Constraining galaxy cluster velocity field with the tSZ-kSZ-kSZ bispectrum

0.3 MB·

by  G. Hurier

Astronomy&Astrophysicsmanuscriptno.sz_bispectre (cid:13)c ESO2017
February1,2017
Constraining galaxy cluster velocity field
with the tSZ-kSZ-kSZ bispectrum.
G.Hurier1,2
1 CentrodeEstudiosdeFísicadelCosmosdeAragón(CEFCA),PlazadeSanJuan,1,planta2,E-44001,Teruel,Spain
2 Institutd’AstrophysiqueSpatiale,CNRS(UMR8617)UniversitéParis-Sud11,Bâtiment121,Orsay,France
e-mail:[email protected]
Received/Accepted
7
1 Abstract
0
2 TheSunyaev-Zel’dovich(SZ)effectsareproducedbytheinteractionofcosmicmicrowavebackground(CMB)photonswiththeion-
izedanddiffusegasofelectronsinsidegalaxyclustersintegratedalongthelineofsight.ThetwomaineffectsarethethermalSZ(tSZ)
n producedbythermalpressureinsidegalaxyclustersandthekineticSZ(kSZ)producebypeculiarmotionofgalaxyclusterscompared
a toCMBrest-frame.ThekSZeffectisparticularlychallengingtomeasureasitfollowsthesamespectralbehaviorastheCMB,and
J
consequentlycannotbeseparatedfromtheCMBusingspectralconsiderations.Inthispaper,weexplorethefeasibilityofdetect-

1 ingthekSZthroughthecomputationofthetSZ-CMB-CMBcross-correlationbispectrumforcurrentandfutureCMBexperiments.
3 WeconcludethatnextgenerationofCMBexperimentswillofferthepossibilitytodetectathighS/NthetSZ-kSZ-kSZbispectrum.
Thismeasurementwillconstraintstheintra-clusterdynamicsandthevelocityfieldofgalaxyclusterthatisextremelysensitivetothe

] growthrateofstructuresandthustodarkenergyproperties.Additionally,wealsodemonstratethatthetSZ-kSZ-kSZbispectrumcan
O
beusedtobreakthedegeneraciesbetweenthemass-observablerelationandthecosmologicalparameterstosettightconstraints,up
C to4%,ontheY−Mrelationcalibration.
h. Keywords.Cosmology:Observations–Cosmicbackgroundradiation–Sunyaev-Zel’dovicheffect
p
-
o
1. Introduction of b = 0.3−0.5. Thus, methodology to break the degeneracy
r
t betweencosmologicalparametersandthemass-observablerela-
s Present CMB experiments such as ACTT (Sievers et al. 2013),
a tionneedstobeinvestigatedfurthertotesttheconsistencyofthe
[ SPT (Bleem et al. 2015), and Planck (Planck Collaboration ΛCDMcosmologicalmodel.
2015ResultsI2015)havemappedthecosmicmicrowaveback-

1 ground (CMB) primary temperature anisotropies with an un- After the tSZ effect, the second dominant sources of
v precedented precision, and constrained cosmological parame- arcminute-scale anisotropies is the kinematic SZ (kSZ) effect
2
ters from the CMB angular power-spectrum analyses (see e.g., (Ostriker & Vishniac 1986), produced by the peculiar motion
7
Sievers et al. 2013; Planck Collaboration 2015 Results XIII of galaxy clusters with respect to CMB rest-frame. The kSZ is
0
9 2015; de Haan et al. 2016). However, the measurement of sec- about ten times fainter than the tSZ effect and contrary to the
0 ondaryanisotropies,suchastheSunyaev-Zel’dovich(SZ)effects tSZ effect, it cannot be separated from the CMB signal using
. (Sunyaev & Zeldovich 1972), is still limited by the noise level spectral considerations. Consequently, the measurement of this
1
andangularresolutionincurrentexperiments. kSZeffectissignificantlyharderthantheoneofthetSZeffect.
0
Thankstoit’sspectralbehavior,thetSZeffectcanbeisolated The kSZ effect is directly related to the peculiar velocity field
7
1 from the CMB and foregrounds emissions (Remazeilles et al. ofthematterdistributionandthebaryonicmatterdensity.Thus,
: 2011;Hurieretal.2013).ThethermalSZ(tSZ)effect,produced this effect presents significantly different cosmological depen-
v by thermal electron in intra-cluster medium, as been shown as danciesthanthetSZeffect,especiallywiththeuniversegrowth
i
X a powerful probe to detect galaxy clusters (Hasselfield et al. rate(Sugiyamaetal.2016),whichisapowerfulprobetounder-
2013; Bleem et al. 2015; Planck Collaboration 2015 Results standtheCMB-LSStensionforcosmologicalparameterestima-
r
a XXVII 2015), and to constrain cosmological parameters us- tion.
ing large-scale structure matter distribution (Hasselfield et al.
2013; Mantz et al. 2015; Planck Collaboration 2015 Results Several approaches have been proposed to recover the kSZ
XXIV2015;PlanckCollaboration2015ResultsXXII2015;de effect:(i)directmeasurementofanisotropiesintheCMBangu-
Haan et al. 2016). However, cosmological constraints obtained lar power spectrum at high-(cid:96) (see Addison et al. 2013; George
via the tSZ effect strongly depend on the mass-observable re- etal.2015,forrecentresults),(ii)thepairwisemomentumesti-
lation. Assuming a value of b = 0.2 (Planck Collaboration mator, using the preferential motion of large-scale structure to-
2015 Results XXIV 2015) for the hydrostatic mass bias, the ward other large scale structure (Peebles 1980; Diaferio et al.
observed number of galaxy clusters on the sky is only half of 2000),(iii)invertingthecontinuityequationrelatingdensityand
thepredictednumberassumingCMB-derivedcosmologicalpa- velocity fields (Ho et al. 2009; Kitaura et al. 2012), (iv) cross-
rameters(PlanckCollaboration2015ResultsXIII2015).CMB- correlation bispectrum between CMB and cosmic shear (Doré
derivedcosmologicalparameter,favoursahydrostaticmassbias etal.2004)orgalaxysurveys(DeDeoetal.2005)
1
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
EvidenceofthekSZeffectangularpowerspectrumhasbeen 3. Powerspectra
found using the CMB angular power spectrum (George et al.
3.1. Generalformalism
2015) and combining the tSZ angular power-spectrum and bis-
pectrum (Crawford et al. 2014). The kSZ detection has been Theangularcrosspowerspectrumbetweentwomapsreads
achieveintwocases:forindividualgalaxyclustersinternaldy-
namics (Sayers et al. 2013; Adam et al. 2016), or for statisti- 1 (cid:88)1(cid:16) (cid:17)
cal sample of galaxy clusters using pairwise momentum (Hand C(cid:96)XY = 2(cid:96)+1 2 x(cid:96)my∗(cid:96)m+x(cid:96)∗my(cid:96)m , (5)
et al. 2012; Hernández-Monteagudo et al. 2015; Soergel et al. m
2016). Recent studies (see e.g., Hernández-Monteagudo et al.
with x and y , the coefficients from the spherical harmonics
2015)achievestaticaldetectionatabout5σsignificancelevel. (cid:96)m (cid:96)m
decomposition of the concerned two maps, and (cid:96) the multipole
The kSZ effect produces a non-gaussian contribution to the
ofthesphericalharmonicexpansion.Inthecontextoflargescale
CMB anisotropies. High order statistics have been shown as a
structuretracers,wemodelthiscross-correlation,aswellasthe
powerful probe to detect the tSZ effect (see e.g., Wilson et al.
auto correlation power spectra, assuming the following general
2012;Hill&Sherwin2013;PlanckCollaboration2015Results
expression
XXII 2015). In this paper, we explore the possibility to detect
thetSZ-kSZ-kSZcross-correlationbispectrumforthenextgen- C(cid:96)XY =C(cid:96)XY−1h+C(cid:96)XY−2h, (6)
eration of CMB experiments. First, in Sect. 2 we present tSZ
andkSZeffects,inSect.3wedetailsthecomputationoftSZand whereC(cid:96)XY−1h is the Poissonian contribution andC(cid:96)XY−2h is the
2-halo term. These terms can be computed considering a mass
kSZ power spectra. Then, in Sect. 4 we present the modeling
function formalism. The mass function, d2N/dMdV, gives the
ofthetSZandkSZbispectra.Finally,inSect.5wepresentour
number of dark matter halos (in this paper we consider the
forecastsfornextgenerationCMBexperiments.
fitting formula from Tinker et al. (2008)) as a function of the
halomassandredshift.
2. TheSZeffects
The Poissonian term can be computed by assuming the
The thermal Sunyaev-Zel’dovich effect is a distortion of the
Fourier transform of normalized halo projected profiles of X
CMBblackbodyradiationthroughinverseComptonscattering. andY,weightedbythemassfunctionandtherespectivefluxes
CMBphotonsreceiveanaverageenergyboostbycollisionwith of the halo for X and Y observable (see e.g. Cole & Kaiser
hot (a few keV) ionized electrons of the intra-cluster medium 1988;Komatsu&Seljak2002,foraderivationofthetSZauto-
(see e.g. Birkinshaw 1999; Carlstrom et al. 2002, for reviews). correlationangularpowerspectrum).
The thermal SZ Compton parameter in a given direction, n, on
(cid:90) (cid:90)
theskyisgivenby dV d2N
CXY−1h=4π dz dM X Y x y , (7)
(cid:90) (cid:96) dzdΩ dMdV 500 500 (cid:96) (cid:96)
k T
y(n)= n B eσ ds (1)
emec2 T whereX500 andY500 aretheaveragefluxesofthehaloinX and
wherek istheBoltzmannconstant,cthespeedoflight,m the Y maps that depend on the critical mass of the galaxy cluster,
B e
ealleocntgrothnemlianses-,oσf-Tsitghhet,Tnh,onmsaonndcTroassre-stehcetieolne,ctdrsonisntuhmedbiesrtadnecne- aMn5d00,dtVh/edrzedsΩhiftt,hze, acnodmcoavninbgevoobltuaimneedewleimthesncta.liTnghereFlaotiuorniesr,
e e
sityandtemperature,respectively.InunitsofCMBtemperature transform of a 3-D profile projected across the line-of-sight on
the contribution of the tSZ effect for a given observation fre- the sphere reads, 4πrs (cid:82)∞dxx2p(x)sin((cid:96)x/(cid:96)s), where p(x) is the
quencyνis halo3-Dprofilein Xls2or0Y maps, x = (cid:96)rx//r(cid:96)s,(cid:96) = D (z)/r ,r is
∆TCMB =g(ν)y, (2) thescaleradiusoftheprofile. s s ang s s
T
CMB
whereT istheCMBtemperature,andg(ν)thetSZspectral Thetwo-halotermcorrespondstolargescalefluctuationsof
CMB
dependance.Neglectingrelativisticcorrectionswehave thedarkmatterfield,thatinducecorrelationsinthehalodistri-
(cid:20) (cid:18)x(cid:19) (cid:21) butionoverthesky.Itcanbecomputedas(seee.g.Komatsu&
g(ν)= xcoth −4 , (3) Kitayama1999;Diego&Majumdar2004;Taburetetal.2011)
2
0w)ith= 2x.72=6±0h.0ν0/(1kBKTC(MMBa).theArtetz al=. 1909,9),wthheeretSTZCMeffB(ezct i=s C(cid:96)XY−2h =4π(cid:90) dzddzdVΩ(cid:32)(cid:90) dMddM2NdVX500x(cid:96)b(M500,z)(cid:33) (8)
negativebelow217GHzandpositiveforhigherfrequencies.
(cid:32)(cid:90) (cid:33)
d2N
× dM Y y b(M ,z) P (k,z)
The kinetic Sunyaev-Zel’dovich effect produces a shift of dMdV 500 (cid:96) 500 X,Y
theCMBblackbodyradiationtemperaturebutwhiteoutspectral
distortion.ContrarytotSZeffect,itthuspossesthesamespectral with b(M ,z) the time dependent linear bias, that relates the
500
energydistribution(SED)thantheCMBprimordialanisotropies powerspectrumbetween X andY distribution, P (k,z),tothe
X,Y
andcannotbeseparatedfromthem.ThekineticSZinducedtem- underlyingdarkmatterpowerspectrum.FollowingMo&White
peratureanisotropiesinagivendirectionontheskyisgivenby (1996);Komatsu&Kitayama(1999)weadopt
(cid:90)
k(n)= n v·nσ ds, (4) b(M500,z)=1+(ν2(M500,z)−1)/δc(z),
e c T
(cid:104) (cid:105)
withvthepeculiarvelocityvectorofthegalaxycluster.Thetotal with ν(M500,z) = δc(z)/ Dg(z)σ(M500) , Dg(z) is the linear
kSZfluxfromaclusterisproportionaltothegasmass, M growthfactorandδ (z)istheover-densitythresholdforspheri-
gas,500 c
ofthecluster,andmodulatedbythescalarproductv·nthatrange calcollapse.
from−vtov.
2
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
Table1.Scaling-lawparametersanderrorbudgetforbothY − By simplicity, to estimate the kSZ flux root-mean-square
500
M (PlanckCollaborationresultsXXIX2013)andY −T (overvelocities),K ,weconsidertherelation
500 500 500 500
(PlanckCollaborationresultsXXIX2013)relations
k T (cid:18)σ (cid:19)−1
Y (cid:39) K B 500 v . (14)
M500−Y500 M500−T500 500 500 mec2 c
logY -0.19±0.02 logT -4.27±0.02
(cid:63) (cid:63)
αsz 1.79±0.08 αT 2.85±0.18 K500 is proportional to Mgas,500, thus the K500 − M500 and
βsz 0.66±0.50 βT 1 Mgas,500 − M500 intrinsic scatters are the identical. The extra-
σ 0.075±0.010 σ 0.14±0.02
logY logT scatter induced by the velocities is accounted by the σ fac-
v
tor. We derived a K − M intrinsic scatter of σ =
500 500 logK
0.03±0.01fromthegalaxyclustersamplepresentedinPlanck
3.2. ThetSZandkSZscalingrelations
Collaborationearlyresults.XI(2011).
A key step in the modeling of the cross-correlation between Additionally,theY500−M500 and K500−M500 intrinsicscatters
tSZ and kSZ is to relate the mass, M , and the redshift, z, arecorrelated.Theintrinsicscattersfollowtherelation
500
of a given cluster to tSZ flux, Y , and kSZ flux, K . The
cross-correlation between tSZ an5d00kSZ effects is thus500highly σ2logYK =σ2logY +σ2logK −2ρσlogYσlogK, (15)
dependent on the M −Y and the M −K relations in
termsofnormalizatio50n0and5s0l0ope.Conseq5u0e0ntly,5w00eneedtouse whereσlogYK = 0.08±0.01istheintrinsicscatteroftheY500−
the relations derived from a representative sample of galaxy K500(equivalenttothescatteroftheY500−Mgas,500)andρisthe
clusters, with careful propagation of statistical and systematic correlationfactorbetweenY500−M500andK500−M500intrinsic
uncertainties. However, such observational constraints are not scatters.Wederivedρ ∈ [0.5,1.0]1.Inthefollowingweassume
available for the kSZ effect. We stress that for power spectrum ρ=0.75.
analysis, the intrinsic scatter of such scaling laws has to be
considered, because it will produce extra power that has to be
3.3. Log-normalscatterandn-pointscorrelationfunctions
accountedforinordertoavoidbiases.
Scalingrelationsshowninsect.3.2presentsanintrinsicphysical
We used the M − Y scaling laws presented in Planck scatter(seetab.1).Thisscatterisgenerallyconsideredasalog-
500 500
CollaborationresultsXXIX(2013), normaldistribution.Whenenteringincorrelationfunctions,this
log-normalscatterwillactasanadditionalsourceofpower.This
E−βsz(z)1D02A−(4zM)Yp50c02=Y(cid:63)(cid:34)0h.7(cid:35)−2+αsz(cid:34)6(1×−1b0)1M4M50(cid:12)0(cid:35)αsz, (9) badetdwiteioenna<lpXonw>er,anfodrXa(cid:63)q,uwanhteirteyXX(cid:63),cisanthbeeloegx-pnroesrmseadlamseaarneloaftitohne
X variable.Inthemostgeneralcase,then-thmomentum, M(n),
withhthedimensionlessHubbleparameter,D theangulardis- expectationofasetofN variableX canbewritten
A i
tance, E(z) = Ωm(1 + z)3 + ΩΛ. The coefficients Y(cid:63), αsz, and
βszfromPlanckCollaborationresultsXXIX(2013),aregivenin (cid:89)N
Table1.Weusedb = 0.2forthebiasbetweenX-rayestimated M(n) =< Xni >
i
massandeffectivemassoftheclusters. i
We also need to have an estimate of the cluster tempera- (cid:90) (cid:89)N
ture, T . In this work, we used the scaling law from Planck < M(n) >= A XnidX
500 norm i i
Collaborationearlyresults.XI(2011): i
E(z)−βTY500 =T(cid:63)(cid:20) T500 (cid:21)αT, (10) exp−(cid:2)log(X)−log(X(cid:63))(cid:3)TC2−S1(cid:2)log(X)−log(X(cid:63))(cid:3),
6keV
(16)
wherethecoefficientsT ,α andβ aregiveninTable1.
(cid:63) T T
Tomodelthe K500−M500 relation,weconsidertherelation whereAnorm isthenormalisationfactorofthelog-normaldistri-
fromDeDeoetal.(2005)forthevelocityfield: bution,C isthescalingrelationscattercovariancematrix,Xis
S
avectorofX variables,X isavectorcontainingthelog-normal
H(z)D dln(D )δ i (cid:63)
vk =i 1+zg dln(ag) kk, (11) evxarpieacbtlaetiXoniXn(cid:63)t,hiefomroemacehnvtuamriaMble(n)Xai,ndansdatnisifiyst(cid:80)heNonrd=erno.fIteaccahn
i i i
beeasilyshownthat
andconsequentlyforthevelocitydispersion
σv(M500,z)= H1(z+)Dzgddlnln((Dag))σ−1, (12) < M(n) >=exp(cid:32)nTC2Sn(cid:33) (cid:89)N X(cid:63)ni,i, (17)
i
with D thegrowthfactorandσ (z)isdefinedforanyinteger j
g j wherenisavectorofn.Thiseffectproduceapowerenhance-
as i
mentof6%forthetSZbispectrum.Whichissignificantforhigh-
σ (M )= 1 (cid:90) dkk2(1+j)P(k)W2(kR), (13) S/N measurement of tSZ bispectrum. For the the tSZ-kSZ-kSZ
j 500 2π2 bispectrum, we derived a power enhancement of 1.8%. In the
following,wecorrectallspectraforthiseffect.
whereW(kR)= 3 (sin(kR)−kRcos(kR))istheFouriertrans-
form of he real(kRs)p3ace top-hat window function, with R = 1 Forconsistency,weusedσlogY =0.10±0.01thathasbeenderived
(cid:113) onthe galaxycluster samplefrom PlanckCollaboration earlyresults.
3M500,whereρ¯ isthecriticaldensityoftheuniverse. XI(2011)
4πρ¯
3
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
Table2.AmplitudeofthedifferenttermscontributiontothetSZ
andkSZpowerspectraandbispectra.P isthepowerspectrum
B
ofthemomentumfieldandP thematterpowerspectrum.
m
1-halo 2-halo 3-halo
CtSZ−kSZ 0 0
(cid:96)
CkSZ−kSZ 1/3 P
(cid:96) B
btSZ−tSZ−kSZ 0 (0,0,0) 0
(cid:96)1(cid:96)2(cid:96)3
btSZ−kSZ−kSZ 1/3 (P ,P ,P /3) B
(cid:96)1(cid:96)2(cid:96)3 B B m k
bkSZ−kSZ−kSZ 0 (0,0,0) 0
(cid:96)1(cid:96)2(cid:96)3
3.4. Pressureanddensityprofiles
ThetSZeffectisdirectlyproportionaltothepressureintegrated
acrossthelineofsight.Inthiswork,wemodelthegalaxyclus-
ter pressure profile by a generalized Navarro Frenk and White
(GNFW, Navarro et al. 1997; Nagai et al. 2007) profile of the
form
P
P(r)= 0 . (18)
(c r)γ[1+(c r)α](β−γ)/α
500 500
Fortheparametersc ,α,β,andγ,weusedthebest-fittingval-
500
uesfromArnaudetal.(2010)presentedinTable.1.Theabsolute
normalizationoftheprofile P issetassumingthescalinglaws
0
Y −M presentedinSect.3.2.
500 500
To model the kSZ profile, we need the density, n (r), pro-
e
file. Thus, we assume a polytropic equation of state (see, e.g.,
Komatsu&Seljak2001),P(r) = n (r)T (r),withn (r) ∝ T (r)δ
e e e e
whereδisthepolytropicindex.ConsideringthatthekSZvaries Figure1. Power density as a function of M500 (top panel) and
withne,thekSZprofileisproportionaltoP(r)(cid:15)p,where(cid:15)p = δ+δ1 redshift (bottom panel) for tSZ power spectrum (dotted blue
rangesfrom0.5to1.0for1.0<δ<∞. line),tSZbispectrum(solidblueline),kSZpowerspectrum(dot-
The overall normalization of kSZ profile is deduced from the tedredline),andtSZ-kSZ-kSZbispectrum(solidredline).
scalinglawK −M presentedinSect.3.2. .
500 500
3.5. tSZandkSZpowerspectra
ThekSZeffectisdependentoftheorientationofthepeculiarve-
locity vector. Consequently, power spectra have to be averaged
over orientations. In Table 2, we present the multiplicative fac- 4.1. cross-correlationbispectra
tors to be applied on kSZ related power spectra, where P is
B
thepowerspectrumofthemomentumfieldandPm isthepower Followingthesamehalomodelapproach,wecaneasilypredict
spectrumofthematterfield.WenoticethatkSZeffectpresenta the SZ bispectra (see Bhattacharya et al. 2012a, for a detailed
2-halo term, induce by the large scale correlations between the description of the tSZ bispectrum). In halo model formalism, a
velocitiesofdifferentclusters. bispectrum can be separated into 3 terms: one-halo, two-halo,
OnFig.1,wepresentthepowerdensityasafunctionofM andthree-halo,as
500
andzfortSZandkSZpowerspectra,tSZbispectrum,andtSZ-
kSZ-kSZcross-bispectrum.WeobservethattSZpowerspectrum
samples higher mass and lower redshift than kSZ power spec- bXYZ =bXYZ−1h+bXYZ−2h+bXYZ−3h, (19)
trum, this effect is a consequence of the slope of Y500 − M500 (cid:96)1(cid:96)2(cid:96)3 (cid:96)1(cid:96)2(cid:96)3 (cid:96)1(cid:96)2(cid:96)3 (cid:96)1(cid:96)2(cid:96)3
(1.7)andK −M (1).Wealsoobservethatbispectragives
500 500
moreweightstoverymassiveandnearbyobjects.
The one-halo term, is produce by the auto-correlation of a
clusterwithhimself,
4. bispectrum
Combining tSZ and kSZ, it is possible to build two auto- (cid:90) (cid:90)
dV d2N
correlationbispectra,andtwocross-correlationbispectra.Inthis bXYZ−1h=4π dz dM X Y Z x y z , (20)
(cid:96)1(cid:96)2(cid:96)3 dzdΩ dMdV 500 500 500 (cid:96)1 (cid:96)2 (cid:96)3
work we aim at predicting the tSZ-kSZ-kSZ bispectrum, thus
ourmodelingonlyaccountsfornon-gaussianobjectsthecorre-
latewiththetSZeffect.Consequently,weonlyconsiderthekSZ
effectfromvirializedhalosandweneglectthekSZcontribution The two-halo involved two point from the same halo and
producedbydiffusebaryonsathigherredshift. thethirdfromanotherone.Asaconsequence,thistermreceives
4
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
threecontributions,
(cid:90) (cid:32)(cid:90) (cid:33)
dV d2N
bXYZ−2h=4π dz P (k,z) dM X Y x y b(M ,z)
(cid:96)1(cid:96)2(cid:96)3 dzdΩ XY,Z dMdV 500 500 (cid:96)1 (cid:96)2 500
(cid:32)(cid:90) (cid:33)
d2N
× dM Z z b(M ,z)
dMdV 500 (cid:96)3 500
(cid:90) (cid:32)(cid:90) (cid:33)
dV d2N
+4π dz P (k,z) dM X Z x z b(M ,z)
dzdΩ XZ,Y dMdV 500 500 (cid:96)1 (cid:96)3 500
(cid:32)(cid:90) (cid:33)
d2N
× dM Y y b(M ,z)
dMdV 500 (cid:96)2 500
(cid:90) (cid:32)(cid:90) (cid:33)
dV d2N
+4π dz P (k,z) dM Y Z y z b(M ,z)
dzdΩ YZ,X dMdV 500 500 (cid:96)2 (cid:96)3 500
(cid:32)(cid:90) (cid:33)
d2N
× dM X x b(M ,z)
dMdV 500 (cid:96)1 500
(21)
The tree-halo involved the correlation of three different ha-
los,
(cid:90) (cid:32)(cid:90) (cid:33)
dV d2N
bXYZ−3h=4π dz B (k ,k ,k ,z) dM X x b (M ,z)
(cid:96)1(cid:96)2(cid:96)3 dzdΩ X,Y,Z 1 2 3 dMdV 500 (cid:96)1 3 500
(cid:32)(cid:90) (cid:33)
d2N
× dM Y y b (M ,z)
dMdV 500 (cid:96)2 3 500
(cid:32)(cid:90) (cid:33)
× dM d2N Z z b (M ,z)Figure2.tSZ(toppanel)andtSZ-kSZ-kSZ(bottompanel)bis-
dMdV 500 (cid:96)3 3 500 pectra1haloterm(blueline),2-haloterm(redline),and3-halo
(22) term(greenline).
.
with B (k ,k ,k ,z)thebispectrumof X,Y,andZ distribu-
X,Y,Z 1 2 3
tion and b (M ,z) the bias that relates dark-matter and halo
3 500
distributions.
4.2. ThetSZ-kSZ-kSZcross-bispectrum
The tSZ-tSZ-kSZ bispectrum have a null expectation due to
similardegeneraciesthanthetSZpowerspectrum,consequently
the average over all the direction for the momentum field of
itcannotbeusedtobreakdegeneraciesbetweencosmologyand
galaxy clusters. Table 2 lists the relative amplitude of each
astrophysicalprocesses.
contributions.
For the prediction of tSZ-kSZ-kSZ 2-halo term, we compute
the momentum field power spectrum similarly to Shaw et al. On Fig. 1, we present the power density for tSZ and tSZ-
(2012). The tSZ-kSZ-kSZ bispectrum 2-halo term involves kSZ-kSZ bispectra. We observe that bispectra are sensitive to
three contributions. Two contributions where the two consid- highermassandlowerredshiftthanpowerspectra.Wenotethat
ered halos receive contribution from the kSZ effect. These tSZ-kSZ-kSZbispectrumandtSZpowerspectrumpresentssim-
terms involve the momentum field power spectrum. The third ilarpowerdensitydistributionasafunctionofmassandredshift.
contributions involves the correlation between a halo weighted thetSZ-kSZ-kSZbispectrumisthussensitivetogalaxyclusters
by his tSZ flux and a halo weighed by the square of the kSZ withmassrangingfrom1014and1015M(cid:12),atz<1.
effect. Consequently, this third contribution involves the matter We present the tSZ and tSZ-kSZ-kSZ bispectra on Fig. 2.
power spectrum that describes the distribution of halos over WeobservedthatforthetSZbispectrum1halotermistwoor-
the sky, weighted by the average over momentum directions derofmagnitudeshigherthen2twohalo.Consequently,forthe
(similarly to the kSZ power spectrum). For the 3-halo term tSZbispectrum2-haloand3-halotermscanbesafelyneglected.
matter-momentum bispectrum we used the prescription from This is consistent with the tSZ angular power spectrum that
(DeDeoetal.2005). only presents significant contribution from the 2-halo term at
verylow-(cid:96).Whenusinghigherorderstatisticswefavourshigher
We studied the cosmological parameter dependancies of mass,lower-zobjectsthatarelessfrequentoverthesky.Indeed,
this cross bispectrum. We found that the tSZ-kSZ-kSZ bispec- the 1-halo term amplitude evolved with the number, N , that
cl
trum is proportional to Ω4m.1σ810.1H02.0, by comparison the tSZ significantly contributes to the spectra, where the 2-halo term
bispectrumisproportionaltoΩ3.9σ12.9H−1.1 et(cid:96) (cid:39) 1000.Then, evolvesasN2 andthe3-halotermasN3.
m 8 0 cl cl
tSZ-kSZ-kSZmightbeapowerfulprobetobreakdegeneracies ContrarytothetSZbispectrum,thetSZ-kSZ-kSZbispectrum2-
between cosmological parameters and scaling relations by haloand3-halotermpresentsignificantcontributionscompared
providing similar cosmological dependancies than tSZ power tothe1-haloterms.Thishighercontributionisexplainedbythe
spectrum with significantly different astrophysical processes factthatthekSZeffectfavourslower-massandhigher-zobjects
dependancies. We also note that the tSZ bispectrum present thanthetSZeffect,asshownonFig.1.
5
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
4.3. uncertaintiesandoptimalestimator 4.4. Foregroundcontamination
TocomputethebispectrumbetweenthreemapsX,Y,andZ,we ThetwomainforegroundscontaminationinSZanalysesarethe
consideredthefollowingformula CIBandextra-galacticradiosources.
CIB leakage in component separated maps is composed by
1 (cid:88) high-z CIB sources that present a spectral behavior than the
b(cid:96)1(cid:96)2(cid:96)3 = f X(cid:96)1Y(cid:96)2Z(cid:96)3, (23) thermaldustinourgalaxythatdrivethecomputationofweights
sky n forILCbasedmethods.Asaconsequence,CIBinsuchmapsis
almost gaussian and will not significantly bias the results. The
where X , Y , Z are the real space map that only contain (cid:96) ,
(cid:96)1 (cid:96)2 (cid:96)3 1 level of CIB in the final bispectrum can be estimated through
(cid:96)2,(cid:96)3multipolesofmapsX,Y,Zrespectively, fskyisthecovered thedifferentshapeofCIBandSZbispectraLacasa(2014).
skyfraction,andnisthedirectionoverthesky.
Radiosourceswithasignificantfluxareinsmallnumberonthe
Thisestimatorisknowntoreducethevarianceofthebispectrum
microwave sky. It in has been shown in tSZ bispectra analyses
withoutbiasingtheexpectationofthebispectrum.Wenotethat,
(Planck Collaboration results XXI 2013) that the results is not
in the case of a cross-correlation bispectrum (cid:96) , (cid:96) , (cid:96) are not
1 2 3 biasedbyradiosources,thatwouldproduceanegativebiasdue
commutativequantities.
tothewayradiosourcesleakintSZmaps(Hurieretal.2013).
Then the bispectrum variance in the weak non-Gaussianity
limitcanbeexpressedas
CXXCYYCZZ 5. Forecasts
<b(cid:96)1(cid:96)2(cid:96)3,b(cid:96)1(cid:48)(cid:96)2(cid:48)(cid:96)3(cid:48) >= f(cid:96)s1kyN(cid:96)(cid:96)21,(cid:96)2,(cid:96)(cid:96)33 δ(cid:96)1(cid:96)1(cid:48)δ(cid:96)2(cid:96)2(cid:48)δ(cid:96)3(cid:96)3(cid:48) 5.1. Cosmicvariancelimitedexperiment
+ C(cid:96)Xf1sXkyCN(cid:96)Y(cid:96)2Z1(cid:96)C2(cid:96)(cid:96)Z33Yδ(cid:96)1(cid:96)1(cid:48)δ(cid:96)2(cid:96)3(cid:48)δ(cid:96)3(cid:96)2(cid:48) IitnedafibrystthsteepskwyeceosntitmaiantiendgthtSeZe,xpkeScZte,danSd/NCiMswBeaanriesojutrsotpliimes-.
IndeedtSZcanbeextractedfromothercomponents(Hurieretal.
CXYCYXCZZ
+ (cid:96)f1skyN(cid:96)(cid:96)21(cid:96)2(cid:96)(cid:96)33 δ(cid:96)1(cid:96)2(cid:48)δ(cid:96)2(cid:96)1(cid:48)δ(cid:96)3(cid:96)3(cid:48) 2p0ri1m3a),rybuatntihseotkrSopZiesisgonralsceacnonndoatrbyeadnisistiontgroupisiehsedthfarotmfoltlhoewCsMthBe
CXYCYZCZX CMBblackbodyemissionlaw.Wealsoconsider fsky =0.5.This
+ (cid:96)f1skyN(cid:96)(cid:96)21(cid:96)2(cid:96)(cid:96)33 δ(cid:96)1(cid:96)2(cid:48)δ(cid:96)2(cid:96)3(cid:48)δ(cid:96)3(cid:96)1(cid:48) crehsoiidcuealosfcaonsmtamallinsaktyiofnrathcatitocnonistammointiavtaetsedtStZoaanvdoiCdMfoBresgigronuanlsd.
CXZCYYCZX WepresentonFig.3theexpectedcumulativeS/Nasafunction
+ (cid:96)f1skyN(cid:96)2(cid:96)1(cid:96)2(cid:96)(cid:96)33 δ(cid:96)1(cid:96)3(cid:48)δ(cid:96)2(cid:96)2(cid:48)δ(cid:96)3(cid:96)1(cid:48) doefttehcetemdufoltripscoalele(cid:96)s.aWboeveob(cid:96)s(cid:39)erv3e0d00t.hat the tSZ-kSZ-kSZ can be
CXZCYXCZY
+ (cid:96)f1skyN(cid:96)2(cid:96)1(cid:96)2(cid:96)(cid:96)33 δ(cid:96)1(cid:96)3(cid:48)δ(cid:96)2(cid:96)1(cid:48)δ(cid:96)3(cid:96)2(cid:48), (24) 5.2. RealisticCMBexperiments
ThePlanckexperimenthaveproducedacosmic-variancelimited
with N , being the number of modes for the ((cid:96) ,(cid:96) ,(cid:96) )
(cid:96)1(cid:96)2(cid:96)3 1 2 3 measurementoftheCMBtemperatureangularpowerspectrum
triangle.
(Planck collaboration results XV 2013). However, secondary
anisotropies(suchasthetSZeffect)measurementarestilldom-
For our purpose, we considered the tSZ-kSZ-kSZ bispec-
inated by instrumental noise. In this section we present the ex-
trum,thuswehavetwopointthatareidentical.Consideringthat
pected signal to noise from the Planck experiment and from a
the expectation of tSZ and kSZ cross-correlation power spec-
futureCOrE+2likeCMBmission.
trum, CtSZ,kSZ, is zero, we can safely neglect this term in the
(cid:96)
computationofuncertainties.InthiscontextEq.24reducesto
5.2.1. Planck-likeexperiment
CXXCXXCZZ
<b(cid:96)1(cid:96)2(cid:96)3,b(cid:96)1(cid:48)(cid:96)2(cid:48)(cid:96)3(cid:48) >= f(cid:96)s1kyN(cid:96)(cid:96)21,(cid:96)2,(cid:96)(cid:96)33 δ(cid:96)1(cid:96)1(cid:48)δ(cid:96)2(cid:96)2(cid:48)δ(cid:96)3(cid:96)3(cid:48) TPolaensctkim-liakteemthiesseixopne,cwteedussigendatlh-teo-nnooisiseelreavteiol fursoimngPdlaatnacfkropmuba-
CXXCXXCZZ licCMB(PlanckCollaboration2015ResultsIX2015)andtSZ
+ (cid:96)f1skyN(cid:96)(cid:96)21(cid:96)2(cid:96)(cid:96)33 δ(cid:96)1(cid:96)2(cid:48)δ(cid:96)2(cid:96)1(cid:48)δ(cid:96)3(cid:96)3(cid:48) (inPglanfsckky C=o0ll.a5b.oFroartiobnoth20C1M5 BReasnudltstSXZXmIIap2s01w5e)umseadpsthaesstuomta-l
(25) measured power-spectra to estimate the noise level in the tSZ-
kSZ-kSZbispectrum.Consequently,thisestimationofthenoise
Ifwehave(cid:96) =(cid:96) then levelaccountsforCIBandpointsourcescontaminationinboth
1 2
tSZandCMBmaps.
In Fig. 3, we present the expected signal-to-noise as a function
CXXCXXCZZ
<b ,b >=2 (cid:96)1 (cid:96)2 (cid:96)3 , (26) of the maximum (cid:96) considered. We observe that Planck is ex-
(cid:96)1(cid:96)2(cid:96)3 (cid:96)1(cid:96)2(cid:96)3 fskyN(cid:96)1(cid:96)2(cid:96)3 pectedtoachievea0.2σmeasurementofthetSZ-kSZ-kSZbis-
pectrum.Thisrulesoutthepossibilitytoextractthissignalfrom
andifwehave(cid:96) (cid:44)(cid:96) then thePlanckdata.ThemainlimitationsarethenoiselevelintSZ
1 2
and CMB map, residual from other astrophysical components
CXXCXXCZZ (mainlytheCIBforthetSZmap),andtheangularresolutionof
<b ,b >= (cid:96)1 (cid:96)2 (cid:96)3 , (27)
(cid:96)1(cid:96)2(cid:96)3 (cid:96)1(cid:96)2(cid:96)3 fskyN(cid:96)1(cid:96)2(cid:96)3 2 http://hdl.handle.net/11299/169642
6
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
Planckmaps,5arcminFWHMfortheCMBmapsand10arcmin Additionally, some components, such as the cosmic-infrared-
forthetSZmaps.TheCMBalsostronglylimitsthesensitivityat background (CIB), cannot be modeled with a single spectral
low-(cid:96).OnFig.3,wecanseelow-(cid:96) oscillationsthatcorrespond lawandcontributeasapartiallycorrelatedcomponentfromfre-
totheCMBacousticoscillationpicscontributingtothenoisefor quencytofrequency.WethusmodeltheCIBcontributiontothe
thetSZ-kSZ-kSZbispectrum. finalvarianceas
V=(cid:16)F [C +C ]−1FT(cid:17)−1, (30)
N CIB
where C is the CIB covariance matrix. We computed the
CIB
CIB covariance matrix using the model presented in (Planck
Collaboration 2015 Results XXIII 2015). For a more realistic
estimation we performed the noise estimation as a function of
themultipole(cid:96)tohaveanestimateofthenoisepower-spectrum
inCMBandtSZmaps.Weverifiedthatthisapproachisrealistic
byapplyingittothePlanckmissionspecificitiesandcomparing
tothenoiselevelobservedinPlancktSZandCMBpublicmaps.
InFig.3,wepresenttheexpectedtSZ-kSZ-kSZbispectrum
S/NforaCOrE+likeexperiment.WeobservethatthetSZ-kSZ-
kSZ S/N is dominated by the cosmic variance up to (cid:96) = 4000.
We can expect a detection up to 200σ when neglecting poten-
tialsystematiceffectintheCMBandtSZmaps.Wealsostress
Figure3. Cumulative S/N for the tSZ-kSZ-kSZ bispectrum as
thatourestimationofthetSZ-kSZ-kSZbispectrummayunder-
afunctionof(cid:96) foracosmicvariancelimitedexperiment(black
estimatetherealsignalathigh-(cid:96),wheretheinternaldynamicsof
line), the Planck experiment (blue line), and a COrE+ like ex-
the gaz inside galaxy clusters will most likely add extra power
periment(redline).Thedashedlineshowsthe5σlevel.
onsmallscales.
.
InFig.4,wepresenttheexpectedconstraintsonΩ ,σ ,and
m 8
Y when combining the measurement of tSZ power spectrum,
(cid:63)
bispectrum,andtSZ-kSZ-kSZbispectrum.
TheLikelihoodfunctioniscomputedasfollows,
5.2.2. FutureCMBexperiments L∝exp(cid:16)−[D−M(σ ,Ω ,Y )]TC [D−M(σ ,Ω ,Y )](cid:17),
8 m (cid:63) SZ 8 m (cid:63)
We now consider a future CMB spacecraft mission assuming (31)
specificities(frequencies,noiselevelpardetector,numberofde-
tectors,beams)foraCOrE+likeexperiment,andaskycoverage Where D is a vector containing the tSZ-tSZ-tSZ and tSZ-kSZ-
afsckcyes=s 0to.5C. FMoBr sourchtSfZutumreapesxpneoriismeelnetv,ewl.eTdhouns’,twheavceodmirpeucttelyd k∆SYZ(cid:63)/bYi(cid:63)spe=ctr0a),fMor iosurafivdecutcoiralcomnotadienlin(gσ8the=tw0.o8,biΩspmec=tra0f.o3r,
theexpectednoiselevelincomponentseparatedmapsobtained parameters (σ8, Ωm, ∆Y(cid:63)/Y(cid:63)), and CSZ is the covariance ma-
through linear combination of multi-frequency intensity maps. trix of the two bispectra in the weakly non-gaussian limit (see
The optimal noise level, V, for a single astrophysical compo- Eq.24).ThecorrelationbetweenthetSZ-tSZ-tSZandtSZ-kSZ-
i
nentisgivenby, kSZbispectraintheweaklynon-gaussianlimitisproportionalto
(cid:16)CtSZ,kSZ(cid:17)2,thuscontraintsfromthetSZ-tSZ-SZbispectrumand
V =(cid:16)FC−1FT(cid:17)−1, (28) fro(cid:96)m the tSZ-kSZ-kSZ bispectrum can be considered as inde-
i i N i
pendent.Figure4showstheexpectedconstraintswhenσ ,Ω ,
8 m
where F is the component spectral behavior in the frequency andY areallowedtovary,wemarginalizedoverH =67.8±0.9
i (cid:63) 0
channels of the experiment, and C−1 is the instrumental noise km/s/Mpcandfixedallotherparameters.
N
covariancematrix.WeassumethatC−1isdiagonal,andthatmul- We observed that σ8 and Ωm present a high degree of de-
N
tipledetectoratagivenfrequencyhaveuncorrelatednoise. generacy, as our three probes present similar degeneracies for
However,thereisseveralcomponentsonthesky.Thus,weuse thistwoparameters.However,weobservedthatwecanachieve
thefollowingequation a precision of 4% on Y(cid:63), without any external prior. Such ap-
proach will allows to calibrate the tSZ scaling relation without
V=(cid:16)FC−1FT(cid:17)−1, (29) theneedofX-rayhydrostaticmass.
N
where F is a rectangular matrix containing the astrophysical 6. Conclusionanddiscussion
componentspectralbehavior.Inthisanalysis,weconsidered:
WehaveproposedanewmethodtodetectkSZeffectusingfu-
• thetSZeffect, ture high-resolution CMB experiments. This method presents
• theCMB, the advantage to be sensitive to the galaxy cluster velocity dis-
• onethermaldustcomponent,thatfollowsamodifiedblack- persion without bias from CMB auto-correlation or from kSZ
bodyspectralenergydistribution(SED)withatemperature, effectproducebythediffusebaryonicgasathighredshift.This
T =20K,andspectralindexβ =1.6, method also allows to constraints the velocity field without se-
d d
• one radio component following a ναr SED with a spectral lection function. By comparison, the kSZ angular power spec-
indexα =−1, trum measured by George et al. (2015), is sensitive to the to-
r
• theCOcomponent, tal kSZ power-spectrum. The constraints achieved on the kSZ
• thespinningdustcomponent. powerspectrumbyCrawfordetal.(2014)whencombiningtSZ
7
G.Hurier:ConstraininggalaxyclustervelocityfieldwiththetSZ-kSZ-kSZbispectrum.
Figure4.LikelihoodforcombinedtSZpowerspectrum,bispectrum,andtSZ-kSZ-kSZbispectrumasafunctionofY ,σ ,andΩ .
(cid:63) 8 m
Darkblue,blue,andlightbluecontoursindicatesthe1,2,and3σlevels.
.
powerspectrumandbispectrumisanindirectconstraints,andis bined with the tSZ power spectrum and bispectrum to set tight
affectedbyastrongcosmicvariance,asitmeasuresthetSZ-tSZ- constraints (4%) on the Y − M relation calibration and thus on
tSZandtSZ-kSZ-kSZbispectraasasinglequantity. thehydrostaticmassbias.Thiswillenablethepossibilitytoset
Themethodproposedinthepresentpaper,relyonadirectmea- cosmological constraints without the need of prior on the hy-
surement of the tSZ-kSZ-kSZ bispectrum after a separation of drostaticmassbias.Whichisacrucialstepconsideringthatthe
the tSZ and kSZ signals. Consequently, it allows to obtain a hydrostaticmass-biasisthemainlimitationforcosmologicalpa-
model-independent and low-cosmic-variance estimation of the rameterestimationfromtSZsurveys.
tSZ-kSZ-kSZbispectrum.
We have presented a complete modeling of the tSZ-kSZ-
Acknowledgements
kSZ bispectrum and deduced the associated cosmological pa-
rameterdependancies.WederivedthedependenciesofthetSZ- The author thanks F.Lacasa for useful discussions. We ac-
kSZ-kSZandtSZbispectrawithrespecttocosmologicalparam- knowledge the support of the French Agence Nationale de la
eters.PreviousworkshavealsodiscussthetSZbispectrumscal- RechercheundergrantANR-11-BD56-015.
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Crawfordetal.2014,seee.g.,),andfoundslightlydifferentscal-
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