⊥r)
×Z dkdk1dk2W˜Θ(k⊥r)W˜Θ1(k1⊥r1)W˜Θ2(k2⊥r2) ××WP˜Θ(1k(k,1⊥η)r)kW1˜⊥Θ2·(kk⊥2⊥Pr)eir((kk⊥·θ+,kη1)⊥,·θ1+k2⊥·θ2) (36)
×ei(kkr+k1kr1+k2kr2+k⊥·rθ+k1⊥·r1θ1+k2⊥·r2θ2) L ⊥ k⊥2k22⊥ g1,m 1⊥
λ˜(k ,η )+H δ˜(k ,η )
×hδ˜g(k,η)δ˜g1(k1,η1) 2 2 k2 2 2 2 i. (33) twiohneroevePrg1k,m isgitvheesgalaxy-matter power spectrum. The integra-
2 2⊥
The configuration-space conditions (32) ensure that we satisfy dlnD
theFourier-spaceconditions(31)andthatwecanapplythecon- ξ3 = −(2π)2ZdηbgIgIg1IISW2 dη Zdk⊥dk1⊥W˜Θ(k⊥r)
sistencyrelations(2)and(9).Thisgives
×W˜Θ1(k1⊥r)W˜Θ2(k1⊥r)PL(k⊥,η)Pg1,m(k1⊥,η)
ξ3 = −Z dηdη1dη2bg(η)Ig(η)Ig1(η1)IISW2(η2) ×eir[k⊥·(θ−θ2)+k1⊥·(θ1−θ2)]kk1⊥2 ·kk2⊥, (37)
1⊥ ⊥
×Z dkdk1dk2W˜Θ(k⊥r)W˜Θ1(k1⊥r1)W˜Θ2(k2⊥r2) andtheintegrationovertheanglesofk⊥ andk1⊥gives
×ei(kkr+k1kr1+k2kr2+k⊥·rθ+k1⊥·r1θ1+k2⊥·r2θ2) (θ−θ )·(θ −θ ) dlnD
×P (k,η)k1·kδ (k+k +k ) ξ3 = |θ−θ22||θ11−θ2|2 (2π)4Z dηbgIgIg1IISW2 dη
L k2 D 1 2 ∞
× hδ˜g1λ˜2+kH2 2δ˜2i′ D(η1D)−(ηD) (η2) ××PZ0(kd,kη⊥)dPk1⊥W(˜kΘ(,kη⊥)rJ)W˜(kΘ1(rk|θ1⊥−r)θW˜|Θ)2(k1⊥r)
2 L ⊥ g1,m 1⊥ 1 ⊥ 2
δ˜ 1 dD ×J (k r|θ −θ |), (38)
+hδ˜ 2i′ (η ) . (34) 1 1⊥ 1 2
g1k2 D(η) dη 2 !
2 whereJ isthefirst-orderBesselfunctionofthefirstkind.
1
Astheexpression(38)arisesfromthekinematicconsistency
Hereweassumedthatonlargescalesthegalaxybiasislin-
ear, relations,itexpressestheresponseofthesmall-scale two-point
correlationhδs (θ )∆s (θ )itoachangeoftheinitialcondition
k→0: δ˜ (k)=b (η)δ˜(k)+ǫ˜(k), (35) associatedwitgh1 th1elaIrSgWe2-sc2alemodeδs(θ).Thekinematiceffect
g g g
Articlenumber,page5of9
A&Aproofs:manuscriptno.ISW_consistency_relations_Jan15
givenattheleadingorderbyEq.(38)isduetotheuniformmo- readsas
tion of the small-scale structuresby the large-scalemode.This
∞
e1x.pl(aθin−sθwh)y⊥th(eθre−suθlt)(.3T8)hveraeniisshaesnionntzheerotwreosfpoolnlosweionfghcδasλesi ξ2 = (2π)2Z dηIg1Iκ2Z0 dk1⊥k1⊥F˜Θ1(k1⊥r)
2 1 2 1 2
ififrsthtedreeriivsaativlieniesanrodnezpeernod.eAncpeosointivδe(θ()neogfahtδiv1eλ)2δi,(θs)olethaadtsittos ×F˜Θ2(k1⊥r)J0(k1⊥r|θ1−θ2|)Pg1,m(k1⊥), (40)
auniformmotionatθ towards(awayfrom)θ,alongthedi- whereweagainusedLimber’sapproximation.Herewedenoted
2
rection(θ−θ ).From the pointofview ofθ andθ , there theangularsmoothingwindowsbyF˜ todistinguishξ fromξ .
2 1 2 2 3
is a reflection symmetry with respect to the axis (θ −θ ). Then,wecanwrite
1 2
For instance, if δ > 0 the density contrastat a position θ
1 3
dftyoisprti∆acnaθcl2ley⊥fdro(eθmc1reθ−asθea2sn)ditnhhetahvpeeomitnheteasnsθa±3wm=iethdθte2hn±esir∆tayθdc2iuoasnrte|θraa3st−ttδhθe1i|sn,aamtnhdee ξ3 = (θ|θ−−θθ2)2|·|θ(1θ1−−θθ2|2)ξ2, (41)
1 3
mean,withtypicallyδ < δ as|θ±−θ | > |θ −θ |.There-
3 2 3 1 2 1 if the angular windowsof the two-pointcorrelation are chosen
fore, the large-scale flow along (θ −θ ) leads to a positive
2 suchthat
λ = −∆δ /∆η independentlyofwhetherthemattermoves
2 2 2
tTohwisarmdesaonrsatwhaatytfhreodmepθe(nhdeerencweeotfohoδk1λa2fiinointeδd(θe)viisatqiounad∆rθa2ti)c. F˜Θ1(k1⊥r)F˜Θ2(k1⊥)=(2π)2IgIIISW2bgddlnηD
[itdoesnotdependonthesignofδ(θ)]andthefirst-orderre- κ2
∞
tsipoonntsoeξf3unvcatnioisnhevsa.n[iFsohresi.nfiTnhietnes,itmhealledaedviinagti-oonrd∆eθr2cwonetrhiabvue- × Z0 dk⊥W˜Θ(k⊥r)J1(k⊥r|θ−θ2|)PL(k⊥,η)!
eλx2tr=em−u∂mδ2o/∂fηth2e=de0n;sbityytchoinstsryasmtmaleotnrgy,tihnetohrethmogeaonnaδl2diisreacn- ×W˜Θ1(k1⊥r)W˜Θ2(k1⊥r)J1(k1⊥r|θ1−θ2|). (42)
tionto(θ −θ ).] k1⊥J0(k1⊥r|θ1−θ2|)
1 2
2. θ = θ . Thisisa particularcaseofthepreviousconfigura-
1 2
tion.Again,bysymmetryfromtheviewpointofδ1,thetwo ThisimpliesthattheangularwindowsF˜Θ1 and F˜Θ2 ofthetwo-
pointsδ(θ2+∆θ2)andδ(θ2−∆θ2)areequivalentandthemean pointcorrelationξ2 haveanexplicitredshiftdependence.
responseassociatedwiththekinematiceffectvanishes. Inpractice,theexpression(42)maynotbeveryconvenient.
Then,tousetheconsistencyrelation(38)itmaybemoreprac-
Thisalso explainswhyEq.(38)changessignwith (θ −θ )
1 2 tical to first measure the power spectra P and P indepen-
and (θ−θ2). Let us consider for simplicity the case where the dently,at the redshiftsneeded for the integLralalongg1,mthe line of
three points are aligned and δ(θ) > 0, so that the large-scale
sight(38),andnextcomparethemeasureofξ withtheexpres-
flowpointstowardsθ.Wealsotakeδ > 0,sothatinthemean 3
1 sion(38)computedwiththesepowerspectra.
the densityispeakedatθ anddecreasesoutwards.Letustake
1
θ close to θ , on the decreasing radial slope, and on the other
2 1
sideofθ1 thanθ.Then,thelarge-scaleflowmovesmatteratθ2 4.3.Lensing-lensing-ISWcorrelation
towardsθ ,sothatthedensityatθ ataslightlylatertimecomes
1 2
frommoreoutwardregions(withrespecttothepeakatθ )with From Eq.(38) we can directly obtain the lensing-lensing-ISW
1
a lower density. This means that λ = −∂δ /∂η is positive so three-pointcorrelation,
2 2 2
thatξ > 0.ThisagreeswithEq.(38),as(θ−θ )·(θ −θ ) > 0
3 2 1 2
inthisgeometry,andweassumetheintegralsoverwavenumbers ξ (κs,κs,∆s )=hκs(θ)κs(θ )∆s (θ )i, (43)
aredominatedbythepeaksof J > 0.Ifweflipθ totheother 3 1 ISW2 1 1 ISW2 2
1 2
sideofθ ,wefindonthecontrarythatthelarge-scaleflowbrings
1 byreplacingthegalaxykernelsb I andI bythelensingcon-
higher-densityregionstoθ ,sothatwehavethechangeofsigns g g g1
2 vergencekernelsI andI ,
λ < 0 and ξ < 0. The same arguments explain the change κ κ1
2 3
ofsignwith(θ−θ2).Infact,itistherelativedirectionbetween (θ−θ )·(θ −θ ) dlnD
((θθ−−θθ2))a·n(dθ(θ−1−θθ)2.)thatmatters,measuredbythescalarproduct ξ3 = |θ−θ22||θ11−θ2|2 (2π)4Z dηIκIκ1IISW2 dη
2 1 2 ∞
tionTtohiξs3gceooumldetprricoavliddeepaesnimdepnlceeteosfttohfethleeacdoinngs-isotrednecrycroenlatrtiiboun-, ×Z0 dk⊥dk1⊥W˜Θ(k⊥r)W˜Θ1(k1⊥r)W˜Θ2(k1⊥r)
withoutevencomputingtheexplicitexpressionintheright-hand ×P (k ,η)P(k ,η)J (k r|θ−θ |)
L ⊥ 1⊥ 1 ⊥ 2
sideofEq.(38). ×J (k r|θ −θ |). (44)
1 1⊥ 1 2
4.2.Three-pointcorrelationintermsofatwo-point As compared with Eq.(38), the advantage of the cross-
correlation correlationwiththeweaklensingconvergenceκisthatEq.(44)
involvesthe matterpower spectrum P(k ) instead of the more
1⊥
Thethree-pointcorrelationξ in Eq.(38)cannotbe writtenasa
3 complicatedgalaxy-mattercrosspowerspectrumP (k ).
productof two-point correlationsbecause there is only one in- g1,m 1⊥
tegral along the line of sight that is left. However, if the linear
power spectrum P (k,z) is already known, we may write ξ in
L 3
5. ConsistencyrelationforthekSZeffect
termsofsometwo-pointcorrelationξ .Forinstance,thesmall-
2
scalecross-correlationbetweenonegalaxydensitycontrastand
In this section we consider cross correlations with the kSZ ef-
oneweaklensingconvergence,
fect.Thisallowsustoapplytheconsistencyrelation(5),which
ξ (δs ,κs)=hδs (θ )κs(θ )i (39) involvesthemomentumpandremainsnonzeroatequaltimes.
2 g1 2 g1 1 2 2
Articlenumber,page6of9
LucaAlbertoRizzo etal.:ConsistencyrelationsfortheISWandkSZeffects
5.1.Galaxy-galaxy-kSZcorrelation we used P (k,η) = D(η)2P (k). If we approximate the three
L L0
linesofsightasparallel,wecanwriten ·k=k ,wherethelon-
Inafashionsimilartothegalaxy-galaxy-ISWcorrelationstudied 2 k
gitudinalandtransversedirectionscoincideforthethreelinesof
in section 4.1, we consider the three-pointcorrelation between
sight.Then,Limber’sapproximation,which correspondsto the
twogalaxydensitycontrastsandonekSZCMBanisotropy,
limitwheretheradialintegrationshaveaconstantweightonthe
ξ (δs,δs ,∆s )=hδs(θ)δs (θ )∆s (θ )i, (45) infinite real axis, gives a Dirac term δD(kk) and ξ3k = 0 [more
3 g g1 kSZ2 g g1 1 kSZ2 2 precisely, as we recalled above Eq.(36), the radial integration
in the squeezed limit given by the conditions (31) in Fourier givesk . H/cwhiletheangularwindowgivesk . H/(cΘ)so
k ⊥
spaceand(32)inconfigurationspace.Theexpressions(15)and thatk ≪k ].Takingintoaccountthesmallanglesbetweenthe
k ⊥
(29)give differentlinesofsight,asforthederivationofEq.(29),theinte-
grationoverk throughtheDiracfactorgivesatleadingorderin
ξ3 =ξ3k+ξ3⊥ (46) theangles 2
with dD
ξ = − dηdη dη b (η)I (η)D(η)I (η )I (η ) (η )
3k Z 1 2 g g g1 1 kSZ2 2 dη 2
ξ = dηdη dη I (η)I (η )I (η ) dkdk dk
3k ××We˜i(ZΘk·2n(rk+2(kn⊥12·)nr121r1)+hkδ˜22g·n(k2gr,2)ηW˜)δ˜Θgg1(1k(k⊥(1n1),rk)ηSW1˜Z)2Θp˜1(e(nk2k2)1()n⊥Z(1k)r21,)η2)i1 2 (47) ×××ZePi[dkkk((rkk−drk2))P⊥+kd⊥k·(1θ(k−kdθk2)1r;2⊥η+kW˜1,k(ηΘr1(−)kir⊥2k)r+k)k+W1˜⊥Θk·(θ1⊥1(−k·θ1(2⊥θ)rr221]−)W˜θΘ)2.(k1⊥r2) (51)
and L0 ⊥ g1,e 1⊥ 1 2 k2
⊥
ξ = −i dηdη dη I (η)I (η )I (η ) dkdk dk WeusedLimber’sapproximationtowriteforinstancePL0(k) ≃
3⊥ Z 1 2 g g1 1 kSZ2 2 Z 1 2 PL0(k⊥),butwe keptthe factor kk in the lastterm, as the trans-
×ei(k·nr+k1·n1r1+k2·n2r2)W˜Θ(k⊥(n)r)W˜Θ1(k1(n⊥1)r1) voefrssieghfatcntoarnkd⊥n·(θ,2is−sθu)p,pdrueesstoedthbeysmthaellsmanagllleabnegtlwee|θen−thθe|l.inWees
2 2
×W˜′ (k(n2)r )hδ˜ (k,η)δ˜ (k ,η )k(2n⊥2)·p˜(en⊥2)(k ,η )i, againsplitξ3k overtwocontributions,ξ3k =ξ3kk+ξ3⊥k,associated
Θ2 2⊥ 2 g g1 1 1 k(n2) 2 2 withthefactorskk andk⊥·(θ2−θ)ofthelastterm.Letusfirst
2⊥ (48) consider the contribution ξ3kk. Writing ikkeikk(r−r2) = ∂∂reikk(r−r2),
weintegratebypartsoverη. Forsimplicitywe assume thatthe
where we split the longitudinal and transverse contributions galaxyselectionfunctionI vanishesatz=0,
g
to Eq.(29). Here {n,n ,n } are the radial unit vectors that
point to the centers {θ,1θ1,2θ2} of the three circular windows, Ig(η0)=0, (52)
and{(kk(n),k(⊥n)),(k1(nk1),k(1n⊥1)),(k2(nk2),k(2n⊥2))}arethelongitudinaland so thatthe boundaryterm atz = 0 vanishes. Then,the integra-
transverse wave numberswith respect to the associated central tionsoverk andk giveafactor(2π)2δ (r−r )δ (r −r ),and
k 1k D 2 D 1 2
linesofsight[e.g.,k(n) =n·k]. wecanintegrateoverηandη .Finally,theintegrationoverthe
k 1
Thecomputationof the transversecontribution(48) is sim- anglesofthetransversewavenumbersyields
ilar to the computationof the ISWthree-pointcorrelation(34),
d dD
usingagainLimber’sapproximation.Atlowestorderweobtain ξk = −(2π)4 dη b I D I I
3k Z dη g g g1 kSZ2 dη
ξ3⊥ = (θ|θ−−θθ1)1|·|θ(θ22−−θθ1|1)(2π)4Z dηbgIgIg1IkSZ2ddlnηD ×Z ∞dk⊥dk1⊥W˜Θ(kh⊥r)W˜Θi1(k1⊥r)W˜Θ2(k1⊥r)
0
∞
×Z0 dk⊥dk2⊥k2⊥W˜Θ(k⊥r)W˜Θ1(k2⊥r)W˜Θ′2(k2⊥r) ×kk1⊥⊥PL0(k⊥)Pg1,e(k1⊥,η)J0(k⊥r|θ−θ2|)
×PL(k⊥,η)Pg1,e(k2⊥,η)J1(k⊥r|θ−θ1|) ×J0(k1⊥r|θ1−θ2|), (53)
×J (k r|θ −θ |), (49)
1 2⊥ 2 1 where J is the zeroth-order Bessel function of the first kind.
0
whereP isthegalaxy-freeelectronscrosspowerspectrum. For the transverse contributionξ⊥ we can proceed in the same
g1,e 3k
The computation of the longitudinal contribution (47) re- fashion,withoutintegrationbypartsoverη.Thisgives
quiresslightlymorecare.Applyingtheconsistencyrelation(5)
dD
gives ξ⊥ = −(2π)4 dηb I I I D
3k Z g g g1 kSZ2 dη
ξ3k = −Z dηdη1dη2bg(η)Ig(η)Ig1(η1)IkSZ2(η2) ×Z ∞dk⊥dk1⊥W˜Θ(k⊥r)W˜Θ1(k1⊥r)W˜Θ2(k1⊥r)
0
×Z dkdk1dk2W˜Θ(k⊥(n)r)W˜Θ1(k1(n⊥1)r1)W˜Θ2(k2(n⊥2)r2) ××kJ1⊥(kPL0r(|kθ⊥)−Pgθ1,e|)(.k1⊥,η)|θ−θ2|J1(k⊥r|θ−θ2|) (54)
0 1⊥ 1 2
dD
×ei(k·nr+k1·n1r1+k2·n2r2)D(η)PL0(k)dη(η2) ComparingEq.(54)withEq.(53),wefindξ3⊥k/ξ3kk ∼ k⊥r|θ−θ2|.
×in2·khδ˜ δ˜ i′ δ (k+k +k ), (50) Ifthecutoffonk⊥issetbytheBesselfunctions,weobtainξ3⊥k ∼
k2 g1 e2 D 1 2 ξk .Forverysmallangles,|θ−θ |→0,thecutoffoverkissetby
3k 2
whereweonlykeptthecontributionthatdoesnotvanishatequal theangularwindowW˜Θ(k⊥r)orbythefalloffofthelinearpower
times,asitdominatestheintegralsalongthelinesofsight,and spectrumP (k ),andξ⊥ ≪ξk .
L0 ⊥ 3k 3k
Articlenumber,page7of9
A&Aproofs:manuscriptno.ISW_consistency_relations_Jan15
In contrast with Eq.(38), the kSZ three-point correlation, The application of the relations above is, unfortunately, a
given by the sum of Eqs.(49), (53) and (54), does not van- nontrivial task in terms of observations: to test those relations
ishfororthogonaldirectionsbetweenthesmall-scaleseparation one would require the mixed galaxy (matter) - free electrons
(θ −θ )andthelarge-scaleseparation(θ−θ ).Indeed,thelead- powerspectrum.Onepossibilitywouldbetodoastackinganal-
1 2 2
ing order contribution in the squeezed limit to the response of ysis of several X-rays observations of the hot ionised gas by
hδ p i to a large-scale perturbation δ factors out as hδ δ iv , measuringthebremsstrahlungeffect.Forinstance,onecouldin-
1 2 1 2 2
where we only take into accountthe contributionthat does not fer n n T−1/2, by making some reasonable assumptions about
e p
vanishatequaltimes(andwe discardthe finite-sizesmoothing theplasmastate,asperformedinFraser-McKelvieetal.(2011),
effects).Theintrinsicsmall-scalecorrelationhδ δ idoesnotde- withtheaimtomeasuren infilaments.Wewouldofcourseneed
1 2 e
pend on the large-scale mode δ, whereas v is the almost uni- to cover a large range of scales. For kpc scales, inside galax-
2
formvelocitydueto the large-scalemode,whichonlydepends ies and in the intergalacticmedium, one coulduse for instance
onthedirectiontoδ(θ)andisindependentoftheorientationof siliconemissionlineratios(Kwitter&Henry1998;Henryetal.
thesmall-scalemode(θ −θ ). 1996). For Mpc scales, or clusters, one may use the SZ ef-
1 2
BecausethemeasurementofthekSZeffectonlyprobesthe fect (Rossettietal. 2016). Nevertheless, all these proposed ap-
radial velocity of the free electrons gas along the line of sight, proachesarequitespeculativeatthisstage.
whichisgeneratedbydensityfluctuationsalmostparalleltothe
lineofsightoverwhichweintegrateandaredampedbythisra-
6. Conclusions
dial integration,the result (53) is suppressedas comparedwith
the ISW result (38) by the radial derivative dln(bgIgD)/dη ∼ In this paper, we have shown how to relate the large-scale
1/r.Also,thecontribution(53),associatedwithtransversefluc- consistency relations with observational probes. Assuming the
tuationsthatarealmostorthogonaltothesecondlineofsight,is standard cosmological model (more specifically, the equiva-
suppressed as compared with the ISW result (38) by the small lenceprincipleandGaussianinitialconditions),nonzeroequal-
angle|θ−θ2|betweenthetwolinesofsight. times consistency relations involve the cross-correlations be-
OnedrawbackofthekSZconsistencyrelation,(49)and(53)- tweengalaxyormatterdensityfieldswiththevelocity,momen-
(54),isthatitisnoteasytoindependentlymeasurethe galaxy- tumortime-derivativedensityfields.Wehaveshownthatthese
freeelectronspowerspectrumPg1,e,whichisneededifwewish relationscan be related to actual measurementsby considering
totestthisrelation.Alternatively,Eqs.(53)-(54)maybeusedas theISWandkSZeffects,whichindeedinvolvethetimederiva-
atestofmodelsforthefreeelectronsdistributionandthecross tive of the matter density field and the free electrons momen-
powerspectrumPg1,e. tumfield.Wefocusedonthelowest-orderrelations,whichapply
tothree-pointcorrelationfunctionsorbispectra,becausehigher-
ordercorrelationsareincreasinglydifficulttomeasure.
5.2.Lensing-lensing-kSZcorrelation
The most practical relation obtained in this paper is prob-
Again, from Eqs.(49) and (53)-(54) we can directly obtain the ably the one associated with the ISW effect, more particularly
lensing-lensing-kSZthree-pointcorrelation, itscross-correlationwithtwocosmicweaklensingconvergence
statistics.Indeed,itallowsonetowritethisthree-pointcorrela-
ξ (κs,κs,∆s )=hκs(θ)κs(θ )∆s (θ )i, (55)
3 1 kSZ2 1 1 kSZ2 2 tionfunctionintermsoftwomatterdensityfieldpowerspectra
byreplacingthegalaxykernelsb I andI bythelensingcon- (linearandnonlinear),whichcanbedirectlymeasured(e.g.,by
vergencekernelsI andI .Thisggivgesξ =g1ξ +ξk +ξ⊥ with two-point weak lensing statistics). Moreover, the result, which
κ κ1 3 3⊥ 3k 3k is the leading-order contribution in the squeezed limit, shows
(θ−θ )·(θ −θ ) dlnD a specific angular dependence as a function of the relative an-
ξ3⊥ = |θ−θ11||θ22−θ1|1 (2π)4Z dηIκIκ1IkSZ2 dη gbuoltahrthpeosaintigounlsarodfetpheentdherenecesmanodotthheedquoabnsteirtavteivdesptarteidstiicctsi.onThperon-,
∞
×Z dk⊥dk2⊥k2⊥W˜Θ(k⊥r)W˜Θ1(k2⊥r)W˜Θ′2(k2⊥r) vpirdinecaiptleestanodftohfepcroinmsoisrtdeinaclyGraeulastsiioann,ittyh.aItfisw,eofctohneseidqeurivinalsetenacde
0
×P (k ,η)P (k ,η)J (k r|θ−θ |) thecross-correlationofthe ISW effectwith two galaxydensity
L ⊥ m,e 2⊥ 1 ⊥ 1
×J (k r|θ −θ |), (56) fields,weobtainasimilarrelationbutitnowinvolvesthemixed
1 2⊥ 2 1
galaxy-matterdensity power spectrum P and the large-scale
g,m
galaxybiasb .Thesetwoquantitiescanagainbemeasured(e.g.,
d dD ∞ g
ξk = −(2π)4 dη [I D]I I dk dk bytwo-pointgalaxy-weaklensingstatistics)andprovideanother
3k Z dη κ κ1 kSZ2 dη Z0 ⊥ 1⊥ testoftheconsistencyrelation.
×W˜Θ(k⊥r)W˜Θ1(k1⊥r)W˜Θ2(k1⊥r)kk1⊥PL0(k⊥) TheTyhdeorenloattioshnoswobatasinimedplweitahngthuelakrSdZeepffenedctenarcee,mwohreicihntwricoautled.
⊥
×P (k ,η)J (k r|θ−θ |)J (k r|θ −θ |), (57) provideasimplesignature,andtheyinvolvethegalaxy-freeelec-
m,e 1⊥ 0 ⊥ 2 0 1⊥ 1 2
tronsormatter-freeelectronspowerspectra.Thesepowerspec-
and traaremoredifficulttomeasure.Onecanestimatethefreeelec-
dD ∞ tron density in specific regions, such as filaments or clusters,
ξ3⊥k = −(2π)4Z dηIκIκ1IkSZ2Ddη Z dk⊥dk1⊥ throughX-ray or SZ observations, or around typical structures
0 bystackinganalysisofclusters.Thiscouldprovideanestimate
×W˜Θ(k⊥r)W˜Θ1(k1⊥r)W˜Θ2(k1⊥r)k1⊥PL0(k⊥) ofthefreeelectronscrosspowerspectraandacheckofthecon-
×P (k ,η)|θ−θ |J (k r|θ−θ |)J (k r|θ −θ |). sistencyrelations.Althoughwecanexpectsignificanterrorbars,
m,e 1⊥ 2 1 ⊥ 2 0 1⊥ 1 2
itwouldbeinterestingtocheckthattheresultsremainconsistent
(58)
withthetheoreticalpredictions.
This now involves the matter-free electrons cross power spec- A violation of these consistency relations would signal ei-
trumP . ther a modification of gravity on cosmological scales or non-
m,e
Articlenumber,page8of9
LucaAlbertoRizzo etal.:ConsistencyrelationsfortheISWandkSZeffects
Gaussianinitialconditions.Weleavetofutureworksthederiva-
tion of the deviationsassociated with various nonstandardsce-
narios.
Acknowledgements. ThisworkissupportedinpartbytheFrenchAgenceNa-
tionaledelaRechercheunderGrantANR-12-BS05-0002.DFMthanksthesup-
portoftheResearchCouncilofNorway.
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