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Appendix A: Mean and covariance of number counts

To avoid introducing numerous Dirac factors, owing to the discreteness of the observed distribution of objects, we follow the simple approach described in Sect. 36 of Peebles (1980) to compute the statistical properties of counts in cells. We illustrate in this section this method for the computation of the mean and covariance of number counts within redshift bins.

We divide the “volume” of the space (z,Ω,lnM), which enters the expression (17) of the angular number density of observed objects in redshift bin i, over small (infinitesimal) cells labeled by the index α, so that Eq. (17) reads as (A.1)where subscript i refers to the redshift bin i. Then, since the cell α_i is infinitesimally small it contains at most one object, whence (Peebles 1980) (A.2)Moreover, by definition its average is given by (A.3)Of course, we recover for the mean number of objects in the redshift bin i the expression (19), which could also be read from Eq. (17) using the average (A.4)We now consider the covariance of the angular number densities . From Eq. (A.1) we have In the second line we used the fact that the redshift bins do not overlap, so that for two “volumes” α_i and α_j to coincide, bins i and j must be the same (and δ_i,j is the Kronecker symbol), while in the third line we used Eq. (A.2). The first term in Eq. (A.7) corresponds to the shot noise, due to the discreteness of the object distribution. The second term includes the nonzero-distance correlation between objects, and reads as (for α_i ≠ α_J) (A.8)where is the “halo” two-point correlation function between “volumes” α_i and α_j, see Peebles (1980). This yields (A.9)using obvious notations where we label the quantities associated with and by the subscripts i and j and we integrate over the bins i and j. This could also be directly obtained from Eq. (17) by writing (A.10)where the second term with the Dirac factors gives the shot-noise contribution.

In this derivation we have assumed in Eq. (A.2) that space can be divided into infinitesimal volumes that contain either zero or one object and that each object only appears in one cell. Even though clusters and dark matter halos are actually extended objects, it is still possible to define a point distribution by associating a single point to each cluster or halo, for instance the halo mass center. Thus, this approach, which follows Peebles (1980), applies to these cases as well and to any distribution of discrete objects, as long as we restrict ourselves to count distributions and do not study the internal structure of these objects.

Appendix B: Finite-size effects

	Fig. B.1 Geometrical illustration of finite-size effects. Close to the survey boundary, part of the sphere of radius r extends beyond the observational cone and should not be counted. The left plot is a transverse view, orthogonal to the central line of sight, whereas the right plot is a view from a point far away on the line of sight.
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As noticed in Sect. 4, in our computations of the mean and covariance of the estimators and we neglect finite-size effects. Indeed, we do not take the fact into account that when a point i gets close to the survey boundaries the available space for points i′ located in the distance bin  [R_i, − ,R_i, + ] , with respect to point i, is only a fraction of this spherical shell since a part of it extends beyond the observational cone. This means that we overestimate the total number of pairs. This has no impact on the mean, , since this effect cancels out between the numerator and denominator in (48), but it means that we slightly overestimate the signal-to-noise ratio.

To estimate the magnitude of this error, we compute the geometrical factor illustrated in Fig. B.1. Approximating the observational cone as a cylinder of radius , a point i at distance ℓ from the central line of sight is the center of a spherical shell of radius r, onto which we count all neighbors i′ to estimate the correlation ξ at this distance r. We denote F(ℓ) as the fraction of this sphere that is enclosed within the observational cone. In our computations elsewhere we used the approximation F = 1, but for R_s − r < ℓ < R_s we actually have F < 1. As in the transverse view shown in the left hand plot of Fig. B.1, the angle θ_ℓ associated with the farthest point of intersection between the cylinder and the sphere satisfies ℓ + rsinθ_ℓ = R, whence (B.1)Next, in the plane of each vertical section (i.e., at fixed θ), shown in the right hand plot of Fig. B.1 that corresponds to a projection along the line of sight, the cylinder appears as a circle of radius R_s, whereas the section of the sphere of center i appears as a circle of radius rsin(θ). Both circles intersect (again for R_s − r < ℓ < R_s) at the symmetric polar angles ϕ _± , with (B.2)Then, the surface of the sphere that extends outside of the observational cylinder writes as (B.3)Thus, for R_s − r < ℓ < R_s the fraction of the sphere that is enclosed within the observational cylinder reads as (B.4)whereas F(ℓ) = 1 for 0 < ℓ < R_s − r. Then, integrating the position of the central point i over the cylinder, the fraction of volume for pairs at distance r, with respect to the approximation F = 1, writes as This gives the ratio of the number of pairs N′, which is measured in the survey, to the number N obtained when we do not take finite-size effects into account. For instance, at z = 1, which corresponds to the angular distance  Mpc, and for a survey angular window of area 50 deg², which corresponds to θ_s ≃ 0.0696 rad, we have  Mpc. Then, we obtain N′/N ≃ 0.91 for a shell at radius r = 30   h^-1 Mpc. This means that the approximation F = 1 overestimates the number of pairs by about 10% and the signal-to-noise ratio by 5%.

Appendix C: Computation of the mean of the estimators and

Defining the 3D Fourier-space top-hat as (C.1)the 3D Fourier-space window of the i-shell reads as (C.2)where the superscript (3) recalls that we consider a 3D radial bin. Then, writing the two-point correlation function in terms of the power spectrum, as in Eq. (3), we obtain for its radial average (54) (C.3)(Here i and i′ refer to the same radial bin; the prime only recalls that we are integrating over a neighbor i′ within a small radial shell with respect to another point in the observational cone.)

Appendix D: Derivation of the covariance of the Peebles & Hauser estimator

We compute here the covariance of the estimators , which is identical to the covariance of the quantities . To simplify the expressions we do not consider mass binning here, but it is straightforward to generalize to the case of several mass bins. From the definition (48) we can write with obvious notations the second moment as (D.1)The average in Eq. (D.1) can be written as in Eq. (A.10), with many Dirac factors for the shot-noise contributions. However, as in Appendix A, it may be easier to follow Peebles (1980) and to divide “volumes” over small (infinitesimal) cells that contain objects, with or 1. Then, we can split the average as (D.2)where we have explicitly written the first “pure sample-variance” contribution and the last six “shot-noise” contributions associated with the Kronecker symbols. The remaining averages with the superscript “(s.v.)” denote “sample-variance” averages, that is, without further shot-noise terms. Here we used the fact that the objects i and i′ are separated by the finite distance r_i′, with r_i′ ≥ R_i, − , so that the elementary “cells” i and i′ cannot coincide and there is no shot-noise contribution of the form δ_i,i′. For the same reason there is no term δ_j,j′. Next, the “sample-variance” averages of Eq. (D.2) read as (Peebles 1980) (D.3)(D.4)(D.5)where ξ^h, ζ^h, and η^h are the two-point, three-point, and four-point correlation functions of the objects. Since we have (D.6)we obtain from Eqs. (D.1)–(D.5) the decomposition (59) of the covariance matrix, with the explicit expressions (60)–(62) of the various “sample-variance” and “shot-noise” contributions. Here we used the symmetries¹⁰  { i ↔ i′ }  and  { j ↔ j′ }  of Eq. (D.1). In Eq. (61) the object “j′” is at the distance r_j′ from the object “i”, since this shot-noise contribution comes from the case where the objects i and j are the same object (or from one of the three remaining cases “i = j′”, “i′ = j”, or “i′ = j′”). The shot-noise contribution (60) comes from the identification “i = j and i′ = j′” (or “i = j′ and i′ = j”). This implies that the distances r_i′ and r_j′ are equal, which gives rise to the Kronecker symbol δ_i,j since we consider the case of nonoverlapping distance bins  [R_i, − ,R_i, + ] .

From Eq. (52) the contribution of Eq. (60) also reads as (D.7)In order to estimate the contributions and we assume that the radial bins  [R_i, − ,R_i, + ]  are restricted to large enough scales to neglect three- and four-point correlation functions, as well as products such as ξ_i;j′ξ_i′;j. Thus, we only keep in this Appendix the contributions that are constant or linear over the two-point correlation function ξ_i;j of the objects, which we recall with the superscripts “1” and “ξ” below. Moreover, we again assume that the two-point correlation function can be factored in as in Eq. (1).

The first contribution to , associated with the factor 1 in the brackets in Eq. (61), reads as (D.8)The contributions that are linear over ξ sum up as (D.9)where and are defined as in Eqs. (54) and (C.3), whereas is defined in Eq. (68) and also writes as (D.10)Next, at this order the contribution (62) to the covariance simplifies as (D.11)where is Limber’s approximation (13) to Eq. (8). Then, collecting all terms, we obtain the expression (67) for the covariance.

Appendix E: Derivation of the mean and covariance of the Landy & Szalay estimator

We can relate the Landy & Szalay estimator defined by Eq. (56) to the Peebles & Hauser estimator (48) by (E.1)where we defined the cross-term by (E.2)We obtain at once, using Eqs. (A.4) and (50), (E.3)which leads to Eq. (57).

From the relation (E.1) we have for the covariance of the estimator , (E.4)where C_i,j is the covariance of the Peebles & Hauser estimator , defined in Eq. (59). To compute the cross-terms in (E.4) we write as in Eq. (D.1), (E.5)Proceeding as in Appendix D, this gives (E.6)with (E.7)(E.8)Next, to compute the last term in Eq. (E.4) we write (E.9)whence (E.10)with (E.11)(E.12)Collecting all terms in Eq. (E.4), which reads as , we obtain the decomposition (63) with the contributions (64)–(66).

Appendix F: Computation of high-order terms for the covariance of

We compute here the high-order terms for the covariance of the Landy-Szalay estimator that we had neglected in Eq. (69).

For numerical computations, it is often more efficient to express the quantities that we encounter in this work in terms of the real-space correlation ξ(x), instead of the Fourier-space power spectra P(k) or Δ²(k), provided ξ(x) is known (e.g., computed in advance on a fine grid¹¹). Indeed, this replaces oscillatory integrals by integrals with slowly-varying factors, which allows faster and more accurate computations. This comes from mostly considering various kinds of volume averages of correlation functions, such as Eq. (8), which are more naturally written in configuration space. This yields integrations over bounded or unbounded domains with typically positive and slowly-varying kernels. In contrast, the transformation to Fourier space yields highly oscillatory kernels as soon as some underlying real-space volumes are finite with a size much larger than some other scales (see for instance the 2D top-hat (12) for a window θ_s that is much broader than the typical angular scale ). On the other hand, intermediate analytical computations are often easier to perform in Fourier space, mostly because of the convolution theorem. Then, a convenient method is to first write expressions in terms of Fourier-space power spectra, perform integrations over angles, and finally go back to the real-space correlation function, using the fact that from Eq. (3), ξ(x) and Δ²(k) are related by (F.1)(F.2)As shown below, this method also allows partial factorization of most integrals.

A first high-order contribution to the covariance arises from the product ξ_i;j′ξ_i′;j in Eq. (66), which also writes as Eq. (75) where we introduced the quantity defined by (F.3)Expressing the two-point correlation functions in terms of the power spectrum, using the flat-sky (small angle) approximation, as well as Limber’s approximation as we did for Eq. (9), we obtain after integration over angles and over the two radial shells, (F.4)Again, the factor 2πδ_D(k_1 ∥  + k_2 ∥ ) comes from the integration over χ_j, which suppresses longitudinal wavelengths. Using the exponential representation of Dirac functions, Eq. (F.4) can be partially factorized as (F.5)and the integration over angles yields (F.6)This also reads as (F.7)where we introduced (F.8)and (F.9)The function A⁽³⁾(y) can be written as where K(k) and E(k) are the complete elliptic integrals of the first and second kinds (Gradshteyn & Ryzhik 1965). One can check that A⁽³⁾(y) is a positive, nonoscillatory, and continuous function (but not analytic at y = 2), with A⁽³⁾(y) ~ 2y for y → 0 and A⁽³⁾(y) ~ 1/y for y → ∞.

Going back to configuration space, by substituting Eq. (F.2), the integral (F.9) can be written as (F.12)with (F.13)which for a > 0 and b > 0 is given by (F.14)Thus, using Eq. (F.12), the quantity of Eq. (F.7) involves slowly varying integrals over real-space variables, which partially factor as three factors within the integrand of Eq. (F.7). This makes it more efficient to use Eq. (F.7) than the Fourier-space expressions (F.4) or (F.6).

To evaluate the two remaining contributions, associated with the factors ζ_i,i′,j′ in Eq. (65) and η_{i,i′;j,j′} in Eq. (66), we use the model for the three- and four-point correlation functions described in Sect. 2.1.2. Thus, using Eq. (4) for the three-point correlation function that enters Eq. (65), this contribution to Eq. (65) reads as (F.15)The first term in the bracket in Eq. (F.15) is given by (F.16)where was defined in Eq. (54), because the integrations over r_i′ and r_j′ are independent. The second term reads as (F.17)which no longer factors. Introducing an auxiliary wavenumber k and the Dirac factor δ_D(k₁ + k₂ − k), which we write under its exponential form as in Eq. (F.5), and using the inverse Fourier transform of the 3D shell (C.2), as well as Eq. (F.9), we obtain (F.20)The third term in Eq. (F.15) is obtained from Eq. (F.20) by exchanging the labels “i” and “j”.

We now turn to the four-point contribution to Eq. (66), using Eq. (6) for the halo four-point correlation function . Thanks to the symmetries  { i ↔ i′ }  and  { j ↔ j′ }  we have two different contributions (a) and (b) associated with the topology of the left diagram in Fig. 2, each with a multiplicity factor 2, and four different contributions (c), (d), (e), and (f), associated with the topology of the right diagram, with multiplicity factors 4,4,2, and 2.

The first contribution (a) reads as (F.21)with (F.22)Proceeding as for Eq. (F.3), we obtain (F.23)the contribution is the symmetric one with respect to  { i ↔ j }  of Eq. (F.21); that is, the product is replaced by .

Next, the contribution reads as (F.24)where the geometrical average writes as (F.25)since integrals over r_i′ and r_j′ can be factored.

The contribution (d) involves where no factorization is possible. Proceeding as for Eq. (F.3) we obtain (F.26)The contribution (e) involves that can be written as (F.27)whereas contribution (f) is obtained from (e) by exchanging the labels “i” and “j”.

Collecting all terms, the high-order contributions to the covariance matrix are given by Eqs. (75)–(77).

Appendix G: Computation of the mean of the estimators ŵ and ŵ^LS

We give here explicit expressions of the average (84) of the correlation function over an angular ring. As in Sect. 2.1.3, using the flat-sky and Limber’s approximations, we obtain (G.1)where we introduced the 2D Fourier-space window of the i-ring, (G.2)and , associated with a full circular window, was defined in Eq. (12). In terms of the two-point correlation function, Eq. (G.1) also writes as (G.3)which avoids introducing oscillatory kernels.

Appendix H: Computation of the covariance of ŵ^LS

The low-order contribution (94) to the covariance matrix of the estimator ŵ^LS involves the angular average (95). Using Limber’s approximation it also reads as (H.1)We now compute the high-order terms of the covariance , which are given in Eqs. (96)–(98). A first contribution (96) ari-ses from the product ξ_i;j′ξ_i′;j in Eq. (92). Using Limber’s approximation and integrating over angles yields (H.2)Introducing a Dirac factor , which we write with the usual exponential representation in a fashion similar to Eq. (F.5), we obtain after integration over angles (H.3)Then, after a rescaling of variables x and y, and defining the quantities (H.4)(H.5)(H.6)we obtain the expression (96), using the property A⁽²⁾(y) = 0 for y > 2. As compared with Eq. (H.2), introducing the Dirac factor and the two auxiliary variables x and y has allowed us to partly factor in the integrals, as seen in Eq. (96), which is convenient for numerical computations. Again, it is useful to express Eq. (H.6) in terms of the real-space two-point correlation function, which yields (H.7)with (H.8)Although there is no explicit expression for the integral (H.8) for arbitrary (a,b), for |a − b| > 1 we can use the properties (H.9)In the band |a − b| < 1 one can check that W₂(a,b) is positive and decays as  ~ b^-2 for large b, so that the real-space expression (H.7) is again more convenient than the Fourier-space expression (H.6).

The second contribution (97) arises from the three-point correlation ζ in Eq. (91). Using Eq. (4) it reads as (H.10)where the three terms in the brackets, which correspond to the three diagrams in Fig. 1, are again geometrical averages along the lines of sight, which we compute with Limber’s approximation. In particular, the first term factors as (H.11)where was defined in Eq. (84), while the second term reads as (H.12)With the same factorization method, and using the inverse Fourier transform of the 2D shell (G.2), we obtain (H.15)where we introduced The third term in Eq. (H.10) is obtained from Eq. (H.15) by exchanging the labels “i” and “j”.

The third contribution (98) arises from the four-point correlation η in Eq. (92). As in Appendix F, we must compute the various terms associated with the diagrams of Fig. 2, with contributions (a) and (b) associated with the left diagram and contributions (c), (d), (e), and (f) associated with the right diagram. The first contribution (a) leads to (H.18)As in Appendix F, this geometrical average factors as (H.19)with Contribution (b) is the symmetric one of (a) with respect to i ↔ j.

Next, contribution (c) involves the geometrical average , which again factors as (H.22)

The contribution (d) involves the average (H.23)which also writes as (H.24)Contribution (e) involves , which reads as (H.25)whereas contribution (f) is obtained from (e) by exchanging the labels “i” and “j”.

Collecting all terms, the high-order contributions to the covariance matrix are given by Eqs. (96)–(98).

Appendix I: Scaling of the number counts signal-to-noise in simulations

	Fig. I.1 Scaling of the number-counts signal-to-noise ratio by as computed in the Horizon simulation, see Sect. 2.2. Different configurations are displayed according to the total surveyed area ΔΩ, the number of subfields , and the mass limit. In the right caption, ΔΩ is expressed in deg2 and the mass unit is h-1   M⊙.
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	Fig. I.2 Same as Fig. I.1 but with a scaling that depends on the number of subfields: , with n =  −0.6
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Fig. J.1 Left panel: cluster mass associated with a 50%, 80%, or 95% detection probability (from bottom to top), for the XXL selection function C1, as a function of redshift. Middle panel: minimum detectable cluster mass, as a function of redshift, for the Planck space mission. Right panel: cluster mass associated with a 50%, 80%, or 95% detection probability (from bottom to top), for the Erosita selection function as a function of redshift (we consider a flux limit of 4 × 10-14 erg   s-1   cm-2 in the  [0.5 − 2]  keV band).

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We present here the result of scaling the number counts with the total surveyed area ΔΩ and the number of subfields . Figures I.1 and I.2 show the scalings expected from Eq. (35) in the shot-noise and sample-variance dominated regimes. Multiple survey configurations are explored by varying the total surveyed area (ΔΩ = 25,50, and 100 deg²), the number of subfields (, and 4), and the mass threshold (M > 2 × 10¹³,10¹⁴, and 5 × 10¹⁴   h^-1   M_⊙).

The weak scatter in those plots shows that (35) provides a valid approximation of the signal-to-noise scaling with respect to ΔΩ and . In agreement with the discussion in Sect. 3.2.1 and Fig. 7, at high redshift and for high mass, the scaling shown in Fig. I.1 is best, as expected for the shot-noise dominated regime, whereas at low redshift and for low mass the scaling shown in Fig. I.2 (with n =  −0.6) is best, as expected for the sample-variance dominated regime.

Appendix J: Selection functions used for various surveys

We give in Fig. J.1 the selection functions that we use for several cluster surveys investigated in Sect. 6. For Planck, the curves shown in the middle panel corresponds to a 100% detection probability.

For the other surveys studied in Sect. 6 we consider simple mass thresholds, rather than detailed selection functions. More precisely, we consider halos above the two thresholds 5 × 10¹³   h^-1   M_⊙ and 5 × 10¹⁴   h^-1   M_⊙ for DES and Euclid, and above 5 × 10¹⁴   h^-1   M_⊙ for SPT.

Appendix K: Dependence on cosmology

In this appendix we investigate the dependence of the results obtained in Sect. 6 on the value of the cosmological parameters. Thus, in addition to the WMAP7 cosmology recalled in the first line of Table K.1, which was used in Sect. 6, we also consider the three modified cosmologies where one among the three parameters h, Ω_m, and σ₈ is changed to the values shown in the second line of Table K.1. They correspond to “” deviations from WMAP7 (Komatsu et al. 2011) and describe current uncertainties. (When we vary Ω_m we keep a flat ΛCDM universe and we change Ω_de according to Ω_de = 1−Ω_m.)

Thus, we compare in Figs. K.1–K.3, the three curves obtained for these three alternative cosmologies with the curve that was obtained in Sect. 6 for the fiducial WMAP7 cosmology. To avoid overcrowding the figures we only consider the all-sky

surveys, Planck, Erosita, and Euclid. We can see that the main features of these figures are not modified when we consider these alternative cosmologies, so our results and conclusions are not sensitive to the precise value of the cosmological parameters. As expected, we can also check that shot-noise effects become less important, with respect to sample-variance contributions, when σ₈ is increased.

Fig. K.1 The ratio of the rms shot-noise contribution to the rms sample-variance contribution , of the covariance of the angular number densities Ni, as in Fig. 34. The fiducial curve that was shown in Fig. 34 is the solid line (mean WMAP7 cosmology), whereas the dashed, dot-dashed, and dotted lines correspond to the three cosmologies where either h, Ωm, or σ8, is changed to the value given in the second line of Table K.1.

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	Fig. K.2 The ratio of the rms contributions and of the covariance matrix of the estimator , as in Fig. 35. The line styles are as in Fig. K.1 and Table K.1.
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	Fig. K.3 The ratio of the rms high-order contribution (75)–(77) to the rms low-order contribution (second term in Eq. (69)) of the sample variance of the correlation ξi, as in Fig. 36. The line styles are as in Fig. K.1 and Table K.1.
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