Astronomy&Astrophysicsmanuscriptno.isotropy (cid:13)c ESO2017 February1,2017 Constraints on the dark energy dipole from large-scale structures. G.Hurier CentrodeEstudiosdeFísicadelCosmosdeAragón(CEFCA),PlazadeSanJuan,1,planta2,E-44001,Teruel,Spain e-mail:ghurier@cefca.es Received/Accepted Abstract Thehigh-significancemeasurementoflarge-scalestructuresignalsenablestestingtheisotropyoftheUniverse.Themeasurement 7 ofcosmologicalparametersthroughthelarge-scaledistributionofmatterisnowamaturedomain.Thisapproachismainlylimited 1 by our knowledge of astrophysical processes that are used to observe the large-scale structure. However, when we assume that 0 theseastrophysicalprocessesarethesameacrosstheUniverse,thenitispossibletotightlyconstraintheisotropyofcosmological 2 parameters across the sky. Particularly the X-SZ cross-correlation has been shown to be a probe of the large scale structures that n hasahighsignal-to-noiseratioandlowbias.Forthisanalysis,weusedalocalizedmeasurementoftheX-SZcross-correlationasa a testofthecosmologicalparameterisotropy.UsingthescatteroftheX-SZcross-correlationacrossthesky,wederivecosmological J constraintsσ8(Ωm/0.28)0.34 = 0.78±0.02andtightisotropyconstraintsonthedarkenergydipole∆ΩΛ < 0.07at95%confidence 1 level. 3 Keywords.large-scalestructures–galaxyclusters–cosmicmicrowavebackground–intraclustermedium ] O C 1. Introduction mericalsimulations.(ii)Wecanusecorrelationfunctions.Both approaches present advantages and drawbacks: (i) The num- h. The cosmological principle is a pillar of current cosmological ber count is highly sensitive to the selection function and the p models(e.g.,ΛCDM).ItassumesthattheUniverseishomoge- purity of the considered sample of objects, but directly allows - neousandisotropicatlargeenoughscales(Dodelson2003). us to explore the mass function in the mass-redshift plane. (ii) o The cosmic microwave background (CMB) is consistent with r Correlationfunctionscanbedirectlyappliedtosignalmaps,but t statistical isotropy (Planck Collaboration 2015 results XIII they imply that we need to consider a redshift-projected signal s 2016; Planck Collaboration 2015 results XVI 2016), but there a andcontaminationinthemapsproducedbyotherastrophysical [ are a few indications of anomalies with respect to ΛCDM objectsorinstrumentalsystematiceffects. prediction(see,e.g.Schwarzetal.2016). Measurement of cosmological parameters through the large- 1 scale structure distribution is now limited by our knowledge v Some observations might imply that the Universe deviates 3 of astrophysical processes and not by statistical uncertainties. 5 from statistical isotropy (see Perivolaropoulos 2014, for a re- EspeciallythethermalSunyaev-Zel’dovich(tSZ)effectandthe view).EveniftheUniverseseemstobeisotropicatthetimeof 0 X-rayemissioncross-correlationhasbeendetectedatveryhigh 9 the recombination, it is possible to induce late-time anisotropy signal-to-noise ratio (S/N) (Hurier et al. 2015) and is less af- 0 forexamplethroughanisotropiesinthedarkenergydistribution. fectedthanautocorrelationangularpowerspectraobtainedwith 1. Thus, testing the isotropy of the local Universe is a powerful otherastrophysicalprocessesandinstrumentalsystematics.The 0 probeofpotentialdarkenergyanisotropies. tSZeffectproducesasmallspectraldistortionintheblackbody PreviousanalyseshavetestedtheisotropyoftheUniverseusing 7 spectrum of the CMB (Sunyaev & Zeldovich 1969, 1972). Its the type I supernovae (SNIa) to measure the large-scale varia- 1 intensity is related to the integral of the pressure along the line : tionofthedistancemodulusacrossthesky(see,e.g.,Antoniou of sight. Its spectral signature can be used to isolate this effect v &Perivolaropoulos2010;Bengalyetal.2015a;Linetal.2016), fromotheremissiononthemicrowaveandsubmillimetersky. i X galaxyclustercatalogs(Bengalyetal.2015b;deMartinoetal. TheionizedgasintheintraclustermediumalsoproducesanX- 2016), the local velocity flow (Kocevski & Ebeling 2006), or r ray emission through bremsstrahlung. This radiation is propor- a theX-raybackgrounddipole(Plionis&Georgantopoulos1999). tionaltothesquareoftheelectrondensity. Previous works derived ∆Ω /Ω < 0.43±0.06 (Antoniou & m m Considering that full-sky catalogs of galaxy clusters derived Perivolaropoulos 2010). Supernovae are a powerful probe of fromtSZ(e.g.,PlanckCollaboration2015resultsXXVII2016) isotropy because they enable the measurement of the Universe and X-ray (e.g., Piffaretti et al. 2011) have complex selection expansionratetowardalargenumberoflinesight. functionsthatstronglyvaryacrossthesky,wechosetousetSZ TheisotropyoftheUniversecanalsobetestedusingthelarge- and X-ray full-sky maps. The choice of using cross-correlation scale structure matter field. The number of galaxy clusters in insteadofautocorrelationismotivatedbecauseweaimtoensure a given volume strongly depends on the cosmological parame- abettercontroloversystematiceffects. ters. There are two approaches to derive cosmological parame- The paper is organized as follows: Sect. 2 briefly presents the ters from the large-scale structure distribution: (i) We can use data, then Sect. 3 describes the approach we used to perform a thenumbercountofgalaxyclustersanddirectlycompareitwith localized estimation of the X-SZ cross-correlation, and finally themassfunction(seee.g.,Tinkeretal.2008)calibratedonnu- 1 G.Hurier:Constraintsonthedarkenergydipolefromlarge-scalestructures. Sect.4presentsthederivedconstraintsoncosmologicalparam- etersandisotropy. 2. Data WemadeuseofaMILCAtSZfullskymap(Hurieretal.2013)at aresolutionof7arcminFWHM.Thismaphasbeenobtainedus- ingPlanckdata(PlanckCollaboration2015resultsI2016)from 100to857GHz.InadditiontothetSZComptonparameter,the main residuals in these maps are radio point sources and high- redshiftcosmicinfraredbackground(CIB,PlanckCollaboration 2015resultsXXIII2016). We also used the ROSAT all-sky survey (RASS) public data1, which cover 99.8% of the sky, including 97% that have an exposure time longer than 100s (Voges et al. 1999). Then, we constructed a full-sky map of the photon count rate in the energy range 0.5-2.0 keV from each ROSAT photon event file and exposure map. Lower energy photons were not considered toreducetheimpactofneutralhydrogenabsorptionthatmayin- Figure1.CorrelationmatrixofX-SZcrossspectracomputedon ducesomesystematicanisotropyatlargescalesifnotconsidered (cid:39)0.1%ofthesky. properly.Finally,weprojectedeacheventandtheassociatedex- posureovertheskyusingtheHEALPix(Górskietal.2005)pix- elizationschemeataresolutionofN =2048. side We constructed a mask to avoid contamination by Galactic Fromthemeasuredcrosspowerspectra,CXY,inaskyregion (cid:96),i or point source emissions in Planck data. Considering that the i,wecanestimatetheCXYcovariancematrixas Planck 857 GHz channel is a good tracer of the thermal dust (cid:96) emission,wechosetomaskallregionsthatpresentanemission     asublotvs.eIX3 K20C1M4B,2foatrtuhniistfcroenqvueenntcioyn(saenedPcloannvcekrsCioonll)a.bWoreamtioanskreed- C(cid:96),(cid:96)(cid:48) = N 1−1(cid:88)NregC(cid:96)X,iYC(cid:96)X(cid:48),Yi − N1 (cid:88)NregC(cid:96)X,iY(cid:88)NregC(cid:96)X(cid:48),Yi. reg i reg i i all sources from the Planck compact source catalog (Planck (1) Collaborationresults.XXVIII2014)detectedinatleastonefre- quency with a S/N above 5. Therefore, we masked all regions Figure1presentsCXYcorrelationmatrix.Weobservetwosepa- withanexposurebelow100secondsintheRASSsurvey.After rateregionsinthis(cid:96)correlationmatrix.Atlow(cid:96) ((cid:96) < 1000),we applyingthesecuts,wekeptabout60%oftheskyfortheanaly- observeaveryhighlevelofcorrelationbetweenmodes,andthis sis. levelofcorrelationisamanifestationofthenon-GaussianX-SZ- X-SZtri-spectrum(four-pointcorrelationfunction).Theampli- tude of this correlation strongly depends on the number of ob- jectsthatsignificantlycontributetothepower.Consequently,as 3. Methodology smallskyfractionsreducethisnumberofobjects,itenhancesthe importance of these correlations. At high (cid:96) ((cid:96) > 1000), the co- The X-SZ cross power spectrum has been detected with a very variancematrixisdominatedbystatisticaluncertaintyproduced highS/Nratio(seeHurieretal.2015,formoredetails).Sucha bythenoiselevelintheMILCAandRASSfull-skymap. high-S/Nprobeoflarge-scalestructuresallowsustoperforma InFig.2wepresenttheX-SZcrosscorrelationpowerspec- localizedanalysistotesttheisotropyofthematterfieldandcos- trameasuredinindependentsmallskyregions,asdemonstrated mologicalparameter. by the study of the correlation matrix (see Fig. 1). We observe WeusedHEALPixpixelizationtosplittheskyinto768(N = a large scatter at low (cid:96) (< 1000) that is induced by the small- side 8) independent sky regions that cover a sky area of roughly numberstatisticsofbrightgalaxyclusters. 7.3 × 7.3 degrees (corresponding to f (cid:39) 0.13%). Then, we sky rejectedallregionsforwhichtheaverageofthemaskwasbelow 40%,leavingN =416regionswithasufficientskycoverage. 4. Results reg We measured the X-SZ cross power spectra in each region and 4.1. Cosmologicalconstraintsfromthecosmicvariance correctedthepowerspectraformode-mixingeffectsinducedby partial sky coverage (Tristram et al. 2005). We computed and For low-(cid:96) ((cid:96) < 1000) modes, testing the isotropy of the matter propagateuncertaintiesaspresentedinHurieretal.(2015),and field implies verifying that the observed scatter for the X-SZ we modeled the X-SZ cross power spectrum through a halo- cross spectrum follows a consistent statistics with what is modelapproachaspresentedinHurieretal.(2014). expected from an isotropic Universe. As this scatter depends on cosmological parameters, testing isotropy implies fitting cosmologicalparameters. 1 ftp://ftp.xray.mpe.mpg.de/rosat/archive/ 2 K is defined as the unit in which a blackbody spectrum at We performed 1000 Monte Carlo simulations of the X-SZ CMB 2.725Kisflatwithrespecttothefrequency. crosspowerspectraforeachsetofcosmologicalparameters.We 2 G.Hurier:Constraintsonthedarkenergydipolefromlarge-scalestructures. Figure2.X-SZcrosscorrelationpowerspectrum,measuredon Figure3. X-SZ cross-correlationamplitude, A , power spec- XSZ 60% of the sky (solid red line) and measured in small sky re- trum. Theblack samplepresents themeasured power spectrum gions, f < 0.13% (solid gray lines). For readability, uncer- from MILCA and RASS full-sky maps, the red line shows the sky taintiesarenotdisplayedinthefigure(seeHurieretal.2015,for predictionwhenisotropyissatisfied. theuncertaintylevelonthetotalX-SZcrosspowerspectrum). used Poissonian realizations of the mass function (Tinker et al. For 1000 < (cid:96) < 2000, we computed AXSZ in each region of 2008) and log-normal realizations of the tSZ and X-ray fluxes the sky through a linear fit of the measured X-SZ cross power consistentwiththescalinglaws.WeobservedthattheX-SZ-X- spectra(presentedinFig.2)withtheHurieretal.(2015)X-SZ SZtri-spectrumamplitudevariesasΩ5.6σ16.5. powerspectrumprediction.Consequently,wederivedafull-sky We note that using different order stamtisti8cs does not allow us mapof AXSZ inHealpix Nside = 8pixellization.Fromthismap, tobreakthedegeneraciesbetweencosmologicalparametersand wecomputedtheAXSZ angularpowerspectrumandcorrectedit astrophysical processes. When using different order statistics, forpartialskycoverageeffects. we changed the weighting applied on the mass function. For the power spectrum, the mass function is weighted by Y L , In Fig. 3 we present the power spectrum of A up to SZ X XSZ whereY andL aretheintegratedComptonparameterandthe (cid:96)=20.Whenisotropyissatisfied,allthefluctuationsweshould SZ X X-rayluminosity,respectively.FortheX-SZ-X-SZtri-spectrum, observe in the A map should be due to noise and cosmic XSZ the mass function is weighted by Y2 L2. Thus, we are sensi- variance, and should be uncorrelated between independent SZ X tivetodifferentregionsofthemassfunctioninthemass-redshift regions of the sky. Consequently, in the isotropic situation we planethathavedifferentcosmologicalparameterdependencies. expecttoobservewhitenoise.Thedipoleof A isconsistent XSZ Consequently, the degeneracies between cosmological and as- with isotropy at 1.5σ. We observe that all modes up to (cid:96) = 20 trophysical parameters are near invariant for power-spectra, bi- are consistent with isotropy for the large-scale structure matter spectra,andtri-spectraanalyses. distribution. However, as power-spectra, bi-spectra, and tri-spectra analyses probedifferentregionsofthemassfunction,theyallowustotest The A power spectrum can be interpreted as a measure- XSZ theconsistencyofthebest-fittingmodelfordifferentregionsof mentoftheisotropyoftheUniverse.ConsideringthattheCMB themass-redshiftplane.Inthepresentcase,theX-SZ-X-SZtri- shows that the matter distribution is isotropic at z (cid:39) 1100, spectra(measuredthroughthescatteroftheX-SZpower-spectra the most natural solution for interpreting an eventual isotropic acrossthesky)leadstoσ (Ω /0.28)0.34 =0.78±0.02.Similarly behavior of the large-scale structure at low-z is through an 8 m to previous tSZ and X-ray analyses, the uncertainties are dom- anisotropic growth factor induced by large-scale dark energy inated by the uncertainties over the relation between the mass anisotropies. Consequently, we translate our constraints on the of the galaxy clusters, tSZ, and X-ray fluxes. These constraints A dipole into constraints on the dark energy large-scale XSZ aredominatedbyobjectsatlowredshift(z<0.5)andwithhigh anisotropies. mass(M500 (cid:39) 1015 M(cid:12)).Thisconstraintisconsistentwithpre- In Fig. 4 we present the distribution on the sky of the 416 vious findings that focused on low-z high-mass galaxy clusters regions used in the analysis (top panel) and the best-fitting ΩΛ (PlanckCollaborationresults.XX2014). dipole(bottompanel).Tofacilitatevisualization,wedisplaythe full-skybest-fittingdipole.Themaximumofthisdipoleisfound in the direction (l,b) = (22.5o,24.6o) with a significance of 4.2. Constraintsonthedarkenergydipole 1.5σ. Considering that the best-fitting dipole maximum is ori- For high (cid:96) ((cid:96) > 1000), we tested the isotropy of the cosmo- entedneartheGalacticcenter,wecannotexcludesomeresidual logical parameters derived from the 416 individual regions on foregroundbiasinthetSZMILCAmap.Thisdipoleisalsoori- the sky. The X-SZ power-spectrum amplitude, A , strongly entedtowardregionsmaskedbyourcutsontheGalacticthermal XSZ depends on cosmological parameters, A ∝ σ8Ω3.5, but the dustemission,thusitsorientationhastobeconsideredcarefully XSZ 8 m shape of the power spectrum is only marginally affected by forpotentialresidualbiasanditsverylowlevelofsignificance. cosmology. Thus, we used the best-fitting cosmology found in Interpretedasanupperlimit,wehave∆ΩΛ < 0.07at95%con- Hurier et al. (2015) to predict the X-SZ cross power spectrum. fidencelevel,with∆ΩΛbeingthedipoleamplitude. 3 G.Hurier:Constraintsonthedarkenergydipolefromlarge-scalestructures. Dodelson,S.2003,Moderncosmology Górski,K.M.,Hivon,E.,Banday,A.J.,etal.2005,ApJ,622,759 Hurier,G.,Aghanim,N.,&Douspis,M.2014,A&A,568,A57 Hurier,G.,Douspis,M.,Aghanim,N.,etal.2015,A&A,576,A90 Hurier,G.,Macías-Pérez,J.F.,&Hildebrandt,S.2013,A&A,558,A118 Kocevski,D.D.&Ebeling,H.2006,ApJ,645,1043 Lin,H.-N.,Wang,S.,Chang,Z.,&Li,X.2016,MNRAS,456,1881 Perivolaropoulos,L.2014,Galaxies,2,22 Piffaretti,R.,Arnaud,M.,Pratt,G.W.,Pointecouteau,E.,&Melin,J.-B.2011, A&A,534,A109 PlanckCollaboration2015resultsI.2016,A&A,594,A1 PlanckCollaboration2015resultsXIII.2016,A&A,594,A13 PlanckCollaboration2015resultsXVI.2016,A&A,594,A16 PlanckCollaboration2015resultsXXII.2016,A&A,594,A22 PlanckCollaboration2015resultsXXIII.2016,A&A,594,A23 PlanckCollaboration2015resultsXXVII.2016,A&A,594,A27 PlanckCollaborationresults.IX.2014,A&A,571,A9 PlanckCollaborationresults.XX.2014,A&A,571,A20 PlanckCollaborationresults.XXVIII.2014,A&A,571,A28 Plionis,M.&Georgantopoulos,I.1999,MNRAS,306,112 Schwarz,D.J.,Copi,C.J.,Huterer,D.,&Starkman,G.D.2016,Classicaland QuantumGravity,33,184001 Sunyaev,R.A.&Zeldovich,Y.B.1969,Nature,223,721 Sunyaev,R.A.&Zeldovich,Y.B.1972,CommentsonAstrophysicsandSpace Physics,4,173 Tinker,J.,Kravtsov,A.V.,Klypin,A.,etal.2008,ApJ,688,709 Tristram,M.,Macías-Pérez,J.F.,Renault,C.,&Santos,D.2005,MNRAS,358, 833 Voges,W.,Aschenbach,B.,Boller,T.,etal.1999,A&A,349,389 Figure4. Top panel: footprint of the sky regions used (in red) and rejected (in gray) from the analysis. Bottom panel: best- fittingdarkenergydipoleinGalacticcoordinates. 5. Discussionandconclusion We have demonstrated that the measurement of a large-scale structure power-spectrum can be achieved on a small sky frac- tion (f < 0.13%) that is limited by cosmic variance up to sky (cid:96)=1000. Weshowedthatthecosmic-variance-dominatedpartoftheSZ-X powerspectrumcovariancematrixcanbeusedtosetcosmolog- ical constraints with the same accuracy level than the SZ-X or SZ power spectra. However, owing to the similar degeneracies between cosmological parameters and astrophysical processes fortheX-SZpowerspectrumandtheX-SZ-X-SZtri-spectrum, suchcosmologicalconstraintsdonotallowustobreakdegenera- cies between cosmology and astrophysical processes. The derived constraints confirm previous results derived from galaxy number counts (e.g., Planck Collaboration results. XX 2014) andpowerspectra(e.g.,PlanckCollaboration2015resultsXXII 2016;Hurieretal.2015). We also present a tight constraint on the dark energy dipole in the local Universe (z < 1), implying that ΩΛ does not vary acrosstheskybymorethan10%at95%confidencelevel.These constraintsarecompetitivecomparedtothosederivedfromthe studyofthedistancemodulusvariationthroughSNIa(Antoniou &Perivolaropoulos2010;Linetal.2016). References Antoniou,I.&Perivolaropoulos,L.2010,J.CosmologyAstropart.Phys.,12, 012 Bengaly,Jr.,C.A.P.,Bernui,A.,&Alcaniz,J.S.2015a,ApJ,808,39 Bengaly,Jr.,C.A.P.,Bernui,A.,Alcaniz,J.S.,&Ferreira,I.S.2015b,ArXiv e-prints deMartino,I.,Martins,C.J.A.P.,Ebeling,H.,&Kocevski,D.2016,ArXiv e-prints 4

Constraints on the dark energy dipole from large-scale structures

Similar Constraints on the dark energy dipole from large-scale structures