Next Generation Cosmology: Constraints from the Euclid Galaxy Cluster Survey
B. Sartoris, A. Biviano, C. Fedeli, J. G. Bartlett, S. Borgani, M. Costanzi, C. Giocoli, L. Moscardini, J. Weller, B. Ascaso, S. Bardelli, S. Maurogordato, P. T. P Viana
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 12 May 2015 (MN L A TEX style file v2.2)
Next Generation Cosmology: Constraints from the
Euclid
GalaxyCluster Survey
B. Sartoris , , A. Biviano , C. Fedeli , , J. G. Bartlett , S. Borgani , , , M. Costanzi ,C. Giocoli , , , , L. Moscardini , , , J. Weller , , , B. Ascaso , S. Bardelli ,S. Maurogordato , and P. T. P. Viana , Dipartimento di Fisica, Sezione di Astronomia, Universit`a di Trieste, Via Tiepolo 11, I-34143 Trieste, Italy INAF / Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34143 Trieste, Italy INAF / Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna, Italy INFN, Sezione di Bologna, Viale Berti Pichat 6 /
2, I-40127 Bologna, Italy APC, AstroParticule et Cosmologie, Universit´e Paris Diderot, CNRS / IN2P3, CEA / lrfu,Observatoire de Paris, Sorbonne Paris Cit´e, 10, rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy Universit¨ats-Sternwarte M¨unchen, Fakult¨at f¨ur Physik, Ludwig-Maximilians Universit¨at M¨unchen, Scheinerstr. 1, D-81679 Mu¨nchen, Germany Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Universit`a di Bologna, viale Berti Pichat, 6 /
2, 40127 Bologna, Italy Aix Marseille Universit´e, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France Excellence Cluster Universe, Boltzmannstr. 2, D-85748 Garching, Germany Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, D-85748 Garching, Germany GEPI, Observatoire de Paris, CNRS, Universit´e Paris Diderot, 61, Avenue de l’Observatoire 75014, Paris France Laboratoire J.-L. LAGRANGE, CNRS / UMR 7293, Observatoire de la Cˆote d’Azur, Universit´e de Nice Sophia-Antipolis, 06304 Nice Cedex 4, France Instituto de Astrof´ısica e Ciˆencias do Espa¸co, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
12 May 2015
ABSTRACT
We study the characteristics of the galaxy cluster samples expected from the European SpaceAgency’s
Euclid satellite and forecast constraints on parameters describing a variety of cos-mological models. The method used in this paper, based on the Fisher Matrix approach, isthe same one used to provide the constraints presented in the
Euclid
Red Book (Laureijs et al.2011). We describe the analytical approach to compute the selection function of the photo-metric and spectroscopic cluster surveys. Based on the photometric selection function, weforecast the constraints on a number of cosmological parameter sets corresponding to dif-ferent extensions of the standard Λ CDM model, including a redshift-dependent Equation ofState for Dark Energy, primordial non-Gaussianity, modified gravity and non-vanishing neu-trino masses. Our results show that
Euclid clusters will be extremely powerful in constrainingthe amplitude of the matter power spectrum σ and the mass density parameter Ω m . The dy-namical evolution of dark energy will be constrained to ∆ w = .
03 and ∆ w a = . Ω k , resulting in a ( w , w a ) Figure of Merit (FoM) of 291. Including the Planck
CMB covariance matrix, thereby information on the geometry of the universe, improves theconstraints to ∆ w = . ∆ w a = .
07 and a FoM = f NL , will be constrained to ∆ f NL ≃ . Euclid clusters alone. Using only
Euclid clusters, the growth factor parameter γ , which signals deviations from General Relativity, will be constrained to ∆ γ = .
02, andthe neutrino density parameter to ∆Ω ν = . ∆ P m ν = . Euclid mission will have a clear advantage in thisrespect, thanks to its imaging and spectroscopic capabilities that will enable internal masscalibration from weak lensing and the dynamics of cluster galaxies. This information willbe further complemented by wide-area multi-wavelength external cluster surveys that willalready be available when
Euclid flies. c (cid:13) B. Sartoris et al.
According to the hierarchical scenario for the formation of cos-mic structures, galaxy clusters are the latest objects to have formedfrom the collapse of high density fluctuations filtered on a typi-cal scale of ∼
10 comoving Mpc (e.g. Kravtsov & Borgani 2012).Since galaxy clusters provide information on the growth historyof structures and on the underlying cosmological model in manyways (see, e.g., Allen, Evrard & Mantz 2011), they have playedan important role in delineating the current standard Λ CDM cos-mological model. As a matter of fact, the number counts andspatial distribution of these objects have a strong dependence ona number of cosmological parameters, especially the amplitudeof the mass power spectrum and the matter content of the Uni-verse. The evolution with redshift of the cluster number den-sity and correlation function can be employed to break the de-generacy between these two parameters, and thus can provideconstraints on the cold Dark Matter (DM henceforth) and DarkEnergy (DE) density parameters (e.g., Wang & Steinhardt 1998;Haiman, Mohr & Holder 2001; Weller, Battye & Kneissl 2002;Battye & Weller 2003; Allen, Evrard & Mantz 2011; Sartoris et al.2012). Furthermore, a number of studies (e.g., Carbone et al. 2012;Costanzi et al. 2013a, 2014) have also shown that clusters can beused to constrain neutrino properties, because massive neutrinoswould directly influence the growth of cosmic structure, by sup-pressing the matter power spectrum on small scales. More gener-ally, since the evolution of the cluster population traces the growthrate of density perturbations, large surveys of clusters extendingover a wide redshift interval have the potential of providing strin-gent constraints on any cosmological model whose deviation from Λ CDM leaves its imprint on this growth.Over the past decade, surveys of galaxy clusters for cos-mological use have been constructed and analysed, based onobservations at di ff erent wavelengths: X-ray (e.g. Borgani et al.2001; Vikhlinin et al. 2009; Clerc et al. 2012; Rapetti et al. 2013);sub-mm, through the Sunyaev & Zeldovich (1972) distortion(SZ henceforth, Staniszewski et al. 2009; Benson et al. 2013;Planck Collaboration et al. 2014b; Burenin & Vikhlinin 2012),and optical (Rozo et al. 2010) bands. Further improvementscan be obtained from the spatial clustering of galaxy clus-ters (Schuecker et al. 2003; H¨utsi 2010; Mana et al. 2013).The resulting cosmological constraints turn out to be com-plementary to those of other cosmological probes such astype Ia supernovae (e.g., Betoule et al. 2014), Cosmic Mi-crowave Background (CMB) radiation (e.g., Hinshaw et al. 2013;Planck Collaboration et al. 2014a), the Baryon Acoustic Oscilla-tions (BAOs; e.g., Anderson et al. 2014), and cosmic shear (e.g.Kitching et al. 2014). These cluster catalogues are however charac-terised either by a large number of objects that cover a relativelysmall redshift range, or rather small samples that span a wide red-shift range. Ideally, in order to exploit the redshift leverage withgood statistics, one should have access to a survey that can providea high number of well characterised clusters over a wide redshiftrange.One future mission that will achieve this goal will be theEuropean Space Agency (ESA) Cosmic Vision mission Euclid (Laureijs et al. 2011). Planned for launch in the year 2020, Euclid will study the evolution of the cosmic web up to redshift z ∼ http: // clustering, Euclid will also provide data usable for other importantcomplementary cosmological probes, such as galaxy clusters. Clus-ter detection will be possible in three di ff erent ways: i ) using pho-tometric data; ii ) using spectroscopic data; and iii ) through gravi-tational (mostly weak) lensing, which may be combined for moree ffi ciency. In this paper, we will perform our analyses by using thephotometric cluster survey (see Section 2), where the cluster detec-tion method is not dissimilar from that used to detect low-redshiftSDSS clusters (Koester et al. 2007). However, thanks to the useof Near Infrared (NIR) bands, Euclid will be capable of detectingclusters at much higher redshifts ( z ∼
2) over a similarly large area.The sky coverage of
Euclid will reach 15 ,
000 deg , almost the en-tire extragalactic celestial sphere. The characteristics of the Euclid spectroscopic survey and its possible use for the calibration of themass-observable relation will be discussed in Appendices A and B,respectively.One fundamental step for the cosmological exploitation ofgalaxy clusters is the definition of the relation between themass of the host DM halo and a suitable observable quan-tity (e.g., Andreon & Hurn 2012; Giodini et al. 2013). Many ef-forts have been devoted to the calibration of the observable-mass scaling relations at di ff erent wave bands (e.g. Arnaud et al.2010; Planck Collaboration et al. 2011; Reichert et al. 2011;Rozo et al. 2011; Ryko ff et al. 2012; Ettori 2013; Rozo et al. 2014;Mantz et al. 2015) and in the definition of mass proxies whichare at the same time precise (i.e. characterised by a small scat-ter in the scaling against cluster mass) and robust (i.e. rel-atively insensitive to the details of cluster astrophysics) (e.g.Kravtsov, Vikhlinin & Nagai 2006). In the case of Euclid , an in-ternal mass calibration will be performed through the exploitationof spectroscopic and WL data of the wide
Euclid survey (see Ap-pendix B), and of the deep
Euclid survey of 40 deg , 2 magnitudesdeeper than the wide survey. The deep survey will be particularlyuseful in adding constraints on the evolution of the observable-massscaling relation at z > Euclid internal data will provide a precise calibra-tion of the relation between cluster richness, which characterisesphotometrically-identified clusters, and their actual mass. Fur-thermore, it will be possible to cross-correlate
Euclid data withdata from other cluster surveys - such as eRosita (Merloni et al.2012),
XCS (Mehrtens et al. 2012), the South Pole Telescope (SPT,Carlstrom et al. 2011), and the Atacama Cosmology Telescope(ACT, Marriage et al. 2011) - to further improve the mass calibra-tion of
Euclid clusters.The aim of this paper is to forecast the strength and thepeculiarity of the
Euclid cluster sample in constraining the pa-rameters describing di ff erent classes of cosmological models thatdeviate from the concordance Λ CDM paradigm. We first con-sider the case of a dynamical evolution of the DE compo-nent, using the two-parameter functional form originally proposedby Chevallier & Polarski (2001) and Linder (2003). The sameparametrisation has been used in the Dark Energy Task Force re-ports (DETF; Albrecht et al. 2006, 2009) to estimate the constrain-ing power of di ff erent cosmological experiments. Second, we al-low for the primordial mass density perturbations to have a non-Gaussian distribution. Third, we explore the e ff ect of deviationsfrom General Relativity (GR) on the linear growth of density per-turbations. Finally, we consider the case of including massive stan-dard neutrinos.The structure of this paper is the following. In Section 2, wedescribe the approach used to estimate the Euclid cluster selectionfunction of the photometric survey. In Section 3, we describe the c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the Euclid
Galaxy Cluster Survey Figure 1.
Number N , c of cluster galaxies within r , c (black curves), and3 σ field where σ field is the rms of the field counts within the same radius, andwithin the adopted 3 ∆ z p cut (red curves). These counts are shown down tothe limiting magnitude of the Euclid survey, H AB =
24, as a function ofredshift for clusters of di ff erent masses, log( M , c / M ⊙ ) = . , . , . Fisher Matrix approach used to derive constraints from the
Euclid cluster survey on cosmological parameters. In Section 4, we brieflydescribe the characteristics of the di ff erent cosmological modelswe consider. In Section 5, we show our results on the number ofclusters that the wide Euclid survey is expected to detect as a func-tion of redshift and the constraints that will be obtained on the cos-mological parameters using the cluster number density and powerspectrum. Finally, we provide our discussion and conclusions inSection 6. We present the analytical derivation of the spectroscopicselection function in Appendix A and the calibration of the clusterobservable-mass relation in Appendix B.
Euclid
PHOTOMETRIC SURVEY
In this Section, we adopt the cosmological parameter values ofthe concordance Λ CDM model from Planck Collaboration et al.(2014a), H =
67 km s − Mpc − for the Hubble constant, Ω m = . Ω k = Euclid photometric survey, we adopt a phenomenological ap-proach. We start by adopting an average universal luminosityfunction (LF hereafter) for cluster galaxies. Lin, Mohr & Stanford(2003) evaluated the K s -band LFs of cluster galaxies out to a radius r , c for several nearby clusters. The radius r ∆ is defined as the ra-dius of the sphere that encloses an average mass density ∆ times thecritical density of the Universe at the cluster redshift. These clus-ter LFs were parametrised using Schechter functions (Schechter1976). We adopt the averages of the normalisations and char-acteristic luminosities listed in Table 1 of Lin, Mohr & Stanford(2003) for the 27 nearby clusters included in that analysis, corre-sponding to φ ⋆ = . − and M ⋆ = − .
85. Also, followingLin, Mohr & Stanford (2003), we use a faint-end slope α = − . z > α (Mancone et al. 2012;Stefanon & Marchesini 2013). Therefore, we assume it to beredshift-invariant. Also, the observed constancy of the richnessvs. mass relation for clusters up to z ≃ . φ ⋆ , apart from the cosmologicalevolution of the critical density, which scales as H ( z ).We assume the M ⋆ parameter to change with z ac-cording to passive evolution models of stellar populations(Kodama & Arimoto 1997). This assumption is justified becauseemission in the K s band is not strongly influenced by young stel-lar generations, and it is supported by observations (Mancone et al.2012, and references therein), at least for clusters more massivethan ∼ M ⊙ . For clusters of lower mass, some high- z surveyshave found evidence for deviation from passive evolution of M ⋆ (Mancone et al. 2010; Tran et al. 2010; Brodwin et al. 2013). How-ever, the current observational evidence does not allow us to pre-cisely parametrise M ⋆ evolution to z > k -correction of Mannucci et al.(2001) to the M ⋆ magnitudes. This correction should be the mostappropriate for galaxies in clusters, which are mostly early-typeeven at relatively high redshifts (Postman et al. 2005; Smith et al.2005). We finally convert the K s magnitudes into the Euclid band H AB using the mean rest-frame colour for cluster galaxies, H − K s = .
26 (we average the values provided by Boselli et al. 1997;de Propris et al. 1998; Ramella et al. 2004), and adopting the trans-formation to the AB-system H AB = H + .
37 (Ciliegi et al. 2005).We thus obtain the cluster LFs in the H AB band at di ff erent redshifts.By integrating these LFs down to the apparent magnitudelimit of the wide Euclid photometric survey ( H AB =
24, seeLaureijs et al. 2011), we then evaluate n , c , namely the redshift-dependent number density of cluster galaxies within r , c . Thenumber of cluster galaxies contained within a sphere of radius r , c (i.e. the cluster richness) is then N , c = π n , c r , c / = / π n , c GM , c / [500 H ( z )], where the last equivalence followsfrom the relation between r , c and M , c , the mass within a meanoverdensity of 500 times the critical density of the universe at thecluster redshifts. Note that the dependence of N , c on H − ( z ) isonly apparent, since φ ⋆ , and hence n , c , scales as H ( z ). The z -dependence only comes in as a result of the fixed magnitude limitof the survey and the passive evolution of galaxies. In Fig. 1 weshow N , c ( z ) for clusters of three di ff erent masses: log M , c = . , . , and 14.5 (black curves). To convert from M , c to M , c we adopt a NFW profile (Navarro, Frenk & White 1997) with amass- and redshift-dependent concentration given by the relationof De Boni et al. (2013, 2 nd relation from top in their Table 5).We then estimate the contamination by field galaxies inthe cluster area. We take the estimate of the number density offield galaxies down to H AB =
24 from the H-band counts ofMetcalfe et al. (2006, see their Table 3), n field ≃
33 arcmin − ,an estimate that is in agreement with the Euclid survey require-ments (Laureijs et al. 2011). Multiplying this density by the areasubtended by a galaxy cluster at any given redshift we obtain thenumber of field galaxies that contaminate the cluster field-of-view, N field = n field π r , c , where r , c is in arcmin.The number of field contaminants can be greatly reduced byusing photometric redshifts, z p . These will be obtained to the re-quired accuracy of ∆ z p ≡ . + z c ), by combining the Euclid pho- c (cid:13) , 000–000 B. Sartoris et al.
Figure 2.
Galaxy cluster mass selection function for the
Euclid photometricsurvey. Solid and dashed lines are for detection thresholds N , c /σ field = tometric survey with auxiliary ground-based data (Laureijs et al.2011). One can safely consider as non-cluster members all thosegalaxies that are more than 3 ∆ z p away from the mean cluster red-shift z c . The mean cluster redshift will be evaluated by averagingthe photometric redshifts of galaxies in the cluster region, and addi-tionally including the (few) spectroscopic galaxy redshifts providedby the Euclid spectroscopic survey (see Appendix A).In order to determine the fraction of field galaxies, f ( z c ), withphotometric redshift z p in the range ± × . + z c ) at any given z c , we need to estimate the photometric redshift distribution of an H AB =
24 limited field survey. To this aim we consider the photo-metric redshift distribution of galaxies with H AB
24 in the cat-alogue of Yang et al. (2014). We find f ( z c ) = . , . , .
34, and0 .
33 at z c = . , . , .
4, and 2 .
0, respectively.Finally, we evaluate the rms , σ field , of the field galaxy counts f ( z c ) N field , by taking into account both Poisson noise and cos-mic variance. For the latter we use the IDL code quickcv ofJohn Moustakas for cosmic variance calculation. In Fig. 1 weshow 3 σ field as a function of redshift, in clusters of log M , c = . , . , and 14.5.The ratio between the cluster galaxy number counts and thefield rms , N , c /σ field , gives the significance of the detection for agiven cluster. The cluster selection function is the limiting clustermass as a function of redshift for a given detection threshold. This isshown in Fig. 2 for two thresholds, N , c /σ field = , and 5. This se-lection function is only mildly dependent on redshift. The limitingcluster mass for the lowest selection threshold ( N , c /σ field = M , c ∼ × M ⊙ , lower than the typical mass of richnessclass 0 clusters in the Abell, Corwin & Olowin (1989) catalogue(Popesso et al. 2012). It is also similar to the limiting mass of theselection function of SDSS clusters identified by the maxBCG al-gorithm (see Fig. 3 in Rozo et al. 2010), and to the typical mass ofthe clusters identified by Brodwin et al. (2007) up to z ∼ . z p in an IR-selected galaxy catalogue. Preliminary tests based onrunning cluster finders on Euclid mocks , show that the mass limit https: // code.google.com / p / idl-moustakas / source / browse / trunk / impro / cosmo / quickcv.pro?r = http: // wiki.cosmos.esa.int / euclid / index.php / EC SGS OU LE3. Accessrestricted to members of the
Euclid
Consortium. M , c ∼ × M ⊙ roughly corresponds to ∼
80% completenessat all redshifts z z ∼ . z ∼ . z < . ∆ c = ∆ c = z < .
5. In reality, observers do not select clusters at given ∆ c , soour estimate of the selection function must be considered only as anapproximation. At the end of Section 5 we comment on the e ff ectof taking a flat selection function out to z = Before presenting our forecasts for the cosmological constraints wenow briefly describe the Fisher Matrix (FM hereafter) formalismthat we use to derive these constraints.The FM formalism is a Gaussian approximation of the likeli-hood around the maximum to second order and it is an e ffi cient wayto study the accuracy of the estimation of a vector of parameters p by using independent data sets. The FM is defined as F αβ ≡ − * ∂ ln L ∂ p α ∂ p β + , (1)where L is the likelihood of the observable quantity (e.g. Dodelson2003). In our FM analysis we combine three di ff erent piecesof information: the galaxy cluster number density, the clus-ter power spectrum, and the prior knowledge of cosmologi-cal parameters as derived from the Planck
CMB experiment(Planck Collaboration et al. 2014a). To quantify the constrainingpower of a given cosmological probe on a pair of joint parameters( p i , p j ) we use the Figure of Merit (FoM henceforth; Albrecht et al.2006) FoM = q det h Cov (cid:16) p i , p j (cid:17)i , (2)where Cov ( p i , p j ) is the covariance matrix between the two param-eters. With this definition, the FoM is proportional to the inverseof the area encompassed by the ellipse representing the 68 per centconfidence level (c.l.) for model exclusion.As described in detail in Sartoris et al. (2010), we follow theapproach of Holder, Haiman & Mohr (2001) and define the FM forthe cluster number counts as F N αβ = X ℓ, m ∂ N ℓ, m ∂ p α ∂ N ℓ, m ∂ p β N ℓ, m . (3)In the previous equation, the sums over ℓ and m run over redshiftand mass intervals, respectively. The quantity N ℓ, m is the number ofclusters expected in a survey with a sky coverage Ω sky , within the ℓ -th redshift bin and m -th bin in observed mass M ob . This can becalculated as (Lima & Hu 2005) N ℓ, m = Ω sky π Z z ℓ + z ℓ dz dVdz Z M ob ℓ, m + M ob ℓ, m dM ob M ob Z + ∞ dM n ( M , z ) p ( M ob | M ) , (4) c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the Euclid
Galaxy Cluster Survey where dV / dz is the cosmology-dependent comoving volume ele-ment per unit redshift interval. The lower observed mass bin isbound by M ob ℓ, m = = M thr ( z ), where M thr ( z ) is defined as the thresholdvalue of the observed mass for a cluster to be included in the sur-vey (see Fig. 2). For the halo mass function n ( M , z ) in equation (4),we assume the expression provided by Tinker et al. (2008). Sincethe Euclid selection function has been computed for masses at ∆ c =
200 with respect to the critical density, we use the Tinker et al.(2008) mass function parameters relevant for an overdensity of ∆ bk = / Ω m ( z ) with respect to the background density. We notethat in equation (4) we have implicitly assumed that the survey skycoverage Ω sky is independent of the observed mass, which may notnecessarily be the case if the sensitivity is not constant over thesurvey area.In equation (4), p ( M ob | M ) is the probability to assign an ob-served mass M ob to a galaxy cluster with true mass M . FollowingLima & Hu (2005), we use a lognormal probability density, namely p ( M ob | M ) = exp[ − x ( M ob )] q πσ M , (5)where x ( M ob ) = ln M ob − ln M bias − ln M q σ M . (6)In the above equation ln M bias is the bias in the mass estimation,which encodes any scaling relation between observable and truemass and should not be confused with the bias in the galaxy distri-bution. σ ln M is the intrinsic scatter in the relation between true andobserved mass (see Section 4). By inserting equation (5) into equa-tion (4), we obtain the expression for the cluster number countswithin a given mass and redshift bin, N ℓ, m = Ω sky π Z z ℓ + z ℓ dz dVdz Z + ∞ dM n ( M , z ) [erfc( x m ) − erfc( x m + )] , (7)where erfc( x ) is the complementary error function and x m = x ( M ob l , m ).The FM for the averaged redshift-space cluster power spec-trum within the ℓ -th redshift bin, the m -th wavenumber bin, and the i -th angular bin can be written as F P αβ = π X ℓ, m , i ∂ ln ¯ P ( µ i , k m , z ℓ ) ∂ p α ∂ ln ¯ P ( µ i , k m , z ℓ ) ∂ p β V e ff ℓ, m , i k m ∆ k ∆ µ (8)(e.g., Tegmark 1997; Feldman, Kaiser & Peacock 1994), where thesums over ℓ , m , i run over bins in redshift, wavenumber, and cosineof the angle between k and the line of sight direction, respectively.The quantity V e ff ( µ i , k m , z ℓ ) represents the e ff ective volume accessi-ble to the survey at redshift z ℓ and wavenumber k (Tegmark 1997;Sartoris et al. 2010), and reads V e ff ( µ i , k m , z ℓ ) = V ( z ℓ ) ˜ n ( z ℓ ) ¯ P ( µ i , k m , z ℓ )1 + ˜ n ( z ℓ ) ¯ P ( µ i , k m , z ℓ ) . (9)In the above equation, V ( z ℓ ) is the total comoving volume con-tained in the unity redshift interval around z ℓ , while ˜ n ( z ℓ ) is the av-erage number density of objects included in the survey at redshift z ℓ , ˜ n ( z ℓ ) = Z + ∞ dM n ( M , z ℓ ) erfc { x [ M thr ( z ℓ ) } . (10) The cluster power spectrum averaged over the redshift bin, appear-ing in equation (8), can be written as¯ P ( µ i , k m , z ℓ ) = S ℓ Z z ℓ + z ℓ dz dVdz ˜ n ( z ) ˜ P ( µ i , k m , z ) , (11)where the normalisation factor S ℓ reads S ℓ = Z z ℓ + z ℓ dz dVdz ˜ n ( z ) . (12)Sartoris et al. (2012) pointed out the importance of taking intoaccount the contribution of cluster redshift space distortions forconstraining cosmological parameters. Following Kaiser (1987),we calculate the redshift-space cluster power spectrum ˜ P ( µ i , k m , z ℓ )in the linear regime according to˜ P ( µ i , k m , z ℓ ) = h b e ff ( z ℓ ) + f ( z ℓ ) µ i P L ( k m , z ℓ ) , (13)where the power spectrum acquires a dependence on the cosine µ ofthe angle between the wavevector k and the line-of-sight direction.In the above equation, b e ff ( z ℓ ) is the linear bias weighted by themass function (see equation 20 in Sartoris et al. 2010), b e ff ( z ℓ ) = n ( z ℓ ) Z + ∞ d M n ( M , z ℓ ) erfc { x [ M thr ( z ℓ )] } b ( M , z ℓ ) . (14)The function f ( a ) = d ln D ( a ) / d ln a is the logarithmic derivativeof the linear growth rate of density perturbations, D ( a ), with re-spect to the expansion factor a . P L ( k m , z ℓ ) is the linear matter powerspectrum in real space, that we calculate using the CLASS code(Blas, Lesgourgues & Tram 2011). For the DM halo bias b ( M , z )we use the expression provided by Tinker et al. (2010).Both the power spectrum and the number counts FMs (equa-tions 3 and 8) are computed in the redshift range defined by the Euclid photometric selection function shown in Fig. 2, namely0 . z
2, with redshift bins of constant width ∆ z = . z c of a cluster is determined in the photometric survey is given by0 . + z c ) / N / , c , where N , c is the total number of galaxies as-signed to the cluster. Therefore, the bin width is always larger thanthe largest error on redshift expected from the Euclid photometricsurvey (see Section 2). In equation (3), the observed mass range ex-tends from the lowest mass limit determined by the photometric se-lection function ( M thr ( z ), see Fig. 2) up to log( M ob / M ⊙ )
16, with ∆ log( M ob / M ⊙ ) = .
2. In the computation of the power spectrumFM (equation 8), we adopt k max = .
14 Mpc − , with ∆ log( k Mpc) = .
1. Finally, the cosine of the angle between k and the line of sightdirection, µ , runs in the range − µ In this Section we discuss the cosmological parameters that havebeen included in the FM analysis in order to predict the constrain-ing power of the
Euclid photometric cluster survey and we describethe peculiarity of all the analysed models. As a starting point, weconsider all the standard cosmological parameters for the concor-dance Λ CDM model, whose fiducial values are chosen by follow-ing Planck Collaboration et al. (2014a): Ω m = .
32 for the present-day total matter density parameter, σ = .
83 for the normalisationof the linear power spectrum of density perturbations, Ω b = . H =
67 km s − Mpc − for theHubble constant, and n S = .
96 for the primordial scalar spectralindex. We also allow for a variation of the curvature parameter,whose fiducial value Ω k = c (cid:13) , 000–000 B. Sartoris et al.
In addition to the Λ CDM parameters, we also include parame-ters describing a dynamical evolution of the DE component. Inthe literature there are a number of models, characterised by dif-ferent parametrisation of the DE Equation of State (EoS hence-forth) evolution (e.g., Wetterich 2004). In this paper we studythe parametrisation originally proposed by Chevallier & Polarski(2001) and Linder (2003) and then adopted in the DETF. We labelthis parametrisation as the CPL DE model, according to which theDE EoS can be written as w ( a ) = w + w a (1 − a ) . (15)We use w = − w a = p = { Ω m , σ , w , w a , Ω k , Ω b , H , n S } . (16)The constraints on the DE dynamical evolution obtained by com-bining Planck
CMB data with WMAP polarisation and withLSS information (Planck Collaboration et al. 2014a), are w = − . + . − . and w a . . Ω k =
0. Currently, the evolution of the cluster number counts alonedoes not constrain the DE equation of state parameters. However,Mantz et al. (2014) were able to obtain: w = − . ± .
18 and w a = − . + . − . (assuming Ω k = ff er-ent redshifts (plus WMAP polarisation; Planck Collaboration et al.2014a).Despite these weak constraints on the CPL DE parametrisa-tion (Vikhlinin et al. 2009), cluster counts are powerful probes ofthe amplitude of the matter power spectrum. For instance, σ isconstrained at the level of ∼ σ and Ω m in CMB datasets, improving theconstraints on the amplitude of the matter power spectrum by a fac-tor of ∼ We extend the standard cosmological model by allowing primor-dial density fluctuations to follow a non-Gaussian distribution(e.g., Bartolo et al. 2004; Desjacques & Seljak 2010; Wang 2014).When this happens, the distribution of primordial fluctuations inBardeen’s gauge-invariant potential Φ cannot be fully describedby a power spectrum - commonly parametrised by a power-law, P Φ ( k ) = Ak n S − (where k ≡ k k k ) - rather we need higher-orderstatistics such as the bispectrum B Φ ( k , k , k ). Di ff erent modelsof inflation are known to produce di ff erent shapes of this bispec-trum. Here we consider only the so-called local shape , where thebispectrum strength is maximised for squeezed configurations, inwhich one of the three momenta k j is much smaller than the othertwo. Within the local shape scenario, we adopt the commonly usedway to parametrise the primordial non-Gaussianity, which allowsus to write Bardeen’s gauge invariant potential as the sum of a lin-ear Gaussian term and a non-linear second-order term that encapsu-lates the deviation from Gaussianity (e.g., Salopek & Bond 1990;Komatsu & Spergel 2001): Φ = Φ G + f NL (cid:16) Φ − h Φ i (cid:17) , (17) where the free dimensionless parameter f NL parametrises thedeviation from the standard Gaussian scenario. We stress thatthere is some ambiguity in the normalisation of equation (17).We adopt the LSS convention (as opposed to the CMB con-vention, see Pillepich, Porciani & Hahn 2010; Grossi et al. 2007;Carbone, Verde & Matarrese 2008a) where Φ is linearly extrapo-lated at z = f NL . The relation betweenthe two normalisations is f NL = D ( z = ∞ )(1 + z ) f CMBNL / D ( z = ≃ . f CMBNL , where D ( z ) is the linear growth factor with respect to theEinstein-de Sitter cosmology.If the density perturbation field is non-Gaussian and has a pos-itively (negatively) skewed distribution, the probability of forminglarge overdensities - and thus large collapsed structures - is en-hanced (suppressed). Thus, the shape and the evolution of the massfunction of DM halos change (e.g., Matarrese, Verde & Jimenez2000; Grossi et al. 2009; LoVerde et al. 2008). Following the pre-scription by LoVerde et al. (2008) one can modify the mass func-tion n ( M , z ) in equation (4) to take into account the non-Gaussiancorrection as follows n ( M , z ) = n (G) ( M , z ) n PS ( M , z ) n (G)PS ( M , z ) . (18)In the previous equation, n (G) ( M , z ) is the mass function in the refer-ence Gaussian model, while n PS ( M , z ) and n (G)PS ( M , z ) represent thePress & Schechter (1974) mass functions in the non-Gaussian andreference Gaussian models, respectively (see the full equations inSartoris et al. 2010).In non-Gaussian scenarios the large-scale clustering of DMhalos also changes. This modification is quite important because italters in a fairly unique way the spatial distribution of tracers ofthe cosmic structure, including galaxy clusters (Dalal et al. 2008;Matarrese & Verde 2008; Giannantonio & Porciani 2010). Specifi-cally, the linear bias acquires an extra scale dependence due to pri-mordial non-Gaussianity, and can be written as (Matarrese & Verde2008) b ( M , z , k ) = b (G) ( M , z ) + h b (G) ( M , z ) − i δ c ( z ) Γ R ( k ) , (19)where Γ R ( k ) encapsulates the dependence on the scale and is givenby an integral over the primordial bispectrum.To summarise, the cosmological parameter vector in this non-Gaussian extension of the Λ CDM model is p = { Ω m , σ , w , w a , Ω k , Ω b , H , n S , f NL } . (20)We assume f NL = Planck
CMB data(Planck Collaboration et al. 2013), − . f NL .
11, for the case ofa local bispectrum shape . Bounds from galaxy cluster abundanceshow consistency with the Gaussian scenario, − . f NL . Euclid spectroscopic galaxies alone is expected to restrict theallowed non-Gaussian parameter space down to ∆ f NL ∼ a few(Carbone, Verde & Matarrese 2008b; Verde & Matarrese 2009;Fedeli et al. 2011). The
Planck
CMB constraints on primordial non-Gaussianity have beenconverted here into the LSS convention.c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the
Euclid
Galaxy Cluster Survey We studied another extension to the standard Λ CDM cosmol-ogy, based on deviations of the law of gravity from GR. Asa matter of fact, a number of non-standard gravity modelshave been proposed in the literature (e.g., Hu & Sawicki 2007;Capozziello & de Laurentis 2011; Amendola et al. 2013) in orderto explain the low-redshift accelerated expansion of the Universewithout need for the DE fluid. Many of these models give rise tomodifications of the late-time linear growth of cosmological struc-ture, which can be parametrised as d ln D ( a ) d ln a = Ω γ m ( a ) , (21)where γ is dubbed the growth index (e.g. Lahav et al. 1991). GRpredicts a nearly constant and scale-independent value of γ ≃ . p = { Ω m , σ , w , w a , Ω k , Ω b , H , n S , γ } , (22)with γ = .
55 taken as our reference value. Using number countsof X-ray clusters alone, Mantz et al. (2015) have found valuesof γ consistent with GR ( γ = . ± . γ = . ± .
28 has beenfound (Bocquet et al. 2014). Lombriser et al. (2012) have directlyconstrained the f ( R ) model by Hu & Sawicki (2007) by exploitingan optically selected cluster sample. In our analysis we also consider the case of massive neutrinos, withthe associated density parameter Ω ν as the relevant parameter tobe constrained. Ω ν is related to the total neutrino mass, P N ν i m ν, i ,through the relation: Ω ν = ρ ν ρ c = P N ν i m ν, i .
14 h eV , (23)where ρ ν and ρ c are the z = N ν is the number of massive neutrinos. Alarger value of Ω ν acts on the observed matter power spectrumin two ways (e.g. Lesgourgues & Pastor 2006; Marulli et al. 2011;Massara, Villaescusa-Navarro & Viel 2014). The peak of the powerspectrum is shifted to larger scale, because a larger value of theradiation density postpones the time of equality. Moreover, sinceneutrinos free-stream over the scale of galaxy clusters, they do notcontribute to the clustered collapsed mass on such a scale. As aconsequence, the halo mass function at fixed value of Ω m will bebelow the one expected in a purely CDM model. Brandbyge et al.(2010) have shown that results from N -body simulations with mas-sive neutrinos can be reproduced in a more accurate way by usingthe Tinker et al. (2008) halo mass function with ρ m → ρ CDM + ρ b = ρ m − ρ ν , (24)where ρ m , ρ CDM , ρ b and ρ ν are the total mass, CDM, and baryonand neutrino densities. Based on the analysis of an extended setof N–body simulations, Castorina et al. (2014) and Costanzi et al.(2013b) have shown that, since neutrinos play a negligible role inthe gravitational collapse, only the contribution of cold dark matterand baryons to the power spectrum has to be used to compute the r.m.s. of the linear matter perturbations, σ ( R ), in the computationof the halo mass function and linear bias: P m → P CDM ( k ) = P m ( k ) " Ω CDM T CDM ( k , z ) + Ω b T b ( k , z )( Ω b + Ω CDM ) T m ( k , z ) . (25)Here T CDM , T b and T m are the CDM, baryon, and total matter trans-fer functions respectively, and P m is the total matter power spec-trum.Hence, the cosmological parameter vector we use in this caseis: p = { Ω m , σ , w , w a , Ω k , Ω b , H , n S , Ω ν } , (26)with a fiducial value of Ω ν = . P m ν = .
06 for three degenerate neutrinos (Carbone et al. 2012;Mantz et al. 2015). Currently, great attention has been devotedto derive constraints on the neutrino mass from the combinationof galaxy clusters with other LSS observables. The analysis ofthe Planck SZ cluster sample resulted in P m ν = . ± .
09 eV(Planck Collaboration et al. 2014c). Mantz et al. (2014), combin-ing cluster, CMB, SN1a and BAO data, found P m ν < .
38 eVat 95.4 per cent c.l. in a wCDM universe. Costanzi et al. (2014)found P m ν < .
15 eV (68 per cent c.l.) in a Λ CDM universe,for a three active neutrino scenario, using cluster counts, CMB,BAO, Lyman- α , and cosmic shear data. In Bocquet et al. (2014) theanalysis of SPT cluster sample resulted in P m ν = . ± .
081 eV.
In our FM analysis, besides the cosmological parameter vectors de-tailed above, we also include four extra parameters to model intrin-sic scatter and bias in the scaling relation between the observed andtrue galaxy cluster masses (see equation 6 above). We assume thefollowing parametrisation for the bias and the scatter, respectively:ln M bias ( z ) = B M , + α ln (1 + z )and σ M ( z ) = σ M , − + (1 + z ) β . (27)We select the following fiducial values p nuisance , F = (cid:8) B M , = , α = , σ ln M , = . , β = . (cid:9) . (28)We refer to these four parameters as nuisance parameters hence-forth. With the fiducial nuisance parameter vector there is no biasin the true mass-observable relation and the value of the scatter at z = ff et al. (2012). Also, the fidu-cial value for β makes the scatter increase with redshift, reach-ing σ ln M ≃ . Euclid survey( z max = Euclid . In the
Euclid survey it will be possible to calibrate such relation with its uncer-tainties thanks to the weak lensing and spectroscopic surveys. Weestimate that
Euclid has the potential to calibrate the scaling rela-tion to
15 per cent accuracy out to z . c (cid:13) , 000–000 B. Sartoris et al.
Redshiftn(z) for N / σ field =3n(z) for N / σ field =5n(>z) for N / σ field =3n(>z) for N / σ field =510 Figure 3.
Number of clusters above a given redshift to be detected withoverdensities N , c /σ f ield > > Euclid photometric survey(dotted blue and solid red lines, respectively). We also show the numberdensity of clusters expected to be detected within redshift bins of width ∆ z = . Here, we present the constraints on the cosmological parametervectors introduced in the previous Section, using the FM formal-ism. As a first result, we plot in Fig. 3 the histograms correspondingto the redshift distributions, n ( z ) = ∆ z dN / dz (equation7), of Euclid photometric galaxy clusters, obtained by adopting the two selectionfunctions, which correspond to the two di ff erent detection thresh-olds N , c /σ f ield > n ( > z ),i.e., the total number of clusters detected above a given redshift. Euclid will detect ∼ × objects with N , c /σ field > ∼ × of them at z >
1. By lowering thedetection threshold down to N , c /σ field =
3, these numbers riseup to ∼ × clusters at all redshifts, with ∼ × of themat z >
1. The large statistics of clusters at z > ∼ . × clusters(with more than 10 bright red-sequence galaxies) and with massesgreater than ∼ × M ⊙ out to z ∼ . eROSITA (Pillepich, Porciani & Reiprich 2012) will detect ∼ . × clusters with masses greater than ∼ × M ⊙ in thesurvey area of 27.000 deg , almost all at z < Euclid photometric clusters on suitable pairs of cosmological pa-rameters. The ellipses in these figures always correspond to the68 per cent c.l. after marginalisation over all other cosmologicalparameters and nuisance parameters. In each of these figures, theblue dotted contours are obtained by combining the number counts(NC) FM (equation 3) and the cluster power spectrum (PS) FM(equation8), assuming no prior information on any of the cosmo-logical and nuisance parameters. Also, the cluster sample is definedby the selection N , c /σ field >
3. The green dash-dotted contoursare obtained in the same way except for the addition of strong priorson the nuisance parameters, i.e. assuming perfect knowledge of thescaling relation between the true and the observed cluster mass (thisis labelled as ” + known SR” in the figures). The magenta solid con-tours have been obtained by further introducing prior informationfrom Planck data (label-ed ” + Planck prior” in the figures). Finally,the cyan solid contours represent the same combination of informa-tion as the magenta solid ones (NC + PS + known SR + Planck prior)obtained from the cluster sample with selection corresponding to N , c /σ field >
5. In the figures, we indicate these contours with thelabel 5 σ .When using the Planck priors, we take for the CPL DE modelthe correlation matrix obtained by combining
Planck
CMB datawith the BAOs from Planck Collaboration et al. (2014a) for theparameters of the Λ CDM cosmology (assuming Ω k = w and w a . For the non-Gaussian case, we use priors from the Planck obtained for the Λ CDM model plus Ω k , parameters . Wealso added a flat prior on the level of non-Gaussianity correspond-ing to − . f CMBNL .
8. Finally, for the modified gravity andthe neutrino scenario we also used priors from the
Planck analysiscarried out over the parameters of the Λ CDM model plus Ω k .In Fig. 4, we show the constraints on Ω m and σ (left panel),as well as those on the two CPL DE parameters w and w a (rightpanel). The contours on the Ω m − σ plane for the combination ofnumber counts and clustering of N , c /σ field > Ω m and σ , with the following constraints: ∆Ω m = . , ∆ σ = . Ω m − σ plane. However, using thecombination of the PS with NC FM, the values of both parametersare constrained to high accuracy: ∆Ω m = . , ∆ σ = . σ , which is more a ff ectedby the nuisance parameters than Ω m . Including information fromthe Planck priors does not improve the forecasted constraints sig-nificantly, in keeping with the expectation that the
Euclid clusterbounds are, by themselves, competitive with CMB bounds.Taking the Λ CDM as a reference model, its parameters willbe constrained with a precision of ∼ − , ∆Ω m = . − , ∆ σ = . − , ∆ h = . − , https: // / reports / proposal-standalone.pdf Available at http://wiki.cosmos.esa.int/planckpla/index.php/Cosmological Parameters PLA / base w wa / planck lowl lowLike BAO PLA / base omegak / planck lowl lowLikec (cid:13) , 000–000 ext Generation Cosmology: Constraints from the Euclid
Galaxy Cluster Survey σ Ω m NCNC+PS+known SR+Planck prior5 σ w a w NCNC+PS+known SR+Planck prior5 σ -1.5-1-0.5 0 0.5 1 1.5-1.15 -1.1 -1.05 -1 -0.95 -0.9 -0.85 Figure 4.
Constraints at the 68 per cent c.l. on the parameters Ω m and σ (left panel) and on the parameters w and w a for the DE EoS evolution (rightpanel). In each panel, we show forecasts for the N , c /σ field > Euclid photometric cluster selection obtained by (i) NC, the FM number counts (reddash-dotted contours), (ii) NC + PS, the combination of FM NC and power spectrum (PS) information (blue dotted contours), (iii) NC + PS + known SR, i.e. byadditionally assuming a perfect knowledge of the nuisance parameters (green dash-dotted contours), and (iv) NC + PS + known SR + Planck prior, i.e. by alsoadding information from
Planck
CMB data (magenta solid contours). With cyan solid lines we show forecasts for the N , c /σ field > Euclid photometriccluster selection in the case NC + PS + known SR + Planck prior (labelled 5 σ ). Planck information includes prior on Λ CDM parameters and the DE EoSparameters. ∆Ω b = . − , ∆ n s = . − (29)thanks to the unprecedented number of clusters that will be de-tected at high redshift. These constraints have been obtained withthe N , c /σ field > strong prior on the nuisance pa-rameters, and no prior from Planck.These results emphasise the importance of exploring the high-redshift clusters in survey mode. Of course a good knowledge of theastrophysical process taking place in clusters is fundamental to cali-brate the mass-observable scaling relations, and also to optimise thedetection algorithms. Hence detailed follow-ups of restricted sam-ples of clusters (such as, e.g., CLASH, CCCP, WtG Postman et al.2012; Rosati et al. 2014; Hoekstra et al. 2012; von der Linden et al.2014) retain a crucial importance.On the other hand, the inclusion of Planck priors shall bringa substantial improvement over the bounds to the DE parameters.This result is expected, since the CMB data provides stringent con-straints on the curvature, thereby breaking the degeneracy between Ω k and the evolution of the DE EoS (Sartoris et al. 2012). Thecontribution of the PS information is less important for ( w , w a )with respect to ( Ω m , σ ): however, the FoM increases from ∼ ∼
73 for NC + PS constraints (see Table 1).For both DE EoS parameters, it is crucial to have a well calibratedscaling relation over the redshift range sampled by the cluster sur-vey (Sartoris et al. 2012). Indeed, by combining NC and PS, andassuming perfect knowledge of the scaling relation increases theFoM to ≃ = ∆ w = .
017 and ∆ w a = .
074 (see Table 1).When we restrict our analysis to the wCDM model (that is characterised by the six free parameters Ω m , σ , h , Ω b , n s , w ), weobtain ∆ w = . w a as a free parameter, we ob-tain ∆ w = .
013 and ∆ w a = . N , c /σ field > strong prior onthe nuisance parameters, and no prior from Planck.In both panels of Fig. 4, the adoption of a more conservativecluster selection ( N , c /σ field >
5) significantly worsens the fore-casted cosmological constraints. For instance, the FoM is degradeddown to 209 in the best-case scenario, as a consequence of the sig-nificantly degraded statistics corresponding to the higher selectionthreshold.In Fig. 5, we show how the FoM depends on the limiting red-shift of the survey. The FoM shown in this figure refers to numbercounts (NC) in the N , c /σ field > Euclid photometric cluster se-lection. The FoM for a survey reaching out to z . z
2. It is there-fore important that the redshift range covered by the survey be largeenough to allow a comparison of the behaviour of DE over a su ffi -ciently long cosmological timescale. In this sense, the Euclid sur-vey will have a unique advantage over other existing and plannedsurveys.In Fig. 6, we show cosmological constraints in the expandedparameter space which includes non-Gaussian primordial densityfluctuations. Specifically, we display the constraints in the f NL − σ plane. Thanks to the peculiar scale-dependence that primordialnon-Gaussianity induces on the linear bias parameter, the powerspectrum of the cluster distribution turns out to be much more sen-sitive to f NL than it is to σ . This is clearly demonstrated by the reddash-dotted contour, which shows forecasted constraints derived c (cid:13) , 000–000 B. Sartoris et al.
Table 1.
Figure of Merit (FoM) and constraints on cosmological parameters as obtained by progressively adding the FM information for di ff erent models,for two di ff erent detection thresholds ( N , c /σ field > N , c /σ field > Euclid photometric cluster selectionParameter arrays: Eqs. 16 & 28 Eqs. 22 & 28 Eqs. 20 & 28 Eqs. 26 & 28Constraints: FoM ∆ w ∆ w a ∆Ω m ∆ σ ∆ γ ∆ f NL ∆Ω ν NC + PS 73 0.037 0.38 0.0019 0.0032 0.023 6.67 0.0015NC + PS + known SR 291 0.034 0.16 0.0011 0.0014 0.020 6.58 0.0013NC + PS + known SR + Planck 802 0.017 0.074 0.0010 0.0012 0.015 4.93 0.0012 N , c /σ field > Euclid photometric cluster selectionNC + PS + known SR + Planck 209 0.034 0.12 0.0022 0.0026 0.034 6.74 0.0020 F o M / F o M z m a x = z max Figure 5.
Relative FoM for number counts in the N , c /σ field > Euclid photometric cluster selection, as a function of the limiting redshift z max ofthe survey, i.e. the ratio between the FoM evaluated over 0 . z z max andthe FoM evaluated over 0 . z . from cluster clustering alone. Quite clearly, σ is basically uncon-strained on the scale of the figure, while f NL is constrained withan uncertainty ∆ f NL ∼ .
4. The addition of cluster number countschanges very little the bounds for primordial non-Gaussianity, how-ever it improves substantially those for the amplitude of the mat-ter power spectrum (see Table 1). This helps to define the degen-eracy between f NL and σ that are both related to the timing ofstructure formation. Interestingly, the estimation of primordial non-Gaussianity is weakly sensitive to the nuisance parameters. Indeed,when a perfect knowledge of the scaling relation between true andobserved cluster mass is assumed, only the constraints on σ im-prove significantly. Planck priors does not a ff ect substantially theconstraints on f NL .When we restrict our analysis to the Λ CDM model plus thenon-Gaussianity parameter f NL , we obtain ∆ f NL = .
44. This con-straint has been obtained with the N , c /σ field > strong prior on the nuisance parameters, and no prior from Planck. σ f NL PSNC+PS+known SR+Planck prior5 σ Figure 6.
Constraints at the 68 per cent c.l. on the f NL − σ parameters. Weshow forecasts for the N , c /σ field > Euclid photometric cluster selec-tion obtained by (i) PS, the FM power spectrum (red dash-dotted contours),(ii) NC + PS, the combination of FM number counts (NC) and PS informa-tion (blue dotted contours), (iii) NC + PS + known SR, i.e. by additionally as-suming a perfect knowledge of the nuisance parameters (green dash-dottedcontours), and (iv) NC + PS + known SR + Planck prior, i.e. by also addinginformation from
Planck
CMB data (magenta solid contours). With cyansolid lines we show forecasts for the N , c /σ field > Euclid photomet-ric cluster selection in the case NC + PS + known SR + Planck prior (labelled5 σ ). Planck information includes prior on Λ CDM +Ω k + f NL parameters. Forecast for eROSITA (Pillepich, Porciani & Reiprich 2012) pre-dict a similar precision, since the narrower redshift range of thissurvey (with respect to
Euclid ) is compensated by its wider area,which allows a better sampling of large scale modes.We point out that in this analysis we are assuming the mostcommonly used parametrisation of non-Gaussianity, where f NL isconsidered scale-invariant. However, there are models that predictotherwise. For these, the combination of clusters and CMB datacomplement each other well, providing tight constraints on the pos-sible scale dependence of f NL .As for the models including GR violation, we show in Fig. c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the Euclid
Galaxy Cluster Survey σ γ NC+PS+known SR+Planck prior5 σ Figure 7.
Constraints at the 68 per cent c.l. on the γ − σ parameterplane. We show forecasts for the N , c /σ field > Euclid photometriccluster selection obtained by (i) NC + PS, the combination of FM numbercounts (NC) and power spectrum (PS) information (blue dotted contours),(ii) NC + PS + known SR, i.e. by additionally assuming a perfect knowl-edge of the nuisance parameters (green dash-dotted contours), and (iii)NC + PS + known SR + Planck prior, i.e. by also adding information from
Planck
CMB data (magenta solid contours). With cyan solid lines we showforecasts for the N , c /σ field > Euclid photometric cluster selection inthe case NC + PS + known SR + Planck prior (labelled 5 σ ). Planck informa-tion includes prior on Λ CDM +Ω k parameters. σ and the growth parameter γ . Similarly tothe constraints on the Ω m – σ plane, the constraints on γ are notstrongly a ff ected by the inclusion of Planck priors, thus implyingthat galaxy clusters are by themselves excellent tools to detect sig-nature of modified gravity through its e ff ect on the growth of pertur-bations. Significant degradation of the constraining power happensif a higher threshold for cluster detection is chosen.Restricting our analysis to the Λ CDM model plus the γ pa-rameter we obtain ∆ γ = . N , c /σ field > strong prior on the nui-sance parameters, and no prior from Planck.Finally, we show in Fig. 8 the joint cosmological constraintson σ and the neutrino density parameter Ω ν . The presence of neu-trinos with masses in the sub-eV range requires higher values of σ : increasing Ω ν at fixed Ω m has the e ff ect of shifting the epochof matter-radiation equality to a later time and to reduce the growthof density perturbations at small scales in the post-recombinationepoch. As a consequence, a larger value of σ is required to com-pensate these e ff ects. We use the Planck prior mainly to add infor-mation on the geometry of the Universe, and the standard Λ CDMparameters. We obtain the constraints ∆Ω ν = . ∆ P m ν = .
01 ). The constraints on the neutrino densityparameter are weakly a ff ected by the inclusion of a prior on thenuisance parameters. However, there is a degradation by a factorof ∼ σ Ω ν σ NC+PS+known SR+Planck prior 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0 0.001 0.002 0.003 0.004 0.005
Figure 8.
Constraints at the 68 per cent c.l. in the Ω ν − σ parameterplane. We show forecasts for the N , c /σ field > Euclid photometriccluster selection obtained by (i) NC + PS, the combination of FM numbercounts (NC) and power spectrum (PS) information (blue dotted contours),(ii) NC + PS + known SR, i.e. by additionally assuming a perfect knowl-edge of the nuisance parameters (green dash-dotted contours), and (iii)NC + PS + known SR + Planck prior, i.e. by also adding information from
Planck
CMB data (magenta solid contours). With cyan solid lines we showforecasts for the N , c /σ field > Euclid photometric cluster selection inthe case NC + PS + known SR + Planck prior (labelled 5 σ ). Planck informa-tion includes prior on Λ CDM +Ω k parameters. To gauge the impact of a particular choice of the selectionfunction on the cosmological constraints, we have so far shown ourresults for both the N , c /σ field > N , c /σ field > Eu-clid photometric cluster selection functions. As a further test, weconsider the e ff ect on the ( w , w a ) constraints of adopting a flatselection function with log( M , c ) = .
9, within 0 . z flat selection function there are less clusters than with the N , c /σ field > ∼ . × vs. ∼ . × ,respectively) and within 0 . . z . .
2. However, the number ofclusters at z > ∼ . × ) for the flat selection func-tion than for the N , c /σ field > ∼ . × ). The e ff ect of alarger number of high- z clusters in the flat selection function sam-ple compensates for the smaller total number of clusters in provid-ing similar constraints on the cosmological parameters to those ob-tained with the N , c /σ field > <
10% on the constraints on the DE parameters). This suggeststhat the precise shape of the selection function has little impact onour results, and in any case much less than its overall normalisation.
In this paper, we presented a comprehensive analysis of the fore-casts on the parameters that describe di ff erent extensions of thestandard Λ CDM model. These were based on the selection func-tion of galaxy clusters from the wide photometric survey to be car-ried out with the
Euclid satellite, a medium-size ESA mission to c (cid:13)000
Euclid satellite, a medium-size ESA mission to c (cid:13)000 , 000–000 B. Sartoris et al. be launched in 2020. We presented the derivation of this selectionfunction and the Fisher Matrix formalism employed to derive cos-mological constraints. This is the same formalism that has beenused in the
Euclid
Red Book (Laureijs et al. 2011). Our main re-sults can be summarised as follows. • Using photometric selection, we found that
Euclid will detectgalaxy clusters at N , c /σ field > ∼ . − × M ⊙ . As a result, the Euclid photometric cluster catalogueshould include ∼ × objects, with about one fifth of them at z > • The
Euclid cluster catalogue has the potential of providingtight constraints on a number of cosmological parameters, such asthe normalisation of the matter power spectrum σ , the total matterdensity parameter Ω m , a redshift-dependent DE equation of state,primordial non-Gaussianity, modified gravity, and neutrino masses(see Table 1). We predict that most of these constraints will be eventighter than current bounds available from Planck . The constrain-ing power of the
Euclid cluster catalogue relies on its unique broadredshift coverage, reaching out to z = • Knowledge of the scaling relation between the true and the ob-served cluster mass turns out to be one of the most important factorsdetermining the constraining power of the
Euclid cluster cataloguefor cosmology. The
Euclid mission will have a distinct advantagein this respect, namely the possibility to calibrate such relation, atleast up to z = .
5, with .
10 and .
30 per cent accuracy, us-ing the weak lensing and spectroscopic surveys, respectively (seeAppendix B). The deep
Euclid survey will allow to extend the cali-bration to even higher redshifts, although with lower precision thanin the wide survey, due to lower number statistics.With the future large surveys, like
Euclid , that will be car-ried out with the next generation of telescopes, the number of de-tected clusters from the individual surveys will range from thou-sands to tens of thousands. As we have shown in this paper, thiswill allow to constrain most cosmological parameters to a preci-sion level of a few per cent. Currently, theoretical halo mass func-tions are defined with an uncertainty of ∼ Λ CDM model (e.g. Tinker et al. 2008), and many e ff orts have beendevoted in the last years to better sample the high mass regime(Watson et al. 2013). To maximally extract cosmological informa-tion from these cluster surveys, it becomes critical to specify thetheoretical halo mass function to better than a few percent accu-racy for a range of cosmologies. A substantial e ff ort is currentlyongoing in this direction (Grossi et al. 2007; Dalal et al. 2008;Cui, Baldi & Borgani 2012; Lombriser et al. 2013; Castorina et al.2014). Moreover, cosmological hydrodynamic simulations willhave to precise the impact of baryons on the shape of the massprofile, which has already been shown to be quite significant(Rudd, Zentner & Kravtsov 2008; Stanek, Rudd & Evrard 2009;Cui et al. 2012; Cui, Borgani & Murante 2014). ACKNOWLEDGMENTS
We thank L. Pozzetti for providing us with her estimates of thenumber densities of H α -emitting galaxies in advance of publica-tion. We acknowledge useful discussions with O. Cucciati, S. Far-rens, A. Iovino, S. Mei, and F. Villaescusa. We thank S. Andreon,M. Brodwin, G. De Lucia, S. Ettori, M. Girardi, T. Kitching, G. Ma-mon, J. Mohr, T. Reiprich for a careful reading of the manuscript.BS acknowledges financial support from MIUR PRIN2010-2011(J91J12000450001) and a grant from “Consorzio per la Fisica - Trieste”. BS and SB acknowledge financial support from the PRIN-MIUR 201278X4FL grant, from a PRIN-INAF / / / INAF n.I / / /
0. JW acknowledges supportfrom the Transregional Collaborative Research Centre TRR 33 -’The Dark Universe’. The authors acknowledge the
Euclid
Collab-oration, the European Space Agency and the support of a numberof agencies and institutes that have supported the development of
Euclid .A detailed complete list is available on the
Euclid web site(http: // REFERENCES
Abell G. O., Corwin, Jr. H. G., Olowin R. P., 1989, ApJS, 70, 1Albrecht A. et al., 2009, ArXiv e-prints, 0901.0721Albrecht A. et al., 2006, ArXiv e-prints, 0609591Allen S. W., Evrard A. E., Mantz A. B., 2011, ARAA, 49, 409Amendola L., et al., 2013, Living Rev.Rel., 16, 6Anderson L. et al., 2014, MNRAS, 441, 24Andreon S., Congdon P., 2014, A&A, 568, A23Andreon S., Hurn M. A., 2012, ArXiv e-prints,1210.6232Arnaud M., Pratt G. W., Pi ff aretti R., B¨ohringer H., Croston J. H.,Pointecouteau E., 2010, A&A, 517, A92Balogh M. L., Couch W. J., Smail I., Bower R. G., Glazebrook K.,2002, MNRAS, 335, 10Bartolo N., Komatsu E., Matarrese S., Riotto A., 2004, Phys.Rept., 402, 103Battye R. A., Weller J., 2003, Phys. Rev. D, 68, 083506Benson B. A. et al., 2013, ApJ, 763, 147Betoule M. et al., 2014, A&A, 568, A22Biviano A., Murante G., Borgani S., Diaferio A., Dolag K., Gi-rardi M., 2006, A&A, 456, 23Blas D., Lesgourgues J., Tram T., 2011, JCAP, 7, 34Bocquet S. et al., 2014, ArXiv e-prints,1407.2942Borgani S. et al., 2001, ApJ, 561, 13Boselli A. et al., 1997, A&A, 324, L13Brandbyge J., Hannestad S., Haugbølle T., Wong Y. Y. Y., 2010,JCAP, 9, 14Brodwin M., Gonzalez A. H., Moustakas L. A., Eisenhardt P. R.,Stanford S. A., Stern D., Brown M. J. I., 2007, ApJ, 671, L93Brodwin M. et al., 2013, ApJ, 779, 138Burenin R. A., Vikhlinin A. A., 2012, Astronomy Letters, 38, 347Capozziello S., de Laurentis M., 2011, Phys. Rept., 509, 167Carbone C., Fedeli C., Moscardini L., Cimatti A., 2012, JCAP, 3,23Carbone C., Verde L., Matarrese S., 2008a, ApJ, 684, L1Carbone C., Verde L., Matarrese S., 2008b, ApJ, 684, L1Carlstrom J. E. et al., 2011, PASP, 123, 568Castorina E., Sefusatti E., Sheth R. K., Villaescusa-Navarro F.,Viel M., 2014, JCAP, 2, 49 c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the Euclid
Galaxy Cluster Survey Chevallier M., Polarski D., 2001, International Journal of ModernPhysics D, 10, 213Ciliegi P. et al., 2005, A&A, 441, 879Clerc N., Sadibekova T., Pierre M., Pacaud F., Le F`evre J.-P.,Adami C., Altieri B., Valtchanov I., 2012, MNRAS, 423, 3561Costanzi M., Sartoris B., Viel M., Borgani S., 2014, JCAP, 10, 81Costanzi M., Sartoris B., Xia J.-Q., Biviano A., Borgani S., VielM., 2013a, JCAP, 6, 20Costanzi M., Villaescusa-Navarro F., Viel M., Xia J.-Q., BorganiS., Castorina E., Sefusatti E., 2013b, JCAP, 12, 12Cui W., Baldi M., Borgani S., 2012, MNRAS, 424, 993Cui W., Borgani S., Dolag K., Murante G., Tornatore L., 2012,MNRAS, 423, 2279Cui W., Borgani S., Murante G., 2014, MNRAS, 441, 1769Dalal N., Dor´e O., Huterer D., Shirokov A., 2008, Phys. Rev. D,77, 123514De Boni C., Ettori S., Dolag K., Moscardini L., 2013, MNRAS,428, 2921de Propris R., Eisenhardt P. R., Stanford S. A., Dickinson M.,1998, ApJ, 503, L45Desjacques V., Seljak U., 2010, Classical and Quantum Gravity,27, 124011Dodelson S., 2003, Modern cosmology. Academic Press, Amster-dam (NL)Elbaz D. et al., 2007, A&A, 468, 33Ettori S., 2013, MNRAS, 435, 1265Fedeli C., Carbone C., Moscardini L., Cimatti A., 2011, MNRAS,414, 1545Feldman H. A., Kaiser N., Peacock J. A., 1994, ApJ, 426, 23Geach J. E. et al., 2010, MNRAS, 402, 1330Giannantonio T., Porciani C., 2010, Phys. Rev. D, 81, 063530Giodini S., Lovisari L., Pointecouteau E., Ettori S., Reiprich T. H.,Hoekstra H., 2013, Space Science Review, 177, 247Grossi M., Dolag K., Branchini E., Matarrese S., Moscardini L.,2007, MNRAS, 382, 1261Grossi M., Verde L., Carbone C., Dolag K., Branchini E., IannuzziF., Matarrese S., Moscardini L., 2009, MNRAS, 398, 321Haiman Z., Mohr J. J., Holder G. P., 2001, in American Instituteof Physics Conference Series, Vol. 586, 20th Texas Symposiumon relativistic astrophysics, Wheeler J. C., Martel H., eds., pp.303–309Hinshaw G. et al., 2013, ApJS, 208, 19Hoekstra H., Mahdavi A., Babul A., Bildfell C., 2012, MNRAS,427, 1298Holder G., Haiman Z., Mohr J. J., 2001, ApJ, 560, L111Hu W., Sawicki I., 2007, Phys. Rev. D, 76, 064004H¨utsi G., 2010, MNRAS, 401, 2477Iglesias-P´aramo J., Boselli A., Cortese L., V´ılchez J. M., GavazziG., 2002, A&A, 384, 383Kaiser N., 1987, MNRAS, 227, 1Kitching T. D. et al., 2014, MNRAS, 442, 1326Kodama T., Arimoto N., 1997, A&A, 320, 41Kodama T., Balogh M. L., Smail I., Bower R. G., Nakata F., 2004,MNRAS, 354, 1103Koester B. P. et al., 2007, ApJ, 660, 239Komatsu E., Spergel D. N., 2001, Phys. Rev. D, 63, 063002Kravtsov A. V., Borgani S., 2012, ARAA, 50, 353Kravtsov A. V., Vikhlinin A., Nagai D., 2006, ApJ, 650, 128Lahav O., Lilje P. B., Primack J. R., Rees M. J., 1991, MNRAS,251, 128Laureijs R. et al., 2011, arXiv:1110.3193Lesgourgues J., Pastor S., 2006, Phys. Rept., 429, 307 Lima M., Hu W., 2005, Phys. Rev. D, 72, 043006Lin Y.-T., Mohr J. J., Gonzalez A. H., Stanford S. A., 2006, ApJ,650, L99Lin Y.-T., Mohr J. J., Stanford S. A., 2003, ApJ, 591, 749Linder E. V., 2003, Physical Review Letters, 90, 091301Linder E. V., 2005, Phys. Rev. D, 72, 043529Lombriser L., Li B., Koyama K., Zhao G.-B., 2013, Phys. Rev. D,87, 123511Lombriser L., Slosar A., Seljak U., Hu W., 2012, Phys. Rev. D,85, 124038LoVerde M., Miller A., Shandera S., Verde L., 2008, Journal ofCosmology and Astro-Particle Physics, 4, 14Mamon G. A., Biviano A., Bou´e G., 2013, MNRAS, 429, 3079Mana A., Giannantonio T., Weller J., Hoyle B., H¨utsi G., SartorisB., 2013, MNRAS, 434, 684Mancone C. L. et al., 2012, ApJ, 761, 141Mancone C. L., Gonzalez A. H., Brodwin M., Stanford S. A.,Eisenhardt P. R. M., Stern D., Jones C., 2010, ApJ, 720, 284Mannucci F., Basile F., Poggianti B. M., Cimatti A., Daddi E.,Pozzetti L., Vanzi L., 2001, MNRAS, 326, 745Mantz A. B., Allen S. W., Morris R. G., Rapetti D. A., Apple-gate D. E., Kelly P. L., von der Linden A., Schmidt R. W., 2014,MNRAS, 440, 2077Mantz A. B. et al., 2015, MNRAS, 446, 2205Marriage T. A. et al., 2011, ApJ, 737, 61Marulli F., Carbone C., Viel M., Moscardini L., Cimatti A., 2011,MNRAS, 418, 346Massara E., Villaescusa-Navarro F., Viel M., 2014, ArXiv e-printsMatarrese S., Verde L., 2008, ApJ, 677, L77Matarrese S., Verde L., Jimenez R., 2000, ApJ, 541, 10Mehrtens N. et al., 2012, MNRAS, 423, 1024Merloni A. et al., 2012, ArXiv e-prints,1209.3114Metcalfe N., Shanks T., Weilbacher P. M., McCracken H. J., FongR., Thompson D., 2006, MNRAS, 370, 1257Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493Pillepich A., Porciani C., Hahn O., 2010, MNRAS, 402, 191Pillepich A., Porciani C., Reiprich T. H., 2012, MNRAS, 422, 44Planck Collaboration et al., 2013, ArXiv e-printsPlanck Collaboration et al., 2014a, A&A, 571, A16Planck Collaboration et al., 2014b, A&A, 571, A20Planck Collaboration et al., 2014c, A&A, 571, A20Planck Collaboration et al., 2011, A&A, 536, A11Poggianti B. M., De Lucia G., Varela J., Aragon-Salamanca A.,Finn R., Desai V., von der Linden A., White S. D. M., 2010,MNRAS, 405, 995Popesso P. et al., 2012, A&A, 537, A58Postman M. et al., 2012, ApJS, 199, 25Postman M. et al., 2005, ApJ, 623, 721Press W. H., Schechter P., 1974, ApJ, 187, 425Ramella M., Boschin W., Geller M. J., Mahdavi A., Rines K.,2004, AJ, 128, 2022Rapetti D., Blake C., Allen S. W., Mantz A., Parkinson D., BeutlerF., 2013, MNRAS, 432, 973Reichert A., B¨ohringer H., Fassbender R., M¨uhlegger M., 2011,A&A, 535, A4Rines K., Geller M. J., 2008, AJ, 135, 1837Rosati P. et al., 2014, The Messenger, 158, 48Rozo E., Evrard A. E., Ryko ff E. S., Bartlett J. G., 2014, MNRAS,438, 62Rozo E., Ryko ff E., Koester B., Nord B., Wu H.-Y., Evrard A.,Wechsler R., 2011, ApJ, 740, 53Rozo E. et al., 2010, ApJ, 708, 645 c (cid:13) , 000–000 B. Sartoris et al.
Rudd D. H., Zentner A. R., Kravtsov A. V., 2008, ApJ, 672, 19Ryko ff E. S. et al., 2012, ApJ, 746, 178Salopek D. S., Bond J. R., 1990, Phys. Rev. D, 42, 3936Sartoris B., Borgani S., Fedeli C., Matarrese S., Moscardini L.,Rosati P., Weller J., 2010, MNRAS, 407, 2339Sartoris B., Borgani S., Rosati P., Weller J., 2012, MNRAS, 423,2503Schechter P., 1976, ApJ, 203, 297Schuecker P., B¨ohringer H., Collins C. A., Guzzo L., 2003, A&A,398, 867Shandera S., Mantz A., Rapetti D., Allen S. W., 2013, JCAP, 8, 4Smith G. P., Treu T., Ellis R. S., Moran S. M., Dressler A., 2005,ApJ, 620, 78Sobral D., Best P. N., Geach J. E., Smail I., Cirasuolo M., GarnT., Dalton G. B., Kurk J., 2010, MNRAS, 404, 1551Stanek R., Rudd D., Evrard A. E., 2009, MNRAS, 394, L11Staniszewski Z. et al., 2009, ApJ, 701, 32Stefanon M., Marchesini D., 2013, MNRAS, 429, 881Sunyaev R. A., Zeldovich Y. B., 1972, Comments on Astrophysicsand Space Physics, 4, 173Tegmark M., 1997, Physical Review Letters, 79, 3806Tinker J., Kravtsov A. V., Klypin A., Abazajian K., Warren M.,Yepes G., Gottl¨ober S., Holz D. E., 2008, ApJ, 688, 709Tinker J. L., Robertson B. E., Kravtsov A. V., Klypin A., WarrenM. S., Yepes G., Gottl¨ober S., 2010, ApJ, 724, 878Tran K.-V. H. et al., 2010, ApJ, 719, L126Umeda K. et al., 2004, ApJ, 601, 805Verde L., Matarrese S., 2009, ApJ, 706, L91Vikhlinin A. et al., 2009, ApJ, 692, 1060von der Linden A. et al., 2014, MNRAS, 439, 2Wang L., Steinhardt P. J., 1998, ApJ, 508, 483Wang Y., 2014, Communications in Theoretical Physics, 62, 109Watson W. A., Iliev I. T., D’Aloisio A., Knebe A., Shapiro P. R.,Yepes G., 2013, MNRAS, 433, 1230Weller J., Battye R. A., Kneissl R., 2002, Physical Review Letters,88, 231301Wetterich C., 2004, Physics Letters B, 594, 17Yang G. et al., 2014, ApJS, 215, 27Ziparo F. et al., 2014, MNRAS, 437, 458
APPENDIX A: THE
EUCLID
SPECTROSCOPIC SURVEY
We use a procedure similar to the one described in Section 2 to de-termine the number of spectroscopic cluster galaxies within r , c ,as a function of both cluster mass and redshift. Since the Euclid spectroscopic survey is flux-limited in the H α line, we consider thecluster H α LF. There are not many determinations of the clusterH α LF in the literature. We use the results of Iglesias-P´aramo et al.(2002, for two nearby clusters, z = . z = .
18 rich cluster), Umeda et al. (2004, for a z = .
25 cluster),and Kodama et al. (2004, for a z = . α LF is (at best) poorlyconstrained, hence we have to make several assumptions for itsthree parameters, the characteristic luminosity L ⋆ , the normalisa-tion φ ⋆ , and the faint-end slope α . We consider two possible evolu-tions. In the first, we assume the z -evolution of L ⋆ to be the sameas the one measured for the field H α LF, i.e. L ⋆ ∝ (1 + z ) . for z .
3, and no further evolution at higher redshift (Geach et al.2010). In the second, we allow L ⋆ to evolve at z > . z -dependence established at lower redshifts. The second sce-nario is based on the idea that the preferred sites for galaxy star- Figure A1.
Number of cluster galaxies with spectroscopic redshifts within r , c expected in the Euclid survey, as a function of redshift for clustersof di ff erent masses, log( M , c / M ⊙ ) = . , . , . , .
5, solid, dot-dashed, dashed, dotted lines, respectively. These numbers are for the caseof an evolving H α luminosity function also beyond z = .
3, i.e. they corre-spond to the solid blue curve in Fig. A2 (top panel).
Figure A2.
Selection function for the
Euclid spectroscopic survey. In thetop panel the solid blue curve indicates the selection function for clusterswith 5 galaxies with measured spectroscopic redshift within r , c . Thiscurve depends on the assumption that L ⋆ continues to evolve beyond z = . z . The dash-dotted curve depends instead on the assumption that thereis no further evolution of L ⋆ beyond z = .
3, consistently with what is ob-served for the field H α LF (Geach et al. 2010). The dashed red curve is anindependent estimate based on the the number densities of H α field galaxiesper redshift bin, estimated by Pozzetti et al. (in prep.). In the lower panelthe solid, dash-dotted and dashed lines show results for clusters with at least5, 10, and 20 galaxies, respectively, with measured spectroscopic redshiftwithin r , c , based on the assumption that L ⋆ continues to evolve beyond z = .
3. The dotted line is the selection function for the
Euclid photometricsurvey (from Fig. 2), shown as a reference.c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the
Euclid
Galaxy Cluster Survey formation tends to shift to higher-density regions at higher redshifts(Elbaz et al. 2007), even if the redshift at which this shift occurs isnot well constrained (Ziparo et al. 2014)The di ff erent cluster LFs we consider have been determinedfor di ff erent overdensities, ∆ . To evaluate the ∆ =
200 value of L ⋆ at z =
0, we perform a regression analysis between log [ L ⋆ / (1 + z c ) . ] and log ∆ . We find L ⋆ z = = . × erg s − . Similarly towhat we did in Section 2 for the K s LF, we assume φ ⋆ ∝ H ( z ).We then take the average of the φ ⋆ values obtained for the di ff erentclusters, after rescaling them for the factor 200 H / [ ∆ H ( z )], andfind φ ⋆ z = = . − . As for α , we fix it to the value − . α LF faint-end.We convert the H α luminosities into fluxes using f H α = L H α / (2 × π D , c ), where D L , c is the cluster luminosity distanceand the factor 1 / Euclid spectroscopic survey (3 × − erg s − cm − ), we fi-nally obtain the expected number density of galaxies within r , c .By multiplying the number density of galaxies within r , c by thevolume of the sphere of radius r , c , we obtain the number of galax-ies in a cluster with H α flux above the Euclid survey limit. Finally,we multiply this number by the expected completeness of the spec-troscopic survey, ∼
80 per cent.In Fig. A1 we show the resulting estimates of the number ofcluster galaxies with spectroscopic redshifts within r , c , as a func-tion of redshift for clusters of di ff erent masses, for the case of anevolving H α LF beyond z = .
3. Note that only the redshift range0 . − . α -linein the Euclid survey, according to the current design baseline .We also consider the following, independent estimate of thecluster selection function in the Euclid spectroscopic survey. Weuse Pozzetti et al.’s (in prep.) estimates of the number densitiesof H α -emitting field galaxies per square degree and redshift bin,that we convert to volume densities, n fd . To estimate the expectednumber density of H α -emitting galaxies in a cluster, we used n cl = n fd b ( z ) ∆ ρ c /ρ m , where ρ c is the critical density and ρ m themass density of the Universe at any given redshift, ∆ is the over-density we want to sample in the cluster, and b ( z ) is the redshift-dependent bias parameter that accounts for the di ff erent distribu-tion of H α galaxies and the underlying matter distribution. Taking ∆ = α galaxies in a cluster of mass M , c is N ( r , c ) = (4 π/ r , c n cl . We estimate the bias b ( z ) from thecomparison of the real-space correlation functions of matter andH α galaxies, b = ( r g / r m ) − γ/ , where γ is the slope of the correla-tion function. We use the correlation lengths of the di ff use matterin our adopted cosmology, and those of H α galaxies with lumi-nosities corresponding to the Euclid flux limit at any given redshift(taken from Sobral et al. 2010). We estimate b ( z = . = . b ( z = . = .
5, and interpolate b ( z ) between these two values atany redshift in the range 0.9–2.0.In Fig. A2, we show the limiting mass M , c of a cluster withat least N z galaxies with measured spectroscopic redshift within r , c as a function of the cluster redshift. This is the selection func-tion of clusters in the Euclid spectroscopic survey, in the sensethat N z concordant redshifts within a region of typical cluster size See the ”
Euclid
GC Interim Science Review” by Guzzo & Percival, athttp: // internal.euclid-ec.org / ?page id = Euclid
Consortium members. w a w NC+PSNC+PS+known SR scat evolNC+PS+known SR evolNC+PS+known SR-1-0.5 0 0.5 1-1.1 -1.05 -1 -0.95 -0.9
Figure B1.
Constraints at the 68 per cent c.l. in the w a − w parameterplane. We show forecasts for the N , c /σ field > Euclid photometric clus-ter survey obtained by (i) combining the FM information for number countsand power spectrum (NC + PS; blue dotted contour), (ii) same as (i) but as-suming perfect knowledge of the evolution of the scatter (see equation 27;orange dashed contour); (iii) same as (i) but assuming perfect knowledgeof the evolution of both the scatter and the bias (black solid contour); (iv)same as (i) but assuming perfect knowledge of all the four nuisance param-eters (green dash-dotted contour). The blue and green curves are the sameof Fig. 4,right panel. Note that the solid black and the green dashed ellipsesare almost coincident. (i.e., r , c ) are required to identify a cluster. The three di ff erent es-timates of the spectroscopic selection function for clusters in the Euclid survey are rather di ff erent, and this reflects the current sys-tematic uncertainties. From Fig. A2 (bottom panel), one can seethat the spectroscopic survey selection function is above the photo-metric survey selection function. Hence, it will prove less e ffi cientto search for clusters in the Euclid spectroscopic survey than in thephotometric survey. Data from the spectroscopic survey will still beuseful to confirm clusters detected in the photometric survey, thusimproving the reliability of the sample.
APPENDIX B: CLUSTER MASS CALIBRATION
The impact of nuisance parameters on cosmological constraintsfrom
Euclid photometric clusters is going to be quite significant.This is especially true for the parameters directly related to thegrowth of structure history like the matter power spectrum normal-isation σ , and for the CLP DE parameter w a , that is particularlysensitive to the level of knowledge of the scaling relation evolu-tion. In Fig. B1, we show how the cosmological constraints on theDE equation of state depend on our knowledge of the scaling re-lation. In particular, we show that strong constraints on the evolu-tion of the scatter and the mass bias, allow to greatly improve theconstraints on the DE EoS parameters. On the other hand, preciseknowledge of these parameters at z = w , w a planein the figure (solid black and dashed green ellipses). c (cid:13) , 000–000 B. Sartoris et al.
To maximise the scientific return of the
Euclid galaxy clus-ter catalogue, it is therefore very important to know the mass scal-ing relation in an as much as possible precise and unbiased way.There are two avenues to obtain this goal. The first one is to cross-correlate the
Euclid cluster sample with samples obtained at dif-ferent wavelengths by di ff erent surveys. For instance, by the time Euclid will fly, the eRosita full-sky X-ray cluster catalogue will beavailable, and will provide an important contribution to the clustertrue mass estimation. Other useful cluster catalogues will includethe SZ samples provided by the South-Pole Telescope (SPT), theAtacama Cosmology Telescope (ACT), and
Planck .The second avenue, that represents the strength of the
Euclid mission, consists in exploiting internal
Euclid data. Many photo-metrically selected clusters will appear as signal-to-noise peaks inthe
Euclid full-sky cosmic shear maps. This weak gravitationallensing signal will permit us to estimate the cluster masses with-out relying on assumptions about dynamical equilibrium. Althoughonly the more massive systems will permit individual mass mea-surements, we can nevertheless statistically calibrate the normali-sation of the cluster scaling relations down to the lowest masses inthe catalogue by stacking. An example is given in Fig. B2, show-ing the level of precision expected on the mean mass of stackedclusters.We first measure the mass of individual clusters with amatched filter, assuming that the mass density profile of all clus-ters follows an NFW profile. We then calculate the uncertainty onthe mean mass of the individual measurements in bins of mass( ∆ log M , c = .
2) and redshift ( ∆ z = . Euclid survey of 15,000 square degrees. The figureonly accounts for shape-noise, with σ = . M , c = × M ⊙ , 2 × M ⊙ , and 1 . × M ⊙ (from top to bottom) as a function of redshift. We do better on thelower mass systems because their larger number compensates fortheir lower individual signal-to-noise measurements. The figuredemonstrates that Euclid has the potential to calibrate the meanmass, and hence scaling relations, to 1% out to redshift unity, andto 10% out to z . . M , c = . × M ⊙ .At the same time, the spectroscopic part of the Euclid surveywill provide velocities for a few cluster members in each clusterdetected with photometric data. Stacking these velocities for manyclusters in bins of richness and redshift will allow a precise calibra-tion of the velocity dispersion vs. richness relation, and from thisof the mass-richness relation.In Fig. B3, we show the number of spectroscopic cluster mem-bers that will be available for stacks of clusters of given mass in binsof ∆ z = . ∆ log M , c = . M , c / M ⊙ = . , . , .
6. The curve for log M , c / M ⊙ = . z .
25 because of our choice of considering only clusters with N z >
5. Note that the curve for log M , c / M ⊙ = . z Figure B2.
Calibrating cluster masses with gravitational shear. The curvesshow the expected precision on the mean mass of clusters in bins of ∆ log M , c = . ∆ z = .
1, centred on masses (from top to bot-tom) of M , c = × M ⊙ (green curve), 2 × M ⊙ (red), and1 . × M ⊙ (blue). We assume a lensing survey of 15,000 sq. deg. ,the Tinker mass function in the base Λ CDM Planck-cosmology, and shapenoise with σ = . statistical noise in the velocity dispersion estimate of a sampleof ∼
500 cluster members is ∼ ∼
27 per cent statistical noise in the mass estimate. A similarfigure has been obtained by Mamon, Biviano & Bou´e (2013) whenusing the full velocity distribution to constrain cluster masses.The value of 500 is displayed in Fig. B3, and it shows that a veryprecise spectroscopic calibration of cluster masses will be possiblefor stacks of clusters with 14 . log M , c / M ⊙ . . z .
2, and even beyond that ( z . . M , c / M ⊙ ≃ .
4. Spectroscopiccalibration of cluster masses at higher redshifts will be feasiblewith reduced precision, but lack of statistics will hamper clustermass calibration at log M , c / M ⊙ < . Euclid survey will allow precise calibration of themass-observable relation out to z . .
6, using gravitational lensingand spectroscopy. The deep
Euclid survey will allow to extend thiscalibration to even higher redshifts, although with a much morelimited statistics on the number of clusters. Overall, by combin-ing
Euclid internal mass calibration with the cross correlation withexternal SZ and X-ray surveys, we should be able to significantlymitigate the degrading e ff ect of the nuisance parameters on cosmo-logical constraints. c (cid:13) , 000–000 ext Generation Cosmology: Constraints from the Euclid
Galaxy Cluster Survey Figure B3.
Calibrating cluster masses with spectroscopy. The curves showthe number of cluster galaxies with redshifts available in stacks of clustersin bins of ∆ log M = . ∆ z = .
1, as a function of redshift, for centralvalues of the mass bins of log M , c / M ⊙ = . , . , . M / M ⊙ = . z .
25. The dotted line shows the value of 500 galaxies as areference.c (cid:13)000