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Mon.Not.R.Astron.Soc.000,000–000(0000) Printed24January2011 (MNLATEXstylefilev2.2) Imprints of primordial non-Gaussianity on the number counts of cosmic shear peaks M. Maturi1, C. Fedeli2,3, and L. Moscardini3,4,5 1Zentrumfu¨rAstronomie,ITA,Universita¨tHeidelberg,Albert-U¨berle-Str.2,D-69120,Heidelberg,Germany 2DepartmentofAstronomy,UniversityofFlorida,312BryantSpaceScienceCenter,Gainesville,FL32611 1 3DipartimentodiAstronomia,Universita` diBologna,ViaRanzani1,I-40127Bologna,Italy 1 4INFN,SezionediBologna,VialeBertiPichat6/2,I-40127Bologna,Italy 0 5INAF-OsservatorioAstronomicodiBologna,ViaRanzani1,I-40127Bologna,Italy 2 n 24January2011 a J 1 ABSTRACT 2 We studied the effect of primordial non-Gaussianity with varied bispectrum shapes on the ] numbercountsofsignal-to-noisepeaksinwidefieldcosmicshearmaps.Thetwocosmolog- O ical contributions to this particular weak lensing statistic, namely the chance projection of C LargeScale Structureandthe occurrenceofreal, cluster-sizeddarkmatterhalos,havebeen modeledsemi-analytically,thusallowingtoeasilyintroducetheeffectofnon-Gaussianinitial . h conditions. We performed a Fisher matrix analysis by taking into account the full covari- p ance of the peak counts in order to forecast the joint constraints on the level of primordial - o non-Gaussianityandtheamplitudeofthematterpowerspectrumthatareexpectedbyfuture r widefieldimagingsurveys.Wefindthatpositive-skewednon-Gaussianityincreasesthenum- t s bercountsofcosmicshearpeaks,moresoathighsignal-to-noisevalues,wherethesignalis a mostlydominatedbymassiveclustersasexpected.Theincrementisatthelevelof 1%for [ f = 10 and 10% for f = 100 for a local shape of the primordialbispectru∼m, while NL NL 1 differentbispec∼trumshapesgivegenericallyasmallereffect.Forafuturesurveyonthemodel v of the proposedESA space mission Euclid and by avoidingthe strongassumption of being 5 capable to distinguish the weak lensing signal of galaxyclusters from chance projectionof 7 LargeScaleStructuresweforecasteda1 σerroronthelevelofnon-Gaussianityof 30 40 1 forthe localandequilateralmodels,and−of 100 200forthe less exploredenfo∼lded−and 4 orthogonalbispectrumshapes. ∼ − . 1 Key words: cosmology: theory - gravitational lensing: weak - cosmological parameters - 0 large-scalestructureoftheUniverse 1 1 : v i X 1 INTRODUCTION Therefore,cosmicshearhaslongbeenrecognizedasanimportant r toolforcosmology.Inthispaperweconsideredoneparticularcos- a Gravitationallensingisoneofthemostpowerful meansofastro- micshear statistic,namely theabundance of signal-to-noise (S/N physical investigation. While baryonic tracers of the dark matter henceforth) ratiopeaks inwidefieldweaklensing maps. Besides distributioncommonlyrelyonstrongsimplifyingassumptions,the the intrinsic ellipticity distribution of background sources, peak gravitationaldeflectionoflight(Bartelmann&Schneider2001)is counts are determined by the cosmological information encapsu- sensitivesolelytotheoverall incidence of any mattercomponent latedbythecosmicmatterdensityfluctuationsfromlineartonon- alongthelineofsight.Inpracticalapplications,theoccurrenceof linearscales,namelyLSSfluctuationsandcluster-sizeddarkmatter stronglydistortedimagesinthecoreofmassivegalaxyclusterscan halos.Countingcosmicshearpeakswithoutaddressingtheirorigin returnimportantinformationabouttheinnerstructureofdarkmat- isnot only easier todo than, for instance, counting weaklensing terhalos,whiletheweaksystematicimagedistortionoflargenum- selectedgalaxyclusters,butalsocontainsmorecosmologicallever- bersofbackgroundgalaxiesallowstoefficientlytracetheoutskirts age(Dietrich&Hartlap2009). ofclustersandtheLargeScaleStructure(LSShenceforth)ingen- eral. InthepresentworkweappliedtheabundanceofS/Npeaksin Cosmic shear, that is weak gravitational lensing on cosmo- cosmicshearmapstotheissueofcosmologicalinitialconditions, logical scales, measures the redshift evolution of the LSS in the andinparticularwhetherornottheprimordial(dark)matterdensity Universeweightedbyaspecificcombination of angular diameter fluctuationsaredistributedaccordingtoaGaussiandistribution.It distances. Henceitcombinescosmological testsbased onthege- isnowwellestablishedthatprimordialnon-Gaussianityhasacon- ometryoftheUniversewithtestsbasedonthegrowthofstructures. siderableimpactondifferentaspectsofstructureformation.Specif- (cid:13)c 0000RAS 2 M.Maturiet al. ically,apositively(negatively) skeweddistributionofinitialmat- standardinflationarymodelthatonehasinmind,aswillbedetailed terfluctuationswouldproducebothamore(less)abundantgalaxy lateron. clusterpopulationandmore(less)biasedstructureswithrespectto A particularly convenient (although not unique) way to de- the underlying matter density field. Therefore, it is expected that scribegeneric deviations fromaGaussian distributionconsistsin the peak statistics would return valuable constraints on the level writingthegauge-invariant Bardeen’spotentialΦasthesumofa andshapeofprimordialnon-Gaussianity. Gaussianrandomfieldandaquadraticcorrection(Salopek&Bond Besides several pioneering works (Messinaetal. 1990; 1990; Ganguietal. 1994; Verdeetal. 2000; Komatsu&Spergel Moscardinietal. 1991; Weinberg&Cole 1992), the problem of 2001),accordingto constraining deviations from primordial Gaussianity by means Φ=Φ + f Φ2 Φ2 . (1) different from the Cosmic Microwave Background (CMB) G NL∗ G−h Gi (cid:16) (cid:17) intrinsic anisotropies has recently attracted renewed attention On sub-horizon scales the Bardeen’s potential equals minus the in the literature, with efforts directed towards the abundance Newtonianpeculiarpotential.Theparameter f inEq.(1)deter- NL of non-linear structures (Matarrese,Verde,&Jimenez 2000; mines the amplitude of non-Gaussianity, and it is in general de- Verdeetal. 2000; Mathis,Diego,&Silk 2004; Grossietal. pendent onthescale.Thesymbol denotes convolution between ∗ 2007, 2009; Maggiore&Riotto 2010c), halo biasing (Dalaletal. functions, and reduces to standard multiplication upon constancy 2008; McDonald 2008; Fedeli,Moscardini,&Matarrese 2009; of f .Inthefollowingweadoptedthelarge-scalestructurecon- NL Fedelietal. 2010), galaxy bispectrum (Sefusatti&Komatsu vention(asopposedtotheCMBconvention,seeAfshordi&Tolley 2007; Jeong&Komatsu 2009), mass density distribution 2008; Carboneetal. 2008; Pillepich,Porciani,&Hahn 2009 and (Grossietal. 2008) and topology (Matsubara 2003; Hikageetal. Grossietal.2009)fordefiningthefundamentalparameter f .Ac- NL 2008), cosmic shear (Fedeli&Moscardini 2010, Pace et al. cordingtothis,theprimordialvalueofΦhastobelinearlyextrap- 2010), integrated Sachs-Wolfe effect (Afshordi&Tolley 2008; olatedatz= 0,andasaconsequencetheconstraintsgivenon f NL Carbone,Verde,&Matarrese2008),Lyαfluxfromlow-densityin- bytheCMBhavetoberaisedby 30percenttocomplywiththis ∼ tergalacticmedium(Vieletal.2009),21 cmfluctuations(Cooray paper’sconvention(seealsoFedeli,Moscardini,&Matarrese2009 − 2006; Pillepich,Porciani,&Matarrese 2007) and reionization foraconciseexplanation). (Crocianietal.2009). Inthecaseinwhich f ,0thepotentialΦisarandomfield NL Asaspecificapplicationofourinvestigationoncosmicshear, with a non-Gaussian probability distribution. Therefore, the field werefertoafuturehalf-skyoptical/near-infraredimaging survey itselfcannot bedescribed by thepower spectrum PΦ(k) = Bkn−4 onthemodeloftheESACosmicVisionproposalEuclid(Laureijs alone,ratherhigher-ordermomentsareneeded,forinstancethebis- 2009). Our aim is to forecast the constraints on the level of pri- pectrumBΦ(k1,k2,k3).ThebispectrumistheFouriertransformof mordialnon-GaussianitythatareexpectedbycountingS/Npeaks thethree-point correlationfunction Φ(k )Φ(k )Φ(k ) anditcan h 1 3 3 i incosmicshearmapsthatshouldbeproducedbyEuclid.Weshall hencebeimplicitlydefinedas alsoinvestigatehowtheseconstraintschangeuponmodificationof severalsurveyparametersanddetectioncriteria,suchastheimag- hΦ(k1)Φ(k3)Φ(k3)i≡(2π)3δD(k1+k2+k3)BΦ(k1,k2,k3). (2) ingdepth,thescaleofoptimalfiltering,andtheS/Nthreshold. As mentioned above understanding the shape of non- ThroughoutthisworkweadoptedforthereferenceGaussian Gaussianityisoffundamental importanceinordertopinpointthe modelthecosmologicalparametersetgivenbythelatestanalysis physicsoftheearlyUniverseandtheevolutionoftheinflatonfield oftheWMAPdata(Komatsuetal.2011),namelythematterden- inparticular.Forthisreason,inthisworkweconsideredfourdiffer- sityparameterΩ =0.272,thecosmologicalconstantdensitypa- entshapesofthepotentialbispectrum,arisingfromdifferentmodi- m,0 rameterΩΛ,0 = 0.728,thebaryondensityparameterΩb,0 = 0.046, ficationsofthestandardinflationaryscenario.Wesummarizethem tmheattHerubpbolweecrosnpsetacntrtuhm≡noHrm0/a(l1i0z0atkiomnss−e1tMbypcσ−1)==00.8.70094.,Tahnedrtehset itnoFtheedefolilleotwali.n(g2,0r1e0fe)rfroinrgfuthrteherreaddeetaritlos.thequotedreferencesand 8 of our paper isorganized asfollows. In Section2 we summarize thevariousnon-Gaussiancosmologiesadopted,whileinSection3 Localshape werecallthemodificationsthattheseintroducetotheclustermass function, the large scale bias, and the matter power spectrum. In Thestandardsingle-fieldinflationaryscenariogeneratesnegligibly Section4wedescribethevariouscontributionstotheweaklensing smalldeviationsfromGaussianity.Thesedeviationsaresaidtobe peaknumbercountsandhowtheyhavebeenmodeled.InSection5 ofthelocalshape,andtherelatedbispectrumoftheBardeen’spo- weshowtheeffectofprimordialnon-Gaussianityontheabundance tentialismaximizedforsqueezedconfigurations,whereoneofthe ofcosmicshearpeaks,whileinSection6wereporttheFisherma- threewavevectorshasmuchsmallermagnitudethantheothertwo. trixanalysisthatweperformedinordertoforecastconstraintson Inthiscasetheparameter f isusuallyassumedtobeaconstant, NL theinitialconditions.FinallyinSection7wesummarizeourcon- anditisexpectedtobeofthesameorderoftheslow-rollparam- clusions. eters(Falk,Rangarajan,&Srednicki 1993), that arevery close to zero. Howevernon-Gaussianitiesofthelocalshapecanalsobegen- eratedinthecaseinwhichanadditional light scalar field,differ- 2 NON-GAUSSIANCOSMOLOGIES entfromtheinflaton,contributestotheobservedcurvaturepertur- Extensions of the most standard model of inflation (Starobinskiˇi bations (Babich,Creminelli,&Zaldarriaga 2004). This happens, 1979; Guth 1981; Linde 1982) can produce substantial de- forinstance,incurvatonmodels(Sasaki,Va¨liviita,&Wands2006; viations from a Gaussian distribution of primordial density Assadullahi,Va¨liviita,&Wands 2007) or in multi-fields models and potential fluctuations (see Bartoloetal. 2004; Chen 2010; (Bartolo,Matarrese,&Riotto2002;Bernardeau&Uzan2002).In Desjacques&Seljak 2010 for recent reviews). The amount and thiscasetheparameter f isallowedtobesubstantiallydifferent NL shape of these deviations depend critically on the kind of non- fromzero. (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 3 Equilateralshape can be obtained as a suitable linear combination of the equilat- eral and enfolded shapes (Senatore,Smith,&Zaldarriaga 2010; In some inflationary models the kinetic term of the inflaton Wagner,Verde,&Boubekeur 2010). Nevertheless, we performed Lagrangian is not standard, containing higher-order derivatives computations for the orthogonal model as well, since it gives a of the field itself. One significant example of this is the DBI differentsignatureontheevolutionoftheLSS. model (Alishahiha,Silverstein,&Tong 2004; Silverstein&Tong 2004, see also Arkani-Hamedetal. 2004; Seery&Lidsey 2005; Li,Wang,&Wang 2008). In this case the primordial bispectrum 3 COSMOLOGICALOBSERVABLES ismaximizedforconfigurationswherethethreewavevectorshave approximatelythesameamplitude,anditiswellrepresentedbythe Primordialnon-Gaussianityproducesmodificationsinthestatistics templateintroducedbyCreminellietal.(2007). of density peaks, resulting in differences in the mass function of Given this, we have the freedom to insert a running cosmicobjectsandthebiasofdarkmatterhaloswithrespecttothe γ(k1,k2,k3) for fNL, since this parameter is not forced to be underlying smooth density field. In the following we summarize constant in this case. The form we chose reads (Chen 2005; howthesemodificationshavebeentakenintoaccountinthepresent LoVerdeetal.2008;Crocianietal.2009) work,andhowtheyreflectintomodificationstothematterpower spectrum. γ(k ,k ,k )= k1+k2+k3 −2κ . (3) 1 2 3 k ! CMB 3.1 Massfunction Inallcalculationsthatfollowweconsideredthisrunningaspartof theequilateralbispectrum.Thisisimportantsincedifferentauthors For the non-Gaussian modification to the mass function of cos- choosedifferent runnings, ornorunning atall,fortheequilateral micobjectsweadopted theprescriptionof LoVerdeetal.(2008). shape.Weadoptedtheexponentκ = 0.2intheremainderofthis The main assumption behind it is that the effect of primor- − work, that increase thelevel of non-Gaussianity at scalessmaller dial non-Gaussianity on the mass function is independent of the thanthat corresponding tok = 0.086h Mpc 1.Thiscoincides prescription adopted to describe the mass function itself. This CMB − withthelargermultipoleusedintheCMBanalysisbytheWMAP means that, if n(G)(M,z) and n (M,z) are the non-Gaussian and PS PS team(Komatsuetal.2009,2011),ℓ 700. Gaussianmassfunctions,respectively,computedaccordingtothe ∼ Press&Schechter(1974)formula,wecandefineacorrectionfac- tor (M,z) n(G)(M,z)/n (M,z). Then, the non-Gaussian mass R ≡ PS PS Enfoldedshape function computed according to an arbitrary prescription, n(M,z) canberelatedtoitsGaussiancounterpartthrough For deviations from Gaussianity evaluated in the regular Bunch- Davies vacuum state, the primordial potential bispectrum is n(M,z)= (M,z)n(G)(M,z). (4) R of local or equilateral shape, depending on whether or not In order to compute n (M,z), and hence (M,z), higher-order derivatives play a significant role in the evolu- PS R LoVerdeetal. (2008) performed an Edgeworth expansion tion of the inflaton field. If the Bunch-Davies vacuum hy- (Blinnikov&Moessner 1998) of the probability distribution for pothesis is dropped, the resulting bispectrum is maximal for the smoothed density fluctuations field, truncating it at the linear squashed configurations (Chenetal. 2007; Holman&Tolley term in σ . The resulting Press&Schechter (1974)-like mass M 2008). Meerburg,vanderSchaar,&Corasaniti (2009) found a functionreads template that describes very well the properties of this enfolded- shapebispectrum(seehoweverCreminellietal.2010foraslightly n (M,z) = 2ρm exp δ2c(z) dlnσM δc(z)+ differenttemplateofthephysicalmodel). PS −rπ M "−2σ2M#" dM σM S σ δ4(z) δ2(z) + 3 M c 2 c 1 + 6 σ4 − σ2 − !! M M Orthogonalshape 1dS δ2(z) + 3σ c 1 . (5) A shape of the bispectrum can be constructed that is 6dM M σ2 − !# M nearly orthogonal to both the local and equilateral forms In the previous equation ρ = 3H2Ω /8πG is the comoving (Senatore,Smith,&Zaldarriaga2010).Constraintsonthelevelof m 0 m,0 matter density in the Universe, σ is the rms of density fluc- non-GaussianitycompatiblewiththeCMBinthelocal,equilateral M tuations smoothed on a scale corresponding to the mass M, and and orthogonal scenarios were recently given by the WMAP team (Komatsuetal. 2011), while constraints on enfolded non- δc(z)=∆c/D+(z).ThefunctionS3(M)≡µ3(M)/σ4Misthereduced skewnessofthenon-Gaussiandistribution,andtheskewnessµ (M) Gaussianity from galaxy bias were given by Verde&Matarrese 3 canbecomputedas (2009) µ (M) = (k ) (k ) (k ) Although there is no theoretical prescription against a 3 ZR9MR 1 MR 2 MR 3 × running of the fNL parameter with the scale in the enfolded and Φ(k )Φ(k )Φ(k ) dk1dk2dk3 . (6) orthogonal shapes, we decided not to include one. The reason × h 1 2 3 i (2π)9 for this is that there is no first principle that can guide one in Thelastthingthatremainstobedefinedisthefunction (k),that the choice of a particular kind of running, and until now no MR relates the density fluctuations smoothed on some scale R to the work has addressed the problem of a running for these shapes respectivepeculiarpotential, (Fergusson&Shellard 2009; Fergusson,Liguori,&Shellard 2010). Moreover, we recall that the four non-Gaussian shapes 2 T(k)k2 (k) W (k), (7) described above are not independent, rather the orthogonal shape MR ≡ 3H2Ω R 0 m,0 (cid:13)c 0000RAS,MNRAS000,000–000 4 M.Maturiet al. whereT(k)isthemattertransferfunctionandW (k)istheFourier withrespecttothemassfunctionandlinearbias.Particularly,we R transformofthetop-hatwindowfunction. shouldhaveareliablewaytoparametrizethefullynon-linearthree- In this work we adopted the Bardeenetal. (1986) matter dimensional power spectrum, since part of the lensing signal is transfer function, with the shape factor correction of Sugiyama given by the integral thereof along the line of sight. In order to (1995). Thisreproduces fairly well the more sophisticated recipe do that we followed the approach of Fedeli&Moscardini (2010) of Eisenstein&Hu (1998) except for the presence of the baryon (andreferencestherein)andmadeuseofthehalomodelinorderto acoustic oscillation, that anyway is not of interest here. We ad- representthematterpowerspectrum. ditionally adopted as reference mass function the prescription The halo model (Seljak 2000; Ma&Fry 2000; of Sheth&Tormen (2002) (see Jenkinsetal. 2001; Warrenetal. Cooray&Sheth 2002) is a physically motivated framework 2006; Tinkeretal. 2008 for alternative prescriptions). Other ap- that allows to compute the correlation function of various LSS proaches alsoexist forcomputing thenon-Gaussian correctionto tracers,includingthedarkmatterparticlesthemselves.Itisbased themassfunction,thatgiveresultsinbroadagreementwiththose onthesimpleconsiderationthatifallthematterintheUniverseis obtained here (Matarrese,Verde,&Jimenez 2000). In computing lockedintohalos,thenthecontributiontothecorrelationfunction thenon-Gaussian corrections tothemassfunction wehavetaken comesfromparticlepairssittinginseparatedhalosatlargescales intoaccount thecorrectiontothecriticaloverdensity forcollapse andfromparticlepairsbelongingtothesamehaloatsmallscales. suggested by Grossietal. (2009) (see also Maggiore&Riotto Accordingly,thepowerspectrumcanbewrittenasthesumoftwo 2010b,c,a),accordingtowhich∆c→∆c√q,withq∼0.8. termsdescribingthetwocontributions,P(k,z)=P1(k,z)+P2(k,z), with 3.2 Halobias P (k,z)= +∞n(M,z) ρˆ(k,M,z) 2dM, (10) Recently much attention has been devoted to the effect of 1 Z0 " ρm # primordial non-Gaussianity on halo bias, and the use thereof and for constraining f (Matarrese&Verde 2008; Dalaletal. 2008; NL Verde&Matarrese 2009; Carbone,Verde,&Matarrese 2008). In P (k,z)= +∞n(M,z)b(M,z,k)ρˆ(k,M,z)dM 2P (k,z). (11) particular,theseworkshaveshownthatprimordialnon-Gaussianity 2 "Z0 ρm # L introducesascaledependenceonthelargescalehalobias.Thispe- culiarityallowstoplacealreadystringentconstraintsfromexisting IntheprevioussetofequationsPL(k,z)isthelinearmatterpower data(Slosaretal.2008;Afshordi&Tolley2008). spectrumandρˆ(k,M,z)istheFouriertransformofthemeandark The non-Gaussian halo bias can be written in a relatively matter halo density profile which is also in principle modified straightforward way in terms of its Gaussian counterpart as by primordial non-Gaussianity, as suggested in Avila-Reeseetal. (Carbone,Mena,&Verde2010) (2003)andSmith,Desjacques,&Marian(2010).Nevertheless,we adopted a standard Navarro,Frenk,&White (1996, 1997) shape b(M,z,k)=b(G)(M,z)+β (k)σ2 b(G)(M,z) 1 2 , (8) (NFWhenceforth) for ρˆ(k,M,z) because thefunctional formsfor R M − h i themodifieddensityprofilesareproventoholdonlyforthelocal wherethefunctionβ (k)encapsulatesallthescaledependenceof R shape,anditisnotclearwhether thisisthecaseforothershapes thenon-Gaussiancorrectiontothebias,andreads aswell.Thereforeweignoredthesechanges,makingouranalysis β (k) = 1 +∞ζ2 (ζ) moreconservative, sinceapositive(negative) fNL impliesamore R 8π2σ2 (k)Z MR × (less)centrallyconcentrateddarkmatterprofile,andconsequently MMR 0 anincrement(decrement)inthenumberofpeaks. 1 BΦ ζ,√α,k whereα× k2Z+−1ζM2+R(cid:16)2√kζαµ(cid:17).Int(cid:16)hPeΦ(skim) pl(cid:17)edcµasedζof,localbispectru(9m) mFeoddeelFlio&wreMprreoafscectriactrhadeliniidnet(e2tar0ei1lss0te)d.oHnreeatrdheeewrteiomjuApslmtemnaoreatne&tathtRiaotenfthreeogfmieatrhs(es20fuh0na4cl)o-; ≡ tionandthehalobias,bothenteringinthehalomodel,aremodified shapeitcanbeshownthatthefunctionβ (k)shouldscaleas k 2 R ∝ − byprimordialnon-GaussianityaccordingtoSections3.1and3.2. atlargescales,sothatasubstantialboost(if f > 0)inthehalo NL biasisexpectedatthosescales.FortheGaussianbiasb(G)(M,z)we adopted the prescription of Sheth,Mo,&Tormen (2001). In this case, since the correction to the Gaussian bias is written in term oftheGaussianbiasitself,theellipsoidalcollapsecorrectionsug- gestedbyGrossietal.(2009)isnotnecessary. It is interesting to note that, while for the first three non- 4 MATTERFLUCTUATIONSWITHCOSMICSHEAR Gaussian shapes introduced in Section 2, a positive f implies NL In this paper we focused on the number counts of weak lensing bothapositiveskewnessofthematterdensityfieldandapositive peaks, that directly reflect the distributionof dark matter fluctua- correctiontothelargescalehalobias,theoppositeistrueforthe tions. Counting directly the peaks allows to minimize the physi- fourthshape,theorthogonalone.Thisisafacttobekeptinmind calassumptionsnecessaryingoingfromtheadoptedcosmological wheninterpretingourresultsinthesubsequentSections. modeltothedataoutcome.Asamatteroffactwebasedthefinal resultsondarkmatterphysicsonly,byobservingquantitiesinsensi- tivetobaryonphysicsandbyrelatingthemodelpredictiondirectly 3.3 Matterpowerspectrum tothemeasuredS/Nratios.Inparticular,thelatterpointavoidsthe Anothercosmologicalobservablethatisrelevantforourpurposes difficulttaskofdisentanglinginaclearwaytheintrinsicnatureof andgetsmodifiedincaseofprimordialnon-Gaussianityisthemat- eachsinglepeak:noisefluctuation,LSSline-of-sightsuperimposi- ter power spectrum. The latter requires a little bit more of care tionoractualgalaxycluster. (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 5 4.1 Lensingformalism P (ℓ)= 1σ2ǫs , (16) ǫ 2 n Thedeflectionpropertiesofisolatedlensesarefullydescribedby g theirtwo-dimensionallensingpotential, determinedbythenumberdensityn ofgalaxiessuitableforweak g lensingmeasurementsandthevarianceσ2 oftheirintrinsicellip- ψ(θ) 2 Dds sΦ(D θ,z)dz, (12) ticitydistribution. ǫs ≡ c2 DdDs Z0 d whereΦistheNewtoniangravitationalpotentialand D, D ,and Sincecontributions (ii)and(iii)arewellrepresented by two s d independentGaussianrandomfields,theirtotalpowerspectrum,is D are the angular-diameter distances between the observer and ds thesumofthetwocontributions,P(ℓ)=P (ℓ)+P (ℓ). thesource,theobserverandthelens,andthelensandthesource, γt ǫ respectively. Finally, s represents the physical distance out tothe sourcesphere. 4.3 Optimallensingfiltering Thepotentialψrelatestheangularpositionsβofasourceand θ of its images on the observer’s sky through the lens equation, A method to measure gravitational lensing signatures in β = θ ψ.Forsourcessuchasdistantbackgroundgalaxiesitis optical/near-infrared catalogues of galaxies is the aperture mass possibl−et∇olinearizethelensequationsuchthattheinducedimage (Schneideretal. 1998). It is a weighted average of the tangential distortionisexpressedbytheJacobianmatrix componentoftheshearγtoverthepositionθonthesky, A=(1−κ) 1−−gg21 1−+gg21 ! . (13) A(θ)=ZR2d2θ′γt(θ′,θ)Q(kθ′−θk), (17) where the radial filter function Q determines the statistical Inthepreviousequation,κ 2ψ/2istheconvergence,responsi- ≡ ∇ properties of the estimate A we are interested in. In this paper blefortheisotropicmagnificationofanimagerelativetoitssource, weaimatconstrainingthedeviationsfromprimordialGaussianity, and g = γ/(1 κ) isthereduced shear, quantifying theobserved − whichmostlyaffectthenon-linearpartofstructureformationand distortion. Here, γ ψ ψ /2 and γ ψ are the two 1 ≡ ,11− ,22 2 ≡ ,12 thusthehighmassendofthemassfunction.Therefore,weadopted components of the com(cid:0)plex shear,(cid:1)where a comma denotes stan- thelinearmatchedfilterdefinedbyMaturietal.(2005)whichisde- darddifferentiationwithrespect tocoordinates onthelensplane. signedtomaximizetheS/Nratioof nonlinearstructuressuchas It is important to recall that since the angular size of the sources galaxyclusters,i.ethosestructureswhicharemostsensitiveto f , isunknown,onlythereducedshearcanbeestimatedstartingfrom NL theobservedellipticityoftheimages. τˆ(ℓ) τˆ(ℓ)2 Qˆ(ℓ)=α , with α−1 = d2ℓ| | . (18) P(ℓ) ZR2 P(ℓ) 4.2 Contributionstothelensingsignal HerethefilterisrepresentedintheFourierspaceforconvenience, τˆ(ℓ)istheFouriertransformoftheexpectedshearprofileofindi- Theabundance ofS/Npeaksincosmicshearmapsisdetermined vidual halos,inour casetheFouriertransformoftheNFWshear bythesumofthreeseparatecontributions,thatwedescribeinthe profile,andP(ℓ)=P (ℓ)+P (ℓ)describedinSection4.2.Notethat following. γt ǫ theshapeoftheoptimalfilterdependsontheassumedparameters (i) The signal due to the occurrence of real non linear struc- fortheNFWdensityprofile,particularlyontheangularsizecorre- turessuchasgalaxyclusters.TheirabundanceisevaluatedinSec- spondingtothescaleradiusθs.Moreover,sincethefilterisonlya tion 3.1 and their shear profile can be evaluated analytically (see radialfunction,itsFouriertransformdependsonlyonℓ= ℓ k k Bartelmann1996;Meneghettietal.2003).Sincethetypicalsepa- WefurtherdefinethevarianceoftheaperturemassestimateA rationbetweentheobservedbackgroundgalaxiesislargerthanthe as typicalradiusofthegalaxyclusters’criticalcurveswecanassume σ2 +∞ ℓdℓP (ℓ)Qˆ(ℓ)2 , (19) γ gthroughout. A≡Z 2π ǫ | | ≃ 0 (ii) The lensing signal due to chance projections of the LSS, sothatitisrelatedonlytothenon-cosmologicalsignalcomponent, givenbytheeffectiveconvergencepowerspectrum i.e. P . It is worth mentioning that in practical applications A is ǫ P (ℓ)= 9H04Ω2m,0 wHdwW¯2(w)P ℓ ,w , (14) approximatedas κ 4c4 Z0 a2(w) fK(w) ! 1 n A(θ)= ǫ (θ)Q( θ θ ), (20) where P(k,z) is the fully non-linear matter power spectrum dis- n t,i k i− k cussed in Section 3.3, Ω is the present-day matter-density pa- Xi=1 m,0 whereǫ (θ)isthetangential ellipticityofagalaxyimagelocated rameter,H istheHubbleconstant,cisthespeedoflight,a(w)is t,i 0 atthepositionθ withrespecttoθ,andwhichprovidesanestimate thescalefactoratthecomovingdistancew,w isthecomovingdis- i H forγ. tancetothehorizon, f isthecomovingangular-diameterdistance, t K andW¯(w)istheweightfunctionencorporatingtheline-of-sightin- tegraloverthedistributionofbackgroundsourcesG(w), 4.4 Expectedshearpeaksnumbercounts W¯(w)= wHdw′G(w′)fK(w′−w) . (15) Oncethelensingsignalsandtheobservational strategyhavebeen Zw fK(w′) defined(seeSections4.2and4.3),theweaklensingnumbercounts Inourapplicationweusedasingle(thetangential)component of can bepredicted asthe sumof twoindependent components. On theshear,forwhichthepowerspectrumreadsP (ℓ)=P (ℓ)/2. onehandthenon-linearstructures,i.e.galaxyclusters,whosede- γt κ (iii) Theobservational noisecontributionfromtheintrinsicel- tection depends on their intrinsic abundance given by the mass lipticityandfinitenumberofbackgroundgalaxiesusedtomeasure function discussed in Section 3.1, and by their observed S/N ra- theshearsignal,whichhasthewhitenoisepowerspectrum tiodependingontheirexpectedsignalA(seeEq.17)andvariance (cid:13)c 0000RAS,MNRAS000,000–000 6 M.Maturiet al. Figure1.Leftpanel.ThedifferentcontributionstotheabundanceofshearS/NpeaksinthereferenceGaussianmodel.Theblackshortdashedlineshowsthe contributiongivenbyLSSprojectionanddatanoise,whilethebluelongdashedlinereferstothecontributionoftherealdarkmatterclumps.Theheavysolid redlineisthetotalcountdistribution.Weadoptedoptimalfilteringwithascaleradiusofθs =2′.Rightpanel.ThetotalnumberofshearS/Npeaksforthree differentfilterscaleradii,aslabeled. σ2 (see Eq. 19). On the other hand the lensing LSS and instru- matterclumpsstartstoberelevantataroundS/N 4anddominates A ∼ mentalnoisewhicharewellapproximatedbyaGaussianrandom athigherS/Nvalues.Weadditionallyshowintherightpanelofthe fieldandforwhichtheirnumbercountsaboveaspecificthreshold same Figure the total number counts expected for three different S/N =y canbeeasilypredicted, scaleradii of the optimal filter.Asit can be noted, the larger the th th scale,thelargerthecontributionofgalaxyclusterswithrespectto 1 σ 2 y y2 n (y )= 1 th exp th , (21) theLSSandthenoise,althoughthelatterremainsalwaysdominant det th 4√2π3/2 σA! σA −2σ2A! at smaller, but stilllarge, S/N.Thisbehavior isexpected because asshownbyMaturietal.(2010).Hereσ ,definedas forlargerfilterscalesthedetectionswillhavelargerareas,thusin- 1 creasingtheirblendingwheretheirnumberdensityishigher,typi- σ2= +∞ ℓ3dℓP(ℓ)Qˆ(ℓ)2 , (22) callyatlowerS/N.Also,theoverallS/NratiovaluesathigherS/N 1 Z0 2π | | areexpectedtobelarger,sincealargerfilterscaleimpliesabetter is the LSS plus noise lensing field variance which depends on statisticforthebackgroundgalaxiesreducingtheirshotnoiseand the observational noise, the convergence power spectrum and the intrinsicellipticity.Thesefeaturesareensuredbytheadoptedop- adopted filter. Note that, by approximating this therm as a Gaus- timalfilterwhich maximizes thecluster signal against theone of sian random field, the primordial non-Gaussianity is accounted LSSandnoise.Intheremainderofthispaperweshallrefermainly through the LSS power spectrum only so that its leverage is not toascaleradiusof2′,unlessexplicitlynotedotherwise. fully included. For our purpose this is a negligible approxima- tion since we anyway expect this effect to be small and put our final predictions on a conservative side. These two contributions 5 SIGNATUREOFPRIMORDIALNON-GAUSSIANITY carrycomplementarycosmologicalinformation,thusthestatistics ofcosmicshearpeaksareinprinciplevaluablecosmologicaltools Theeffectofprimordialnon-Gaussianityonthestatisticsofshear (Dietrich&Hartlap2009). peaksisexemplifiedinFigure2forthefourdifferentnon-Gaussian Thetwocosmologicalcontributionstotheabundanceofcos- shapes that were discussed in Section 2, and for levels of non- micshearpeaksareexemplifiedintheleftpanelofFigure1forthe Gaussianity f = 50 and f = 100. As it can be seen, the NL ± NL ± referenceGaussiancosmologicalmodel.ForthisFigureandinthe effectofnon-GaussianityismoreevidentathighvaluesofS/Nra- restoftheworkweadoptedthesourceredshiftdistributionobser- tio,whereitismostlygivenbytheeffectonthemassfunctionof vationally derived by Benjaminetal. (2007), while their average dark matter halos. We remind the reader that our predictions re- surface number density is n = 40 arcmin 2, a level achieved by gardingtheimpactofprimordialnon-GaussianityattheS/Nratios g − actualweaklensingsurveysandwhichshouldbereachedbyEU- dominatedbytheLSSareconservativeasdiscussedinSection4.4. CLID(apredictionforasampleofexistingandfutureweaklens- Also,asexpected,theeffectislargestforlocalnon-Gaussianityfor ingsurveysisgivenbyMaturietal.2010).Asexpected,thechance which,athighS/N,thecountscanbemodifiedbyupto 7%for alignmentoftheLSSdominatesthecountsatsmallS/Nratioval- f = 100andsmallestfortheorthogonalshape,where∼itbarely NL ± ues,whilethecontributioncomingfromtheoccurrenceofrealdark reaches 1%.Fortheorthogonalshapepositivevaluesof f pro- ∼ NL (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 7 Figure2.Theratioofthepeaknumbercountsindifferentmodelswithprimordialnon-GaussianitytothesamequantitiesinthereferenceGaussianΛCDM cosmology,asafunctionoftheS/Nratio.Fourdifferentnon-Gaussianshapesandfourdifferentvaluesof fNLareshown,aslabeledintheplot. videadecrementinthenumbercountsofshearpeaksathighS/N InFigure 3weshow the ratioof thepeak number counts in values,whiletheoppositeistrueforallothermodels.Thisagrees non-Gaussian cosmologies with local shape (f = 10) to the NL ± withthebehavioroftheskewnessthathasbeendiscussedinSec- samequantitiesinthereferenceGaussiancaseadoptingthreedif- tion2.Althoughtheeffectofnon-Gaussianity issubstantiallyre- ferentvaluesof thepower spectrumamplitude σ for allmodels. 8 ducedforshapesdifferentfromthelocalone,itislikelythatuse- Weconsidered 2%fluctuations inthevalue of σ , comparable ∼ 8 fulconstraintscanbeputon f fortheseshapesaswell.Forin- withthe1 σuncertaintyaroundthefiducialvaluederivedbythe NL − stanceintheorthogonal case, f isconstrained onlyat thelevel WMAP-7release.Quiteunexpectedly,theeffectofprimordialnon- NL of a few hundreds by CMB data (Komatsuetal. 2011). Our pre- Gaussianityishigherforlowervaluesofσ .Weinterpretthisfact 8 dictioniscompatiblewiththerecentworkofMarianetal.(2010), asduetothedelayedstructureformationimpliedbyalowernor- whofoundaneffectontheshearpeaksnumbercountsatthelevel malization. This allows the effect of primordial non-Gaussianity of 10%forthehighestsignificancedetections.Althoughadirect onthematterdensityfieldtoberetainedforalongertimebythe ∼ comparisonofthetwoworksisnotpossibleduetodifferentsource LSS, with the consequence that the S/N peak number counts are redshiftdistributions,differentdetectionschemes,etc.,weactually slightlymoreaffected.Inotherwords,sincewithalowerσ theon- 8 verifiedthatbyreducingthescaleradiusoftheoptimalfilterdown setofnon-linear evolutionisdelayed, gravitationalclusteringhas toθ =1 theeffectisraisedto 10%.Thus,weconcludethatthe lesstimetoerasethecosmologicalinitialconditions. s ′ ∼ twoworksareinbroadagreementforthelocalnon-Gaussianmodel concerning the magnitude of the effect. This is reassuring, since thetworesultsarebasedonquitedifferentpremisesandadoptdif- ferentmethodologies:weoptedforsemi-analyticmodeling,while Marianetal. (2010) resorted to the use of cosmological simula- tions. (cid:13)c 0000RAS,MNRAS000,000–000 8 M.Maturiet al. 200 ’tabCovariance.gnu’ 150 100 50 0 -50 3 2.5 2 1.5 al 1 on 0.5 ag Numerical 0 Di Figure 3. The ratio of the peak number counts in two different non- Numerical Poiss. -0.5 Analytic Poiss. -1 Gaussian models with local bispectrum shape to the same quantities in -1.5 TthheereufpepreerncseetGoafucssuiravnesΛrCeDfeMr tocofsNmLol=og1y,0,aswahifluentchteiolnowoferthseetSr/Neferrastitoo. 0.0625 0.125 0.25 0.5 0.75 1 1.25 1.5 2S/N2.5 3 4 5 6 7 8 9 10 fNL = −10.Differentcolorsrefertodifferentvaluesofthematterpower Figure 4. Weak lensing number counts covariance matrix for a ΛCDM spectrumnormalizationσ8,aslabeled. model.Theoff-diagonaltermsaresmallbutnotnegligibleonlyatlowS/N ratios.Thebottompanelshowsthecomparisonbetweenthediagonalcom- ponentsofthefullcovariancematrixderivednumerically(showninthetop 6 COSMOLOGICALCONSTRAINTS panel)andwhatwouldbeexpectedforidealnumbercountswithPoisson InthissectionweuseaFisher-matrixanalysistoestimatetheconfi- statisticandstatisticallyindependentS/Nbins.ForhighS/Nthecovariance dencelevelintheσ f plane,asobtainedwiththeweaklensing isdiagonalandPoissonianbyconstruction. 8− NL numbercounts. low S/N ratios up to S/N 3, where we expect to have a small ∼ 6.1 Fishermatrixanalysis butnotnegligiblecovariancecontributionsincedifferentS/Nbins canbecorrelated.Toaccount forthenon-diagonal covariance el- TheFishermatrixisgivenby ements, werealizedaset of 1000 numerical realizations of weak lensingdata,modeledwithaGaussianrandomfieldwhosepower ∂2 Fij =−*∂ξ∂Lξ +, (23) spectrum, P(ℓ),isdiscussedinSection4.2.Asexpectedthenum- i j ber counts derived fromthesimulationagrees withtheanalytical where is the logarithm of the likelihood function and ξ = predictionsexceptforverylowvaluesoftheS/Nratio(S/N<0.2) L (σ8,fNL) are the free parameters of the number counts model. In wheretheanalyticapproximationfailsasdiscussedbyMaturietal. caseofamultivariateGaussianlikelihood,theFishermatrixcanbe (2010).Inanycasetheseverylow S/Narecompletelynegligible writtenas foranycosmologicalanalysis. 1 Thetotalcovariancematrixofweaklensingnumbercountsis Fij = 2Tr AiAj+C−1Mij , (24) shown inFigure 4. Asexpected, ithas astrong diagonal compo- h i where A = C 1C , M = 2(∂µ/∂ξ)(∂µ/∂ξ ), C is the data co- nentanditsoff-diagonaltermsaresmallbutnonnegligibleonlyat i − ,i ij i j lowS/Nratios.Bytreatingthetwocomponents,LSS+noiseand variancematrix,andµistheassumedmodel,i.e.thepeaksnumber galaxyclustersasindependent,wedonotaccountfortheirmutual countsfordifferentS/N.Hereacommadenotesdifferentiationwith influenceonthefinalnumbercounts,butthiseffectisexpectedto respecttotherelevantcosmologicalparameter.SinceChasavery besmallbecausethesignatureofeachclusteraffectsonlyfewpix- weakdependenceonthemodelparametersσ and f thefirstterm 8 NL els. Inany case, thisapproach isconservative withrespect tothe inEq.24isnegligiblebecauseofC .WeevaluatedtheFisherma- trixatthefiducialpointσ = 0.809,i and f = 0asmeasuredby finalcosmologicalconstraintsbecauseweneglecttheimpactofthe WMAP-7forastandardΛ8CDMmodel(KoNmLatsuetal.2011). detectionswithhighS/Nratios,carriersofthenon-Gaussianitysig- nal,onthosewithlowerS/Nratios. As discussed in Section 4.4, the number counts model µ is givenbytwocomponents. Ononehand,thenonlinearstructures whosenumbercountsareindependent fordifferentS/Nratiobins 6.2 Cosmologicalconstraintson f and whose covariance contribution is therefore diagonal with a NL Poissonian amplitude. On the other hand, the contribution given InthisSectionweassumetheexpectedperformancesoftheweak bytheLSSandnoiseforwhichthenumbercountsarerelevantfor lensingsurveyEuclid(Laureijs2009),i.e.afieldofviewof20,000 (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 9 Figure5.Cosmological1 σconfidencelevelsasobtainedforthelocal Figure 6. Confidence contours for the local shape non-Gaussian model, − shapenon-Gaussianmodelusingweaklensingnumbercounts,byassum- withdifferentlowerS/Ncut-offlimits.Thisshowstheinformationlostby ingbothPoissonianstatisticalindependence(Poissonian),andbyusingthe cuttingthedataatdifferentS/Nratios. numericalsimulationsincludingthefullcovariancematrix(numerical). thefulldatacovariancehasbeenusedonlyforthefinalresultsre- portedbelow.Theresultsareevaluatedforasurveyskycoverage ng ∆σ8 ∆fNL θs ∆σ8 ∆fNL of20,000deg2 andscalewiththesquarerootofthesurveyarea. 20 0.0023 60 1’ 0.0016 57 Theconstraintsonboth fNL andσ8 improvewithincreasingnum- 40 0.0015 45 2’ 0.0015 45 berdensityofbackgroundsources.Howeveronlyamarginalben- 80 0.0012 41 4’ 0.0015 38 efitisobtainedbyincreasingthenumberdensityover40arcmin 2. − In other words, in terms of joint constraints for the level of non- Table1.Dependenceofthe1 σjointboundsonthebackgroundgalaxy Gaussianity and amplitude of the matter power spectrum, it does − numberdensityng(left)andonthefilterscaleradius,θs(right).Thefiducial notpayofftoincreasethedepthofthesurveybeyondwhatisal- modelwithσ8=0.809and fNL=0isadopted. ready planned for Euclid. Concerning the scale radius of the op- timal filter,by increasing it we get asignificant tightening of the constraintson f ,whilethoseonσ arealmostinsensitivetoit. NL 8 deg2, an average background galaxy number density of n = 40 Nowthatwehaveanoverallpictureofthedependenceonthe g arcmin 2withanintrinsicellipticityrmsofσ =0.3,andaredshift survey characteristics, we used the numerical simulations includ- − ǫs distributionsimilartotheonediscussedbyBenjaminetal.(2007). ingthefullcovariancematrixtoobtainourfinalresults.InFigure6 TheconfidencelevelsareevaluatedwiththeFishermatrixanalysis we show the 1 σ confidence levels in the σ f plane ob- − 8 − NL discussed in Section 6.1. In Figure 5 we compare the confidence tainedforthenon-Gaussianmodelwithlocalshapewhendifferent regions for the local shape non-Gaussian model obtained first by minimumS/Nratiosareconsidered.Itisinfactcommonpractice assumingpurelyPoissonianindependentstatistic,i.e.adatacovari- foractualapplicationstouseonlythosedetectionswithS/Nlarger ancehavingapurediagonalcomponentwithPoissonianamplitude, thenS/N = 3 5intheattempttoisolategalaxyclustersfrom min − andthenbyusingthefullcovariancematrixobtainedfromthenu- all other signal components. Herewe show how the cut-off level merical simulations (see Section 6.1). As can be seen, the 1 σ affectsthefinalresults.Therelatedlossofinformationhappensbe- − confidence ellipse is slightly tilted in the latter case with respect cause the presence of non linear structures affects the statisticof totheformer,implyingasomewhatdifferentdirectionofdegener- weaklensingmapsalsoatlowersignaltonoiseratioswherethey acybetweenthetwoparametersunderconsideration.Overall,how- cannotbeseenasclearsingledetectionsbutwheretheirpresenceis ever,theconstraintsonthenon-Gaussianitylevel f arebasically stillvisible.Moreover,althoughthiseffectisnegligibleforthe f NL NL unchanged, whilethoseonthenormalizationofthematterpower estimates, chance projection of theLSSdominates at thesesmall spectrumσ areonlyveryslightlyloosened byusingthesimpler S/Nvalues, andstillcontains cosmological information. Figure6 8 Poissonmodel. Asaconsequence, it issafetostatethat thefinal shows that adopting a conservative choice for the minimum S/N differencesbetweenthetwoapproachesarenegligibleandthatfor value can results in a loss of constraining power of a factor 3 ∼ afirstfastanalysisthesimplifiedapproachcanbeused. on f andafactorof 4onσ .Thus,particularcarehavetobe NL ∼ 8 Giventhat,weusedthefirstapproachtoestimatetheCrame´r- usedwhendefiningthedetectionselectioncriteriasincedatacanbe Rao bounds on σ and f at 1 σ level for different survey fullyexploitedonlyiflowerS/Nlevels,e.g.S/N 2,areused. 8 NL − min ∼ depths,i.e.differentgalaxiesnumberdensities,andfordifferentfil- Thiswouldbepossibleonlyifadeepstatisticalunderstanding of terscalesasshowninTable1.Inordertosavecomputationaltime, dataisachieved together withadetailedmodeling. Atentativeto (cid:13)c 0000RAS,MNRAS000,000–000 10 M.Maturiet al. Figure7.Confidenceregionsforthefouradoptedmodelswithnon-Gaussianinitialconditions.Wereportresultsforthelocal(topleft),equilateral(topright), enfolded(bottom left)anqdorthogonal(bottom right)primordialbispectrum shapes.Thethreecontours ineachpanelreferto68.3%,95.4%,and99.7% confidencelevels. gointhisdirectionisprovidedbytheanalyticpredictionforweak lensing-selected clusters it is possible to push the bounds on f NL lensingnumbercountsproposedbyMaturietal.(2010)butwhich belowwhatwouldbeachievedwithoutredshiftinformation.Inad- stillneedstoimprovesomeoftheadoptedapproximations. dition, it is likely that a combination of these diverse probes can reduce the error on f substantially. We plan to investigate this FinallyweshowinFigures7theconstraintsfor68.3%,95.4% NL and 99.7% confidence levels with S/N = 0 for the four dif- synergyinafuturework. min ferent primordial bispectrum shapes discussed in Section 2. We The constraints on the amplitude of the matter power spec- adopted the optimal filter with scale of θs = 2′. As one could trum,∆σ8 ∼ 10−3 arethemselvesquitecompetitive.Forinstance, naively expect, while the constraints on σ8 are rather insensitive recentlyWangetal.(2010)forecastedanerroronσ8of∼7×10−2 totheadoptedshapeoftheprimordialbispectrum,theconstraints byusingtheshapeandpositionoftheBaryonAcousticOscillation on f varywidelywithit.Particularly,whilethe1 σerror∆f measuredbyEuclid.DespitethefactthatWangetal.(2010)letall isatNtLhelevelofafewtensforthelocalandequila−teralshapes,NiLt thecosmologicalparametersfreetovary,whilewelimitouranal- growsupto 100 200fortheenfoldedandorthogonal shapes. ysisto fNL andσ8,thesenumbersargueforourconstraintsonthe Theseconstra∼intsar−enotcompetitivewithwhatisexpectedbyfu- amplitudeofthematterpowerspectrumbeingsound. tureCMBandgalaxyclusteringprobes,andareonlycomparable The degeneracy between the level of primordial non- withtheexpectedperformanceoftheweaklensingpowerspectrum Gaussianityandtheamplitudeofthematterpowerspectrumisalso (Fedeli&Moscardini2010)analysis.Themainreasonforthisre- easy to understand, since increases in both f and σ bring an NL 8 sultsisthatinthisworkwedidnotusethefullredshiftinformation incrementinthelargescalematterpowerspectrumandtheoccur- whichisotherwiseincludedintheotherworksandwhichcouldbe rence of massive dark matter halos. The only exception tothisis includedbytheuseoflensingtomography. Recently,Fedelietal. givenbytheorthogonalmodel,becauseinthiscaseapositive f NL (2010) have shown that using the correlation function of weak impliesadecrementinboththeabundanceofcosmicstructuresand (cid:13)c 0000RAS,MNRAS000,000–000

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Preview Imprints of primordial non-Gaussianity on the number counts of cosmic shear peaks

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed24January2011 (MNLATEXstylefilev2.2) Imprints of primordial non-Gaussianity on the number counts of cosmic shear peaks M. Maturi1, C. Fedeli2,3, and L. Moscardini3,4,5 1Zentrumfu¨rAstronomie,ITA,Universita¨tHeidelberg,Albert-U¨berle-Str.2,D-69120,Heidelberg,Germany 2DepartmentofAstronomy,UniversityofFlorida,312BryantSpaceScienceCenter,Gainesville,FL32611 1 3DipartimentodiAstronomia,Universita` diBologna,ViaRanzani1,I-40127Bologna,Italy 1 4INFN,SezionediBologna,VialeBertiPichat6/2,I-40127Bologna,Italy 0 5INAF-OsservatorioAstronomicodiBologna,ViaRanzani1,I-40127Bologna,Italy 2 n 24January2011 a J 1 ABSTRACT 2 We studied the effect of primordial non-Gaussianity with varied bispectrum shapes on the ] numbercountsofsignal-to-noisepeaksinwidefieldcosmicshearmaps.Thetwocosmolog- O ical contributions to this particular weak lensing statistic, namely the chance projection of C LargeScale Structureandthe occurrenceofreal, cluster-sizeddarkmatterhalos,havebeen modeledsemi-analytically,thusallowingtoeasilyintroducetheeffectofnon-Gaussianinitial . h conditions. We performed a Fisher matrix analysis by taking into account the full covari- p ance of the peak counts in order to forecast the joint constraints on the level of primordial - o non-Gaussianityandtheamplitudeofthematterpowerspectrumthatareexpectedbyfuture r widefieldimagingsurveys.Wefindthatpositive-skewednon-Gaussianityincreasesthenum- t s bercountsofcosmicshearpeaks,moresoathighsignal-to-noisevalues,wherethesignalis a mostlydominatedbymassiveclustersasexpected.Theincrementisatthelevelof 1%for [ f = 10 and 10% for f = 100 for a local shape of the primordialbispectru∼m, while NL NL 1 differentbispec∼trumshapesgivegenericallyasmallereffect.Forafuturesurveyonthemodel v of the proposedESA space mission Euclid and by avoidingthe strongassumption of being 5 capable to distinguish the weak lensing signal of galaxyclusters from chance projectionof 7 LargeScaleStructuresweforecasteda1 σerroronthelevelofnon-Gaussianityof 30 40 1 forthe localandequilateralmodels,and−of 100 200forthe less exploredenfo∼lded−and 4 orthogonalbispectrumshapes. ∼ − . 1 Key words: cosmology: theory - gravitational lensing: weak - cosmological parameters - 0 large-scalestructureoftheUniverse 1 1 : v i X 1 INTRODUCTION Therefore,cosmicshearhaslongbeenrecognizedasanimportant r toolforcosmology.Inthispaperweconsideredoneparticularcos- a Gravitationallensingisoneofthemostpowerful meansofastro- micshear statistic,namely theabundance of signal-to-noise (S/N physical investigation. While baryonic tracers of the dark matter henceforth) ratiopeaks inwidefieldweaklensing maps. Besides distributioncommonlyrelyonstrongsimplifyingassumptions,the the intrinsic ellipticity distribution of background sources, peak gravitationaldeflectionoflight(Bartelmann&Schneider2001)is counts are determined by the cosmological information encapsu- sensitivesolelytotheoverall incidence of any mattercomponent latedbythecosmicmatterdensityfluctuationsfromlineartonon- alongthelineofsight.Inpracticalapplications,theoccurrenceof linearscales,namelyLSSfluctuationsandcluster-sizeddarkmatter stronglydistortedimagesinthecoreofmassivegalaxyclusterscan halos.Countingcosmicshearpeakswithoutaddressingtheirorigin returnimportantinformationabouttheinnerstructureofdarkmat- isnot only easier todo than, for instance, counting weaklensing terhalos,whiletheweaksystematicimagedistortionoflargenum- selectedgalaxyclusters,butalsocontainsmorecosmologicallever- bersofbackgroundgalaxiesallowstoefficientlytracetheoutskirts age(Dietrich&Hartlap2009). ofclustersandtheLargeScaleStructure(LSShenceforth)ingen- eral. InthepresentworkweappliedtheabundanceofS/Npeaksin Cosmic shear, that is weak gravitational lensing on cosmo- cosmicshearmapstotheissueofcosmologicalinitialconditions, logical scales, measures the redshift evolution of the LSS in the andinparticularwhetherornottheprimordial(dark)matterdensity Universeweightedbyaspecificcombination of angular diameter fluctuationsaredistributedaccordingtoaGaussiandistribution.It distances. Henceitcombinescosmological testsbased onthege- isnowwellestablishedthatprimordialnon-Gaussianityhasacon- ometryoftheUniversewithtestsbasedonthegrowthofstructures. siderableimpactondifferentaspectsofstructureformation.Specif- (cid:13)c 0000RAS 2 M.Maturiet al. ically,apositively(negatively) skeweddistributionofinitialmat- standardinflationarymodelthatonehasinmind,aswillbedetailed terfluctuationswouldproducebothamore(less)abundantgalaxy lateron. clusterpopulationandmore(less)biasedstructureswithrespectto A particularly convenient (although not unique) way to de- the underlying matter density field. Therefore, it is expected that scribegeneric deviations fromaGaussian distributionconsistsin the peak statistics would return valuable constraints on the level writingthegauge-invariant Bardeen’spotentialΦasthesumofa andshapeofprimordialnon-Gaussianity. Gaussianrandomfieldandaquadraticcorrection(Salopek&Bond Besides several pioneering works (Messinaetal. 1990; 1990; Ganguietal. 1994; Verdeetal. 2000; Komatsu&Spergel Moscardinietal. 1991; Weinberg&Cole 1992), the problem of 2001),accordingto constraining deviations from primordial Gaussianity by means Φ=Φ + f Φ2 Φ2 . (1) different from the Cosmic Microwave Background (CMB) G NL∗ G−h Gi (cid:16) (cid:17) intrinsic anisotropies has recently attracted renewed attention On sub-horizon scales the Bardeen’s potential equals minus the in the literature, with efforts directed towards the abundance Newtonianpeculiarpotential.Theparameter f inEq.(1)deter- NL of non-linear structures (Matarrese,Verde,&Jimenez 2000; mines the amplitude of non-Gaussianity, and it is in general de- Verdeetal. 2000; Mathis,Diego,&Silk 2004; Grossietal. pendent onthescale.Thesymbol denotes convolution between ∗ 2007, 2009; Maggiore&Riotto 2010c), halo biasing (Dalaletal. functions, and reduces to standard multiplication upon constancy 2008; McDonald 2008; Fedeli,Moscardini,&Matarrese 2009; of f .Inthefollowingweadoptedthelarge-scalestructurecon- NL Fedelietal. 2010), galaxy bispectrum (Sefusatti&Komatsu vention(asopposedtotheCMBconvention,seeAfshordi&Tolley 2007; Jeong&Komatsu 2009), mass density distribution 2008; Carboneetal. 2008; Pillepich,Porciani,&Hahn 2009 and (Grossietal. 2008) and topology (Matsubara 2003; Hikageetal. Grossietal.2009)fordefiningthefundamentalparameter f .Ac- NL 2008), cosmic shear (Fedeli&Moscardini 2010, Pace et al. cordingtothis,theprimordialvalueofΦhastobelinearlyextrap- 2010), integrated Sachs-Wolfe effect (Afshordi&Tolley 2008; olatedatz= 0,andasaconsequencetheconstraintsgivenon f NL Carbone,Verde,&Matarrese2008),Lyαfluxfromlow-densityin- bytheCMBhavetoberaisedby 30percenttocomplywiththis ∼ tergalacticmedium(Vieletal.2009),21 cmfluctuations(Cooray paper’sconvention(seealsoFedeli,Moscardini,&Matarrese2009 − 2006; Pillepich,Porciani,&Matarrese 2007) and reionization foraconciseexplanation). (Crocianietal.2009). Inthecaseinwhich f ,0thepotentialΦisarandomfield NL Asaspecificapplicationofourinvestigationoncosmicshear, with a non-Gaussian probability distribution. Therefore, the field werefertoafuturehalf-skyoptical/near-infraredimaging survey itselfcannot bedescribed by thepower spectrum PΦ(k) = Bkn−4 onthemodeloftheESACosmicVisionproposalEuclid(Laureijs alone,ratherhigher-ordermomentsareneeded,forinstancethebis- 2009). Our aim is to forecast the constraints on the level of pri- pectrumBΦ(k1,k2,k3).ThebispectrumistheFouriertransformof mordialnon-GaussianitythatareexpectedbycountingS/Npeaks thethree-point correlationfunction Φ(k )Φ(k )Φ(k ) anditcan h 1 3 3 i incosmicshearmapsthatshouldbeproducedbyEuclid.Weshall hencebeimplicitlydefinedas alsoinvestigatehowtheseconstraintschangeuponmodificationof severalsurveyparametersanddetectioncriteria,suchastheimag- hΦ(k1)Φ(k3)Φ(k3)i≡(2π)3δD(k1+k2+k3)BΦ(k1,k2,k3). (2) ingdepth,thescaleofoptimalfiltering,andtheS/Nthreshold. As mentioned above understanding the shape of non- ThroughoutthisworkweadoptedforthereferenceGaussian Gaussianityisoffundamental importanceinordertopinpointthe modelthecosmologicalparametersetgivenbythelatestanalysis physicsoftheearlyUniverseandtheevolutionoftheinflatonfield oftheWMAPdata(Komatsuetal.2011),namelythematterden- inparticular.Forthisreason,inthisworkweconsideredfourdiffer- sityparameterΩ =0.272,thecosmologicalconstantdensitypa- entshapesofthepotentialbispectrum,arisingfromdifferentmodi- m,0 rameterΩΛ,0 = 0.728,thebaryondensityparameterΩb,0 = 0.046, ficationsofthestandardinflationaryscenario.Wesummarizethem tmheattHerubpbolweecrosnpsetacntrtuhm≡noHrm0/a(l1i0z0atkiomnss−e1tMbypcσ−1)==00.8.70094.,Tahnedrtehset itnoFtheedefolilleotwali.n(g2,0r1e0fe)rfroinrgfuthrteherreaddeetaritlos.thequotedreferencesand 8 of our paper isorganized asfollows. In Section2 we summarize thevariousnon-Gaussiancosmologiesadopted,whileinSection3 Localshape werecallthemodificationsthattheseintroducetotheclustermass function, the large scale bias, and the matter power spectrum. In Thestandardsingle-fieldinflationaryscenariogeneratesnegligibly Section4wedescribethevariouscontributionstotheweaklensing smalldeviationsfromGaussianity.Thesedeviationsaresaidtobe peaknumbercountsandhowtheyhavebeenmodeled.InSection5 ofthelocalshape,andtherelatedbispectrumoftheBardeen’spo- weshowtheeffectofprimordialnon-Gaussianityontheabundance tentialismaximizedforsqueezedconfigurations,whereoneofthe ofcosmicshearpeaks,whileinSection6wereporttheFisherma- threewavevectorshasmuchsmallermagnitudethantheothertwo. trixanalysisthatweperformedinordertoforecastconstraintson Inthiscasetheparameter f isusuallyassumedtobeaconstant, NL theinitialconditions.FinallyinSection7wesummarizeourcon- anditisexpectedtobeofthesameorderoftheslow-rollparam- clusions. eters(Falk,Rangarajan,&Srednicki 1993), that arevery close to zero. Howevernon-Gaussianitiesofthelocalshapecanalsobegen- eratedinthecaseinwhichanadditional light scalar field,differ- 2 NON-GAUSSIANCOSMOLOGIES entfromtheinflaton,contributestotheobservedcurvaturepertur- Extensions of the most standard model of inflation (Starobinskiˇi bations (Babich,Creminelli,&Zaldarriaga 2004). This happens, 1979; Guth 1981; Linde 1982) can produce substantial de- forinstance,incurvatonmodels(Sasaki,Va¨liviita,&Wands2006; viations from a Gaussian distribution of primordial density Assadullahi,Va¨liviita,&Wands 2007) or in multi-fields models and potential fluctuations (see Bartoloetal. 2004; Chen 2010; (Bartolo,Matarrese,&Riotto2002;Bernardeau&Uzan2002).In Desjacques&Seljak 2010 for recent reviews). The amount and thiscasetheparameter f isallowedtobesubstantiallydifferent NL shape of these deviations depend critically on the kind of non- fromzero. (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 3 Equilateralshape can be obtained as a suitable linear combination of the equilat- eral and enfolded shapes (Senatore,Smith,&Zaldarriaga 2010; In some inflationary models the kinetic term of the inflaton Wagner,Verde,&Boubekeur 2010). Nevertheless, we performed Lagrangian is not standard, containing higher-order derivatives computations for the orthogonal model as well, since it gives a of the field itself. One significant example of this is the DBI differentsignatureontheevolutionoftheLSS. model (Alishahiha,Silverstein,&Tong 2004; Silverstein&Tong 2004, see also Arkani-Hamedetal. 2004; Seery&Lidsey 2005; Li,Wang,&Wang 2008). In this case the primordial bispectrum 3 COSMOLOGICALOBSERVABLES ismaximizedforconfigurationswherethethreewavevectorshave approximatelythesameamplitude,anditiswellrepresentedbythe Primordialnon-Gaussianityproducesmodificationsinthestatistics templateintroducedbyCreminellietal.(2007). of density peaks, resulting in differences in the mass function of Given this, we have the freedom to insert a running cosmicobjectsandthebiasofdarkmatterhaloswithrespecttothe γ(k1,k2,k3) for fNL, since this parameter is not forced to be underlying smooth density field. In the following we summarize constant in this case. The form we chose reads (Chen 2005; howthesemodificationshavebeentakenintoaccountinthepresent LoVerdeetal.2008;Crocianietal.2009) work,andhowtheyreflectintomodificationstothematterpower spectrum. γ(k ,k ,k )= k1+k2+k3 −2κ . (3) 1 2 3 k ! CMB 3.1 Massfunction Inallcalculationsthatfollowweconsideredthisrunningaspartof theequilateralbispectrum.Thisisimportantsincedifferentauthors For the non-Gaussian modification to the mass function of cos- choosedifferent runnings, ornorunning atall,fortheequilateral micobjectsweadopted theprescriptionof LoVerdeetal.(2008). shape.Weadoptedtheexponentκ = 0.2intheremainderofthis The main assumption behind it is that the effect of primor- − work, that increase thelevel of non-Gaussianity at scalessmaller dial non-Gaussianity on the mass function is independent of the thanthat corresponding tok = 0.086h Mpc 1.Thiscoincides prescription adopted to describe the mass function itself. This CMB − withthelargermultipoleusedintheCMBanalysisbytheWMAP means that, if n(G)(M,z) and n (M,z) are the non-Gaussian and PS PS team(Komatsuetal.2009,2011),ℓ 700. Gaussianmassfunctions,respectively,computedaccordingtothe ∼ Press&Schechter(1974)formula,wecandefineacorrectionfac- tor (M,z) n(G)(M,z)/n (M,z). Then, the non-Gaussian mass R ≡ PS PS Enfoldedshape function computed according to an arbitrary prescription, n(M,z) canberelatedtoitsGaussiancounterpartthrough For deviations from Gaussianity evaluated in the regular Bunch- Davies vacuum state, the primordial potential bispectrum is n(M,z)= (M,z)n(G)(M,z). (4) R of local or equilateral shape, depending on whether or not In order to compute n (M,z), and hence (M,z), higher-order derivatives play a significant role in the evolu- PS R LoVerdeetal. (2008) performed an Edgeworth expansion tion of the inflaton field. If the Bunch-Davies vacuum hy- (Blinnikov&Moessner 1998) of the probability distribution for pothesis is dropped, the resulting bispectrum is maximal for the smoothed density fluctuations field, truncating it at the linear squashed configurations (Chenetal. 2007; Holman&Tolley term in σ . The resulting Press&Schechter (1974)-like mass M 2008). Meerburg,vanderSchaar,&Corasaniti (2009) found a functionreads template that describes very well the properties of this enfolded- shapebispectrum(seehoweverCreminellietal.2010foraslightly n (M,z) = 2ρm exp δ2c(z) dlnσM δc(z)+ differenttemplateofthephysicalmodel). PS −rπ M "−2σ2M#" dM σM S σ δ4(z) δ2(z) + 3 M c 2 c 1 + 6 σ4 − σ2 − !! M M Orthogonalshape 1dS δ2(z) + 3σ c 1 . (5) A shape of the bispectrum can be constructed that is 6dM M σ2 − !# M nearly orthogonal to both the local and equilateral forms In the previous equation ρ = 3H2Ω /8πG is the comoving (Senatore,Smith,&Zaldarriaga2010).Constraintsonthelevelof m 0 m,0 matter density in the Universe, σ is the rms of density fluc- non-GaussianitycompatiblewiththeCMBinthelocal,equilateral M tuations smoothed on a scale corresponding to the mass M, and and orthogonal scenarios were recently given by the WMAP team (Komatsuetal. 2011), while constraints on enfolded non- δc(z)=∆c/D+(z).ThefunctionS3(M)≡µ3(M)/σ4Misthereduced skewnessofthenon-Gaussiandistribution,andtheskewnessµ (M) Gaussianity from galaxy bias were given by Verde&Matarrese 3 canbecomputedas (2009) µ (M) = (k ) (k ) (k ) Although there is no theoretical prescription against a 3 ZR9MR 1 MR 2 MR 3 × running of the fNL parameter with the scale in the enfolded and Φ(k )Φ(k )Φ(k ) dk1dk2dk3 . (6) orthogonal shapes, we decided not to include one. The reason × h 1 2 3 i (2π)9 for this is that there is no first principle that can guide one in Thelastthingthatremainstobedefinedisthefunction (k),that the choice of a particular kind of running, and until now no MR relates the density fluctuations smoothed on some scale R to the work has addressed the problem of a running for these shapes respectivepeculiarpotential, (Fergusson&Shellard 2009; Fergusson,Liguori,&Shellard 2010). Moreover, we recall that the four non-Gaussian shapes 2 T(k)k2 (k) W (k), (7) described above are not independent, rather the orthogonal shape MR ≡ 3H2Ω R 0 m,0 (cid:13)c 0000RAS,MNRAS000,000–000 4 M.Maturiet al. whereT(k)isthemattertransferfunctionandW (k)istheFourier withrespecttothemassfunctionandlinearbias.Particularly,we R transformofthetop-hatwindowfunction. shouldhaveareliablewaytoparametrizethefullynon-linearthree- In this work we adopted the Bardeenetal. (1986) matter dimensional power spectrum, since part of the lensing signal is transfer function, with the shape factor correction of Sugiyama given by the integral thereof along the line of sight. In order to (1995). Thisreproduces fairly well the more sophisticated recipe do that we followed the approach of Fedeli&Moscardini (2010) of Eisenstein&Hu (1998) except for the presence of the baryon (andreferencestherein)andmadeuseofthehalomodelinorderto acoustic oscillation, that anyway is not of interest here. We ad- representthematterpowerspectrum. ditionally adopted as reference mass function the prescription The halo model (Seljak 2000; Ma&Fry 2000; of Sheth&Tormen (2002) (see Jenkinsetal. 2001; Warrenetal. Cooray&Sheth 2002) is a physically motivated framework 2006; Tinkeretal. 2008 for alternative prescriptions). Other ap- that allows to compute the correlation function of various LSS proaches alsoexist forcomputing thenon-Gaussian correctionto tracers,includingthedarkmatterparticlesthemselves.Itisbased themassfunction,thatgiveresultsinbroadagreementwiththose onthesimpleconsiderationthatifallthematterintheUniverseis obtained here (Matarrese,Verde,&Jimenez 2000). In computing lockedintohalos,thenthecontributiontothecorrelationfunction thenon-Gaussian corrections tothemassfunction wehavetaken comesfromparticlepairssittinginseparatedhalosatlargescales intoaccount thecorrectiontothecriticaloverdensity forcollapse andfromparticlepairsbelongingtothesamehaloatsmallscales. suggested by Grossietal. (2009) (see also Maggiore&Riotto Accordingly,thepowerspectrumcanbewrittenasthesumoftwo 2010b,c,a),accordingtowhich∆c→∆c√q,withq∼0.8. termsdescribingthetwocontributions,P(k,z)=P1(k,z)+P2(k,z), with 3.2 Halobias P (k,z)= +∞n(M,z) ρˆ(k,M,z) 2dM, (10) Recently much attention has been devoted to the effect of 1 Z0 " ρm # primordial non-Gaussianity on halo bias, and the use thereof and for constraining f (Matarrese&Verde 2008; Dalaletal. 2008; NL Verde&Matarrese 2009; Carbone,Verde,&Matarrese 2008). In P (k,z)= +∞n(M,z)b(M,z,k)ρˆ(k,M,z)dM 2P (k,z). (11) particular,theseworkshaveshownthatprimordialnon-Gaussianity 2 "Z0 ρm # L introducesascaledependenceonthelargescalehalobias.Thispe- culiarityallowstoplacealreadystringentconstraintsfromexisting IntheprevioussetofequationsPL(k,z)isthelinearmatterpower data(Slosaretal.2008;Afshordi&Tolley2008). spectrumandρˆ(k,M,z)istheFouriertransformofthemeandark The non-Gaussian halo bias can be written in a relatively matter halo density profile which is also in principle modified straightforward way in terms of its Gaussian counterpart as by primordial non-Gaussianity, as suggested in Avila-Reeseetal. (Carbone,Mena,&Verde2010) (2003)andSmith,Desjacques,&Marian(2010).Nevertheless,we adopted a standard Navarro,Frenk,&White (1996, 1997) shape b(M,z,k)=b(G)(M,z)+β (k)σ2 b(G)(M,z) 1 2 , (8) (NFWhenceforth) for ρˆ(k,M,z) because thefunctional formsfor R M − h i themodifieddensityprofilesareproventoholdonlyforthelocal wherethefunctionβ (k)encapsulatesallthescaledependenceof R shape,anditisnotclearwhether thisisthecaseforothershapes thenon-Gaussiancorrectiontothebias,andreads aswell.Thereforeweignoredthesechanges,makingouranalysis β (k) = 1 +∞ζ2 (ζ) moreconservative, sinceapositive(negative) fNL impliesamore R 8π2σ2 (k)Z MR × (less)centrallyconcentrateddarkmatterprofile,andconsequently MMR 0 anincrement(decrement)inthenumberofpeaks. 1 BΦ ζ,√α,k whereα× k2Z+−1ζM2+R(cid:16)2√kζαµ(cid:17).Int(cid:16)hPeΦ(skim) pl(cid:17)edcµasedζof,localbispectru(9m) mFeoddeelFlio&wreMprreoafscectriactrhadeliniidnet(e2tar0ei1lss0te)d.oHnreeatrdheeewrteiomjuApslmtemnaoreatne&tathtRiaotenfthreeogfmieatrhs(es20fuh0na4cl)o-; ≡ tionandthehalobias,bothenteringinthehalomodel,aremodified shapeitcanbeshownthatthefunctionβ (k)shouldscaleas k 2 R ∝ − byprimordialnon-GaussianityaccordingtoSections3.1and3.2. atlargescales,sothatasubstantialboost(if f > 0)inthehalo NL biasisexpectedatthosescales.FortheGaussianbiasb(G)(M,z)we adopted the prescription of Sheth,Mo,&Tormen (2001). In this case, since the correction to the Gaussian bias is written in term oftheGaussianbiasitself,theellipsoidalcollapsecorrectionsug- gestedbyGrossietal.(2009)isnotnecessary. It is interesting to note that, while for the first three non- 4 MATTERFLUCTUATIONSWITHCOSMICSHEAR Gaussian shapes introduced in Section 2, a positive f implies NL In this paper we focused on the number counts of weak lensing bothapositiveskewnessofthematterdensityfieldandapositive peaks, that directly reflect the distributionof dark matter fluctua- correctiontothelargescalehalobias,theoppositeistrueforthe tions. Counting directly the peaks allows to minimize the physi- fourthshape,theorthogonalone.Thisisafacttobekeptinmind calassumptionsnecessaryingoingfromtheadoptedcosmological wheninterpretingourresultsinthesubsequentSections. modeltothedataoutcome.Asamatteroffactwebasedthefinal resultsondarkmatterphysicsonly,byobservingquantitiesinsensi- tivetobaryonphysicsandbyrelatingthemodelpredictiondirectly 3.3 Matterpowerspectrum tothemeasuredS/Nratios.Inparticular,thelatterpointavoidsthe Anothercosmologicalobservablethatisrelevantforourpurposes difficulttaskofdisentanglinginaclearwaytheintrinsicnatureof andgetsmodifiedincaseofprimordialnon-Gaussianityisthemat- eachsinglepeak:noisefluctuation,LSSline-of-sightsuperimposi- ter power spectrum. The latter requires a little bit more of care tionoractualgalaxycluster. (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 5 4.1 Lensingformalism P (ℓ)= 1σ2ǫs , (16) ǫ 2 n Thedeflectionpropertiesofisolatedlensesarefullydescribedby g theirtwo-dimensionallensingpotential, determinedbythenumberdensityn ofgalaxiessuitableforweak g lensingmeasurementsandthevarianceσ2 oftheirintrinsicellip- ψ(θ) 2 Dds sΦ(D θ,z)dz, (12) ticitydistribution. ǫs ≡ c2 DdDs Z0 d whereΦistheNewtoniangravitationalpotentialand D, D ,and Sincecontributions (ii)and(iii)arewellrepresented by two s d independentGaussianrandomfields,theirtotalpowerspectrum,is D are the angular-diameter distances between the observer and ds thesumofthetwocontributions,P(ℓ)=P (ℓ)+P (ℓ). thesource,theobserverandthelens,andthelensandthesource, γt ǫ respectively. Finally, s represents the physical distance out tothe sourcesphere. 4.3 Optimallensingfiltering Thepotentialψrelatestheangularpositionsβofasourceand θ of its images on the observer’s sky through the lens equation, A method to measure gravitational lensing signatures in β = θ ψ.Forsourcessuchasdistantbackgroundgalaxiesitis optical/near-infrared catalogues of galaxies is the aperture mass possibl−et∇olinearizethelensequationsuchthattheinducedimage (Schneideretal. 1998). It is a weighted average of the tangential distortionisexpressedbytheJacobianmatrix componentoftheshearγtoverthepositionθonthesky, A=(1−κ) 1−−gg21 1−+gg21 ! . (13) A(θ)=ZR2d2θ′γt(θ′,θ)Q(kθ′−θk), (17) where the radial filter function Q determines the statistical Inthepreviousequation,κ 2ψ/2istheconvergence,responsi- ≡ ∇ properties of the estimate A we are interested in. In this paper blefortheisotropicmagnificationofanimagerelativetoitssource, weaimatconstrainingthedeviationsfromprimordialGaussianity, and g = γ/(1 κ) isthereduced shear, quantifying theobserved − whichmostlyaffectthenon-linearpartofstructureformationand distortion. Here, γ ψ ψ /2 and γ ψ are the two 1 ≡ ,11− ,22 2 ≡ ,12 thusthehighmassendofthemassfunction.Therefore,weadopted components of the com(cid:0)plex shear,(cid:1)where a comma denotes stan- thelinearmatchedfilterdefinedbyMaturietal.(2005)whichisde- darddifferentiationwithrespect tocoordinates onthelensplane. signedtomaximizetheS/Nratioof nonlinearstructuressuchas It is important to recall that since the angular size of the sources galaxyclusters,i.ethosestructureswhicharemostsensitiveto f , isunknown,onlythereducedshearcanbeestimatedstartingfrom NL theobservedellipticityoftheimages. τˆ(ℓ) τˆ(ℓ)2 Qˆ(ℓ)=α , with α−1 = d2ℓ| | . (18) P(ℓ) ZR2 P(ℓ) 4.2 Contributionstothelensingsignal HerethefilterisrepresentedintheFourierspaceforconvenience, τˆ(ℓ)istheFouriertransformoftheexpectedshearprofileofindi- Theabundance ofS/Npeaksincosmicshearmapsisdetermined vidual halos,inour casetheFouriertransformoftheNFWshear bythesumofthreeseparatecontributions,thatwedescribeinthe profile,andP(ℓ)=P (ℓ)+P (ℓ)describedinSection4.2.Notethat following. γt ǫ theshapeoftheoptimalfilterdependsontheassumedparameters (i) The signal due to the occurrence of real non linear struc- fortheNFWdensityprofile,particularlyontheangularsizecorre- turessuchasgalaxyclusters.TheirabundanceisevaluatedinSec- spondingtothescaleradiusθs.Moreover,sincethefilterisonlya tion 3.1 and their shear profile can be evaluated analytically (see radialfunction,itsFouriertransformdependsonlyonℓ= ℓ k k Bartelmann1996;Meneghettietal.2003).Sincethetypicalsepa- WefurtherdefinethevarianceoftheaperturemassestimateA rationbetweentheobservedbackgroundgalaxiesislargerthanthe as typicalradiusofthegalaxyclusters’criticalcurveswecanassume σ2 +∞ ℓdℓP (ℓ)Qˆ(ℓ)2 , (19) γ gthroughout. A≡Z 2π ǫ | | ≃ 0 (ii) The lensing signal due to chance projections of the LSS, sothatitisrelatedonlytothenon-cosmologicalsignalcomponent, givenbytheeffectiveconvergencepowerspectrum i.e. P . It is worth mentioning that in practical applications A is ǫ P (ℓ)= 9H04Ω2m,0 wHdwW¯2(w)P ℓ ,w , (14) approximatedas κ 4c4 Z0 a2(w) fK(w) ! 1 n A(θ)= ǫ (θ)Q( θ θ ), (20) where P(k,z) is the fully non-linear matter power spectrum dis- n t,i k i− k cussed in Section 3.3, Ω is the present-day matter-density pa- Xi=1 m,0 whereǫ (θ)isthetangential ellipticityofagalaxyimagelocated rameter,H istheHubbleconstant,cisthespeedoflight,a(w)is t,i 0 atthepositionθ withrespecttoθ,andwhichprovidesanestimate thescalefactoratthecomovingdistancew,w isthecomovingdis- i H forγ. tancetothehorizon, f isthecomovingangular-diameterdistance, t K andW¯(w)istheweightfunctionencorporatingtheline-of-sightin- tegraloverthedistributionofbackgroundsourcesG(w), 4.4 Expectedshearpeaksnumbercounts W¯(w)= wHdw′G(w′)fK(w′−w) . (15) Oncethelensingsignalsandtheobservational strategyhavebeen Zw fK(w′) defined(seeSections4.2and4.3),theweaklensingnumbercounts Inourapplicationweusedasingle(thetangential)component of can bepredicted asthe sumof twoindependent components. On theshear,forwhichthepowerspectrumreadsP (ℓ)=P (ℓ)/2. onehandthenon-linearstructures,i.e.galaxyclusters,whosede- γt κ (iii) Theobservational noisecontributionfromtheintrinsicel- tection depends on their intrinsic abundance given by the mass lipticityandfinitenumberofbackgroundgalaxiesusedtomeasure function discussed in Section 3.1, and by their observed S/N ra- theshearsignal,whichhasthewhitenoisepowerspectrum tiodependingontheirexpectedsignalA(seeEq.17)andvariance (cid:13)c 0000RAS,MNRAS000,000–000 6 M.Maturiet al. Figure1.Leftpanel.ThedifferentcontributionstotheabundanceofshearS/NpeaksinthereferenceGaussianmodel.Theblackshortdashedlineshowsthe contributiongivenbyLSSprojectionanddatanoise,whilethebluelongdashedlinereferstothecontributionoftherealdarkmatterclumps.Theheavysolid redlineisthetotalcountdistribution.Weadoptedoptimalfilteringwithascaleradiusofθs =2′.Rightpanel.ThetotalnumberofshearS/Npeaksforthree differentfilterscaleradii,aslabeled. σ2 (see Eq. 19). On the other hand the lensing LSS and instru- matterclumpsstartstoberelevantataroundS/N 4anddominates A ∼ mentalnoisewhicharewellapproximatedbyaGaussianrandom athigherS/Nvalues.Weadditionallyshowintherightpanelofthe fieldandforwhichtheirnumbercountsaboveaspecificthreshold same Figure the total number counts expected for three different S/N =y canbeeasilypredicted, scaleradii of the optimal filter.Asit can be noted, the larger the th th scale,thelargerthecontributionofgalaxyclusterswithrespectto 1 σ 2 y y2 n (y )= 1 th exp th , (21) theLSSandthenoise,althoughthelatterremainsalwaysdominant det th 4√2π3/2 σA! σA −2σ2A! at smaller, but stilllarge, S/N.Thisbehavior isexpected because asshownbyMaturietal.(2010).Hereσ ,definedas forlargerfilterscalesthedetectionswillhavelargerareas,thusin- 1 creasingtheirblendingwheretheirnumberdensityishigher,typi- σ2= +∞ ℓ3dℓP(ℓ)Qˆ(ℓ)2 , (22) callyatlowerS/N.Also,theoverallS/NratiovaluesathigherS/N 1 Z0 2π | | areexpectedtobelarger,sincealargerfilterscaleimpliesabetter is the LSS plus noise lensing field variance which depends on statisticforthebackgroundgalaxiesreducingtheirshotnoiseand the observational noise, the convergence power spectrum and the intrinsicellipticity.Thesefeaturesareensuredbytheadoptedop- adopted filter. Note that, by approximating this therm as a Gaus- timalfilterwhich maximizes thecluster signal against theone of sian random field, the primordial non-Gaussianity is accounted LSSandnoise.Intheremainderofthispaperweshallrefermainly through the LSS power spectrum only so that its leverage is not toascaleradiusof2′,unlessexplicitlynotedotherwise. fully included. For our purpose this is a negligible approxima- tion since we anyway expect this effect to be small and put our final predictions on a conservative side. These two contributions 5 SIGNATUREOFPRIMORDIALNON-GAUSSIANITY carrycomplementarycosmologicalinformation,thusthestatistics ofcosmicshearpeaksareinprinciplevaluablecosmologicaltools Theeffectofprimordialnon-Gaussianityonthestatisticsofshear (Dietrich&Hartlap2009). peaksisexemplifiedinFigure2forthefourdifferentnon-Gaussian Thetwocosmologicalcontributionstotheabundanceofcos- shapes that were discussed in Section 2, and for levels of non- micshearpeaksareexemplifiedintheleftpanelofFigure1forthe Gaussianity f = 50 and f = 100. As it can be seen, the NL ± NL ± referenceGaussiancosmologicalmodel.ForthisFigureandinthe effectofnon-GaussianityismoreevidentathighvaluesofS/Nra- restoftheworkweadoptedthesourceredshiftdistributionobser- tio,whereitismostlygivenbytheeffectonthemassfunctionof vationally derived by Benjaminetal. (2007), while their average dark matter halos. We remind the reader that our predictions re- surface number density is n = 40 arcmin 2, a level achieved by gardingtheimpactofprimordialnon-GaussianityattheS/Nratios g − actualweaklensingsurveysandwhichshouldbereachedbyEU- dominatedbytheLSSareconservativeasdiscussedinSection4.4. CLID(apredictionforasampleofexistingandfutureweaklens- Also,asexpected,theeffectislargestforlocalnon-Gaussianityfor ingsurveysisgivenbyMaturietal.2010).Asexpected,thechance which,athighS/N,thecountscanbemodifiedbyupto 7%for alignmentoftheLSSdominatesthecountsatsmallS/Nratioval- f = 100andsmallestfortheorthogonalshape,where∼itbarely NL ± ues,whilethecontributioncomingfromtheoccurrenceofrealdark reaches 1%.Fortheorthogonalshapepositivevaluesof f pro- ∼ NL (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 7 Figure2.Theratioofthepeaknumbercountsindifferentmodelswithprimordialnon-GaussianitytothesamequantitiesinthereferenceGaussianΛCDM cosmology,asafunctionoftheS/Nratio.Fourdifferentnon-Gaussianshapesandfourdifferentvaluesof fNLareshown,aslabeledintheplot. videadecrementinthenumbercountsofshearpeaksathighS/N InFigure 3weshow the ratioof thepeak number counts in values,whiletheoppositeistrueforallothermodels.Thisagrees non-Gaussian cosmologies with local shape (f = 10) to the NL ± withthebehavioroftheskewnessthathasbeendiscussedinSec- samequantitiesinthereferenceGaussiancaseadoptingthreedif- tion2.Althoughtheeffectofnon-Gaussianity issubstantiallyre- ferentvaluesof thepower spectrumamplitude σ for allmodels. 8 ducedforshapesdifferentfromthelocalone,itislikelythatuse- Weconsidered 2%fluctuations inthevalue of σ , comparable ∼ 8 fulconstraintscanbeputon f fortheseshapesaswell.Forin- withthe1 σuncertaintyaroundthefiducialvaluederivedbythe NL − stanceintheorthogonal case, f isconstrained onlyat thelevel WMAP-7release.Quiteunexpectedly,theeffectofprimordialnon- NL of a few hundreds by CMB data (Komatsuetal. 2011). Our pre- Gaussianityishigherforlowervaluesofσ .Weinterpretthisfact 8 dictioniscompatiblewiththerecentworkofMarianetal.(2010), asduetothedelayedstructureformationimpliedbyalowernor- whofoundaneffectontheshearpeaksnumbercountsatthelevel malization. This allows the effect of primordial non-Gaussianity of 10%forthehighestsignificancedetections.Althoughadirect onthematterdensityfieldtoberetainedforalongertimebythe ∼ comparisonofthetwoworksisnotpossibleduetodifferentsource LSS, with the consequence that the S/N peak number counts are redshiftdistributions,differentdetectionschemes,etc.,weactually slightlymoreaffected.Inotherwords,sincewithalowerσ theon- 8 verifiedthatbyreducingthescaleradiusoftheoptimalfilterdown setofnon-linear evolutionisdelayed, gravitationalclusteringhas toθ =1 theeffectisraisedto 10%.Thus,weconcludethatthe lesstimetoerasethecosmologicalinitialconditions. s ′ ∼ twoworksareinbroadagreementforthelocalnon-Gaussianmodel concerning the magnitude of the effect. This is reassuring, since thetworesultsarebasedonquitedifferentpremisesandadoptdif- ferentmethodologies:weoptedforsemi-analyticmodeling,while Marianetal. (2010) resorted to the use of cosmological simula- tions. (cid:13)c 0000RAS,MNRAS000,000–000 8 M.Maturiet al. 200 ’tabCovariance.gnu’ 150 100 50 0 -50 3 2.5 2 1.5 al 1 on 0.5 ag Numerical 0 Di Figure 3. The ratio of the peak number counts in two different non- Numerical Poiss. -0.5 Analytic Poiss. -1 Gaussian models with local bispectrum shape to the same quantities in -1.5 TthheereufpepreerncseetGoafucssuiravnesΛrCeDfeMr tocofsNmLol=og1y,0,aswahifluentchteiolnowoferthseetSr/Neferrastitoo. 0.0625 0.125 0.25 0.5 0.75 1 1.25 1.5 2S/N2.5 3 4 5 6 7 8 9 10 fNL = −10.Differentcolorsrefertodifferentvaluesofthematterpower Figure 4. Weak lensing number counts covariance matrix for a ΛCDM spectrumnormalizationσ8,aslabeled. model.Theoff-diagonaltermsaresmallbutnotnegligibleonlyatlowS/N ratios.Thebottompanelshowsthecomparisonbetweenthediagonalcom- ponentsofthefullcovariancematrixderivednumerically(showninthetop 6 COSMOLOGICALCONSTRAINTS panel)andwhatwouldbeexpectedforidealnumbercountswithPoisson InthissectionweuseaFisher-matrixanalysistoestimatetheconfi- statisticandstatisticallyindependentS/Nbins.ForhighS/Nthecovariance dencelevelintheσ f plane,asobtainedwiththeweaklensing isdiagonalandPoissonianbyconstruction. 8− NL numbercounts. low S/N ratios up to S/N 3, where we expect to have a small ∼ 6.1 Fishermatrixanalysis butnotnegligiblecovariancecontributionsincedifferentS/Nbins canbecorrelated.Toaccount forthenon-diagonal covariance el- TheFishermatrixisgivenby ements, werealizedaset of 1000 numerical realizations of weak lensingdata,modeledwithaGaussianrandomfieldwhosepower ∂2 Fij =−*∂ξ∂Lξ +, (23) spectrum, P(ℓ),isdiscussedinSection4.2.Asexpectedthenum- i j ber counts derived fromthesimulationagrees withtheanalytical where is the logarithm of the likelihood function and ξ = predictionsexceptforverylowvaluesoftheS/Nratio(S/N<0.2) L (σ8,fNL) are the free parameters of the number counts model. In wheretheanalyticapproximationfailsasdiscussedbyMaturietal. caseofamultivariateGaussianlikelihood,theFishermatrixcanbe (2010).Inanycasetheseverylow S/Narecompletelynegligible writtenas foranycosmologicalanalysis. 1 Thetotalcovariancematrixofweaklensingnumbercountsis Fij = 2Tr AiAj+C−1Mij , (24) shown inFigure 4. Asexpected, ithas astrong diagonal compo- h i where A = C 1C , M = 2(∂µ/∂ξ)(∂µ/∂ξ ), C is the data co- nentanditsoff-diagonaltermsaresmallbutnonnegligibleonlyat i − ,i ij i j lowS/Nratios.Bytreatingthetwocomponents,LSS+noiseand variancematrix,andµistheassumedmodel,i.e.thepeaksnumber galaxyclustersasindependent,wedonotaccountfortheirmutual countsfordifferentS/N.Hereacommadenotesdifferentiationwith influenceonthefinalnumbercounts,butthiseffectisexpectedto respecttotherelevantcosmologicalparameter.SinceChasavery besmallbecausethesignatureofeachclusteraffectsonlyfewpix- weakdependenceonthemodelparametersσ and f thefirstterm 8 NL els. Inany case, thisapproach isconservative withrespect tothe inEq.24isnegligiblebecauseofC .WeevaluatedtheFisherma- trixatthefiducialpointσ = 0.809,i and f = 0asmeasuredby finalcosmologicalconstraintsbecauseweneglecttheimpactofthe WMAP-7forastandardΛ8CDMmodel(KoNmLatsuetal.2011). detectionswithhighS/Nratios,carriersofthenon-Gaussianitysig- nal,onthosewithlowerS/Nratios. As discussed in Section 4.4, the number counts model µ is givenbytwocomponents. Ononehand,thenonlinearstructures whosenumbercountsareindependent fordifferentS/Nratiobins 6.2 Cosmologicalconstraintson f and whose covariance contribution is therefore diagonal with a NL Poissonian amplitude. On the other hand, the contribution given InthisSectionweassumetheexpectedperformancesoftheweak bytheLSSandnoiseforwhichthenumbercountsarerelevantfor lensingsurveyEuclid(Laureijs2009),i.e.afieldofviewof20,000 (cid:13)c 0000RAS,MNRAS000,000–000 Shearpeakcounts 9 Figure5.Cosmological1 σconfidencelevelsasobtainedforthelocal Figure 6. Confidence contours for the local shape non-Gaussian model, − shapenon-Gaussianmodelusingweaklensingnumbercounts,byassum- withdifferentlowerS/Ncut-offlimits.Thisshowstheinformationlostby ingbothPoissonianstatisticalindependence(Poissonian),andbyusingthe cuttingthedataatdifferentS/Nratios. numericalsimulationsincludingthefullcovariancematrix(numerical). thefulldatacovariancehasbeenusedonlyforthefinalresultsre- portedbelow.Theresultsareevaluatedforasurveyskycoverage ng ∆σ8 ∆fNL θs ∆σ8 ∆fNL of20,000deg2 andscalewiththesquarerootofthesurveyarea. 20 0.0023 60 1’ 0.0016 57 Theconstraintsonboth fNL andσ8 improvewithincreasingnum- 40 0.0015 45 2’ 0.0015 45 berdensityofbackgroundsources.Howeveronlyamarginalben- 80 0.0012 41 4’ 0.0015 38 efitisobtainedbyincreasingthenumberdensityover40arcmin 2. − In other words, in terms of joint constraints for the level of non- Table1.Dependenceofthe1 σjointboundsonthebackgroundgalaxy Gaussianity and amplitude of the matter power spectrum, it does − numberdensityng(left)andonthefilterscaleradius,θs(right).Thefiducial notpayofftoincreasethedepthofthesurveybeyondwhatisal- modelwithσ8=0.809and fNL=0isadopted. ready planned for Euclid. Concerning the scale radius of the op- timal filter,by increasing it we get asignificant tightening of the constraintson f ,whilethoseonσ arealmostinsensitivetoit. NL 8 deg2, an average background galaxy number density of n = 40 Nowthatwehaveanoverallpictureofthedependenceonthe g arcmin 2withanintrinsicellipticityrmsofσ =0.3,andaredshift survey characteristics, we used the numerical simulations includ- − ǫs distributionsimilartotheonediscussedbyBenjaminetal.(2007). ingthefullcovariancematrixtoobtainourfinalresults.InFigure6 TheconfidencelevelsareevaluatedwiththeFishermatrixanalysis we show the 1 σ confidence levels in the σ f plane ob- − 8 − NL discussed in Section 6.1. In Figure 5 we compare the confidence tainedforthenon-Gaussianmodelwithlocalshapewhendifferent regions for the local shape non-Gaussian model obtained first by minimumS/Nratiosareconsidered.Itisinfactcommonpractice assumingpurelyPoissonianindependentstatistic,i.e.adatacovari- foractualapplicationstouseonlythosedetectionswithS/Nlarger ancehavingapurediagonalcomponentwithPoissonianamplitude, thenS/N = 3 5intheattempttoisolategalaxyclustersfrom min − andthenbyusingthefullcovariancematrixobtainedfromthenu- all other signal components. Herewe show how the cut-off level merical simulations (see Section 6.1). As can be seen, the 1 σ affectsthefinalresults.Therelatedlossofinformationhappensbe- − confidence ellipse is slightly tilted in the latter case with respect cause the presence of non linear structures affects the statisticof totheformer,implyingasomewhatdifferentdirectionofdegener- weaklensingmapsalsoatlowersignaltonoiseratioswherethey acybetweenthetwoparametersunderconsideration.Overall,how- cannotbeseenasclearsingledetectionsbutwheretheirpresenceis ever,theconstraintsonthenon-Gaussianitylevel f arebasically stillvisible.Moreover,althoughthiseffectisnegligibleforthe f NL NL unchanged, whilethoseonthenormalizationofthematterpower estimates, chance projection of theLSSdominates at thesesmall spectrumσ areonlyveryslightlyloosened byusingthesimpler S/Nvalues, andstillcontains cosmological information. Figure6 8 Poissonmodel. Asaconsequence, it issafetostatethat thefinal shows that adopting a conservative choice for the minimum S/N differencesbetweenthetwoapproachesarenegligibleandthatfor value can results in a loss of constraining power of a factor 3 ∼ afirstfastanalysisthesimplifiedapproachcanbeused. on f andafactorof 4onσ .Thus,particularcarehavetobe NL ∼ 8 Giventhat,weusedthefirstapproachtoestimatetheCrame´r- usedwhendefiningthedetectionselectioncriteriasincedatacanbe Rao bounds on σ and f at 1 σ level for different survey fullyexploitedonlyiflowerS/Nlevels,e.g.S/N 2,areused. 8 NL − min ∼ depths,i.e.differentgalaxiesnumberdensities,andfordifferentfil- Thiswouldbepossibleonlyifadeepstatisticalunderstanding of terscalesasshowninTable1.Inordertosavecomputationaltime, dataisachieved together withadetailedmodeling. Atentativeto (cid:13)c 0000RAS,MNRAS000,000–000 10 M.Maturiet al. Figure7.Confidenceregionsforthefouradoptedmodelswithnon-Gaussianinitialconditions.Wereportresultsforthelocal(topleft),equilateral(topright), enfolded(bottom left)anqdorthogonal(bottom right)primordialbispectrum shapes.Thethreecontours ineachpanelreferto68.3%,95.4%,and99.7% confidencelevels. gointhisdirectionisprovidedbytheanalyticpredictionforweak lensing-selected clusters it is possible to push the bounds on f NL lensingnumbercountsproposedbyMaturietal.(2010)butwhich belowwhatwouldbeachievedwithoutredshiftinformation.Inad- stillneedstoimprovesomeoftheadoptedapproximations. dition, it is likely that a combination of these diverse probes can reduce the error on f substantially. We plan to investigate this FinallyweshowinFigures7theconstraintsfor68.3%,95.4% NL and 99.7% confidence levels with S/N = 0 for the four dif- synergyinafuturework. min ferent primordial bispectrum shapes discussed in Section 2. We The constraints on the amplitude of the matter power spec- adopted the optimal filter with scale of θs = 2′. As one could trum,∆σ8 ∼ 10−3 arethemselvesquitecompetitive.Forinstance, naively expect, while the constraints on σ8 are rather insensitive recentlyWangetal.(2010)forecastedanerroronσ8of∼7×10−2 totheadoptedshapeoftheprimordialbispectrum,theconstraints byusingtheshapeandpositionoftheBaryonAcousticOscillation on f varywidelywithit.Particularly,whilethe1 σerror∆f measuredbyEuclid.DespitethefactthatWangetal.(2010)letall isatNtLhelevelofafewtensforthelocalandequila−teralshapes,NiLt thecosmologicalparametersfreetovary,whilewelimitouranal- growsupto 100 200fortheenfoldedandorthogonal shapes. ysisto fNL andσ8,thesenumbersargueforourconstraintsonthe Theseconstra∼intsar−enotcompetitivewithwhatisexpectedbyfu- amplitudeofthematterpowerspectrumbeingsound. tureCMBandgalaxyclusteringprobes,andareonlycomparable The degeneracy between the level of primordial non- withtheexpectedperformanceoftheweaklensingpowerspectrum Gaussianityandtheamplitudeofthematterpowerspectrumisalso (Fedeli&Moscardini2010)analysis.Themainreasonforthisre- easy to understand, since increases in both f and σ bring an NL 8 sultsisthatinthisworkwedidnotusethefullredshiftinformation incrementinthelargescalematterpowerspectrumandtheoccur- whichisotherwiseincludedintheotherworksandwhichcouldbe rence of massive dark matter halos. The only exception tothisis includedbytheuseoflensingtomography. Recently,Fedelietal. givenbytheorthogonalmodel,becauseinthiscaseapositive f NL (2010) have shown that using the correlation function of weak impliesadecrementinboththeabundanceofcosmicstructuresand (cid:13)c 0000RAS,MNRAS000,000–000

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