A NEW METHOD TO SEARCH FOR HIGH-REDSHIFT CLUSTERS USING PHOTOMETRIC REDSHIFTS

In this section we report a sequential list of logical statements that clarify the theory the PPM procedure is based on. We refer to the Appendix for the proofs.

• 1.  
  Since high photometric redshift uncertainties affect any high-z cluster search, the redshifts and projected coordinates of the galaxies are considered separately. The field is tessellated with concentric regions of equal area centered at the projected coordinates of the beacon, i.e., the radio galaxy in our case (see Section A.1).
• 2.  
  Sources with photometric redshifts within the redshift bin Δz centered at the redshift centroid z_centroid are selected. The values of Δz and z_centroid densely span the ranges of our interest (see Section A.2).
• 3.  
  The probability of the null hypothesis (i.e., no clustering) is calculated for each region, given the values of Δz and z_centroid (see Section A.4).
• 4.  
  Starting from the projected coordinates of the beacon, the first consecutive regions for which the null hypothesis is rejected at a level ⩾70% are selected. Then, the regions are merged to form a new one (see Section A.4). This procedure aims at selecting the region in the projected space where the overdensity is present.
• 5.  
  The probability $1-\mathcal {P}$ of the null hypothesis is calculated for the new region, for each value of Δz and z_centroid. The null hypothesis is rejected with a probability $\mathcal {P}$. Photometric redshift uncertainties are implicitly neglected (see Sections A.3 and A.4).
• 6.  
  Fluctuations of $\mathcal {P}$ on scales δΔz and δz_centroid smaller than the typical statistical photometric redshift uncertainties are not physical and ultimately due to noise. They are locally removed by convolving $\mathcal {P}$ with a Gaussian filter, i.e., $\overline{\mathcal {P}} = \mathcal {W}\ast \mathcal {P}$.⁹ $\overline{\mathcal {P}}$ is an effective mean field defined on the space of (z_centroid; Δz); see Section A.5.
• 7.  
  We apply a variational approach and show that the filter $\mathcal {W}$ simultaneously suppresses (in linear approximation) the variations of $\mathcal {P}$ both in the space of (z_centroid; Δz) and in the ensemble of all the possible redshift realizations of the galaxies in the field (see Section A.5). The variations of $\mathcal {P}$ in the ensemble originated from the fact that photometric redshift uncertainties are neglected when calculating $\mathcal {P}$ (see Section A.3).
• 8.  
  $\overline{\mathcal {P}}$ is a good estimate for (1) the probability that the null hypothesis is rejected, where photometric redshift uncertainties are not neglected and (2) the significance of the number count excess in the field. In fact, the significance of the number count excess is decreased by the filtering procedure to take the additional variance due to the photometric redshift uncertainties into account (see Sections A.3 and A.5).
• 9.  
  We conservatively fix a 2σ wide redshift bin Δz = 0.28 and we apply the peak finding algorithm we developed for our discrete case. Such a procedure belongs to a more general context known as Morse theory (see Section A.6). The algorithm allows us to select the cluster candidates in the field within the redshift range of our interest and, for each overdensity, it provides (1) an estimate for its redshift, (2) an estimate for the cluster core size, and (3) a rough estimate for the cluster richness (see Section A.7).
• 10.  
  The association of the cluster candidates detected in the field with the beacon (i.e., in our case the radio galaxy) is performed by using the cluster redshift estimate and the redshift of the radio galaxy, as well as the corresponding uncertainties (see Section A.8).
• 11.  
  A generalization of the method to other data sets and surveys is provided in Section A.10.

• 1.  
  In Figure 3 we show the PPM plots (as in Figure 1, see Section 3) for F062 (left) and F126 (right). We adopt the following color code: ⩾2σ, ⩾3σ, and ⩾4σ points are plotted in cyan, blue, and red, respectively. The abscissa of the vertical solid line indicates the redshift of the cluster candidate.In Table 1 we report the PPM results for F062 and F126 (top part). We also report the PPM results of our simulations, when these two clusters are added to the fields of COSMOS-FRI 66 and COSMOS-FRI 70 of the Chiaberge et al. (2009) sample (middle part). In the bottom part of the table, we report the PPM results where the clusters are shifted to z = 1.5. In the first four columns we list the cluster ID number (i.e., F062, F126), the cluster redshift, the cluster redshift as estimated by the PPM, and the cluster detection significance. In the fifth column we report the distance r_max from the location of the radio galaxy in the projected space at which the overdensity formally ends. For the above quantity, the average, the rms dispersion around the average, and the median value (inside parentheses) in units of arcsec are reported, as estimated by the PPM procedure. The rms dispersion and the median value are not reported where the former is null, i.e., where the estimated r_max is maximally stable with respect to z_centroid, i.e., where the rms dispersion is null. In the sixth column we report the field to which the cluster is added; 66 and 70 denote that the cluster members are added to the fields of COSMOS FR I 66 and COSMOS FR I 70, respectively. The symbol "—" denotes that the PPM is applied to the fields of F062 and F126, where the cluster members are not subtracted.Concerning the PPM results for F062 and F126 (top part), they are detected with significance levels of 3.8σ and 4.3σ, respectively. The estimated redshifts are z = 0.86 and z = 0.96, respectively. In addition to the cluster candidate at z ∼ 1, the PPM detects another 2.7σ overdensity in the field of F126 at z = 0.64. This is a clear example of a projection effect.Note that our redshift estimates fully agree with the actual redshifts of the two cluster candidates (i.e., z = 0.88 and z = 1.0 for F062 and F126, respectively) and that the PPM effectively finds systems whose masses are compatible to those of rich groups and, therefore, they are even below the typical cluster mass cutoff ∼1 × 10¹⁴ M_☉, as is the case with F062 and F126. Hereinafter we do not estimate the redshift uncertainties following the PPM procedure prescription. This is mainly because we know the redshift of the cluster in our simulations. Therefore, we can directly compare our estimates with the original cluster redshifts to derive the statistical uncertainties. Conversely, in Paper II we estimate redshift uncertainties (following the procedure described above) for the overdensities we find within the C09 sample.
• 2.  
  We then apply the PPM using increasing offsets θ_off between the cluster center of the PPM tessellation and the actual center as measured from the X-ray emission. This is done to find the required accuracy in the coordinates of the cluster center in order to detect megaparsec-scale overdensities with the PPM. We keep the right ascension of the center of the PPM tessellation fixed and we change its declination from θ_off =10 up to 500 arcsec.We find that F062 and F126 are detected up to θ_off = 150 and 500 arcsec, respectively. The clusters are detected with a fairly constant significance (between ∼3.2–3.8σ and ∼3.7–4.8σ for F062 and F126, respectively). However, a mild trend of decreasing significance for increasing offsets is observed. F062 is detected with significances of 3.8σ, 3.2σ, and 3.2σ at θ_off = 0, 75, and 150 arcsec, respectively. In fact, F126 is detected with significances of 4.3σ, 3.9σ, 3.7σ, and 3.7σ at θ_off = 0, 100, 300, and 500 arcsec, respectively.A clear trend between r_max and θ_off is observed for F062. The estimated size increases up to r_max ≃ 150 arcsec for θ_off = 125 arcsec. While the estimated size for θ_off = 0 arcsec is r_max = 72.6 ± 5.1 arcsec. Conversely, no trend is observed for F126.These results suggest that the PPM is effective to detect megaparsec-scale overdensities even if the projected coordinates of the cluster center are known with an accuracy of only ∼100 arcsec. This implies that the PPM can be efficiently applied even if the cluster center coordinates are not accurately known.
• 3.  
  We want to shift these two groups to redshifts higher than z ∼ 1, thus we select the fiducial cluster members of both F062 and F126. We select those sources that fall within circular regions of radius 70.7 and 165.8 arcsec centered at the coordinates of the cluster center, for F062 and F126, respectively. These are the regions in the projected space within which the clusters are detected by the PPM.We conservatively select sources with photometric redshifts within a redshift slice Δz = 4σ_z(z_c) centered around the redshift z_c of the cluster, where σ_z(z_c) = 0.054(1 + z_c) is the 1σ statistical photometric redshift uncertainty of faint galaxies with i⁺ ∼ 24 and 1.5 < z < 3 sources (see Table 3 of Ilbert et al. 2009), typical of the cluster galaxies we consider. The redshift slice considered here is bigger than that adopted throughout the PPM procedure (i.e., Δz = 0.28) to make sure that the large majority of the sources at the redshift of the cluster are included in the bin, even if the accuracy of their photometric redshifts is poor. This is not a cluster membership assignment. In fact, the cluster members will be selected among these sources by using an (I − K)_AB color criterion, as we will describe in the following.¹¹Since red and passively evolving galaxies constitute the majority among the cluster core galaxies at z ∼ 1, we also adopt a color selection criterion to define the fiducial cluster members. We sort the selected galaxies according to their (I − K)_AB color, from redder to bluer. This specific color criterion has been chosen because the rest frame ∼4000 Å absorption feature typical of the spectra of early type galaxies falls just between the K and I bands at redshift z ∼ 1.The cluster members are then removed from the field starting from the reddest source until the average COSMOS number density within the selected 4σ_z bin is reached.According to the outlined procedure we select as cluster members 57 and 249 galaxies down to (I − K)_AB = 1.12 and 1.30 magnitudes for F062 and F126, respectively. As expected, these cluster members are faint, since their I-band magnitudes are between I ∼ 21.9–25.3 and I ∼ 21.1–25.7 for F062 and F126, respectively.We want to make sure that the cluster membership is not biased toward preferentially selecting sources that are located in certain regions in the projected space around the cluster center. In order to do this we verify whether the differential radial number counts are consistent with a constant—no clustering—once the cluster members are removed from the field.In Figure 4 we plot the differential number counts of the sources in the fields of F062 (left panel) and F126 (right panel), as a function of the distance from the projected cluster center coordinates. Sources are counted within the 4σ_z redshift slice adopted throughout the cluster membership procedure and within regions of equal areas (i.e., 2.18 arcmin²), analogously to what was done for the PPM. The areas are chosen to be equal among each other in order to have a constant mean field density per region.The galaxy number counts along with the corresponding 1σ Poisson uncertainties for the error are plotted as blue symbols. Number counts, once the cluster members are subtracted from the fields of both F062 and F126, are plotted as black points, along with the 1σ uncertainties estimated according to the Skellam distribution.¹² The uncertainty in the radial coordinate corresponds to the half-width of each region.The vertical red dashed lines show the radial interval within which the cluster members are selected. By construction, according to the cluster membership procedure, black and blue points coincide outside of this interval. The horizontal line shows the mean COSMOS number counts of ∼30 galaxies associated with a 2.18 arcmin² area around which the black points are scattered.The radial profiles of both F062 and F126 clearly show that the number count excess (blue points) is limited within the projected area defined within the vertical dashed lines. Furthermore, once the cluster members are subtracted, such a number count excess disappears. In fact, the values associated with the black points are consistent with the mean COSMOS number density within the reported 1σ uncertainties.In Figure 5 we report the PPM plots of the fields of F062 and F126, where the cluster members are subtracted. The adopted color code is analogous to that of Figure 3. We apply the PPM and we verify that neither F062 nor F126 are now detected. In fact, as is clear from visual inspection, the high significance pattern at the redshift of the cluster completely disappears in the case of F062, while a residual ≳ 2σ feature is still present in the case of F126 at its redshift. According to the PPM procedure, such a feature is interpreted as noise because it is not enough extended to be detected as overdensity, i.e., it is less than δz_centroid = 0.1 long on the redshift axis z_centroid at fixed Δz = 0.28.The other megaparsec-scale overdensity that was previously detected by the PPM in the field of F126 at z = 0.64 with a significance of 2.7σ is still detected with similar significance (2.6σ) and redshift (z = 0.61). This confirms that the specific cluster membership assigned here combined with the PPM is efficient at removing the degeneracy resulting from the projection effect.As explained above, for our simulations we perform the cluster membership by selecting fiducial cluster members within a circular region centered at the X-ray coordinates of F062 and F126. The radius of the region corresponds to the projected size of the cluster, as estimated by the PPM. In the following we reconsider our estimates by using the PPM plots and we compare the fiducial sizes estimated with the PPM with those obtained by previous work.In Figure 6 we report the PPM plots for F062 (left panel) and F126 (right panel), where only the points corresponding to ⩾3σ overdensities are plotted. This is because the two clusters are detected with significance higher than 3σ. Since we are interested in the cluster size, we plot the cluster size (r_max) associated with each point in the plot, as estimated by the PPM for specific z_centroid and Δz. We refer to the legend in Figure 6 for the specific color code adopted. As clear from visual inspection of the plots, the values of r_max are stable with respect to the Δz parameter.By averaging among the values associated with the points at Δz ≃ 0.28 that define the overdensities, we estimate the sizes r_max = 72.6 ± 5.1 arcsec and r_max = 181.3 ± 33.4 arcsec for F062 and F126, respectively, according to the PPM procedure. Here we report the average value and the rms dispersion around the average. The cluster sizes 70.7 arcsec and 165.8 arcsec that are assumed when performing the cluster membership correspond to the median values of r_max for F062 and F126, respectively.Based on the X-ray surface brightness, FGH07 estimated a core size r₅₀₀ = 48 arcsec for both F062 and F126. By assuming spherical symmetry and a β-model density profile for the cluster matter distribution (Cavaliere & Fusco-Fermiano 1978) we estimate r₂₀₀ = 76 arcsec for both F062 and F126.¹³George et al. (2011) estimated core sizes of r₂₀₀ = 73 arcsec and 81 arcsec and core masses M₂₀₀ = 5.25 × 10¹³ M_☉ and 8.32 × 10¹³ M_☉, for F062 and F126,¹⁴ respectively, on the basis of the mass versus X-ray luminosity relation given in Leauthaud et al. (2010). Using spectroscopic redshift information, Knobel et al. (2012) estimated a size of 659 kpc (i.e., ∼84 arcsec at the redshift of the cluster) for F062.We find that our size estimates are in good agreement with those reported by previous work. However, for F126 our estimate is higher than previous work.Since we want to shift the cluster members of both F062 and F126 to higher redshifts we select two fields where no overdensity is detected with the PPM in the redshift range z ∼ 1–2. We prefer to shift the cluster members of both F062 and F126 into other fields because we want to make sure than no overdensity is detected by the PPM in the redshift range z ∼ 1–2 for the considered field. We note in fact that this is not the case of F062, i.e., a 2.4σ overdensity is detected at z = 2.00. Furthermore, the choice of the same field for both F062 and F126 allows us to directly compare the results we obtain with the PPM for the two clusters, once the cluster members are added to such a field.In Figure 7 we report the PPM plots for the fields of COSMOS-FRI 66 (left panel) and COSMOS-FRI 70 (right panel). As clear for visual inspection of the plots, no high significance pattern is detected in these plots within the redshift range z ∼ 1–2. A weak 2σ overdensity is detected by the PPM at redshift z = 1.60 in the field of COSMOS-FRI 66. However, such a feature is not detected if a slightly different redshift bin (i.e., Δz = 0.24) is adopted throughout the PPM procedure. All of the other isolated ≳ 2σ patterns clearly visible in the plots are interpreted as noise. This is because either they are not located around the y-axis value Δz = 0.28 that is relevant for the overdensity detection or they are not extended enough to be detected as an overdensity (i.e., they are less than δz_centroid = 0.1 long on the redshift axis z_centroid at fixed Δz = 0.28), according to the PPM procedure.Since no clear overdensity is detected in the fields of COSMOS-FRI 66 and COSMOS-FRI 70 we use them as empty control fields (ECFs). Note that we cannot exclude the presence of underdense or dense regions that are not detected by the PPM but are still present in these two ECFs at the redshifts of our interest.If a cluster is superimposed on an underdense region, the PPM might underestimate the detection significance or it might detect no overdensity. Conversely, if the cluster is added to an overdense region, the PPM tends to overestimate the overdensity significance. The reason to choose two ECFs instead of one is to see whether these two scenarios occur. In particular, we will compare our results obtained from each ECF separately to look for a possible mismatch.We add to each ECF the fiducial cluster members of F062 and F126, separately, and we apply the PPM at the coordinates of COSMOS-FRI 66 and COSMOS-FRI 70.¹⁵ In Figure 8 we report the resulting PPM plots, where the clusters F062 (left panel) and F126 (right panel) are in the field of COSMOS-FRI 70. As is clear from visual inspection of the plots, both F062 and F126 are still detected at their true redshift with significances between ∼3–4σ. In Table 1 (middle part) we report the PPM results of our simulations. The estimated redshifts for F062 and F126 are z = 0.86 and z = 1.0, respectively, where the cluster members are added to the fields of COSMOS-FRI 70. Therefore, the estimated redshifts fully agree with those of the clusters. The estimated sizes of F062 and F126 are r_max = 74.3 ± 6.7 arcsec and r_max = 92.0 ± 10.0 arcsec, respectively, where the cluster members are added to the field of COSMOS-FRI 70. These size estimates agree, independently of the ECF adopted (within the errors), with those previously obtained where the cluster members are not subtracted from their own fields (see Table 1, top). These results suggest that the cluster properties estimated by the PPM are not affected by the applied cluster membership and by the fact that the cluster members are added to a field different from that original of the cluster.In order to shift the cluster members of both F062 and F126 to higher redshifts (i.e., z_c = 1.5 and 2.0) we need to address the problem of the detection limit. The COSMOS number density drops off rapidly with increasing redshift. In fact, the number density per unit redshift is, on average, dn/dz/dΩ ≃ 25, 10, and 3 arcmin⁻² at redshifts z ∼1, 1.5, and 2.0, respectively (see Ilbert et al. 2009). Therefore we expect some of the selected cluster members would not be detected if they were located at higher redshifts since their flux would be lower than the survey threshold.In order to address this problem we estimate the I-band magnitude each cluster member would have if it were located at a higher redshift. Then we reject all the sources with I ⩾ 25, that is the magnitude cut applied to the Ilbert et al. (2009) catalog.We assume that each cluster member is located at the redshift of the clusters F062 and F126, i.e., z_c = 0.88 and 1.0, respectively. Then, we estimate the simulated I-band magnitude each cluster member would have if shifted to z_{c, sim} = 1.5 and 2.0. Practically, we perform the K-correction by using the spectral energy distribution (SED) of each object, i.e., we linearly interpolate the flux measurements reported in the I09 catalog, and we correct the apparent magnitude for the luminosity distance.Then, as outlined above, we reject all the members for which I_AB ⩾ 25. This procedure reduces the number of the cluster members from 57 to 9 sources (z_{c, sim} = 1.5) and to 1 source (z_{c, sim} = 2.0) in the case of F062 and from 249 to 58 (z_{c, sim} = 1.5) and 9 galaxies (z_{c, sim} = 2.0) for F126. We note that the magnitude cut (I < 25) is applied in the Ilbert et al. (2009) to the I(auto) magnitude, that corresponds to the Subaru i⁺ band magnitude obtained with SExtractor (Bertin & Arnouts 1996). Therefore, the I(auto) magnitudes should be considered instead of the I (Subaru or CFHT) magnitudes. However, we prefer to adopt the I magnitude instead of the I(auto) magnitude because the latter is automatically estimated by SExtractor and, therefore, the former is more reliable for our simulations. However, we verify that the I(auto) magnitudes of the selected cluster members are, on average, only 0.3 ± 0.4 and 0.3 ± 0.2 lower than the corresponding I magnitudes for F062 and F126, respectively. The reported uncertainty is the rms dispersion around the average. Therefore, the I magnitudes are consistent within ∼1σ with the I(auto) magnitudes for the selected cluster members. This suggests that the results of our simulations would not change if we chose the I(auto) instead of the I magnitudes.In the following we will address the problem of assigning coordinates to the cluster members of both F062 and F126 when they are located at z_{c, sim} ⩾ 1.5. The K-correction applied here ignores any contribution from possible evolution.
• 4.  
  Having addressed the problem of cluster membership, we assign fiducial coordinates to each of the cluster members, when the overdensity is shifted to a higher redshift. We assume that the coordinates in the projected space of each galaxy remain unchanged when the overdensity is shifted to higher redshift. Therefore, projection effects and the peculiar motions of the galaxies are neglected. This approximation is good enough because a high accuracy of the projected coordinates of the cluster members is not required in order to apply the PPM. In fact, each area of the PPM tessellation has a projected size of a few ∼100 kpc. Such a size is much larger than the projected positional uncertainty resulting from our approximation.Concerning the galaxy redshifts, we assume that all the selected members are at the same distance to the observer, corresponding to redshift z_{c, sim} = 1.5, 2.0, equal to that of the simulated cluster. Then, we assign to each cluster member a photometric redshift accordingly to a Gaussian probability distribution centered at the redshift of the cluster z_{c, sim} and a standard deviation σ_{c, sim} = 0.054(1 + z_{c, sim}).This value corresponds to the 1σ statistical photometric redshift uncertainty of i⁺ ∼ 24 and 1.5 < z < 3 sources (see Table 3 of Ilbert et al. 2009), typical of the cluster galaxies we consider.
• 5.  
  We shift both F062 and F126 to higher redshift, i.e., z_{c, sim} = 1.5, where the cluster members are added to the fields of COSMOS-FRI 66 and COSMOS-FRI 70, separately. In Figure 9 we report the corresponding PPM plots for both F062 (left panel) and F126 (right panel). The vertical solid line is located at the redshift of the overdensity. As clear from visual inspection of the PPM plots, both F062 and F126 are still detected if they are located at z_{c, sim} = 1.5. In Table 1 (bottom part) we report the PPM results for these simulations. F062 and F126 are detected with 2.9σ and 2.5σ significance levels; the estimated redshifts are z = 1.56 and z = 1.59, respectively, if the cluster members are added to the field of COSMOS-FRI 70. This suggests that the PPM is effective in finding high-redshift groups at z ≃ 1.5, albeit with lower significance than at z ∼ 1 (i.e., ∼2.5–3σ). The estimated sizes for both F062 and F126 are consistent, within the reported errors, with those previously obtained for these two clusters at their true redshift (see Table 1, top part) The results are quite independent of the ECF considered. Neither F062 nor F126 is detected at z_{c, sim} = 2.0.

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In Table 2 we summarize the results for all 54 simulated clusters. Each entry of the table shows the fraction of ECFs in which the cluster with specific values of N_c, R_c, and z_c is detected. For example, the fraction 4/6 means that the cluster is detected in four out of the six ECFs.

Our simulations suggest that the majority, i.e., 47, 44, and 41 out of the 54 simulated clusters, are detected at least in three, four, and five ECFs. For the 41 clusters that are detected in at least five out of the six ECFs, the redshift z_PPM estimated by the PPM is fully consistent with the input simulated cluster redshift z_c. In fact, the average difference for the 41 clusters is 〈z_PPM − z_c〉 = 0.02 ± 0.05, where all the detections for the 41 clusters are considered and the reported uncertainty is the rms dispersion around the average. Therefore, the statistical 1σ uncertainty for our redshift estimates is ∼0.05. It is estimated by Gaussian propagation of the mean offset with the rms dispersion. Interestingly, this is fully consistent with the independently estimated redshift uncertainties (∼0.06–0.09) described throughout the PPM procedure

The 41 overdensities that are found in at least 5 ECFs are detected with significances spanning from ≳ 2.7σ up to ∼12σ, depending on the adopted parameters, and a median value of 5.2σ. At a fixed richness (N_c) and size (R_c), the clusters are more easily detected for increasing redshifts. This is because the mean COSMOS number density rapidly drops with increasing redshifts. At fixed richness (N_c) and redshift (z_c), more compact clusters are more easily detected with higher significance than more extended overdensities. This is because compact clusters have higher number densities than more extended overdensities.

Furthermore, clusters with low values for N_c are more easily detected at redshifts higher than at z_c = 1. This is due to the decreasing mean number density for increasing redshifts. In fact, at redshifts z ⩾ 1.5, only 10 cluster members seem to be sufficient (see also the results outlined in Paper II). In fact, among the six clusters with N_c = 10 and z ⩾ 1.5, five are detected in at least three ECFs. However, only one of them is detected in at least five ECFs.

The reported trends are clearly due the fact that we consider the cluster parameters N_c, R_c, and z_c as independent. In fact, we do not change the cluster parameters N_c and R_c when we shift the cluster to higher redshift. This is motivated by the fact that the statistics at z ≳ 1 is poor and we prefer to investigate whether the PPM is able to detect overdensities over a wide range of adopted parameters. However, the results of these simulations are clearly dependent on all the simplifications we made (e.g., spherical symmetry; N_c, R_c, and z_c are considered independent). We note that this is a different approach to that adopted for the simulations in Section 4, where we simulate how the cluster would be observed if it were located at higher redshift. Given all the assumptions we make, the accuracy of our simulations is reasonably good for our purpose of detecting high-redshift overdensities on the basis of number counts and photometric redshifts, given the specific properties of both real clusters and the adopted survey.

The general relationship among the richness parameter N_c, the size R_c of the cluster, the cluster mass, and the significance of the cluster detection is complex (i.e., it depends on the depth of the photometric catalog, the redshifts, the evolution of luminosity function), especially at the redshift of our interest (z ∼ 1–2), where the properties of the cluster galaxy population in terms of luminosity and segregation within the cluster are expected to evolve and are not fully understood.

We will address problems of completeness and purity of the cluster catalogs derived with the PPM in a forthcoming paper (G. Castignani et al., in preparation).

5.1.1. Projected Cluster Sizes

5.1.2. The Case of a 2 Mpc Comoving Size Cluster

As outlined in Table 2, at z_c = 1, 11 out of 18 simulated clusters are detected in at least 5 ECFs. Among the 18 simulated clusters, apart from the clusters with R_c ⩾ 2 Mpc and N_c = 10, the remaining 16 simulated clusters are all detected at z_c ⩾ 1.5 in at least four ECFs.

The simulated cluster with N_c = 30 cluster members and R_c = 2.0 Mpc is one of the four clusters that are detected in at least four ECFs only if located at z_c ⩾ 1.5

In Figure 11 we report the PPM plots for this specific cluster in the case where it is located in the field of COSMOS-FRI 70 and where the offset θ = 0 arcsec. The abscissa of the vertical dashed line is equal to the input redshift of the simulated cluster: z_c = 1.0 (left panel), z_c = 1.5 (middle panel), and z_c = 2.0 (right panel). We adopt the same color code as in Figure 1.

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In particular, the cluster at z_c = 1.5 and z_c = 2.0 is detected with significances of 3.7σ and 4.7σ, respectively. The estimated redshifts are z = 1.56 and z = 2.00, respectively. The estimated sizes are r_max = 86.6 arcsec and r_max = 70.7 arcsec, which correspond to 1.9 Mpc and 1.8 Mpc (comoving), at the estimated cluster redshifts, respectively.

Interestingly, both the redshift and size of these overdensities fully agree with the input parameters of our simulations. All these results further confirm that the PPM is very effective in finding clusters and rich groups and in estimating their properties such as redshift and size, if the projected cluster coordinates are known. In the following we will test the PPM against simulations in the case where the projected cluster coordinates are known with an accuracy of 100 and 200 arcsec.

We repeat all 54 simulations for increasing offsets θ between the input coordinates of the PPM and the center of the simulated cluster. In order to do so, we shift the coordinates of cluster members by θ = 100 arcsec. We keep unchanged both the PPM input cluster equatorial coordinates and the equatorial coordinates of the adopted ECF. As explained above, this is because the surroundings of the ECFs might host dense regions in the redshift range of our interest. In Table 3 we summarize our results, where θ = 100 arcsec, analogous to what is reported in Table 2 for the case of null offset θ.

For θ > 100 arcsec the PPM becomes highly inefficient mainly because of the constraint applied to an angular separation of ∼2 arcmin from the coordinates at which the PPM tessellation is centered. In fact, for θ = 200 arcsec, the simulated clusters are all detected in less than five ECFs. Only four high-redshift (z_c = 1.5), rich (N_c ⩾ 60), and extended (R_c ⩾ 2 Mpc) clusters are detected in at least three ECFs. Among the four, only the cluster with z_c = 1.5, N_c = 200, and R_c = 3 Mpc is detected in four ECFs. This is not surprising. In fact, for such a high value of θ, very extended and rich structures have more chances to be detected.

Note that by considering all the possible combinations of the parameters (i.e., the different ECFs, the offsets, and the different values for N_c, R_c, and z_c), 972 clusters are simulated as part of this work.

5.2.1. General Results and Trends

5.2.2. Detection Significances

5.2.3. Projected Cluster Sizes

We tessellate the projected space with a circle centered at the coordinates of the beacon (in our specific case this is the location of the FR I radio galaxy) and 49 consecutive adjacent annuli. These regions are concentric and have the same area, i.e., 2.18 arcmin². This is done in order to have the same average field density for each of the regions. The inner radius of the i-th annulus is equal to $\arccos [1+i(-1+\cos 50\, {\rm arcsec})] \simeq 50\times \sqrt{i}$ arcsec. This means that the radius of the circle centered at the coordinates of the beacon is equal to 50 arcsec. Such an angular separation is consistent with that adopted in previous work focused on high-z clusters (e.g., Santos et al. 2009; Adami et al. 2010, 2011; Durret et al. 2011; Galametz et al. 2012; Spitler et al. 2012). In fact, it corresponds to 427 kpc at redshift z = 1.5, which is typical of the cluster core size at redshift z ∼ 1. If we chose a smaller scale, we would be highly affected by shot noise. In fact, on average, for a fixed area of 2.18 arcmin², the differential number counts (dN/dz) per unit redshift in the COSMOS field are quite small and equal to ∼55, 22, and 7, at redshifts z ≃ 1, 1.5, and 2.0, respectively (Ilbert et al. 2009). Conversely, if we chose a greater scale, we would characterize the cluster environment of the FR Is in our sample with rough (megaparsec-scale) accuracy only.

In general, due to the specific tessellation of the PPM, the method is effective in detecting megaparsec-scale rich groups and clusters (as also discussed in Paper II). Conversely, it might be less efficient in finding poor groups and larger, i.e., a few megaparsec-scale, diffuse structures. Therefore, a more detailed treatment of the projected space typical of sophisticated tessellations such as Voronoi–Delaunay (see, e.g., Ebeling & Wiedenmann 1993) correlation estimators (e.g., Adami et al. 2011), wavelet analysis (e.g., Eisenhardt et al. 2008), filter techniques, and adaptive kernels (e.g., Scoville et al. 2007a) might be ineffective and difficult to apply for cluster searches with the use of photometric redshifts only (see also Scoville et al. 2013, for further discussion). On the other hand, such methods might be more useful to study spectroscopically confirmed clusters or groups (e.g., Jelić et al. 2012).

The typical size inspected by our tessellation changes at most ∼6% within the entire redshift range of our interest. In fact, 1 arcmin corresponds to a physical size of 482, 512, and 509 kpc at redshift z = 1.0, 1.5, and 2.0, respectively. Depending on the adopted cosmology, the angular distance assumes a maximum between z ∼ 1–2. These considerations imply that our tessellation is effective at characterizing megaparsec-scale overdensities with the required accuracy, independently of redshift, under the assumption that cluster core size does not dramatically change for increasing redshifts.

Furthermore, our approach implicitly assumes azimuthal symmetry around the axis oriented at the coordinates of the beacon. We do not exclude the possibility of detecting non-circularly symmetric systems, since we extend the tessellation up to ∼6 arcmin (i.e., ∼3 Mpc at z = 1.5) from the coordinates of the beacon (Postman et al. 1996). Moreover, our method is also flexible enough to find clusters even if the coordinates of the cluster center are known within ∼100 arcsec only (as tested with simulations in Section 5.2).

However, we note that the great majority of low-power radio sources in clusters or groups are found within ∼200 kpc from the core center up to z ≃ 1.3 (Ledlow & Owen 1995; Smolčić et al. 2011). Therefore, this suggests that low-power radio galaxies in cluster environments are preferentially hosted within the central regions of the core, at least at low or intermediate redshifts. The results presented in the companion paper (see discussion in Section 8.8 of Paper II) for the z ∼ 1–2 cluster candidates within the Chiaberge et al. (2009) sample suggest that this is generally true also at higher redshifts. Therefore, all these results support the specific projected space tessellation method described in this section and adopted for our cluster search.

As discussed in Scoville et al. (2007a), identifying large-scale structures on the basis of 2D number densities requires a careful selection of those galaxies that are at the redshift of the structure. This is because, especially in the case of high-z clusters, foreground galaxies contaminate the field. Despite this, the contamination from foreground sources is limited by the smaller angular size of high-z clusters with respect to the contamination of those at lower redshifts.

As pointed out in Scoville et al. (2007a), three different criteria are commonly adopted to discriminate among the galaxies at different distances by using (1) color selections (e.g., Papovich 2008; Gladders & Yee 2005), (2) spectroscopic redshifts (e.g., Knobel et al. 2009, 2012; Diener et al. 2013), or (3) photometric redshifts (e.g., Adami et al. 2010; Durret et al. 2011).

(1) Color selection is not used here since it might be biased toward large-scale structures with specific properties in terms of galaxy colors. This is particularly important especially at redshift z ≳ 1.5, where the properties of the cluster galaxy population and their changes with redshift in terms of galaxy morphologies, types, masses, colors (e.g., Bassett et al. 2013; McIntosh et al. 2014), and star formation content (e.g., Zeimann et al. 2012; Santos et al. 2013; Strazzullo et al. 2013; Gobat et al. 2013) are still debated. (2) Spectroscopic redshifts are preferred to the photometric redshifts. However, spectroscopic redshift catalogs (e.g., Lilly et al. 2007) are limited to a small fraction of the galaxies in the COSMOS field. (3) We adopt the photometric redshift catalog of Ilbert et al. (2009), which was obtained considering sources with AB magnitude I < 25.

We consider M ≫ 1 redshift bins, i.e., the closed intervals $[z_l^i, z_r^i]$, with $z_r^i>z_l^i$, i = 1, 2,..., M, and M = 22,500. We define the length and the centroid of each interval as $\Delta z^i = z_r^i - z_l^i$ and $z_{\rm centroid}^i = (z_r^i+z_l^i)/2$, respectively. The subscripts l and r stand for left and right, respectively. Both the redshift lengths Δzⁱ ∈ [0.02; 0.4] and the redshift centroids $z_{\rm centroid}^i\in [0.4;4.0]$ are randomly and independently chosen assuming a uniform distribution. Since M ≫ 1, the considered ranges are densely spanned in terms of both the redshift bin and the redshift centroid. The redshift range of our interest is z ∼ 1–2, while the typical statistical photometric redshift uncertainties at those redshifts are σ_z ∼ 0.1–0.2 (Ilbert et al. 2009). Therefore, both Δzⁱ and $z_{\rm centroid}^i$ are conservatively selected over wider intervals than those of our interest. This is done in order to avoid spurious boundary effects that might derive from our selection. Before describing in detail our method in the following section we will discuss its theoretical framework.

We denote by n the galaxy number density within a given projected area of the sky subtended by a solid angle Ω. The total variance in the number counts n is given by Peebles (1980)

where

$$\begin{equation} \sigma _v^2 = \frac{1}{\Omega ^2}\int\!\! \int \omega (\theta)\hspace{2.84544pt}d\Omega _1d\Omega _2 \end{equation} \tag{ A2 }$$

is the sampling variance due to source clustering. As pointed out in, e.g., Massardi et al. (2010), $\sigma _v^2$ adds a significant contribution to the uncertainties in the case of small-area fields. As pointed out in Peebles (1980), assuming ergodicity, the average denoted by the brackets $\langle \hspace{5.69046pt}\rangle$ is either the ensemble average (i.e., the average among all the field realizations) or the volume average (i.e., the average among different areas in the survey, each of them is subtended by a solid angle Ω). The clustering term in Equation (A2) is expressed as the integral over the field of the projected two-point correlation function ω(θ), where θ is the angular separation between the solid angle elements dΩ₁ and dΩ₂.

We note that n would be Poisson distributed if the clustering term were not present. Then, we ask what is the probability that the null hypothesis (i.e., no clustering) occurs for the given field. This is equivalent to set $\sigma _v^2=0$ and to assume that n is Poisson distributed. According to our formalism, we estimate the probability of not clustering as the probability to have a number density n' equal to or higher than the observed value n:

$$\begin{equation} \mathcal {P}_{{\rm Poisson}}(n^{\prime } \ge n) = \sum _{k= n\times \Omega }^{\infty }\frac{(\langle n\rangle \times \Omega)^k}{k!}e^{-\langle n\rangle \times \Omega }. \end{equation} \tag{ A3 }$$

As for Equation (A1), 〈n〉 and Ω are the average number density for the survey considered and the solid angle subtended by the selected field for which we estimate the null hypothesis probability, respectively. Therefore, the null hypothesis is rejected with a probability

$$\begin{equation} \mathcal {P} = 1 -\mathcal {P}_{{\rm Poisson}}(n^{\prime } \ge n) = \sum _{k= 0}^{(n-1)\times \Omega }\frac{(\langle n\rangle \times \Omega)^k}{k!}e^{-\langle n\rangle \times \Omega }. \end{equation} \tag{ A4 }$$

The probability $\mathcal {P}$ is higher in those fields where the sources are more clustered, i.e., where $\sigma _v^2$ is non-negligible with respect to the shot noise term 1/〈n〉. Therefore, $\mathcal {P}$ can be independently considered in this paper as the probability that an overdensity is present in the field.

So far, our formalism implicitly assumes that the sample selection is a negligible source of uncertainty (i.e., it does not contribute to the total variance of Equation (A1)). However, the cluster membership selection (based on, e.g., fluxes, colors, or redshifts) always contributes to the total number count variance because of the observable uncertainties. Equivalently, observable uncertainties imply that the total variance in Equation (A1) is underestimated. Consequently, Equation (A4) overestimates the probability $\mathcal {P}$ that the null hypothesis (i.e., no overdensity) is rejected.

Limiting the analysis to the PPM, our method is based on number counts and photometric redshifts. According to the PPM procedure, for each redshift bin Δzⁱ, only those sources within the redshift interval $[z_l^i, z_r^i]$ are considered (see below, Section A.4). Photometric redshift uncertainties are significantly higher than those of any other observable (e.g., flux, projected coordinates) associated with our sample.

This implies that we can estimate the additional term in the number count variance in Equation (A1) due to observable uncertainties by considering photometric redshift uncertainties only. We denote such a term as $\sigma ^2_{{\rm ph}}$, where the notation ph stands for photometric redshifts. This term can be independently estimated as the number count variance obtained by averaging over all the possible realizations of the photometric redshifts of the galaxies in the selected field. These realizations are ideally drawn from the redshift probability distributions of the galaxies in the field. Hence, we have

where $\langle \hspace{5.69046pt}\rangle _{{\rm ph}}$ denotes the average over the ensemble constituted by all the possible redshift realizations. The net effect of increasing the number count variance in Equation (A1) by the amount $\sigma _{{\rm ph}}^2$ is similar to that described in, e.g., Sheth (2007) in the context of luminosity functions estimated by adopting photometric redshifts. Sheth (2007) showed that photometric redshift uncertainties (i.e., distance errors) have the effect of scattering objects to the low-luminosity and high-luminosity ends of the luminosity function (i.e., toward higher and lower luminosities). Similarly, in our case, photometric redshift uncertainties have the effect of scattering the number counts over a wider range.

Given the additional term $\sigma _{{\rm ph}}^2$ in the right hand side of Equation (A1), a rigorous calculation of the probability of the null hypothesis (i.e., no clustering, $\sigma _{v}^2=0$) would require us to consider the redshift probability distribution associated with each galaxy in the field. Estimating the correct expression of the null hypothesis probability might be done with simulations, i.e., adopting different realizations of the redshifts, which are drawn from the redshift probability distribution associated with each galaxy in the field. However, this procedure would be enormously demanding in terms of time and would not add a significant contribution to the PPM. We prefer to adopt a different approach. We neglect any additional terms in the number count variance due to observational uncertainties. We prefer to take into account such an approximation by correcting our final estimate (see Section A.5). Our correction will decrease the probability $\mathcal {P}$. In the next sections we will describe how our method works.

First, we consider each pair defined by the redshifts $z_l^i$ and $z_r^i$, as in Section A.2, and the sources in the Ilbert et al. (2009) catalog that have redshifts within the interval $z_l^i\le z < z_r^i$ and that fall within the largest circle that can be inscribed in the COSMOS survey. Such a circle subtends a ∼1.25 deg² solid angle. Then, we estimate the average mean density 〈n〉_i as the ratio of the number of these sources to the solid angle subtended by the circle. We test if cosmic variance affects our analysis by adopting different choices in estimating such an average number density (i.e., we also selected four disjoint quadrants in the COSMOS survey and we estimated the average number density for each quadrant, separately). We verified that the results are independent of the choice adopted.

For a given beacon, in our case the projected coordinates of the radio galaxy, we consider each of the 50 projected regions defined in Section A.1. We denote them with the index t = 1, ..., 50, where t = 1 corresponds to the central circle, while the adjacent annuli are denoted with progressively higher indexes. We denote by $N_t^i$ the observed number of galaxies within the chosen redshift interval $z_{l}^i\le z< z_{r}^i$ that fall within the t-th area. By construction, each region subtends a solid angle ΔΩ_t = 2.18 arcmin². We also define the observed number density for the selected region and redshift bin as $n_t^i = N_t^i/\Delta \Omega _t$. The probability of a null hypothesis (i.e., no clustering) for the t-th region, consistent with Equation (A3), is

$$\begin{equation} \mathcal {P}_{{\rm Poisson}}^{i,t}\left({n^{\prime }}_t^i\ge n_t^i \right) = \sum _{k= N_t^i}^{\infty }\frac{(\langle n\rangle _i\times \Delta \Omega _t)^k}{k!}e^{-\langle n\rangle _i\times \Delta \Omega _t}, \end{equation} \tag{ A6 }$$

which corresponds to the probability of having a number density ${n^{\prime }}_t^i$ higher than or equal to the observed one, according to the Poisson statistics. Note that, because of the low number counts, the Gaussian statistics is not a good approximation of the Poisson statistics, and therefore the latter is required.

Starting from the central region corresponding to t = 1 we select the first adjacent regions, denoted with the indexes $\lbrace \overline{t}_i,\overline{t}_i+1, \ldots, \overline{t}_i + h_i\rbrace$, for which the probability of the null hypothesis in Equation (A6) is ⩽30%. Here $\overline{t}_i\ge 1$ and h_i ⩾ 0 are both integer numbers. Note that we do not exclude the possibility of selecting one single region, i.e., h_i = 0. The central circle may be selected or not, depending on whether the threshold criterion is satisfied or not by that region.

According to our prescription, we have selected only those regions for which the null hypothesis is rejected at a level of ⩾70%, i.e., down to about 1σ. Note that the adopted ∼1σ threshold is not a tight constraint. A 70% threshold is also adopted by Gomez et al. (1997) and is similar to the values adopted for other selection criteria applied by previous papers that were focused on high-z clusters and that used photometric redshift information (e.g., Papovich et al. 2010; Finkelstein et al. 2010).

According to the procedure, the probabilities are always estimated according to Poisson statistics. Even if the Gaussian statistics are not adopted we often refer to the probability in terms of σ (e.g., we refer to 68.27%, 95.45%, and 99.73% probabilities as 1σ, 2σ, and 3σ significances, respectively). We adopt this notation for practical reasons, for sake of convenience.

Similar to what was done in Gomez et al. (1997), in order to determine the true significance of the number count excess in the field, we merge together all the h_i + 1 adjacent regions to form a (larger) circle or an annulus, depending on whether the central circle is included or not, respectively. Then, we define the total observed number count N_i for the new region as

$$\begin{equation} N_i = \sum _{t=\overline{t}_i}^{\overline{t}_i+h_i}N^i_t\hspace{2.84544pt}\rm {,} \end{equation} \tag{ A7 }$$

and the corresponding number density n_i as

$$\begin{equation} n_i=\frac{N_i}{(h_i+1)\Delta \Omega _t}\hspace{2.84544pt}\rm {.} \end{equation} \tag{ A8 }$$

We stress that the goal of the procedure described so far is not to quantify the number count excess associated with the field. Merging the h_i + 1 regions aims at selecting an area in the projected space for which there is indication of a number count excess and that is most likely associated with the projected coordinates of the beacon.

Note that if we chose a more constraining (i.e., >1σ) threshold criterion, we would select only those regions that show a significant number count excess. Therefore, we might be biased toward selecting those regions (1) that are associated with highly overdense substructures in the cluster or (2) show high shot noise fluctuations. These scenarios might occur because of the small area subtended by each region (i.e., ∼2 arcmin²) and the small number densities that occur at the redshifts of our interest. This discussion suggests that a more constraining (i.e., >1σ) criterion might not be effective in selecting properly the cluster field in the projected space.

Then, analogously to what was done for each of the 50 regions (see Equation (A6)), we estimate the probability of a null hypothesis (i.e., no clustering) for the new area as

$$\begin{equation} \mathcal {P}_{{\rm Poisson}}^{i}({n^{\prime }}_i\ge n_i) = \sum _{k= N_i}^{\infty }\frac{(\langle n\rangle _i\times (1+h_i)\Delta \Omega _t)^k}{k!} e^{-\langle n\rangle _i\times (1+h_i)\Delta \Omega _t}\hspace{2.84544pt}\rm {,} \end{equation} \tag{ A9 }$$

which is the probability of having a number density n'_i higher than or equal to that observed, n_i, according to the Poisson statistics. For the sake of clarity, hereafter we omit the argument of $\mathcal {P}_{{\rm Poisson}}^{i}$. We also define $r_{\rm min}^i$ and $r_{\rm max}^i$ as the minimum and maximum projected distances, respectively, from the coordinates of the beacon (in our case it is the radio galaxy) within which the number count excess is detected. These radii are equal to $r_{{\rm min}}^i=\arccos [1+(\overline{t}_i-1)\times (-1+\cos 50\, {\rm arcsec})]\simeq \sqrt{\overline{t}_i-1}\times 50$ arcsec and $r_{{\rm max}}^i=\arccos [1+(\overline{t}_i+h_i)\times (-1+\cos 50\, {\rm arcsec})] \simeq \sqrt{\overline{t}_i+h_i}\times 50$ arcsec, because of the specific tessellation, consistent with the adopted procedure.

According to what was discussed in Section A.3, the null hypothesis is rejected with a probability $\mathcal {P}^i = 1- \mathcal {P}_{{\rm Poisson}}^{i}$. We set $\mathcal {P}^i\equiv 0$ if the overdensity starts to be detected from r_min ≳ 132 arcsec, i.e., if $\overline{t}_i\ge 8$. This is done to reject those overdensities that are detected only at large angular separation from the location of the source. In fact, since 132 arcsec corresponds to ∼1.1 Mpc at z = 1.5, these overdensities might not be associated with our beacon. However, note that such a constraint does not exclude the possibility of detecting structures that are extended up 132 arcsec or even higher. These extended structures are detected if they start at a distance lower than 132 arcsec.

The specific projected distance of 132 arcsec corresponds to 0.8 h⁻¹ Mpc (h = 0.71), which is the scale where the amplitude of the correlation function between radio loud AGNs and luminous red galaxies is reduced to a few percent (∼4%) of the value at its maximum, up to z ≃ 0.8 (e.g., Donoso et al. 2010; Worpel et al. 2013).

Limiting the number counts to the h_i + 1 overdense regions between the radii $r_{\rm min}^i$ and $r_{\rm max}^i$ does not bias our results toward overestimating the number count excess because we are allowed to select slightly overdense regions, down to ∼1σ number count excess.

Similarly, the probability of detecting those fields that show low number count excess or shot-noise fluctuations is negligible. This is because the number of selected galaxies does not decrease when the h_i + 1 regions are merged and then the number count excess probability is re-estimated for the new (larger) region delimited by the radii $r_{\rm min}^i$ and $r_{\rm max}^i$. We will describe in the following sections the noise mitigation procedure and the peak finding algorithm adopted to detect overdensities. As will be clarified below, both of these procedures further suppress the probability of detecting as overdensity any number count excess that is simply due to shot noise fluctuations.

In Figure 12, left panel, we plot $\mathcal {P}$ as a function of the redshift bin Δz and its centroid z_centroid. We omit the index i corresponding to the specific redshift interval. We will introduce such an index only where necessary. Red, blue, and cyan colors refer to points with significances ⩾4σ, ⩾3σ, and ⩾2σ, respectively. We plot as a vertical solid line the spectroscopic redshift of the source.

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Isolated high significance spiky patterns are clearly visible in the plot. They occur because of the presence of a significant source number excess at specific redshifts and redshift bins. However, the fact that such patterns are spiky and extended over scales that are smaller than the typical statistical photometric redshift uncertainties (σ_z ∼ 0.1–0.2) suggests that they are not physical and are ultimately due to noise fluctuations.

In order to eliminate such high frequency noisy patterns, we apply a Gaussian filter to the function $\mathcal {P}^i$ as follows:

$$\begin{equation} \overline{\mathcal {P}}^i = \frac{\sum _{j}\mathcal {W}^{ij}\mathcal {P}^i}{ \sum _j \mathcal {W}^{ij}}\hspace{2.84544pt}\rm {,} \end{equation} \tag{ A10 }$$

where the kernel

$$\begin{equation} \mathcal {W}^{ij}=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}e^{-\frac{{\zeta _{ij}}^2}{2\sigma _w^2}} &{\rm if\hspace{5.69046pt}\zeta _{ij}}\le 7.5 \sigma _w \\ \ms 0 & {\rm otherwise} \end{array}\right., \end{equation} \tag{ A11 }$$

and

$$\begin{equation} {\zeta _{ij}}^2 =\left(z_{\rm centroid}^i-z_{\rm centroid}^j \right)^2+ \left({\Delta z}^i - {\Delta z}^j \right)^2 \hspace{2.84544pt}. \end{equation} \tag{ A12 }$$

For practical reasons, the sum over j is extended only to those points that are at most 7.5σ_w from ($z_{\rm centroid}^i$; Δzⁱ), so that the kernel has a compact domain and acts as a weighted local average. The newly defined function, $\overline{\mathcal {P}}$, is simply the convolution between the Gaussian filter and the previously defined probability $\mathcal {P}$.

In practice, patterns that are extended over scales of the order of ≲ σ_w with respect to z_centroid or Δz are removed from the plots. We choose σ_w = 0.02, which is much lower than the typical statistical photometric redshift uncertainty σ_z ∼ 0.1–0.2 of the Ilbert et al. (2009) catalog. Therefore, our choice conservatively removes those patterns that are clearly due to noise.

Isolated local maxima are removed from the plots since they are suppressed by the surrounding low significance points. Conversely, local maxima that belong to extended high significance patterns still remain associated with high significance patterns, even if their significance is decreased because of the average procedure.

A heuristic physical interpretation may be provided. Averaging $\mathcal {P}$ among those points that belong to the neighborhood of z_centroid and Δz mimics the presence of the redshift uncertainties at fixed z_centroid and Δz. This is because changing z_centroid and Δz by small amounts has the net effect of including some sources and excluding others when the redshift interval is changed. To understand better such a heuristic equivalence, in the following we will adopt a variational approach.

We note that $\overline{\mathcal {P}}$ can be naturally interpreted as an effective mean field defined on the space of ($z_{\rm centroid}^i$; Δzⁱ). This is because the incoherent fluctuations of $\mathcal {P}$ due to noise and at small scales are locally suppressed by the filter $\mathcal {W}$.

There is no theoretical and observational reason to prefer one specific form for the kernel $\mathcal {W}$. In the effective field theory context, since the width of the filter is related to the statistical photometric redshift uncertainties, it is more relevant than the specific shape of the filter $\mathcal {W}$. In fact, the Gaussian filter is chosen because its exponential declining assures that noisy features associated with scales (that are smaller than the typical statistical photometric redshift uncertainty) are conservatively removed.

A more physical interpretation of the filtering procedure described in this section may be provided by using a variational approach. Several methods of analysis of functionals of Poisson processes, including variational calculus, have been developed in the context of stochastic analysis (e.g., Privault 1994; Albeverio et al. 1996; Molchanov & Zuyev 2000). We stress that the following discussion does not intend to be a formal proof but simply an argument that shows how we can assign a physical meaning to the $\overline{\mathcal {P}}$ values and, thus, better explain the abovementioned heuristic equivalence that arises when the filtering procedure is reconsidered. A more detailed analysis will be performed in a forthcoming paper.

We adopt a compact notation and we define the vector $\boldsymbol {x}=(z_{\rm centroid}$; Δz) whose i-th component is given by ($z_{\rm centroid}^i$; Δzⁱ). We consider the galaxies that are in the field of our beacon (i.e., the radio galaxy in our case) and whose redshifts belong to a neighborhood of the redshift z_centroid corresponding to the specific $\boldsymbol {x}$. The values of Δz and z_centroid can be expressed as the redshift range spanned by the selected galaxies and the redshift centroid of that redshift range, respectively. This implies that the value of $\mathcal {P}$ at $\boldsymbol {x}$ ultimately depends on the set {z_j} of the redshifts of the galaxies in the field of the beacon, where each set {z_j} is selected in such a way that the corresponding (z_centroid; Δz) belongs to a neighborhood of $\boldsymbol {x}$. This argument shows that $\boldsymbol {x}$ is a function of {z_j} and, therefore, $\mathcal {P}$ is a function of {z_j}, i.e., $\mathcal {P}(\boldsymbol {x})=\mathcal {P}(\boldsymbol {x}(\lbrace z_j\rbrace))=\mathcal {P}(\lbrace z_j\rbrace)$.

Each set {z_j} is a specific realization of the photometric redshifts. Each set belongs to the ensemble constituted by all the possible redshift realizations. These realizations are ideally drawn from the redshift probability distributions of the galaxies in the field of the beacon.

Since $\mathcal {P}$ is not an analytic function, first we assume that $\mathcal {P}$ is defined on a discrete domain, given by the points in Figure 12 (left panel), then we consider a local analytic first-order approximation of $\mathcal {P}$ at $\boldsymbol {x}$. By construction, the analytic approximation has a local continuous domain. For the sake of simplicity, in the following we do not distinguish $\mathcal {P}$ from its analytic approximation. Adopting the analytic approximation allows us to use a variational approach¹⁶ and expand $\mathcal {P}$ in Taylor series as follows:

$$\begin{equation} \delta \mathcal {P} = \boldsymbol {\nabla } \mathcal {P} \cdot \delta \boldsymbol {x} + o\left(\frac{|\delta \boldsymbol {x}|}{\Delta z} \right)\;, \end{equation} \tag{ A13 }$$

where $\delta \mathcal {P}$ is the variation of $\mathcal {P}$ induced by the variation $\delta \boldsymbol {x}$.

The first term in the right hand side (rhs) of the equation is the scalar product of the variation $\delta \boldsymbol {x}$ and the gradient $\boldsymbol {\nabla }$ of $\mathcal {P}$ with respect to $\boldsymbol {x}$.

We stress that it is not straightforward to provide an explicit expression for all the terms in the Taylor series.¹⁷ In fact, the explicit expression depends on the specific number counts in the field, the specific redshift z_centroid, and redshift bin Δz. We checked that finite differences $\Delta \mathcal {P}$ of $\mathcal {P}$ corresponding to small displacements $|\Delta \boldsymbol {x}|$ in the PPM plots (e.g., in Figure 12, left panel) satisfy the relation $|\Delta \mathcal {P}|/|\Delta \boldsymbol {x}|\lesssim 1 /\Delta z$ for almost all of the ⩾2σ points in the plot (except for a negligible set). This argument leads to the term $o(|\delta \boldsymbol {x}|/\Delta z)$ in Equation (A13) and implicitly implies that only those points associated with high significance (i.e., ≳ 2σ) patterns are considered. This constraint does not affect our analysis. In fact, as outlined below, only these patterns may be ultimately associated with cluster detections.

The variation $\delta \mathcal {P}$ at $\boldsymbol {x}$ can be alternatively estimated as follows. Limiting our analysis to a neighborhood of $\boldsymbol {x}$ and to the field of the beacon, analogously to what was done in Equation (A13), we estimate the variation $\delta _{{\rm ph}}\mathcal {P}$ of $\mathcal {P}$ induced by the variations of {z_j} within the ensemble as follows:

$$\begin{equation} \delta _{{\rm ph}}\mathcal {P} = {\sum _j} \frac{\partial \mathcal {P}}{\partial z_j}\delta z_j + o\left(\frac{\sqrt{\sum _j \delta z_j^2 / \sum _j 1}}{\Delta z}\right)\;, \end{equation} \tag{ A14 }$$

where the sum over j is restricted to a neighborhood of $\boldsymbol {x}$ as specified above and to the galaxies in the field of the beacon. The subscript ph stands for photometric redshifts and it is introduced to distinguish the variation $\delta _{{\rm ph}}\mathcal {P}$ from $\delta \mathcal {P}$. The first and second terms in the rhs of the equation are the first term and the higher order terms of the Taylor expansion, respectively. The second term reported in Equation (A14) is estimated similarly to what was done in Equation (A13). The chain rule is also applied to express the derivatives with respect to z_j into derivatives with respect to Δz and z_centroid.

We stress that the variations $\delta \mathcal {P}$ and $\delta _{{\rm ph}}\mathcal {P}$ are different; $\delta \mathcal {P}$ is the variation of $\mathcal {P}$ due to small variations δ(Δz) and δz_centroid of the redshift bin and the redshift centroid, respectively. Conversely, $\delta _{{\rm ph}}\mathcal {P}$ is the variation of $\mathcal {P}$ due to the photometric redshift uncertainties of the redshifts of the galaxies in the field of the beacon. Equivalently, $\delta _{{\rm ph}}\mathcal {P}$ is the variation of $\mathcal {P}$ in the ensemble of all possible redshift realizations of the galaxies in the field of the beacon. Combining Equation (A13) and Equation (A14) we estimate the difference between the two variations of $\mathcal {P}$ as follows:

$$\begin{equation} \delta \mathcal {P} - \delta _{{\rm ph}}\mathcal {P} = o\left(\frac{|\delta \boldsymbol {x}| + \sqrt{\sum _j \delta z_j^2 / \sum _j 1}}{\Delta z }\right)\;, \end{equation} \tag{ A15 }$$

where the same notation and the chain rule adopted above is used. The equation, combined with the Equations (A13) and (A14), shows that the two variations $\delta \mathcal {P}$ and $\delta _{{\rm ph}}\mathcal {P}$ are equal up to the first order in the perturbations.

By construction, the filter $\mathcal {W}$ removes the fluctuations of $\mathcal {P}$ on small scales. Because of the exponential declining of the Gaussian filter, it is effective on scales $|\delta \boldsymbol {x}|\lesssim 3\sigma _w=0.06$. As will be explained below, cluster candidates are selected at the fixed redshift bin Δz = 0.28. This redshift bin corresponds to the estimated statistical 2σ photometric redshift uncertainty at z ∼ 1.5 for dim galaxies (i.e., with AB magnitude i⁺ ∼ 24; Ilbert et al. 2009).

Therefore, $|\delta \boldsymbol {x}| /\Delta z\simeq 2\%\ll 1$. This inequality implies that the scales are sufficiently small to assure that the filter $\mathcal {W}$ suppresses the variation $\delta \mathcal {P}$ up to the first order, as in Equation (A13), i.e., $\delta \mathcal {P}=0$.

Similarly, the condition $\sqrt{\vphantom{A^A}\smash{{\sum _j \delta z_j^2/ \sum _j 1}}}\lesssim \Delta z$ is reasonably satisfied because $\sqrt{\vphantom{A^A}\smash{{\sum _j \delta z_j^2/ \sum _j 1}}}$ approximately corresponds to the quadratic average of the photometric redshift uncertainties of the galaxies in the field. Such an average is reasonably smaller than the selected redshift bin Δz = 0.28. Therefore, the variation $\delta _{{\rm ph}}\mathcal {P}$ in Equation (A14) is well approximated, for our purposes, by the linear expansion.

Resuming, Equations (A13) and (A14), combined with the two reported inequalities, suggest that in our case both $\delta \mathcal {P}$ and $\delta _{{\rm ph}}\mathcal {P}$ can be expressed in linear theory. Similarly, Equation (A15), combined with the two inequalities, tells that the two variations are equal in first-order approximation. Therefore this argument suggests that the filter $\mathcal {W}$ simultaneously suppresses (in linear approximation) both the variation $\delta \mathcal {P}$ and variation $\delta _{{\rm ph}}\mathcal {P}$ due to the photometric redshift uncertainties.

This discussion suggests that $\overline{\mathcal {P}}$ is a good estimate of the number count excess probability. In fact, as discussed in Section A.3, since the photometric redshift uncertainties add a significant contribution to the total number count variance, $\mathcal {P}$ represents an overestimate of the true detection significance. Our procedure takes into account a posteriori the initial overestimation: the significance of the local maxima is decreased when the filter is applied. Equivalently, the procedure reasonably removes, as required, the excess of $\mathcal {P}$ that is ultimately due to the photometric redshift uncertainties and the corresponding number count variance expressed in Equation (A1).

Note that the parameters adopted here for the Gaussian kernel and those used in the following for the cluster detection procedure are chosen because of the properties of the photometric redshifts of our sample and of the Ilbert et al. (2009) catalog. In particular, the parameters Δz (= 0.28) and σ_w (=0.02) are finetuned in such a way that the linear perturbation theory (see Equations (A13) and (A14) is reasonably correct in both the two spaces (that of $\boldsymbol {x}$ and the ensemble of all the redshift realizations). In general, all the parameters are adapted to the specific data set used. We verified that our results are independent of a slightly different choice of these parameters. This is ultimately due to the fact that the procedure is not performed on physical observables (e.g., on the density field), but acts directly on the PPM plots as those reported in Figure 12.

In Figure 12, right panel, we plot the values of $\overline{\mathcal {P}}$ as a function of z_centroid and Δz. The same color code for the left panel is adopted. When the Gaussian filter is applied, isolated noisy patterns are substantially removed from the plot, as is clear from visual comparison of the left panel with the right panel in the figure. Furthermore, we observe that triangular-shape high significance (i.e., ≳ 2σ) patterns are still clearly present in the plot. They are stable with respect to different values of Δz, i.e., the patterns are extended along the Δz axis. In particular, they tend to increase their width in the z_centroid direction for increasing Δz. This is due to the fact that for increasing Δz, we are including more and more objects that are far from z_centroid. Therefore, we still detect a number count excess at values of z_centroid that are increasingly far from the true redshift of the overdensity, even if with lower significance. In fact, lower significance is associated with the boundaries of the triangular-shape patterns than that related to the central regions of these patterns.

In this section we will describe our procedure to detect and characterize the overdensities we find in the considered field by using the PPM plots. These goals will be achieved by finding the local maxima of the function $\overline{\mathcal {P}}$. The peak finding algorithm we will describe is a specific procedure we developed for our discrete case and it belongs to a more general context known as Morse theory. Such a theory can be used to find and characterize the critical points of differentiable functions defined on a manifold (see, e.g., Guest 2001, for a review). Notably, there are many applications of Morse theory in the context of differential topology (Bott 1960; Milnor 1963) and in quantum field theory (Witten 1982, and following work).

First, since the high significance patterns in the PPM plots are stable with respect to the Δz axis we simplify the problem to a one-dimensional (1D) case as follows. We consider only those points $p_k = (z_{\rm centroid}^k;\Delta z^k)$ such that the redshift bins Δz^k fall within a tiny δ(Δz) = 0.01 wide interval centered at Δz = 0.28. This redshift bin corresponds to the estimated statistical 2σ photometric redshift uncertainty at z ∼ 1.5 for dim galaxies (i.e., with AB magnitude i⁺ ∼ 24; Ilbert et al. 2009). These magnitudes are typical of the galaxies we expect to find in clusters in the redshift range of our interest. We verified that our results are stable with respect to a sightly different choice of the redshift bin Δz.

Then, we sort the points {p_k} in increasing order of $z^k_{\rm centroid}$, and we redefine the ordering of the points in such a way that $z_{\rm centroid}^{k+1}\ge z_{\rm centroid}^{k}$. Hence, our problem is reduced to finding the maxima of $\overline{\mathcal {P}}$ defined on the 1D discrete domain $\lbrace z_{{\rm centroid}}^k\rbrace$. Having reduced the dimensionality of the domain is a great simplification, since saddle points are not present in the 1D case, where critical points are of only two types: maxima and minima.

Starting from the significance s = 2σ, we select those intervals of consecutive points that have significances ⩾s. We merge consecutive intervals that are separated by δz_centroid = 0.02 or less along the z_centroid axis. We also reject those intervals that are shorter than δz_centroid = 0.1 along the z_centroid axis. The first condition merges those intervals that are separated by a tiny separation along the z_centroid axis. The minimum allowed separation between two consecutive intervals is set equal to the dispersion σ_w adopted for the filtering procedure. In fact, fluctuations that occur on these scales may not be physical since they occur for redshift separations that are well below those of typical statistical redshift uncertainties. Similarly, the second condition is applied in order to detect only high significance features whose length along the z_centroid axis is at least comparable with the typical statistical photometric redshift uncertainty of the Ilbert et al. (2009) sample. Given the significance s, this procedure gives a set of intervals that we define as s intervals.

Then, we increase the significance threshold by a tiny amount ds = 0.1σ and we repeat the above outlined procedure. We note that each (s + ds) interval is entirely included within an s interval. We retain those s intervals that do not contain any (s + ds) interval, whereas we reject all the other s intervals.

The significance s is increased and the procedure is repeated until no s interval is found. The final set of s intervals represents the local maxima of $\overline{\mathcal {P}}$. These intervals have different significances s and they are centered at different redshifts z_centroid. Each s interval corresponds to a cluster candidate detection and is associated with a number count excess found in the given field and around a specific redshift. In Figure 13 we show a visual representation of the peak finding algorithm adopted. In the following section we will describe how the method estimates cluster properties such as the redshift and size.

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The significance s of a given s interval is interpreted as the detection significance of the corresponding cluster candidate. In this section we describe our procedure that provides (1) an estimate for the redshift of the overdensity, (2) an estimate for the cluster core size, and (3) a rough estimate for the cluster richness.

We estimate the size of each cluster candidate in terms of the minimum and maximum distances from the beacon (in our case the FR I) at which the overdensity is detected. According to our procedure, the points of the s interval are associated with different values of $r_{\rm min}^i$ and $r_{\rm max}^i$. We estimate a minimum and a maximum projected radius of the overdensity as the average (and the median) of the minimum ($r_{\rm min}^i$) and maximum distances ($r_{\rm max}^i$) associated with all of the points of the s interval, respectively. The uncertainty is estimated with the rms dispersion around the average. The maximum projected radius also provides an estimate for the cluster core size, as further tested in this work and in Paper II.

In Figure 14 we show the PPM plots of source 01, (after having applied the Gaussian filter) where the radial information concerning r_min (left panel) and r_max (right panel) is considered. Analogously to what was done for Figure 12, we only plot points that are associated with ⩾2σ overdensities (see the legend in the panels for the color code adopted). The vertical solid line in each panel is located at the redshift of source 01. The values of $r_{\rm max}^i$ and $r_{\rm min}^i$ associated with the high significance patterns of our plots are very stable with respect to Δz (see Figure 14). Therefore, the particular choice Δz = 0.28 does not affect the results concerning the projected space analysis.

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We also estimate the redshift of the cluster as the middle point of the s interval. To estimate the fiducial uncertainty of the cluster redshift we consider all sources located within the median value of the minimum distance and the median value of the maximum distance from the coordinates of the FR I beacon within which the overdensity is detected in the projected space. We also limit to the sources that have photometric redshifts within a redshift bin Δz = 0.28 centered at the estimated cluster redshift. This is done consistently with our detection procedure. The cluster richness is roughly estimated by the number of selected sources. Then, the cluster redshift is estimated at the 1σ level by the rms dispersion around the average of the redshifts of the selected sources.

In particular, if N ≫ 1 sources were uniformly distributed within the redshift bin Δz = 0.28 we would obtain an rms dispersion of 0.08. We expect the estimated redshift uncertainty to be around this value.

The method described here detects megaparsec-scale overdensities within the entire redshift range spanned by the z_centroid values. Then, we associate with the radio galaxy only those overdensities that are detected in the field, at a redshift compatible with that of the source, i.e., when the interval centered at the redshift estimated for the overdensity and with a half-width equal to two times the fiducial redshift error intersects the redshift range defined within the radio galaxy redshift uncertainties. Multiple overdensity associations are not excluded.

We point out that the redshift of the FR I beacon is considered only during the last step of the procedure, when we perform the association between the detected overdensities and the radio galaxy. This is primarily motivated by the fact that we do not have spectroscopic information for most of our FR Is. Therefore, our approach is necessarily different from previous studies which select cluster members using photometric redshifts for the majority of the galaxies in the field, but also knowing the spectroscopic redshifts of some of the cluster members (e.g., Papovich et al. 2010).

Furthermore, our choice implies that the high significance patterns in our plots (see, e.g., Figure 12) have a typical width along the z_centroid axis comparable to the statistical photometric redshift uncertainty σ_z ∼ 0.054(1 + z_centroid). Such an uncertainty corresponds to sources with i⁺ ∼ 24 and 1.5 < z < 3 sources (see Table 3 of Ilbert et al. 2009), which we expect to find in our clusters. Therefore, our method estimates the cluster redshifts with similar accuracy. On the other hand, including the (photometric) redshift information of the FR I beacon from the beginning of our procedure would imply an increase of the intrinsic scatter due to the FR I redshift uncertainty. If we sum in quadrature the redshift uncertainty associated with each FR I to the statistical photometric redshift uncertainty σ_z ∼ 0.1–0.2, the uncertainty increases up to ∼0.2–0.5, depending on the redshift of the radio galaxy and its uncertainty. This effect would make our method ineffective to search for high-z cluster candidates around radio galaxies.

In this section we describe how the PPM can be generalized for application to other data sets and photometric redshift catalogs, whose statistical redshift accuracy is possibly comparable to or better than that of the Ilbert et al. (2009) catalog. Surveys with depth similar to or higher than that of COSMOS are preferred. In particular, the PPM might be applied to both present and future wide field surveys such as SDSS Stripe 82, LSST, and Euclid.

The parameters should be adapted to take into account the different mean field number density of galaxies in the survey and the statistical photometric redshift uncertainties of the redshift catalog. Therefore, it is not straightforward to provide precise rules.

In general, the redshift bin Δz at which the overdensities are evaluated in the PPM plots should be set equal to the 2σ statistical photometric redshift uncertainty of the galaxies at the redshifts of our interest and magnitudes typical of the sources we expect to find in clusters at high redshifts.

Consequently, the length of the s intervals should be at least about one-third of the specific redshift bin Δz adopted to select the overdensities in the PPM plots. We remind readers that the s intervals are the high significance intervals in the PPM plots (at the specific Δz) that are associated with overdense regions. Similarly, the Gaussian dispersion σ_w used to remove the noise from the PPM plots should be equal to one-fifth of the abovementioned minimum length for the s interval. As described and motivated in the procedure description (see Section A.6), when defining the s intervals, consecutive intervals that are separated by an amount σ_w (=0.02 in this paper) or less along the z_centroid axis should be always merged. Such scaling relations are motivated by the fact that the adopted parameters should ultimately rescale linearly with the typical statistical photometric redshift accuracy.

Furthermore, according to the procedure description, we do not change the PPM tessellation with increasing cluster redshift. This is mainly because we are looking for overdense regions with sizes typical of those of the cluster cores and the angular distance D_A(z) at the redshift z is fairly constant between z ∼ 1–2. However, it might be appropriate to rescale linearly the size of the PPM tessellation by a factor of ∼D_A(z = 1.5)/D_A(z) in the case where the PPM is used to search for diffuse protoclusters at redshifts significantly higher than z ∼ 1–2 (e.g., at z ≃ 8, Trenti et al. 2012). This leads to a correction ∼21%, 46%, and 51% at redshifts z ∼ 4, z ∼ 6, and z ∼ 8, respectively.