Issue 
A&A
Volume 665, September 2022



Article Number  A100  
Number of page(s)  8  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202243538  
Published online  14 September 2022 
AMICO galaxy clusters in KiDSDR3: Constraints on cosmological parameters and on the normalisation of the massrichness relation from clustering^{⋆}
^{1}
Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
email: giorgio.lesci2@unibo.it
^{2}
INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, 40129 Bologna, Italy
^{3}
Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth PO1 3FX, UK
^{4}
INFN – Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
^{5}
Dipartimento di Fisica, Università degli Studi Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy
^{6}
Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany
^{7}
ITP, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
^{8}
INAF – Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy
^{9}
Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, C.U. di Monte Sant’Angelo, via Cintia, 80126 Napoli, Italy
^{10}
INAF – Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131 Napoli, Italy
^{11}
INFN – Sezione di Napoli, via Cintia, 80126 Napoli, Italy
Received:
14
March
2022
Accepted:
24
June
2022
Aims. We analysed the clustering of a photometric sample of galaxy clusters selected from the Third Data Release of the KiloDegree Survey, focusing on the redshiftspace twopoint correlation function (2PCF). We compared our measurements to theoretical predictions of the standard Λ cold dark matter (ΛCDM) cosmological model.
Methods. We measured the 2PCF of the sample in the clustercentric radial range r ∈ [5, 80] h^{−1} Mpc, considering 4934 galaxy clusters with richness λ^{*} ≥ 15 in the redshift range z ∈ [0.1, 0.6]. A Markov chain Monte Carlo analysis has been performed to constrain the cosmological parameters Ω_{m}, σ_{8}, and S_{8} ≡ σ_{8}(Ω_{m}/0.3)^{0.5}, assuming Gaussian priors on the massrichness relation given by the posteriors obtained from a joint analysis of cluster counts and weak lensing. In addition, we constrained the normalisation of the massrichness relation, α, with fixed cosmological parameters.
Results. We obtained Ω_{m} = 0.28^{+0.05}_{−0.04}, σ_{8} = 0.82^{+0.14}_{−0.12}, and S_{8} = 0.80^{+0.08}_{−0.08}. The constraint on S_{8} is consistent within 1σ with the results from WMAP and Planck. Furthermore, by fixing the cosmological parameters to those provided by Planck, we obtained α = 0.12^{+0.06}_{−0.06}, which is fully consistent with the result obtained from the joint analysis of cluster counts and weak lensing performed for this sample.
Key words: galaxies: clusters: general / cosmology: observations / largescale structure of Universe / surveys
Some supplementary data are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/cat/J/A+A/665/A100
© G. F. Lesci et al. 2022
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
According to the standard Λcold dark matter (ΛCDM) cosmological model, galaxy clusters are the largest gravitationally bound systems in the Universe, lying in the highest peaks of the matter density field (Kaiser 1984). Through their abundances and clustering, these objects trace the statistical properties of the matter density field and its growth (Allen et al. 2011), providing stringent constraints on cosmological parameters. However, it is difficult to fully exploit the clustering properties of galaxy clusters because collecting large homogeneous cluster samples is a complex task. Surveys of galaxies, probing increasingly larger volumes of the Universe, have played a primary role in this field (see e.g., Guzzo et al. 2013; de Jong et al. 2015; Aihara et al. 2018). Ongoing and forthcoming wide extragalactic surveys, from the lowest to the highest frequencies, will provide complete and pure galaxy cluster samples up to high redshifts and down to low masses. These surveys include the Kilo Degree Survey^{1} (KiDS, de Jong et al. 2017; Kuijken et al. 2019), the Dark Energy Survey^{2} (The Dark Energy Survey Collaboration 2005; Abbott et al. 2021), the Vera C. Rubin Observatory LSST^{3} (LSST Dark Energy Science Collaboration 2012, 2021), and Euclid^{4} (Amendola et al. 2018; Euclid Collaboration 2019, 2020; Scaramella et al. 2022) in the optical and nearinfrared, the South Pole Telescope^{5} (Bayliss et al. 2016; Chown et al. 2018), the Atacama Cosmology Telescope^{6} (Naess et al. 2020; OrlowskiScherer et al. 2021), and the Simons Observatory^{7} (Ade et al. 2019; Xu et al. 2021) surveys at highradio frequencies, and eROSITA^{8} (Brunner et al. 2022; Liu et al. 2022) in Xrays.
Although it is observationally expensive to build up complete and pure samples of galaxy clusters covering wide enough ranges of masses and redshifts, there are numerous advantages in exploiting clusters as cosmic tracers. Galaxy clusters are more clustered than galaxies, with a clustering signal that is progressively stronger for richer systems (see e.g., Moscardini et al. 2001, and references therein). Furthermore, clusters are relatively unaffected by nonlinear dynamics at small scales, and the redshiftspace anisotropies on large comoving scales have a minor impact on the cluster clustering compared to galaxies because the bias is larger (see e.g., Kaiser 1987; Hamilton 1992). Having smaller theoretical uncertainties over the description of nonlinear dynamics and redshiftspace distortions, the modelling of the cluster clustering signal is potentially less affected by systematics than in the galaxy clustering case. Large galaxy cluster samples have been exploited to provide strong cosmological constraints from both secondorder and thirdorder statistics (see e.g., Estrada et al. 2009; Veropalumbo et al. 2014, 2016; Marulli et al. 2018, 2021; Moresco et al. 2021; Lindholm et al. 2021, and references therein). These constraints are even more robust when cluster clustering is combined with cluster number counts (e.g., Mana et al. 2013; Salvati et al. 2018).
Furthermore, measuring cluster masses gives the opportunity of predicting the effective bias of the cluster sample as a function of the cosmological model. It is thus crucial to accurately estimate the cluster masses, possibly with multiple independent probes, such as Xray emission or the SunyaevZel’dovich effect, and to calibrate massobservable scaling relations. In view of the ongoing and nextgeneration large photometric surveys, which explore the Universe in visible and nearinfrared wavelengths, it is decisive to estimate the cluster massobservable scaling relations involving quantities measured in these bands. Currently, the most reliable mass measurements are provided by weak gravitational lensing (e.g., Bardeau et al. 2007; Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al. 2015; Schrabback et al. 2018; Stern et al. 2019; Sereno et al. 2017; Giocoli et al. 2021; Ingoglia et al. 2022). Combining this information with the estimate of some optical cluster properties, such as the richness, it is possible to derive cluster massobservable scaling relations.
The goal of this paper is to present a cosmological analysis based on the monopole of the redshiftspace twopoint correlation function (2PCF), measured in the catalogue of galaxy clusters developed by Maturi et al. (2019). This catalogue has been built up through the use of the Adaptive Matched Identifier of Clustered Objects (AMICO) algorithm (Bellagamba et al. 2018) in the Third Data Release of the Kilo Degree Survey (KiDSDR3, de Jong et al. 2017). In this work, we analyse the AMICO KiDSDR3 catalogue in two redshift bins, assuming a flat ΛCDM cosmological model, including in the modelling the nonnegligible effects of the errors of photometric redshifts (photozs) on the 2PCF shape. In particular, our reference cosmological model is given by Planck Collaboration VI (2020, Table 2, TT, TE, and EE+lowE). Through this analysis, we constrain the matter density parameter, Ω_{m}, the amplitude of the density fluctuations, σ_{8}, and the cluster normalisation parameter, S_{8} ≡ σ_{8}(Ω_{m}/0.3)^{0.5}. Furthermore, by fixing the cosmological parameters, we infer the normalisation of the cluster massrichness scaling relation, finding results consistent with those derived from the joint analysis of cluster counts and weak lensing by Lesci et al. (2022) in KiDSDR3.
The statistical analyses presented in this paper are performed with the CosmoBolognaLib^{9} (CBL; Marulli et al. 2016), a set of free software C++/Python numerical libraries for cosmological calculations. Specifically, both the measurements and the statistical Bayesian analyses are performed with the CBL V5.3.
The paper is organised as follows. In Sect. 2 we present the AMICO KiDSDR3 cluster catalogue. In Sect. 3 we describe the methods used to measure and model the 2PCF of this sample. The results of our analysis are presented in Sect. 4, leading to our conclusions, which are discussed in Sect. 5.
2. Data
KiDS (de Jong et al. 2013) is an ESO publicimaging survey that will cover 1350 square degrees when it is completed, exploiting the OmegaCAM widefield camera (Kuijken 2011) on the Very Large Telescope (VLT) survey telescope (Capaccioli et al. 2012, VST) with a resolution of 0.21 arcsec pixel^{−1}. This work is based on the catalogue of galaxy clusters detected in KiDSDR3 (Maturi et al. 2019) by applying the AMICO algorithm (Bellagamba et al. 2018, discussed in Sect. 2.1). The KiDSDR3 catalogue provides the 2 arcsec aperture photometry in the bands u, g, r, i, as well as photometric redshifts for all the detected galaxies down to the 5σ magnitude limits of 24.3, 25.1, 24.9, and 23.8 for the aforementioned bands, respectively, over a total area of 438 deg^{2}. The effective area amounts to 377 deg^{2}, obtained by excluding the regions of the sky affected by satellite tracks and haloes produced by bright stars, or falling in the secondary or tertiary halo masks used for the weaklensing analysis (de Jong et al. 2015; Kuijken et al. 2015).
2.1. AMICO algorithm
AMICO (Bellagamba et al. 2018; Maturi et al. 2019) is an algorithm for the detection of galaxy clusters in photometric surveys, based on the optimal matchedfiltering technique (see Maturi et al. 2005, for a detailed discussion). Specifically, AMICO adopts an iterative approach for the extraction of cluster candidates from the observed galaxy distribution, aiming at maximising the signaltonoise ratio (S/N) of the overdensity detections, exploiting the statistical properties of both field galaxies and member galaxies, which are described by an arbitrary number of observed quantities.
In particular, the detection process adopted in this run of AMICO relies on angular coordinates, rband magnitudes, and photozs of galaxies. The model adopted to describe the clusters is a convolution of a Schechter luminosity function (Schechter 1976) and a NavarroFrenkWhite profile (Navarro et al. 1997). Differently from other clusterdetection algorithms based on the socalled red sequence, colours are not considered by AMICO. This has been done to minimise the dependence of the cluster sample on the colour properties of the cluster members. Therefore, AMICO is expected to be effective also at higher redshifts, where the red sequence may be not prominent yet. AMICO has been applied on realistic mock catalogues reproducing the expected characteristics of the future Euclid photometric survey, achieving remarkable purity and completeness levels compared to other algorithms (Euclid Collaboration 2019). AMICO is one of the two algorithms for cluster identification officially adopted by the Euclid mission.
2.2. AMICO KiDSDR3 catalogue
We analysed the clustering properties of the AMICO KiDSDR3 cluster catalogue (Maturi et al. 2019), which contains 7988 galaxy clusters down to S/N = 3.5 that lie in the redshift range z ∈ [0.10, 0.80]. We accounted for the bias described in Maturi et al. (2019) that affects cluster redshifts, that is, we defined the corrected redshift as z_{corr} = z − 0.02 (1 + z). This bias corresponds to what was found in de Jong et al. (2017) by comparing the KiDS photozs to the GAMA spectroscopic redshifts (see their Table 8). In a small redshift range around z ∼ 0.32, the photoz errors are higher and harder to model because of the shape of the g and r filter transmissions. In the following cosmological analysis (Sect. 4), we also considered the photoz range z ∈ [0.10, 0.60] because we assumed priors on the massrichness relation estimated from a weaklensing analysis performed in this redshift range (see Bellagamba et al. 2019). Consequently, we decided to model the 2PCF in two separate redshift bins: z ∈ [0.10, 0.30] and z ∈ [0.35, 0.60].
In the analysis, we used the intrinsic richness, λ^{*}, as mass proxy. It is defined as
where P_{i}(j) is the probability assigned by AMICO to the ith galaxy of being a member of a given detection j (see Maturi et al. 2019). The intrinsic richness thus represents the sum of the membership probabilities, that is, the weighted number of visible galaxies belonging to a detection, under the conditions given by Eq. (1). The sum of the membership probabilities is an excellent estimator of the true number of member galaxies, as shown in Bellagamba et al. (2018) by running the AMICO algorithm on mock catalogues (see Fig. 8 in the reference). In particular, in Eq. (1), z_{j} is the redshift of the jth detected cluster, m_{i} is the magnitude of the ith galaxy, and R_{i} corresponds to the distance of the ith galaxy from the centre of the cluster. The parameter R_{max}(z_{j}) represents the radius enclosing a mass M_{200} = 10^{14}h^{−1} M_{⊙}, such that the corresponding mean density is 200 times the critical density of the Universe at the given redshift z_{j}. Lastly, m^{*} is the typical magnitude of the Schechter function in the cluster model assumed in the AMICO algorithm. We use the term ’intrinsic richness’ as opposed to ’apparent richness’, both defined in Maturi et al. (2019). In particular, because the threshold in magnitude is always brighter than the survey limit thanks to the redshift dependence of m^{*}, there is no dependence of λ^{*} on the survey limit. Conversely, the apparent richness is a quantity that includes all visible galaxies and therefore is redshift dependent.
We considered only the clusters with λ^{*} > 15, which ensures a purity higher than 97% over the whole sample (see Maturi et al. 2019). Consequently, the sample consists of 1019 clusters for z ∈ [0.10, 0.30] and 3915 clusters for z ∈ [0.35, 0.60]. Fig. 1 shows the redshift and richness distributions of the objects considered in the analysis.
Fig. 1. Redshift and intrinsic richness distributions of the clusters in the sample. Top panel: redshift distribution after the correction for the redshift bias, including only objects with λ^{*} > 15. The grey shaded areas represent the redshift ranges that were not used in the analysis. Bottom panel:λ^{*} distribution of the objects in the redshift ranges z ∈ [0.10, 0.30] (black hatched histogram) and z ∈ [0.35, 0.60] (blue histogram). 
3. Method
3.1. Twopoint correlation function estimator
We estimated the redshiftspace 2PCF with the Landy & Szalay (1993) (LS) estimator,
where DD(r), RR(r), and DR(r) are the number of datadata, randomrandom, and datarandom pairs with separation r ± Δr, respectively, while N_{DD}, N_{RR}, and N_{DR} are the total number of datadata, randomrandom, and datarandom pairs, respectively. This estimator is widely exploited in clustering studies as it is unbiased and with minimum variance for an infinitely large random sample and when ξ≪1 (Hamilton 1992; Kerscher et al. 2000; Labatie et al. 2012; Keihänen et al. 2019).
In order to build up the random catalogue, we extracted random (RA, Dec) cluster positions within the survey tiles by accounting for the same masks as were used by Maturi et al. (2019) for the construction of the cluster catalogue. Regarding the redshifts, we shuffled the observed photozs. Namely, to each object of the random sample, we assigned the photoz randomly extracted from the AMICO KiDSDR3 catalogue. Hence, the random sample has the same redshift distribution as the real sample by construction. We verified that the observed cluster distribution does not depend on the sky position, which is expected as Maturi et al. (2019) imposed the strict magnitude cut at r = 24, corresponding to the depth of the shallowest tile. The generated random catalogue is 30 times larger than the AMICO KiDSDR3 cluster sample to limit shot noise effects. The observed coordinates were then converted into comoving ones by assuming the ΛCDM model, with cosmological parameters from Planck18 (Planck Collaboration VI 2020).
3.2. Twopoint correlation function model
The observed redshift, z_{ob}, can be expressed as
where z_{c} is the cosmological redshift, σ_{z} is the error on the redshift measurements, and v_{∥} is the component of the peculiar velocity along the line of sight. Therefore, using z_{ob} to estimate the comoving distance creates distortions in the measures of the 2PCF, not only because of the error on the measurements (σ_{z}), but also because z_{ob} encodes information on the peculiar motions along the line of sight (v_{∥}). The peculiar motions cause the socalled dynamical distortions, an effect commonly known also as redshiftspace distortions (RSD).
Since our whole analysis is performed on a catalogue extracted from photometric data, it is crucial to take the errors on the observed cluster redshifts into account. Following the approach presented in Sereno et al. (2015), we modelled the redshiftspace 2D power spectrum as follows:
where P_{DM}(k) is the dark matter power spectrum, , with k_{∥} and k_{⊥} being the wavevector components parallel and perpendicular to the line of sight, respectively, μ ≡ k_{∥}/k, b_{eff} represents the effective bias factor (see Sect. 3.3), f is the growth rate, and the fμ^{2} term parametrises the coherent motions of largescale structure, enhancing the clustering signal at all scales (Kaiser 1987). The exponential cutoff term describes the random perturbations affecting redshift measurements, caused by both nonlinear stochastic motions and redshift errors. This is a Gaussian damping term, which causes a scaledependent effect by removing the signal over a typical scale k ∼ 1/σ, where σ represents the displacement along the line of sight due to random perturbations of cosmological redshifts, defined as
where H(z_{m}) is the Hubble function computed at the mean redshift of the cluster distribution in the bin, z_{m}, and σ_{z} is the typical cluster redshift error, expressed as
where σ_{z, 0} is a free parameter in the analysis. We derived σ_{z, 0} from the mock catalogue described in Maturi et al. (2019). In particular, we measured the conditional probability P(z_{ob}z_{tr}), where z_{ob} and z_{tr} are the observed and true redshifts, respectively, in several bins of z_{tr}, namely Δz_{tr}. We described the standard deviation of P(z_{ob}z_{tr}) through Eq. (6), where z_{m} is the mean value of z_{tr} within Δz_{tr}. Given the input galaxy photozs, AMICO provides unbiased estimates of redshift (see Maturi et al. 2019). Then we performed a statistical MCMC analysis assuming a common flat prior on σ_{z, 0} in all the Δz_{tr} bins, obtaining σ_{z, 0} = 0.02 with an uncertainty of ∼2 × 10^{−4}, namely equal to ∼1%.
To derive the monopole of the correlation function, we integrated Eq. (4) over μ and computed the inverse Fourier transform of the result. The solution can be written as a function of b_{eff} as follows:
where ξ′(s) is the inverse Fourier transform of the monopole P′(k), that is,
and ξ″(s) and ξ‴(s) are the inverse Fourier transform of P″(k) and P‴(k), respectively,
We neglected geometric distortions, which appear when a fiducial cosmology is assumed (in our case, Planck Collaboration VI 2020, Table 2, TT, TE, and EE+lowE) to convert observed coordinates into physical ones, because their effect is negligible with respect to dynamic distortions and photoz errors (see Marulli et al. 2012).
3.3. Effective bias and massrichness relation
The cosmological model of structure formation and evolution predicts that the dark matter halo bias, b, primarily depends on halo mass and redshift. Specifically, at a fixed redshift, the bias increases with the tracer mass, while for a given mass, the bias is an increasing function of the redshift (e.g., Sheth & Tormen 1999). We derived the effective bias in the ith bin of redshift, namely Δz_{i}, as
where N_{i} is the number of clusters in the ith redshift bin, j is the cluster index, and b is the halo bias, for which the model by Tinker et al. (2010) is assumed. As discussed in Sect. 4.1, the results do not significantly change when the halo bias model by Sheth et al. (2001) is assumed. In addition, P(Mλ^{*}, z) is a lognormal distribution whose mean is given by the massrichness scaling relation and the standard deviation (rms) is given by the intrinsic scatter, σ_{intr}, set as a free parameter of the model,
where
where E(z)≡H(z)/H_{0}, and we set and z_{piv} = 0.35 following Bellagamba et al. (2019). In addition, the intrinsic scatter is expressed as
In Eq. (10), P(zz_{ob, j}) and are Gaussian distributions, whose mean is the jth cluster’s observed redshift, z_{ob, j}, and richness, , respectively. The rms of P(zz_{ob, j}) is expressed as σ_{z, 0}(1 + z_{ob, j}), where σ_{z, 0} was derived from the mock catalogue developed by Maturi et al. (2019) as described in Sect. 3.2. Analogously, we derived an uncertainty on λ^{*} amounting to ∼17% from the mock catalogue.
3.4. Likelihood
For the cosmological Bayesian analysis performed in this work, a standard Gaussian likelihood was considered,
with
where N is the number of comoving separation bins in which the 2PCF is computed, d and m indicate data and model, respectively, and is the inverse of the covariance matrix. The covariance matrix measures the variance and the correlation between the different bins of the 2PCF. It is estimated from the data with the jackknife technique (Norberg et al. 2009),
where is the value of the correlation function at the ith bin for the kth subsample, is the mean value of the subsample, and N_{sub} is the number of resamplings of our cluster catalogue. In particular, the survey tiles are set as the subsample regions for the jackknife.
4. Results
Based on the methods outlined in Sect. 3, we performed a cosmological analysis of the redshiftspace 2PCF of the AMICO KiDSDR3 cluster sample. Our analysis was based on two fully independent redshift bins, z ∈ [0.10, 0.30] and z ∈ [0.35, 0.60], and we considered galaxy clusters with λ^{*} > 15. The 2PCF of this cluster sample was estimated in the spatial range s ∈ [5, 80] h^{−1} Mpc. At larger scales, the clustering signal starts to be weak and is dominated by the errors, while at smaller scales, the signal is negligible as the cluster size sets the minimum cluster separation. In Sect. 4.1 we present the clustering measurements and cosmological analysis. The aim of this analysis is to constrain the matter density parameter, Ω_{m}, the square root of the mass variance computed on a scale of 8 h^{−1} Mpc, σ_{8}, and the cluster normalisation parameter, S_{8} ≡ σ_{8}(Ω_{m}/0.3)^{0.5}, by assuming Gaussian priors for the parameters of the massrichness relation. In addition, as described in Sect. 4.2, we investigated a method to infer the normalisation of the cluster massobservable relation from cluster clustering measurements.
4.1. Constraints on cosmological parameters
We exploited the methods described in Sect. 3 to constrain the main parameters of the ΛCDM model. We assumed large flat priors for σ_{8} and Ω_{m}, while for the parameters of the massrichness relation (Eqs. (12) and (13)), α, β, γ, σ_{intr, 0}, and σ_{intr, λ*}, we considered Gaussian priors with the mean and standard deviation given by the posteriors derived from the joint analysis of cluster counts and weak lensing performed by Lesci et al. (2022). We also assumed a Gaussian prior on σ_{z, 0}, entering the damping factor of P(k) accounting for the uncertainties on the photozs (Eqs. (5) and (6)), with the mean equal to 0.02 and the standard deviation equal to 2 × 10^{−4} (see Sect. 3.2). Lastly, we assumed Gaussian priors for the baryon density, Ω_{b}, the primordial spectral index, n_{s}, and the normalised Hubble constant, h, assuming the same mean values derived by Planck Collaboration VI (2020, Table 2, TT, TE, and EE+lowE). With regard to the standard deviation of such priors, for Ω_{b} and n_{s} we imposed a standard deviation equal to five times the 1σ error derived by Planck, while for h we assumed a standard deviation equal to 0.1. We verified that the results do not significantly change if we instead use the 1σ errors derived from Planck as the standard deviations for the priors on Ω_{b}, n_{s,} and h. In Table 1 we show the priors and the posteriors of the free parameters of the model. In Fig. 2 we compare our 2PCF measurements in the two selected redshift ranges to the bestfit model. The statistical analysis was performed by assuming a standard Gaussian likelihood, defined in Eq. (14).
Fig. 2. Redshiftspace 2PCF (black dots) of the AMICO KiDSDR3 clusters in the spatial range s ∈ [5, 80] h^{−1} Mpc, and redshift ranges z ∈ [0.10, 0.30] (top panel) and z ∈ [0.35, 0.60] (bottom panel). In both panels, the grey band represents the 68% confidence level derived from the multivariate posterior of the free parameters considered in the cosmological analysis, described in Sect. 4.1. The model computed by assuming the cosmological parameters derived by Planck (Planck Collaboration VI 2020, Table 2, TT, TE, and EE+lowE; blue lines) and WMAP (Hinshaw et al. 2013, Table 3, WMAPonly Nineyear; red lines) is represented by the solid blue lines and by the dashed red lines, respectively. In both cases, the median values of the scaling relation parameters derived by Lesci et al. (2022) are assumed. 
Parameters considered in the cosmological analysis.
We obtained , , and S_{8}=, where we quote the median and 16th and 84th percentiles, as shown in Fig. 3 and Table 1. These constraints agree within 1σ with the WMAP (Hinshaw et al. 2013, Table 3, WMAPonly Nineyear) and Planck results (Planck Collaboration VI 2020, Table 2, TT, TE, and EE+lowE). In addition, Fig. 4 shows that our constraint on S_{8} agrees within 1σ with the results obtained from the joint analysis of cluster counts and weak lensing performed in KiDSDR3 by Lesci et al. (2022). The agreement within 1σ also holds for the cluster count analyses performed by Costanzi et al. (2019), based on SDSSDR8 data, and by Bocquet et al. (2019), based on the 2500 deg^{2} SPTSZ survey data, as well as for the results derived from the cosmic shear analyses performed by Amon et al. (2022) and Secco et al. (2022) on DESY3 data, Hikage et al. (2019) on HSCY1 data, and Asgari et al. (2021) on KiDSDR4 data. In addition, our result on S_{8} agrees with the constraint by Lindholm et al. (2021), namely , derived from the autocorrelation of Xray selected CODEX clusters. We also performed the analysis by assuming the halo bias model by Sheth et al. (2001), obtaining . This result agrees well with the result derived from the previously described analysis.
Fig. 3. Constraints obtained from the cosmological analysis, compared to WMAP and Planck results. In the left panel, we show the 68% and 95% confidence levels in the Ω_{m} − σ_{8} parameter space, along with the 1D marginalised posteriors with the relative intervals between the 16th and 84th percentiles, in the case of the cluster clustering analysis of the AMICO KiDSDR3 catalogue (grey lines). In the same panel, we also display the results from WMAP (Hinshaw et al. 2013, Table 3, WMAPonly Nineyear; red lines) and Planck (Planck Collaboration VI 2020, Table 2, TT, TE, and EE+lowE; blue lines). In the right panel, we show the posteriors for the parameter S_{8}, where the bands show the intervals between 16th and 84th percentiles. The colours are the same as in the left panel. 
4.2. Massobservable scaling relation
Cluster clustering might provide robust constraints on the normalisation of the massobservable relation, α, if a cosmological model is assumed (see e.g., Chiu et al. 2020). Based on the 2PCF measures used in the cosmological analysis detailed in the previous section, we also performed the analysis by assuming a flat prior on α and Gaussian priors on β, γ, σ_{intr, 0}, and σ_{intr, λ*}, given by the posteriors derived by Lesci et al. (2022), and the same prior on σ_{z, 0} assumed in Sect. 4.1. In addition, we fixed the cosmological parameters to the values derived by Planck Collaboration VI (2020, Table 2, TT, TE, and EE+lowE). We obtained , which agrees within 1σ with the result obtained by Lesci et al. (2022), as shown in Fig. 5. It should be noted that the median value of α corresponds within 2σ to the prior considered in Sect. 4.1. This is due to the adopted prior on S_{8}, whose value is higher than what was predicted by the clustering signal (see Fig. 4). A higher value of α is required to compensate for a higher S_{8}.
Fig. 4. Comparison of the constraints on S_{8} ≡ σ_{8}(Ω_{m}/0.3)^{0.5} obtained, from top to bottom, from the cluster clustering analysis in the AMICO KiDSDR3 catalogue (black dot), from the joint analysis of cluster counts and weak lensing in KiDSDR3 performed by Lesci et al. (2022, pink dot), from the results obtained by Planck Collaboration VI (2020, blue dots), Hinshaw et al. (2013, red dots), Costanzi et al. (2019, green dot), Bocquet et al. (2019, brown dot), Amon et al. (2022) and Secco et al. (2022, magenta dot), Hikage et al. (2019, orange dot), and Asgari et al. (2021, cyan dot). The median and the 16th and 84th percentiles are shown. 
Fig. 5. Constraint on the normalisation of the massrichness relation, α. The result obtained from the cluster clustering analysis is shown in grey, and the constraint derived from the joint analysis of counts and weak lensing (Lesci et al. 2022) is shown in blue. The bands show the intervals between the 16th and 84th percentiles. 
5. Conclusions
We presented a study of the clustering properties of a photometric sample of galaxy clusters detected by applying the AMICO algorithm on KiDSDR3 data. The catalogue consists of 4934 clusters in the redshift bins z ∈ [0.1, 0.3], z ∈ [0.35, 0.6], with intrinsic richness λ^{*} > 15. We measured the monopole of the 2PCF and performed a cosmological statistical analysis. The clustering model considered includes a damping of the power spectrum to account for the uncertainties on the photozs. In addition, we derived the effective bias in each redshift bin (Eq. (10)) from the massrichness scaling relation (Eq. (12)).
We performed a cosmological analysis by assuming flat priors on Ω_{m} and σ_{8}, and Gaussian priors on the parameters of the massrichness relation given by the constraints derived by Lesci et al. (2022), from the joint analysis of cluster counts and weak lensing in KiDSDR3. In addition, we marginalised our posteriors over the other cosmological parameters and over the damping of the power spectrum caused by the uncertainties on photozs. From this modelling, we derived , , and , which are consistent within 1σ with the results obtained from CMB experiments and from the most recent analyses of the evolved Universe. In addition, by fixing the cosmological parameters to the values derived by Planck Collaboration VI (2020, Table 2, TT, TE, and EE+lowE) and assuming Gaussian priors on the parameters β and γ of the massrichness relation, we derived a robust constraint on the normalisation α. In particular, we obtained , which agrees within 1σ with the result obtained by Lesci et al. (2022).
We expect more accurate constraints on the cosmological parameters and on the massrichness scaling relation from the analysis of the latest KiDSDR4 (Kuijken et al. 2019), which covers an area of ∼1000 square degrees (more than twothirds of the final KiDS area and a factor ∼2.5 larger than DR3). KiDSDR4 photometry also extends to the nearinfrared (ugriZYJHK_{s}), joining the data from KiDS and VIKING (Edge et al. 2013) surveys, which will lead to a better estimate of the photometric redshifts (Wright et al. 2019): this will be of great help for the application of the AMICO code (Maturi et al. in prep.), and will lead to the joint analysis of cluster weak lensing, counts, and clustering (Lesci et al., in prep.).
Acknowledgments
Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A3016, 177.A3017 and 177.A3018, and on data products produced by Target/OmegaCEN, INAFOACN, INAFOAPD and the KiDS production team, on behalf of the KiDS consortium. The authors acknowledge the use of computational resources from the parallel computing cluster of the Open Physics Hub (https://site.unibo.it/openphysicshub/en) at the Physics and Astronomy Department in Bologna. F.M. and L.M. acknowledge the grants ASI n.I/023/12/0 and ASI n.201823HH.0. L.M. acknowledges support from the grant PRINMIUR 2017 WSCC32. M.S. acknowledges financial contribution from contracts ASIINAF n.201714H.0 and INAF mainstream project 1.05.01.86.10. We thank Shahab Joudaki for his valuable advice.
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All Tables
All Figures
Fig. 1. Redshift and intrinsic richness distributions of the clusters in the sample. Top panel: redshift distribution after the correction for the redshift bias, including only objects with λ^{*} > 15. The grey shaded areas represent the redshift ranges that were not used in the analysis. Bottom panel:λ^{*} distribution of the objects in the redshift ranges z ∈ [0.10, 0.30] (black hatched histogram) and z ∈ [0.35, 0.60] (blue histogram). 

In the text 
Fig. 2. Redshiftspace 2PCF (black dots) of the AMICO KiDSDR3 clusters in the spatial range s ∈ [5, 80] h^{−1} Mpc, and redshift ranges z ∈ [0.10, 0.30] (top panel) and z ∈ [0.35, 0.60] (bottom panel). In both panels, the grey band represents the 68% confidence level derived from the multivariate posterior of the free parameters considered in the cosmological analysis, described in Sect. 4.1. The model computed by assuming the cosmological parameters derived by Planck (Planck Collaboration VI 2020, Table 2, TT, TE, and EE+lowE; blue lines) and WMAP (Hinshaw et al. 2013, Table 3, WMAPonly Nineyear; red lines) is represented by the solid blue lines and by the dashed red lines, respectively. In both cases, the median values of the scaling relation parameters derived by Lesci et al. (2022) are assumed. 

In the text 
Fig. 3. Constraints obtained from the cosmological analysis, compared to WMAP and Planck results. In the left panel, we show the 68% and 95% confidence levels in the Ω_{m} − σ_{8} parameter space, along with the 1D marginalised posteriors with the relative intervals between the 16th and 84th percentiles, in the case of the cluster clustering analysis of the AMICO KiDSDR3 catalogue (grey lines). In the same panel, we also display the results from WMAP (Hinshaw et al. 2013, Table 3, WMAPonly Nineyear; red lines) and Planck (Planck Collaboration VI 2020, Table 2, TT, TE, and EE+lowE; blue lines). In the right panel, we show the posteriors for the parameter S_{8}, where the bands show the intervals between 16th and 84th percentiles. The colours are the same as in the left panel. 

In the text 
Fig. 4. Comparison of the constraints on S_{8} ≡ σ_{8}(Ω_{m}/0.3)^{0.5} obtained, from top to bottom, from the cluster clustering analysis in the AMICO KiDSDR3 catalogue (black dot), from the joint analysis of cluster counts and weak lensing in KiDSDR3 performed by Lesci et al. (2022, pink dot), from the results obtained by Planck Collaboration VI (2020, blue dots), Hinshaw et al. (2013, red dots), Costanzi et al. (2019, green dot), Bocquet et al. (2019, brown dot), Amon et al. (2022) and Secco et al. (2022, magenta dot), Hikage et al. (2019, orange dot), and Asgari et al. (2021, cyan dot). The median and the 16th and 84th percentiles are shown. 

In the text 
Fig. 5. Constraint on the normalisation of the massrichness relation, α. The result obtained from the cluster clustering analysis is shown in grey, and the constraint derived from the joint analysis of counts and weak lensing (Lesci et al. 2022) is shown in blue. The bands show the intervals between the 16th and 84th percentiles. 

In the text 
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