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Appendix A: The cluster-mass bias

The ratios r_ν(L) defined in Sect. 2.4.1 can be predicted under the assumption that, on large scales, dark matter halos of a given mass M are biased tracers of the underlying dark matter distribution. In the so-called local bias model, to zeroth order in perturbation theory (e.g., Cooray & Sheth 2002), this is expressed as P(k;M) = b²(M)P_mat(k), where P_mat(k) denotes the power spectrum of the dark matter and b(M) is a scale-independent halo-mass bias. Given a halo mass function n(M)dM (i.e., the number of dark matter halos with masses between M and M + dM per unit comoving volume) and a selection function (i.e., the probability of having a cluster with mass M given some selection criteria), the real-space marked power spectra can be written as (e.g., Cooray & Sheth 2002; Sheth 2005), where (A.1)and (A.2)is the effective cluster-matter bias. For a given bin of X-ray luminosity, the selection function φ(M,L) can be expressed as the average of the scaling relation given by Eq. (1) in that luminosity bin. The ratios r_{1, 2, 3}(L) in real space can therefore be interpreted as estimates of b(L)/b(L_ref), b_w(L)b(L) /(b(L_ref)b_w(L_ref)), and b_w(L)/b_w(L_ref) respectively. Also, the information contained in the ratio r₄(L) is the same as contained in the ratio r₅(L), since these are estimates of the ratio (b_w(L) /b(L))^1/2 and b_w(L)/b(L) respectively. Accordingly, the luminosity power spectrum can be written as (A.3)from which the predictions for the ratios r_{6, 7}(L) can be readily obtained. According to Eq. (1), the moments of the luminosity L are linked to the scaling relation via . Therefore, the bias is directly sensitive to the mean of the scaling relation, yet indirectly (only through ) to the intrinsic scatter.

Fig. A.1 Ratio r3(L) (top panels) defined in Eq. (7), and luminosity bias b(L) (bottom panels) obtained in the range 0.02 ≤ k/(h Mpc-1) ≤ 0.08. Results are shown in real (right panel) and redshift (left panel) space. For readability, we only show the standard deviation obtained from the N-body simulations with shaded regions. The lines represent the predictions presented in Sect. 2.4.1 using expressions for the halo-mass bias as reported by different authors, MW: Mo & White (1996); ST: Sheth & Tormen (1999); T: Tinker et al. (2010); and P: Pillepich et al. (2010).

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The redshift-space estimates of the ratios r_ν(L) can be obtained in a similar way. Under the plane-parallel approximation, the large-scale signal of the redshift-space cluster power spectrum P^s(k;L) can be described by the so-called Kaiser effect (e.g., Kaiser 1987; Hamilton 1998) P^s(k;L) = (1 + 2β/3 + β²/5)P(k;L), where and f ≡ dlnD(a)/dlna is the growth index (D(a) represents the growth factor) (e.g., Peebles 1980). Given the cosmology and redshift output of the L-BASICC simulations that we used, f = 0.44. The Kaiser effect can be generalized to the marked power spectra (e.g., Skibba et al. 2006) as (A.4)and (A.5)where . We have checked whether these expressions describe the ratios r_ν(L). To this end, the halo abundance n(M) is taken to be described by the fitting formulae of Jenkins et al. (2001), which is suitable for simulations such as the L-BASICC. A crucial step is to choose the halo-mass bias b(M). In Fig. A.1 we show predictions for the ratio r₃(L), obtained using some examples of prescriptions for this quantity: Mo & White (1996), Sheth & Tormen (1999), Tinker et al. (2010). and Pillepich et al. (2010). To witness the performance of these prescriptions, the bottom panels in Fig. A.1 show the luminosity bias obtained similar to what is described by Eq. (7), using the estimates of the matter power spectrum of the L-BASICC II simulation. This figure shows that i) as established by a number of studies, a scale-independent halo-mass bias is a fair modeling of the cluster power spectrum on large scales; ii) the Kaiser effect is a good description of the redshift-space power spectra, at least within the range of masses and scales probed by our analysis (see, e.g., Bianchi et al. 2012, for a broader discussion on this subject); iii) when assessing the ability of a model to retrieve either cosmological or astrophysical information from one or two-point statistics, extreme caution is required in view of the discrepancies observed between models and the simulations. In particular, more accurate models of halo-mass function and halo-mass bias are demanded, given the small statistical errors expected from forthcoming surveys (see, e.g., Cunha & Evrard 2010; Wu et al. 2010; Smith et al. 2012) with volumes comparable to that of the L-BASICC simulation. The differences between the different fitting formulae presented in Fig. A.1 can be caused by several effects, namely, the difference cosmological models and/or parameters used in the N-body simulations used to fit each of them, the characteristics of the halo-identification algorithm (which introduces systematic effects both in the mass function and the halo-mass bias), and the way the biases are measured (i.e, either from the correlation function, the power spectrum, or by means of count-in cells experiments). It is beyond the scope of this work to analyze these differences in detail; however, we note that different fitting formulae can provide a good description of the measured luminosity bias at different ranges of X-ray luminosity. In particular, the results from Tinker et al. (2010) generate a fair description for luminosities above ~3 × 10⁴³ h^-2 erg s^-1. Lower luminosities correspond to halos defined by a relatively low number of dark matter particles, where resolution effects can be relevant. Finally, iv) the discrepancies between the different prescription of halo-mass bias are slightly diminished when we work with estimates of relative biases instead of absolute biases.

Appendix B: Halo exclusion

Dark matter halos are not point-like objects. As a consequence, the idea that their spatial distribution can be described as a Poisson-point process drawn from a realization of an underlying continuous field with a positive correlation function (Peebles 1980) is, strictly speaking, not a realistic assumption. So far, a statistical description of halo distribution that takes their finite size into account is not fully accomplished. Instead, simple geometrical approaches have been developed to empirically model the exclusion effect within the context of the two-point correlation function. Here we briefly illustrate how this model works. Assuming that we can assign a radius to each spherical halo, the correlation function can be written as (e.g., Porciani & Giavalisco 2002) (B.1)where , with ⟨R|M ⟩ denoting the expected radius of a cluster with mass M. In this expression, b(M) denotes the dark-matter halo scale-independent bias, B(r) denotes a possible scale-dependency in the halo-matter bias (e.g., Tinker et al. 2005), while ξ_mat(r) is the full nonlinear matter correlation function. For halos with masses within an infinitesimally narrow range, this expression predicts a sharp transition towards ξ_h(r) =  −1 at a scale equal to twice the radius of the halo. This transition becomes smoother when halos with different masses (and thus sizes) are included. In Fourier space, the exclusion is translated to a lack of power on small scales, which goes counter to the effect of the nonlinear clustering. Since the exclusion effect is more evident when more massive halos are considered, such lack of power displays a clear trend with the characteristic mass (or X-ray luminosity) of the sample. The halo power spectrum from Eq. (B.1) can be separated into three components: (B.2)The first term is the halo power spectrum with a scale-independent bias. The second term is the Fourier transform of the −1 in Eq. (B.1), where j₁(x) denotes the spherical Bessel function of first order. The third contribution can be written as (B.3)

where the kernel is given by , with (B.4)and (B.5)In the limit (i.e., no halo exclusion), both the second term on the right hand side of Eq. (B.3) and the kernel G₂ go to zero. In that case a nonlinear contribution to the cluster power spectrum relies on the behavior of the scale-dependent halo-mass bias B(r). In the limit of an homogeneous distribution, we end up with a power spectrum of the form . Thus, to obtain an unbiased estimation of the cluster power spectrum (of spherically symmetric nonoverlapping clusters), this last term would need to be subtracted from the raw estimates (as in Eq. (3)), together with the white shot noise . The combination of these two effects can be regarded as scale-dependent shot-noise.

Fig. B.1 Halo exclusion: ratio between the measured cluster power spectrum described in Sect. 2.3 and the expected linear cluster power spectrum b2(L)Pmat(k), for clusters in four different bins of X-ray luminosity characterized by a luminosity Li, with L3 > L2 > L1. The luminosity bias b2(L) is that measured from the simulations. The shaded regions and the solid line represent the standard deviation and the mean, respectively, obtained from the N-body simulations.

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Figure B.1 depicts the halo exclusion effect as measured from the ensemble of halos presented in Sect. 2.1. As pointed out above, the strength of the effect scales with the luminosities (or masses) of the objects considered in the analysis. Therefore, this signal can be used to retrieve information on the underlying scaling relation (e.g., mass-X-ray luminosity when analyzing cluster samples or the mass-number of hosted galaxies when analyzing a galaxy redshift survey) (e.g., Porciani & Giavalisco 2002). Finally, the exclusion effect is attenuated when observed in redshift space, simply because pairs of halos are observed to be closer due to their peculiar velocities. For instance, exploring the power spectrum obtained from the full luminosity sample shows that in real space exclusion sets in at k ~ 0.2 h Mpc^-1, while this value shifts to ~0.3 h Mpc^-1 in redshift space.

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